# Large-time Limit of Nonlinearly Coupled Measure-valued Equations that Model
Many-Server Queues with Reneging

Rami Atar  Viterbi Faculty of Electrical and Computer Engineering
Technion
Haifa, Israel<EMAIL_ADDRESS>,  Weining Kang  Department of Mathematics
and Statistics,
University of Maryland, Baltimore County<EMAIL_ADDRESS>,  Haya Kaspi
Department of Industrial Engineering and Management
Technion
Haifa, Israel<EMAIL_ADDRESS>and  Kavita Ramanan  Division of Applied
Mathematics
Brown University
Providence, RI 02118, USA<EMAIL_ADDRESS>
(Date: July 12, 2021)

###### Abstract.

The large-time behavior of a nonlinearly coupled pair of measure-valued
transport equations with discontinuous boundary conditions, parameterized by a
positive real-valued parameter $\lambda$, is considered. These equations
describe the hydrodynamic or fluid limit of many-server queues with reneging
(with traffic intensity $\lambda$), which model phenomena in diverse
disciplines, including biology and operations research. For a broad class of
reneging distributions with finite mean and service distributions with finite
mean and hazard rate function that is either decreasing or bounded away from
zero and infinity, it is shown that if the fluid equations have a unique
invariant state, then the Dirac measure at this state is the unique random
fixed point of the fluid equations, which implies that the stationary
distributions of scaled $N$-server systems converge to the unique invariant
state of the corresponding fluid equations. Moreover, when $\lambda\neq 1$, it
is shown that the solution to the fluid equation starting from any initial
condition converges to this unique invariant state in the large time limit.
The proof techniques are different under the two sets of assumptions on the
service distribution. When the hazard rate function is decreasing, a
reformulation of the dynamics in terms of a certain renewal equation is used,
in conjunction with recursive asymptotic estimates. When the hazard rate
function is bounded away from zero and infinity, the proof uses an extended
relative entropy functional as a Lyapunov function. Analogous large-time
convergence results are also established for a system of coupled measure-
valued equations modeling a multiclass queue.

###### Key words and phrases:

many-server queues, GI/G/N+G queue, fluid limits, reneging, abandonment,
measure-valued processes, renewal equation, large-time behavior, stationary
distribution, call centers, enzymatic processing networks, transport equation,
age-structured population models

###### 2010 Mathematics Subject Classification:

Primary: 60F17, 60K25, 90B22; Secondary: 60H99, 35D99.

KR is supported by the U.S. Army Research Office via Grant W911NF2010133 and
RA is supported by the Israel Science Foundation grant 1035/20.

##  1\. Introduction

###  1.1. Background, Motivation and Results

The focus of this work is the analysis of the large-time behavior of a
nonlinearly coupled pair of measure-valued transport equations with
discontinuous boundary conditions that describe the hydrodynamic or fluid
limit of a many-server queue with reneging. Many-server queues with reneging
arise in a range of applications, including as models of computer networks,
telephone call centers or (more general) customer contact centers [17, 30,
37], and enzymatic processing networks in biology, where reneging seeks to
model the phenomenon of dilution (see, e.g., [31]). A basic model, also
referred to as the GI/G/N$+$G queue, consists of a system with $N$ identical
servers, to which jobs arrive with independent and identically distributed
(i.i.d.) service requirements that are drawn from a general distribution, with
each job also being equipped with an i.i.d. patience time drawn from another
general distribution. Depending on the application, the servers represent
processors, call center agents or enzymes, and the jobs represent packets,
customers with tasks or proteins. Arriving jobs enter service immediately if
there is an idle server available, else they join the back of the queue. As
servers become available, jobs from the queue start service in the order of
arrival. Once a job completes service, it departs the system. In addition,
jobs also renege from the queue if the amount of time they have been in queue
exceeds their patience time. Important system performance measures of interest
include the stationary waiting time and queue distributions. In the special
case when arrivals are Poisson and the service distribution is exponential,
but the abandonment distribution is general, explicit formulas for the scaled
steady-state distributions were obtained in [10], and their asymptotics as
$N$, the number of servers, goes to infinity, were studied in [42]. However,
the case of general service distributions, which is relevant for many
applications, is more challenging and it appears not feasible to obtain exact
analytical expressions for these quantities for general service and
abandonment distributions. Instead, one often resorts to obtaining asymptotic
approximations that are exact in the limit as the number of servers goes to
infinity.

In [23] the state of an $N$-server queue at time $t$ is represented in terms
of two coupled measures, the queue measure and the server measure. The queue
measure encodes jobs currently in the queue and has a unit Dirac delta mass at
the amount of time elapsed since that job entered the system, whereas the
server measure $\nu^{N}$ keeps track of jobs currently in service and has a
unit Dirac delta mass at the age of each such job, where the age is the amount
of time elapsed since the job entered service. For analytical purposes, it
turns out that the queue measure itself is more conveniently represented in
terms of a potential queue measure $\eta^{N}$, which keeps track of the times
elapsed since entry into the system of all jobs (whether or not they have
abandoned or entered service), and not only of jobs currently in the queue, as
well as the total number $X^{N}$ of jobs in system. Under fairly general
conditions on the service and patience distributions (see Assumption 2.1), it
was shown in [23] that when the average arrival rate or traffic intensity
converges to $\lambda>0$, the rescaled state descriptor
$(X^{(N)},\nu^{(N)},\eta^{(N)})$ converges to a deterministic limit
$(X,\nu,\eta)$, where $\nu$ and $\eta$ are characterized as the unique weak
solutions to a nonlinearly coupled system of deterministic measure-valued
transport equations, subject to discontinuous boundary conditions, which we
refer to as the fluid equations (see Definition 2.3).

In this work we study the long-time behavior of the solution $(X,\nu,\eta)$ to
the measure-valued fluid equations obtained in [23] under Assumption 2.1 and
the assumption that the fluid equations admit a unique invariant state
(equivalently, fixed point). We study the subcritical, critical and
supercritical regimes, characterized, respectively, by the regions where
$\lambda<1$, $\lambda=1$ and $\lambda>1$, additionally assuming in the
subcritial and supercritical regimes that the hazard rate function of the
service time distribution is either decreasing or bounded away from zero and
infinity. (Here, and in the sequel, we will say decreasing to mean non-
increasing.) Our main results are summarized in Theorem 3.2. Specifically, we
show that when $\lambda\neq 1$, from any initial condition, the solution to
the fluid equations converges to the unique invariant state in the large time
limit and when $\lambda=1$ and the hazard rate function is decreasing then the
total mass of $\nu$ (which represents the mass of busy servers in the fluid
system) converges to $1$. In all cases above, we show that the fluid equations
have a unique invariant distribution (or equivalently, unique random fixed
point, to use a term introduced later in this paper; see Definition 2.10).
This crucially implies that the stationary distributions of the $N$-server
dynamics converge to the invariant state of the fluid dynamics, the proof of
which was one of the motivations of this work. In particular, as elaborated in
Remark 3.3, it is the uniqueness of the invariant distribution (or random
fixed point) for the fluid dynamics, rather than just the uniqueness of an
invariant state, that is relevant for the convergence of stationary
distributions of the $N$-server dynamics. In the absence of reneging, such
long-time convergence results were established for a single-class system in
Proposition 6.1 of [25] for the subcritical regime and in Theorem 3.9 of [25]
for the critical regime, with the latter requiring an additional finite second
moment assumption. In the presence of reneging, although the system is in a
sense more stable (e.g., the system is also stable in the supercritical
regime, making it of particular interest), certain monotonicity properties are
lost and the fluid equation dynamics are considerably more complicated, making
the analysis significantly more challenging.

The proof of convergence in the subcritical regime is obtained via a direct
analysis of the fluid equations (see Section 4.1). The proofs in the critical
and supercritical cases are considerably more subtle, and rely on rather
different arguments under the two sets of assumptions. When the hazard rate
function of the service distribution is decreasing, we use a reformulation in
terms of renewal equations, in conjunction with certain recursive estimates,
and the convergence of the measure-valued state processes is with respect to
the weak topology (see Section 4.3). These arguments are inspired by those
used in the work [26], which studies the long-time behavior of fluid equations
for the GI/G/N+G model under the assumption that the service time distribution
has a concave or convex renewal function (which is implied by decreasing
hazard rate functions). However, the fluid equations of [26] are based on a
different measure-valued state representation, involving residual service
times of customers rather than ages, and moreover, convergence is established
in [26] for the queue process, not the measure-valued process. Thus the
results of [26] do not directly apply. Moreover, we also need to establish
additional estimates to prove convergence of the measure-valued process $\nu$
in the supercritical case.

The arguments used when the hazard rate function $h^{s}$ of the service
distribution is bounded away from zero and infinity are of a completely
different nature. These results address a class of distributions not covered
by related results in the literature for many-server systems. They are based
on the analysis of weak solutions to partial differential equations, and
entail showing that an extended relative entropy functional (that takes in
arguments that are not necessarily probability measures) serves as a Lyapunov
functional for the dynamics. As a result, the convergence of the measure-
valued state processes is with respect to the stronger total variation
topology. The analysis here is somewhat reminiscent of the study of age-
structured population models arising in biology (see, e.g., [13, 36, 32]).
Indeed, although the server measure $\nu$ need not have a density, and in fact
will typically not have a continuous density, a purely formal derivation (see
Section 4.2.1) shows that the density of $\nu$ satisfies a partial
differential equation (PDE) that is similar to age-structured population
models. However, our fluid equation differs crucially from such models in
several aspects that make it in some ways harder to analyze. One issue, as
elaborated in Remark 4.6 is that the service time hazard rate function, which
appears as a coefficient in the PDE, is not integrable on $[0,\infty)$, and
thus the relative entropy method developed in [32] does not apply. But, more
significantly, a key complicating factor is the fact that the boundary
condition for $\nu$ is discontinuous and assumes a different form depending on
whether the total mass of $\nu$ is less than or equal to, $1$. In fact, a
significant challenge in the analysis of the critical and supercritical
regimes is the to control the oscillations of the total mass of $\nu$ below
$1$.

Finally, we also analyze the long-time behavior of fluid equations for a
multiclass model under a nonpreemptive priority policy, which was formulated
in [9] and used therein to establish asymptotic optimality of the policy when
the reneging distribution is exponential (see Definition 5.1). In the case
that service time distribution is class-independent and satisfies the same
conditions as above, reneging times are exponential, but possibly class-
dependent, and the fluid equations have a unique invariant state, we establish
(in Theorem 5.2) uniqueness of the random fixed points and analogous long-time
convergence results in the supercritical regime. It should be mentioned here
that in the queueing context, other works that have studied long-time behavior
of measure-valued fluid equations using Lyapunov functionals include [35, 19,
38]. All of these works focus on the dynamics of residual times for jobs in
bandwidth sharing and processor sharing models, which have a different
structure from the measure-valued equations arising from our fluid equations.

An interesting open problem for future investigation would be to determine
precisely the full class of service distributions for which such long-time
convergence holds, and also whether there is a unified proof for all cases, at
least in the supercritical regime. In addition, in the critical regime, a more
complete study of the convergence of the state process even under the
conditions imposed here would also be of interest. Moreover, the techniques
developed here may be potentially used to establish such convergence results
for more general many-server systems, including load-balancing systems with
general service distributions, where the fluid limits are described in terms
of a system of coupled measure-valued equations [2] or partial differential
equations [5], and uniqueness of the fixed point holds under general
conditions [1]. For the multiclass model, it is of interest to investigate
broader conditions, such as class-dependent service distributions and less
restrictive assumptions on the hazard rate, under which convergence holds.
This would also allow to treat asymptotic optimality of the aforementioned
index rule in broader settings.

###  1.2. Ramifications for stationary distributions of $N$-server queues

The results of this paper also shed insight into the (law-of-large-numbers)
scaled limit of stationary distributions of $N$-server queues for a much
broader class of service distributions. More precisely, it follows from
Theorem 3.2 and Theorem 7.1 of [24] that the measure-valued state dynamics
$(X^{(N)},\eta^{(N)},\nu^{(N)})$ for each $N$-server system describe an
ergodic Feller process with a unique stationary distribution, whereas Theorem
3.3 of [24] shows that the sequence of scaled stationary distributions
$\pi^{(N)}$ of the normalized state is tight. Moreover, the latter theorem
also states that any subsequential limit of $\pi^{(N)}$ must coincide with the
(deterministic) invariant state of the fluid equations, whenever the latter is
unique. However, there is a gap in the proof of this statement in [24]. A
priori one only knows that any subsequential limit of the scaled stationary
distributions of $N$-server queues with reneging is a random fixed point of
the fluid equations (see Definition 2.10), and not that it is necessarily
equal to a deterministic fixed point. However, as shown in Proposition 4.16 of
the present paper, when there is convergence of the fluid equations to a
unique invariant state from any initial condition (or, when $\lambda=1$, just
convergence of $\eta_{t}$ and the fraction of busy server servers), it follows
that the set of random fixed points is in fact equal to the Dirac delta
measure at the unique invariant state, thus closing the gap in Theorem 3.3 of
[24].

Our work in the multi-class setting also closes an exactly analogous gap
present in Theorem 4.4 of [9]. Indeed, one of the auxiliary goals of this work
is to (partially) fix the gaps in these proofs, under the additional
assumptions on the service distribution imposed herein (see Remark 3.3 for
further elaboration of this point). In the case of [9], the gap also affects
the validity of Theorem 5.1 there, regarding the asymptotic optimality of an
index policy, referred to as the $c\mu/\theta$ rule, which was introduced in
[7]. The results obtained in this paper validate the asymptotic optimality
result, Theorem 5.1 of [9] under the additional assumption that the service
time distributions do not depend on the class. (Note, however, that there is
no problem with the validity of the asymptotic optimality results of the
$c\mu/\theta$ rule stated in [7] and [8], which deal with the case of
exponential service time distributions. Also, note recent developments on this
policy under various additional settings in [27]). Finally, we note that
limits of stationary distributions of many-server systems in the (so-called
Halfin-Whitt) diffusive regime have been considered in [20, 3, 4] in the
absence of abandonment, and in [22, 16, 21] in the presence of abandonment.

###  1.3. Organization of the rest of the Paper

In Section 2.1 we introduce the fluid equations in the single-class setting,
and in Section 2.2 define their invariant states. In Section 3 we state our
assumptions and the main results, and provide the proofs in Section 4.
Finally, in Section 5 we introduce the multiclass fluid equations and
establish our convergence results in that setting. First, in Section 1.4, we
introduce common notation that is used throughout the paper.

###  1.4. Common Notation and Terminology

The following notation will be used throughout the paper. ${\mathbb{Z}}$ is
the set of integers, ${\mathbb{N}}$ is the set of strictly positive integers,
${\mathbb{R}}$ is set of real numbers, ${\mathbb{R}}_{+}$ the set of non-
negative real numbers. For $a,b\in{\mathbb{R}}$, $a\vee b$ denotes the maximum
of $a$ and $b$, $a\wedge b$ the minimum of $a$ and $b$ and the short-hand
$a^{+}$ is used for $a\vee 0$. Also, given a set $A$, we will use $1_{A}$ to
denote the indicator function, which is $1$ on $A$ and zero otherwise.

Given any metric space $E$, ${\mathcal{C}}_{b}(E)$ and ${\mathcal{C}}_{c}(E)$
are, respectively, the space of bounded, continuous functions and the space of
continuous real-valued functions with compact support defined on $E$, while
${\mathcal{C}}^{1}(E)$ is the space of real-valued, once continuously
differentiable functions on $E$, and ${\mathcal{C}}^{1}_{c}(E)$ is the
subspace of functions in ${\mathcal{C}}^{1}(E)$ that have compact support. The
subspace of functions in ${\mathcal{C}}^{1}(E)$ that, together with their
first derivatives, are bounded, will be denoted by ${\mathcal{C}}^{1}_{b}(E)$.
For $H\leq\infty$, let ${\mathcal{L}}^{1}[0,H)$ and
${\mathcal{L}}^{1}_{loc}[0,H)$, respectively, represent the spaces of
integrable and locally integrable functions on $[0,H)$, where a locally
integrable function $f$ on $[0,H)$ is a measurable function on $[0,H)$ that
satisfies $\int_{[0,a]}f(x)dx<\infty$ for all $a<H$. Given any càdlàg, real-
valued function $f$ defined on $[0,\infty)$, we define
$\left\|f\right\|_{T}:=\sup_{s\in[0,T]}|f(s)|$ for every $T<\infty$, and let
$\left\|f\right\|_{\infty}:=\sup_{s\in[0,\infty)}|f(s)|$, which could possibly
take the value $\infty$. In addition, the support of a function $f$ is denoted
by ${\rm supp}(f)$. Given a nondecreasing function $f$ on $[0,\infty)$,
$f^{-1}$ denotes the inverse function of $f$, defined precisely as

(1.1) $f^{-1}(y)=\inf\\{x\geq 0:f(x)\geq y\\}.$

For each differentiable function $f$ defined on ${\mathbb{R}}$, $f^{\prime}$
denotes the first derivative of $f$. For each function $f(t,x)$ defined on
${\mathbb{R}}\times{\mathbb{R}}^{n}$, we will use both $f_{x}$ and
$\partial_{x}f$ to denote the partial derivatives of $f$ with respect to $x$,
and likewise, both $f_{t}$ and $\partial_{t}f$ to denote the partial
derivatives of $f$ with respect to $t$. We use ${\bf 1}$ to denote the
function that is identically equal to $1$. We will mostly be interested in the
case when $E=[0,H)$ and $E=[0,H)\times{\mathbb{R}}_{+}$, for some
$H\in(0,\infty]$. To distinguish these cases, we will usually use $\psi$ to
denote generic functions on $[0,H)$ and $\varphi$ to denote generic functions
on $[0,H)\times{\mathbb{R}}_{+}$. By some abuse of notation, given $\psi$ on
$[0,H)$, we will sometimes also treat it as a function on
$[0,H)\times{\mathbb{R}}_{+}$ that is constant in the second variable.

We use ${\mathcal{P}}(E)$ and ${\mathcal{M}}(E)$ to denote, respectively, the
space of Radon measures on a metric space $E$, endowed with the Borel
$\sigma$-algebra, and let ${\mathcal{M}}_{F}(E)$ denote the subspace of finite
measures in ${\mathcal{M}}(E)$, and ${\mathcal{M}}^{c}_{F}(E)$ the subspace of
continuous measures (i.e., measures that do not charge points) in
${\mathcal{M}}_{F}(E)$. The symbol $\delta_{x}$ will be used to denote the
measure with unit mass at the point $x$ and, with some abuse of notation, we
will use ${\bf 0}$ to denote the identically zero Radon measure on $E$. When
$E$ is an interval, say $[0,H)$, for notational conciseness, we will often
write ${\mathcal{M}}_{F}[0,H)$ or ${\mathcal{M}}_{F}^{c}[0,H)$ instead of
${\mathcal{M}}_{F}([0,H))$ or ${\mathcal{M}}_{F}^{c}([0,H))$, respectively.
For any Borel measurable function $\psi:[0,H)\rightarrow{\mathbb{R}}$ that is
integrable with respect to $\xi\in{\mathcal{M}}[0,H)$, we often use the short-
hand notation

$\langle\psi,\xi\rangle:=\int_{[0,H)}\psi(x)\,\xi(dx),$

and likewise, for any Borel measurable function
$\varphi:[0,H)\times[0,\infty)\rightarrow{\mathbb{R}}$ and $t>0$ such that
$x\mapsto\varphi(\cdot,t)$ is integrable with respect to
$\xi\in{\mathcal{M}}[0,H)$, we often use the short-hand notation

$\langle\varphi(\cdot,t),\xi\rangle:=\int_{[0,H)}\varphi(\cdot,t)d\xi=\int_{[0,H)}\varphi(x,t)\,\xi(dx).$

We also let ${\mathcal{P}}(E)$ denote the space of probability measures on
$E$, equipped with the Borel $\sigma$-algebra.

For any measure $\mu\in{\mathcal{M}}_{F}[0,H)$, we define

(1.2) $F^{\mu}(x):=\mu[0,x],\quad x\in[0,H),$

and we define $(F^{\mu})^{-1}$ to be its right-continuous inverse:

(1.3) $(F^{\mu})^{-1}(y)=\inf\\{x>0:F^{\mu}(x)\geq y\\}.$

Also, given $\mu,\mu_{t},t\in[0,\infty),$ in ${\mathcal{M}}_{F}[0,H)$, we will
use the notation $\mu_{t}\Rightarrow\mu$ to denote weak convergence:

$\lim_{t\rightarrow\infty}\langle\psi,\mu_{t}\rangle=\langle\psi,\mu\rangle,\quad\forall\psi\in{\mathcal{C}}_{b}[0,H).$

We will also on occasion use the total variation distance on
${\mathcal{M}}_{F}[0,H)$, denote by
$d_{\text{TV}}(\mu,\nu):=2\sup_{A\in{\mathcal{F}}}|\mu(A)-\nu(A)|$, where
${\mathcal{F}}$ is the Borel $\sigma$-algebra on $[0,H)$.

Given a Polish space $\mathcal{H}$, Let
${\mathcal{D}}_{\mathcal{H}}[0,\infty)$ denote the space of
$\mathcal{H}$-valued, càdlàg functions on $[0,\infty)$ and
${\mathcal{I}}_{{\mathbb{R}}_{+}}[0,\infty)$ denote the subset of non-
decreasing functions $f\in{\mathcal{D}}_{{\mathbb{R}}_{+}}[0,\infty)$ with
$f(0)=0$. Let ${\mathcal{D}}^{+}_{{\mathbb{R}}^{J}}({\mathbb{R}}_{+})$ denote
the subset of functions in
${\mathcal{D}}_{{\mathbb{R}}^{J}}({\mathbb{R}}_{+})$ that are nonnegative and
nondecreasing componentwise.

##  2\. Fluid Equations and Random Fixed Points

###  2.1. Fluid Equations

We now describe the fluid equations introduced in [24]. Let $G^{s}$ and
$G^{r}$ denote the cumulative distribution functions of the service time and
patience time distributions, respectively. Throughout, we make the following
standing assumptions on $G^{s}$ and $G^{r}$ and let $\bar{G}^{s}=1-G^{s}$ and
$\bar{G}^{r}=1-G^{r}$ denote the corresponding complementary cumulative
distribution functions. Recall that we abbreviate lower semicontinuous as lsc.

######  Assumption 2.1.

The cumulative distribution functions $G^{r}$ and $G^{s}$ satisfy
$G^{r}(0+)=G^{s}(0+)=0$, and are both absolutely continuous on $[0,\infty)$
with densities $g^{r}$ and $g^{s}$ that satisfy the following properties:

  1. (1)

The mean patience and service times are finite: in particular,

(2.1)
$\theta^{r}:=\int_{[0,H^{r})}xg^{r}(x)\,dx=\int_{[0,\infty)}\bar{G}^{r}(x)\,dx<\infty,$

and, we normalize units so that

(2.2) $\int_{[0,\infty)}xg^{s}(x)\,dx=\int_{[0,H^{s})}\bar{G}^{s}(x)\,dx=1,$

where

(2.3) $\displaystyle H^{s}$ $\displaystyle:=$
$\displaystyle\sup\\{x\in[0,\infty):G^{s}(x)<1\\},$ (2.4) $\displaystyle
H^{r}$ $\displaystyle:=$ $\displaystyle\sup\\{x\in[0,\infty):G^{r}(x)<1\\},$

denote the right-end of the supports of the measures corresponding to $G^{s}$
and $G^{r}$, respectively.

  2. (2)

There exists $\bar{H}^{s}<H^{s}$ such that $h^{s}:=g^{s}/\bar{G}^{s}$ is
either bounded or lsc on $(\bar{H}^{s},H^{s})$, and likewise, there exists
$\bar{H}^{r}<H^{r}$ such that $h^{r}:=g^{r}/\bar{G}^{r}$ is either bounded or
lsc on $(\bar{H}^{r},H^{r})$.

######  Remark 2.2.

Strictly speaking, $g^{s}$ and $g^{r}$ (and thus $h^{s}$ and $h^{r}$) are
determined only almost everywhere. The convention implicitly adopted in the
above statement is that $h^{s}$ (respectively, $h^{r}$) is almost everywhere
(a.e.) equal to a function from ${\mathbb{R}}_{+}$ to itself that is bounded
or lsc.

At any time $t\geq 0$, the state of the fluid system is represented by a
triplet $(X(t),\nu_{t},\eta_{t})$, where $X(t)$ represents the total mass of
jobs in system at time $t$, including those in queue and those in service,
$\nu_{t}$ is the fluid age measure, which is a sub-probability measure on
$[0,H^{s})$ that assigns to any interval $(a,b)$ the (limiting) fraction of
servers for whom the job currently in service has been in service for a number
of time units lying in that interval, and $\eta_{t}$ is the fluid potential
queue measure, which is a finite measure on $[0,H^{r})$ that to any interval
$(a,b)\subset[0,\infty)$ assigns the mass (or normalized limit) of jobs that
have arrived by time $t$ and whose patience lies in that interval
(irrespective of whether or not they have entered service or departed the
system by time $t$). Note that the total fraction of busy servers at time $t$
is $1-\langle{\bf 1},\nu_{t}\rangle$, which is zero if $X(t)\geq 1$ and
$1-X(t)$, otherwise. This is captured succinctly by the relation
$1-\langle{\bf 1},\nu_{t}\rangle=[1-X(t)]^{+}$.

The input data for the fluid equations includes the arrival rate $\lambda$,
and the initial conditions, consisting of the total initial mass in system,
and the initial (fluid) age and potential queue measures. Then the space of
possible initial conditions for the fluid equations is given by

(2.5)
${\mathfrak{S}}:=\left\\{\begin{array}[]{cc}&(\tilde{x},\tilde{\nu},\tilde{\eta})\in{\mathbb{R}}_{+}\times{\mathcal{M}}_{F}[0,H^{s})\times{\mathcal{M}}_{F}[0,H^{r}):\\\
&1-\langle{\bf 1},\tilde{\nu}\rangle=[1-\tilde{x}]^{+}\end{array}\right\\}.$

We now give a precise formulation of the fluid equations introduced in [24]
with $E(t)=E^{\lambda}(t):=\lambda t$ for $t\geq 0$ therein. These equations
will also involve the queue process $Q(t)$, which represents the total mass in
queue (awaiting service) at time $t$, and the non-decreasing processes
$D(t),K(t)$, $S(t)$ and $R(t)$ represent, respectively, the cumulative mass of
departures from the queue, entry into service, and respectively, the
potentially reneged and actually reneged jobs from the queue in the interval
$[0,t]$.

######  Definition 2.3.

(Fluid Equations) Given hazard rate functions $h^{r}$ and $h^{s}$, the càdlàg
function $(X,\nu,\eta)$ defined on $[0,\infty)$ and taking values in
${\mathbb{R}}_{+}\times{\mathcal{M}}_{F}[0,H^{s})\times{\mathcal{M}}_{F}[0,H^{r})$
is said to solve the fluid equations with arrival rate $\lambda\geq 0$ and
initial condition $(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$ if for every
$t\in[0,\infty)$, we have

(2.6) $S(t):=\int_{0}^{t}\langle h^{r},\eta_{s}\rangle\,ds<\infty,\qquad
D(t):=\int_{0}^{t}\langle h^{s},\nu_{s}\rangle\,ds<\infty,$

and the following relations are satisfied: for every
$\varphi\in{\mathcal{C}}^{1}_{c}([0,H^{s})\times{\mathbb{R}}_{+})$,

$\displaystyle\displaystyle\langle\varphi(\cdot,t),\nu_{t}\rangle$
$\displaystyle=$
$\displaystyle\displaystyle\langle\varphi(\cdot,0),\nu_{0}\rangle+\int_{0}^{t}\langle\varphi_{s}(\cdot,s)+\varphi_{x}(\cdot,s),\nu_{s}\rangle\,ds$
$\displaystyle\quad-\displaystyle\int_{0}^{t}\langle
h^{s}(\cdot)\varphi(\cdot,s),\nu_{s}\rangle\,ds+\int_{0}^{t}\varphi(0,s)\,dK(s),$

where

(2.8) $K(t)=\langle{\bf 1},\nu_{t}\rangle-\langle{\bf 1},\nu_{0}\rangle+D(t);$

for every $\varphi\in{\mathcal{C}}^{1}_{c}([0,H^{r})\times{\mathbb{R}}_{+})$,

$\displaystyle\displaystyle\langle\varphi(\cdot,t),\eta_{t}\rangle$
$\displaystyle=$
$\displaystyle\displaystyle\langle\varphi(\cdot,0),\eta_{0}\rangle+\int_{0}^{t}\langle\varphi_{s}(\cdot,s)+\varphi_{x}(\cdot,s),\eta_{s}\rangle\,ds$
$\displaystyle\quad-\displaystyle\int_{0}^{t}\langle
h^{r}(\cdot)\varphi(\cdot,s),\eta_{s}\rangle\,ds+\lambda\int_{0}^{t}\varphi(0,s)\,ds;$

with the non-idling constraint

(2.10) $1-\langle{\bf 1},\nu_{t}\rangle=[1-X(t)]^{+},$

where

(2.11) $\displaystyle X(t)$ $\displaystyle=$ $\displaystyle X(0)+\lambda
t-D(t)-R(t),$

with

(2.12) $\displaystyle R(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\left(\int_{0}^{Q(s)}h^{r}((F^{\eta_{s}})^{-1}(y))dy\right)ds,$

where recall $F^{\eta_{t}}(x):=\eta_{t}[0,x],$ and $(F^{\eta_{t}})^{-1}$
denotes the right-continuous inverse defined in (1.3), and

(2.13) $\displaystyle Q(t)$ $\displaystyle=$ $\displaystyle X(t)-\langle{\bf
1},\nu_{t}\rangle,$

with $Q$ also satisfying the inequality constraint

(2.14) $\displaystyle Q(t)$ $\displaystyle\leq$ $\displaystyle\langle{\bf
1},\eta_{t}\rangle.$

######  Remark 2.4.

Note that if $(X,\nu,\eta)$ solves the fluid equations with arrival rate
$\lambda$, and initial condition $(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$,
then we also have $(X(t),\nu_{t},\eta_{t})\in{\mathfrak{S}}$ for every $t>0$.
It is also true that if $\eta_{0}\in{\mathcal{M}}_{F}^{c}[0,H^{r})$, then we
also have $\eta_{t}\in{\mathcal{M}}_{F}^{c}[0,H^{r})$ for every $t>0$ (this
follows from the expression for $\eta_{t}$ in (2.18) below, from which it is
clear that if $\eta_{0}$ does not charge points then neither does $\eta_{t}$).

Also, note from (2.13) and (2.10) that for each $t\in[0,\infty)$,

(2.15) $Q(t)=[X(t)-1]^{+}.$

For future use, we also observe that (2.8), (2.13) and (2.11), when combined,
show that for every $t\in[0,\infty)$,

(2.16) $Q(0)+\lambda t=Q(t)+K(t)+R(t),$

which is simply a mass conservation equation. In addition, we will find it
convenient to define

(2.17) $B(t):=\langle{\bf 1},\nu_{t}\rangle,\quad t\geq 0,$

which represents the limiting fraction of busy servers.

######  Remark 2.5.

Given a solution $(X,\nu,\eta)$, we will refer to $(D,K,R,S,Q,B)$ as auxiliary
processes.

We now provide an informal, intuitive explanation for the form of the fluid
equations. Note that $\nu_{s}(dx)$ represents the amount of mass (or fraction
of servers) that are processing jobs whose ages lie in the range $[x,x+dx)$ at
time $s$, and $h^{s}(x)$ represents the conditional mean rate at which the
mass of jobs with age in $[x,x+dx)$ completes service at time $s$. Hence, in
(2.6), $\langle h^{s},\nu_{s}\rangle$ represents the departure rate of mass
from the fluid system due to services at time $s$, and its integral, $D(t)$,
is the cumulative departure rate due to service completion in the interval
$[0,t]$. By an exactly analogous reasoning, the other quantity
$S(t)=\int_{0}^{t}\langle h^{r},\eta_{s}\rangle ds$ in (2.6) represents the
cumulative potential reneging from the system in the interval $[0,t]$.
However, the actual reneging rate is restricted to abandonments of those in
queue. Since entries into the queue take place in the order of arrival, the
age of the oldest (equivalently, head-of-the line) mass in the fluid queue is
$\bar{a}_{s}:=(F^{\eta_{s}})^{-1}(Q(s))$, so that
$\eta_{s}[0,\bar{a}_{s}]=Q(s)$. Here, recall $F^{\eta_{s}}$ represents the
cumulative distribution function of $\eta_{s}$. Thus, the actual reneging rate
at any time $s$ only counts the mass reneging from the potential queue measure
$\eta_{s}$ whose age lies in the restricted interval $[0,\bar{a}_{s}]$, rather
than the entire interval $[0,\infty)$. A standard change of variable then
yields the expression in (2.12). Next, recalling the interpretations of the
quantities $K$, $R$ and $Q$ stated prior to Definition 2.3, note that
equations (2.8), (2.13) and (2.11) are simply mass conservation equations, and
(2.10) represents a non-idling condition that ensures that no server can idle
when there is work in the queue. Moreover, the inequality (2.14) expresses the
constraint that at any time $t$, the fluid queue is bounded by the total mass
of the fluid potential queue measure, since the latter also includes mass that
may have already gone into service (and possibly also departed the system) by
that time, provided its patience time exceeds the total time elapsed since
arrival. Finally, equations (2.3) and (2.3) govern the evolution of the fluid
age measure $\nu$ and potential queue measure $\eta$, respectively. In
particular, the second term on the right-hand-side of (2.3) represents the
change in $\langle\varphi,\nu\rangle$ over the interval $[0,t]$ due to
transport or shift of the ages at unit rate to the right, the third term
accounts for changes due to departure of mass from the system due to service,
and the last term captures changes due to new entry into system, which are
driven by the function $K$, the cumulative entry into service. The equation
(2.3) is exactly analogous, but with $h^{r}$ and the cumulative arrivals
$E^{\lambda}$ into the system in place of $h^{s}$ and $K$, respectively, and
the third term on the right-hand side now representing departure of mass from
the system due to potential reneging.

We now state a result that was proved in [23, 25]. Recall the definition of
the space ${\mathfrak{S}}$ given in (2.5).

######  Theorem 2.6.

Suppose Assumption 2.1 holds and fix $\lambda\geq 0$, and
$(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$. Then there is at most one solution
to the fluid equations with arrival rate $\lambda$ and initial condition
$(X(0),\nu_{0},\eta_{0})$, and if $\eta_{0}\in{\mathcal{M}}_{F}^{c}[0,H^{r})$,
then there also exists a continuous solution
$(X,\nu,\eta)=\\{(X(t),\nu_{t},\eta_{t}),t\geq 0\\}$ with arrival rate
$\lambda$ and initial condition $(X(0),\nu_{0},\eta_{0})$. Moreover, given any
solution $(X,\nu,\eta)=\\{(X(t),\nu_{t},\eta_{t}),t\geq 0\\}$ to the fluid
equations associated with $\lambda$ and
$(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$, the following properties hold:
(i) for any bounded or nonnegative measurable function $\psi$ on $[0,\infty)$
and for $\psi=h^{r}$, for every $t\geq 0$,

(2.18)
$\int_{[0,H^{r})}\psi(x)\eta_{t}(dx)=\int_{[0,H^{r})}\psi(x+t)\frac{\bar{G^{r}}(x+t)}{\bar{G^{r}}(x)}\eta_{0}(dx)+\int_{0}^{t}\psi(s)\bar{G^{r}}(s)\lambda
ds;$

(ii) for any bounded or nonnegative measurable function $\psi$ on $[0,\infty)$
and for $\psi=h^{s}$, for every $t\geq 0$,

(2.19)
$\int_{[0,H^{s})}\psi(x)\nu_{t}(dx)=\int_{[0,H^{s})}\psi(x+t)\frac{\bar{G^{s}}(x+t)}{\bar{G^{s}}(x)}\nu_{0}(dx)+\int_{0}^{t}\psi(t-s)\bar{G^{s}}(t-s)dK(s),$

with $K$ equal to the auxiliary process defined in (2.8) of the fluid
equations;
(iii) if $Q$ and $B$ are the associated auxiliary processes defined in (2.13)
and (2.17) respectively, then $K$ is an absolutely continuous function and for
a.e. $t\geq 0$, the derivative $K^{\prime}$ of $K$ satisfies

(2.20) $K^{\prime}(t)=k(t):=\left\\{\begin{array}[]{rl}\lambda&\mbox{ if
}B(t)<1,\\\ \lambda=\langle h^{s},\nu_{t}\rangle&\mbox{ if }B(t)=1\mbox{ and
}Q(t)=0,\\\ \langle h^{s},\nu_{t}\rangle&\mbox{ if }B(t)=1\mbox{ and
}Q(t)>0.\end{array}\right.$

###### Proof.

Uniqueness of the solution to the fluid equations follows from Theorem 3.5 of
[23] since $(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$ implies
$(E^{\lambda},X(0),\nu_{0},\eta_{0})$ lies in the space ${\mathcal{S}}_{0}$
therein, where recall $E^{\lambda}(t)=\lambda t$. Likewise, existence of a
solution with arrival rate $\lambda$ and initial condition
$(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$ with
$\eta_{0}\in{\mathcal{M}}_{F}^{c}[0,H^{r})$ can be deduced from Theorem 3.6 of
[23], once we justify that the conditions of that theorem are satisfied in the
present setting. First, it is not hard to see that for any arrival rate
$\lambda\geq 0$ and initial condition
$(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$ with
$\eta_{0}\in{\mathcal{M}}_{F}^{c}[0,H^{r})$ one can construct a sequence of
$N$-server systems with Poisson $(N\lambda)$ arrival process $E^{N}$ and
initial condition $(X^{N}(0),\nu^{(N)}_{0},{\eta}^{(N)}_{0})$ such that
Assumption 3.1 of [23] is satisfied. Second, note that since $\eta_{0}$ is a
continuous measure, and $E^{\lambda}$ is continuous, Assumption 3.2 of [23] is
also satisfied. Finally, Assumption 3.3 of [23] is a direct consequence of
Assumption 2.1 of this paper, and thus the application of Theorem 3.6 of [23]
is justified.

We now turn to estabishing the properties of any solution $(X,\nu,\eta)$ to
the fluid equations. First, note that the forms of both (2.3) and (2.3) are
analogous to that of (4.2) in [25], and therefore the integrability conditions
in (2.6) imply that (4.1) of [25] holds. Thus, (2.18) and (2.19) for
$\psi\in{\mathcal{C}}_{c}[0,H^{r})$ and $\psi\in{\mathcal{C}}_{c}[0,H^{s})$,
respectively, follow from Theorem 4.1 of [25]. By using a standard
approximation argument, namely representing indicators of finite open
intervals in ${\mathbb{R}}_{+}$ as monotone limits of continuous functions
with compact support and appealing to the monotone class theorem, it follows
that both equations in fact hold for any bounded measurable or nonnegative
measurable $\psi$. In particular, these equations also hold with $\psi=h^{r}$
in (2.18) and $\psi=h^{s}$ in (2.19). The latter fact is used several times in
this paper.

We now turn to the proof of property (iii), which is similar in spirit to
(3.12) of [25] and Corollary 3.7 of [23]. We supply the details for
completeness. First, note that $D,S$ and $R$ are absolutely continuous by
definition. Thus, (2.11) shows that $X$ is absolutely continuous, which then
implies that $B(t)=\langle{\bf 1},\nu_{t}\rangle=\min(X(t),1)$ is also
absolutely continuous. In turn, by (2.8) and (2.13), this implies that $K$ is
absolutely continuous. Further, (2.8), (2.16), (2.6) and (2.17) show that for
a.e. $t>0$,

(2.21)
$K^{\prime}(t)=\lambda-Q^{\prime}(t)-\int_{0}^{Q(t)}h^{r}((F^{\eta_{t}})^{-1}(y))dy,\quad\mbox{
and }\quad K^{\prime}(t)=B^{\prime}(t)+\langle h,\nu_{t}\rangle.$

We now recall the following standard fact that given an absolutely continuous
function and a set $A$ on which it is constant, the derivative of the function
is zero for almost every $t\in A$. Thus, for almost every $t$ in the set where
$B$ is constant, we have $B^{\prime}(t)=0$, and (2.21) implies
$K^{\prime}(t)=\langle h,\nu_{t}\rangle$. On the other hand, for almost every
$t$ when $Q(t)=0$, it follows that $Q^{\prime}(t)=0$ and hence, by (2.21) that
$K^{\prime}(t)=\lambda$. Thus, when both $B(t)=1$ and $Q(t)=0$,
$K^{\prime}(t)=\lambda=\langle h,\nu_{t}\rangle$. The remaining claims in
(2.20) then follow from the observations that when $B(t)<1$, one has $Q(t)=0$
and when $Q(t)>0$, one has $B(t)$ equal to the constant one, both of which are
easily deduced from (2.10) and (2.13). ∎

We now state a simple result on the action of time-shifts on solutions to the
fluid equations. To state the result, which was formulated as Lemma 3.4 of
[24], we will need the following notation: for any $t\in[0,\infty)$, define

$\begin{array}[]{rclrclrclr}K^{[t]}&:=&K(t+\cdot)-K(t),\qquad\qquad
X^{[t]}&:=&X(t+\cdot),\qquad\qquad\nu^{[t]}&:=&\nu_{t+\cdot},\\\
R^{[t]}&:=&R(t+\cdot)-R(t),\qquad\qquad\quad\eta^{[t]}&:=&\eta_{t+\cdot},\qquad\qquad\quad
Q^{[t]}&:=&Q(t+\cdot).\end{array}$

######  Lemma 2.7 (Lemma 3.4 of [24]).

Suppose Assumption 2.1 holds. Suppose
$(X,\nu,\eta)=\\{(X(s),\nu_{s},\eta_{s}),s\geq
0\\}\in{\mathcal{D}}_{{\mathfrak{S}}}[0,\infty)$ solves the fluid equations
with arrival rate $\lambda$ and initial condition
$(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$, then for any $t>0$,
$(X^{[t]},\nu^{[t]},\eta^{[t]})$ solves the fluid equations with arrival rate
$\lambda$ and initial condition $(X(t),\nu_{t},\eta_{t})\in{\mathfrak{S}}$,
but with $K,R$ and $Q$ replaced with $K^{[t]},R^{[t]}$ and $Q^{[t]}$,
respectively.

As in [24], we leave the proof to the reader, since it can be verified by just
rewriting the fluid equations and invoking the uniqueness result stated in
Theorem 2.6.

###  2.2. Invariant States and Random Fixed Points of the Fluid Equations

Let $\nu_{*}$ and $\eta_{*}$ be Borel probability measures on $[0,\infty)$
defined as follows:

(2.22) $\displaystyle\nu_{*}[0,x)$ $\displaystyle:=$
$\displaystyle\int_{0}^{x}\bar{G}^{s}(y)\,dy,\qquad x\in[0,H^{s}),$ (2.23)
$\displaystyle\eta_{*}[0,x)$ $\displaystyle:=$
$\displaystyle\int_{0}^{x}\bar{G}^{r}(y)\,dy,\qquad x\in[0,H^{r}).$

Note that $\nu_{*}$ and $\eta_{*}$ are well defined due to Assumption 2.1. For
$\lambda\geq 1$, define the set ${\mathcal{X}}_{\lambda}$ as follows:

(2.24)
${\mathcal{X}}_{\lambda}:=\left\\{x\in[1,\infty):G^{r}\left(\left(F^{\lambda\eta_{*}}\right)^{-1}\left((x-1)^{+}\right)\right)=\frac{\lambda-1}{\lambda}\right\\},$

and let

$x_{l}^{\lambda}:=\inf\left\\{x\in[1,\infty):\
x\in{\mathcal{X}}_{\lambda}\right\\}\qquad\mbox{and}\qquad
x_{r}^{\lambda}:=\sup\left\\{x\in[1,\infty):\
x\in{\mathcal{X}}_{\lambda}\right\\}.$

By (2.23), the map $x\to\eta_{*}[0,x)$ is strictly increasing on $[0,H^{r})$,
and therefore $(F^{\lambda\eta_{*}})^{-1}$ is continuous. Since $G^{r}$ is
also continuous, we have
${\mathcal{X}}_{\lambda}=[x_{l}^{\lambda},x_{r}^{\lambda}]$ is non-empty. Let
${\mathcal{I}}_{\lambda}$ be the invariant manifold for the fluid equations,
defined by

(2.25)
${\mathcal{I}}_{\lambda}:=\left\\{\begin{array}[]{ll}\left\\{(\lambda,\lambda\nu_{*},\lambda\eta_{*})\right\\}&\mbox{
if }\lambda<1,\\\
\left\\{(x_{*},\nu_{*},\lambda\eta_{*}):x_{*}\in{\mathcal{X}}_{\lambda}\right\\}&\mbox{
if }\lambda\geq 1.\end{array}\right.$

Our study of the critical and super-critical regimes will be carried out under
the following additional assumption on the invariant manifold
${\mathcal{I}}_{\lambda}$.

######  Assumption 2.8.

The set ${\mathcal{I}}_{\lambda}$ has a single element, which we express as
$z_{*}^{\lambda}=(x_{*}^{\lambda},(\lambda\wedge 1)\nu_{*},\lambda\eta_{*})$,
where $x_{*}^{\lambda}=\lambda$ when $\lambda<1$ and $x_{*}^{\lambda}$ is the
unique element of ${\mathcal{X}}_{\lambda}$ when $\lambda\geq 1$.

Note that Assumption 2.8 imposes a non-trivial restriction only when
$\lambda\geq 1$. As stated in Lemma 3.1 of [24], a sufficient condition for
Assumption 2.8 to hold when $\lambda>1$ is for the equation
$G^{r}(x)=(\lambda-1)/\lambda$ to have a unique solution.

Whereas Assumption 2.8 guarantees a unique deterministic fixed point for the
fluid equations, to understand the large-time limits of the fluid equations,
it turns out to be important to also understand the collection of random fixed
points, defined below. We first introduce the notion of a solution to the
fluid equations when the input data is random.

######  Definition 2.9.

Given any ${\mathfrak{S}}$-valued random element $(X(0),\nu_{0},\eta_{0})$
defined on some probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$, we
say the càdlàg ${\mathfrak{S}}$-valued stochastic process
$(X,\nu,\eta)=\\{(X(t),\nu_{t},\eta_{t}),t\geq 0\\}$ is a solution to the
fluid equations with arrival rate $\lambda$ and random initial condition
$(X(0),\nu_{0},\eta_{0})$ if for each $\omega\in\Omega$, the function
$(X(\omega),\nu(\omega),\eta(\omega))=\\{(X(t,\omega),\nu_{t}(\omega),\eta_{t}(\omega)),t\geq
0\\}$ solves the fluid equations with arrival rate $\lambda$ and initial
condition $(X(0,\omega),\nu_{0}(\omega),\eta_{0}(\omega))$.

######  Definition 2.10.

For $\lambda>0$, a probability measure $\mu$ on ${\mathfrak{S}}$ is said to be
a random fixed point of the fluid equations with arrival rate $\lambda$ if
given any ${\mathfrak{S}}$-valued random element
$(\tilde{X},\tilde{\nu},\tilde{\eta})$ whose law is $\mu$, there exists a
solution $(X,\nu,\eta)$ to the fluid equations with arrival rate $\lambda$ and
random initial condition $(\tilde{X},\tilde{\nu},\tilde{\eta})$ such that for
each $t\geq 0$, the law of $(X(t),\nu_{t},\eta_{t})$ is equal to $\mu$.

######  Remark 2.11.

Under our assumptions, a random fixed point always exists. Indeed, it follows
from Theorem 5.5 of [24] that the set ${\mathcal{I}}_{\lambda}$ in (2.25)
describes the so-called invariant manifold (or collection of deterministic
fixed points) of the fluid equations. Since for $\lambda\geq 0$,
${\mathcal{X}}_{\lambda}$ is always non-empty, an immediate consequence is
that for any $z\in{\mathcal{I}}_{\lambda}$, the measure $\delta_{z}$ is a
random fixed point of the fluid equations with arrival rate $\lambda$.
Moreover, under Assumption 2.8, $\delta_{z^{\lambda}_{*}}$ is the only random
fixed point that is degenerate (i.e., which concentrates all its mass on one
point). A key question we address in this article is to determine conditions
under which this is in fact the only random fixed point with arrival rate
$\lambda$. As shown in Proposition 4.16 below, a sufficient condition for this
to hold is that any solution $(X,\nu,\eta)$ to the fluid equations with
arrival rate $\lambda$ and initial condition
$(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$ and auxiliary process
$B=\langle{\bf 1},\nu\rangle$ satisfies $\eta_{t}\Rightarrow\lambda\eta_{*}$
and $B_{t}\rightarrow\lambda\wedge 1$, as $t\rightarrow\infty$.

##  3\. Assumptions and Main Results

We now state our main results, which require the following additional
condition on the service distribution.

######  Assumption 3.1.

The cumulative distribution function $G^{s}$ of the service distribution has a
density $g^{s}$ and the hazard rate function $h^{s}=g^{s}/\bar{G}^{s}$
satisfies one of the following:

  1. (1)

The quantities $\varepsilon_{h}:=\operatorname*{ess\,inf}_{x\geq 0}h^{s}(x)>0$
and $c_{h}:=\operatorname*{ess\,sup}_{x\geq 0}h^{s}(x)<\infty$.

  2. (2)

The function $h^{s}$ is decreasing.

The second part of the above assumption should be understood in the sense of
Remark 2.2, namely $h^{s}$ is a.e. equal to a decreasing function from
$[0,H^{s})$ to ${\mathbb{R}}_{+}$. Note that under both parts of the
assumption, the hazard rate function $h^{s}$ has a finite essential supremum.
Since the hazard rate function of any distribution is only locally integrable
and never integrable on its support, both Assumptions 3.1(1) and 3.1(2) imply
$H^{s}=\infty$.

######  Theorem 3.2.

Suppose Assumption 2.1 holds, and $\nu_{*}$ and $\eta_{*}$ are as defined in
(2.22) and (2.23). Also, suppose $(X,\nu,\eta)$ solves the fluid equations
with arrival rate $\lambda$ and initial condition
$(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$, with auxiliary processes
$(D,K,R,S,Q,B)$ as in Remark 2.5. Then the following is true:

  1. (1)

When $\lambda<1$, it follows that
$(X_{t},\nu_{t},\eta_{t})\rightarrow(\lambda,\lambda\nu_{*},\lambda\eta_{*})$
as $t\rightarrow\infty$. In particular, $\delta_{z_{*}^{\lambda}}$, with
$z_{*}^{\lambda}=(\lambda,\lambda\nu_{*},\lambda\eta_{*})$, is the unique
random fixed point of the fluid equations with arrival rate $\lambda$.

  2. (2)

When $\lambda>1$, and Assumption 3.1 is also satisfied, then
$\eta_{t}\Rightarrow\lambda\eta_{*}$ as $t\to\infty$ and

    1. (a)

there exists $T<\infty$ such that

(3.1) $B(t)=\langle{\bf 1},\nu_{t}\rangle=1,\quad\mbox{ for all }t\geq T,$

and

(3.2) $\nu_{t}\Rightarrow\nu_{*}\mbox{ and }\langle h^{s},\nu_{t}\rangle\to
1\quad\mbox{ as }t\rightarrow\infty,$

with the convergence in (3.2) also holding in total variation when Assumption
3.1(1) holds;

    2. (b)

if, in addition, Assumption 2.8 is also satisfied (with $z_{*}^{\lambda}$ as
defined therein), then $\delta_{z_{*}^{\lambda}}$ is the unique random fixed
point of the fluid equations with arrival rate $\lambda$.

  3. (3)

If $\lambda=1$ and Assumption 3.1(2) is satisfied, then
$\eta_{t}\Rightarrow\lambda\eta_{*}$ and $B(t)\to 1$ as $t\to\infty$. If, in
addition, Assumption 2.8 holds (with $z_{*}^{1}$ as defined therein), then
$\delta_{z_{*}^{1}}$ is the unique random fixed point of the fluid equations
with arrival rate $1$.

###### Proof.

The proof of statement (1) is given in Section 4.1, and for all $\lambda\geq
0$, the (weak) convergence of $\eta_{t}$ to $\lambda\eta_{*}$ under Assumption
3.1 follows from Lemma 4.1. Now, when $\lambda>1$, (3.1) and (3.2) follow from
Proposition 4.8 and Remark 4.7 when Assumption 3.1(1) holds, and from
Proposition 4.12 and Lemma 4.15 when Assumption 3.1(2) holds. Further, when
$\lambda=1$ the convergence $B(t)\rightarrow 1$ under Assumption 3.1(2) also
follows from Proposition 4.12. Lastly, the uniqueness results for the random
fixed point stated in (2b) and (3) follow from the convergence results in (2a)
and (3) and Proposition 4.16.

∎

######  Remark 3.3.

The main application of Theorem 3.2 is to characterize the limit of the scaled
stationary distributions of the sequence of $N$-server measure-valued state
processes, and thereby (partially) fix a technical flaw in the convergence
result stated in Theorem 3.3 of [24].

To explain this in greater detail, let
$\bar{Z}^{(N)}_{*}:=(\bar{X}^{(N)}_{*},\bar{\nu}^{(N)}_{*},\bar{\eta}^{(N)}_{*})$
have the law of the stationary distribution of the measure-valued $N$-server
state dynamics of an $N$-server queue with reneging introduced in [23], when
the scaled arrival process is given by $\bar{E}^{(N)}$ (e.g., Poisson with a
scaled arrival rate $\lambda^{(N)}>0$). Existence of such a stationary
distribution was established in Theorem 7.1 of [24]. Also, let
$\bar{Z}^{(N)}:=(\bar{X}^{(N)},\bar{\nu}^{(N)},\bar{\eta}^{(N)})$ represent
the dynamics of the fluid-scaled measure-valued state representation of the
$N$-server queue with initial condition $\bar{Z}^{(N)}(0)=\bar{Z}^{(N)}_{*}$.
Then, under the assumption that $\bar{E}^{(N)}$ converges weakly to
$E^{\lambda}$ for some $\lambda>0$, tightness of the sequence
$\\{\bar{Z}^{(N)}_{*}:=(\bar{X}^{(N)}_{*},\bar{\nu}^{(N)}_{*},\bar{\eta}^{(N)}_{*})\\}_{N\in{\mathbb{N}}},$
was established in Theorem 6.2 of [24]. Let
$\bar{Z}=(\bar{X},\bar{\nu},\bar{\eta})$ denote any subsequential limit. We
now claim that then (the law of) $\bar{Z}$ must be a random fixed point of the
fluid equations with arrival rate $\lambda$. To see why the claim is true, we
invoke the fluid limit theorem established in Theorem 3.6 of [23], to conclude
that for any $t>0$, the $N$-server fluid-scaled state process
$\bar{Z}^{(N)}(t)$ (initialized at the stationary distribution
$\bar{Z}^{(N)}_{*}$) converges weakly to $Z(t)$, where $Z:=(X,\nu,\eta)$
solves the fluid equations with arrival rate $\lambda$ and initial condition
$\bar{Z}$. However, for any $t>0$, since by stationarity $\bar{Z}^{(N)}(t)$
has the same law as $\bar{Z}^{(N)}_{*}$, it follows that the laws of their
corresponding weak limits, $Z(t)$ and $\bar{Z}$, must also coincide. By
Definition 2.10, this proves the claim that $\bar{Z}$ is a random fixed point.

In the proof of Theorem 3.3 (of Section 6.2) in [24], it was assumed without
justification that $\bar{Z}$ is deterministic, and that was used to conclude
that $\bar{Z}$ must belong to the invariant manifold ${\mathcal{I}}_{\lambda}$
(see Remark 2.11). When combined with Assumption 2.8, this leads to the
conclusion that $\bar{Z}=z_{*}^{\lambda}$, thus showing that all subsequential
limits coincide, and hence, that $z_{*}^{\lambda}$ is the weak limit of the
original stationary sequence $(\bar{Z}^{(N)}_{*})_{N\in{\mathbb{N}}}$.
However, one cannot assume a priori that $\tilde{Z}$ is deterministic, and, as
argued above, one only knows that any subsequential limit is a random fixed
point. To make this argument complete, which was one of the main motivations
of this paper, one needs to show that there is precisely one random fixed
point, namely the one concentrated at $z_{*}^{\lambda}$. Theorem 3.2 does
precisely this for the class of service distributions satisfying Assumption
3.1, thus closing the gap in the proof of the convergence result in [24] (for
service distributions in that class). However, this still leaves the open
question of whether this result remains true for a larger class of service
distributions, in particular the entire class considered in [24].

######  Remark 3.4.

Further, a related ancillary goal of this work is to determine whether the
diagram in Figure 1 below commutes under general convergence conditions on the
initial states (essentially Assumption 3.1 of [23]). Referring to the same
notation as used in Remark 3.3, the top horizontal arrow in Figure 1 holds due
to ergodicity of the $N$-server state dynamics, which was established in
Theorem 7.1 of [24] under some additional conditions on the service and
reneging distributions (see Assumption 7.1 therein). On the other hand, as
already mentioned in Remark 3.3, the left vertical arrow follows from the
fluid limit theorem Theorem 3.6 of [23] (under suitable convergence
assumptions on the initial data).

$\begin{array}[]{lcr}\bar{Z}^{(N)}(t)=(\bar{X}^{(N)}(t),\bar{\nu}^{(N)}_{t},\bar{\eta}^{(N)}_{t})&\stackrel{{\scriptstyle\mbox{\tiny{Thm
7.1 of
\cite[cite]{[\@@bibref{}{KanRam12}{}{}]}}}}}{{\Longrightarrow}}&\qquad\bar{Z}^{(N)}_{*}=(\bar{X}^{(N)}_{*},\bar{\nu}^{(N)}_{*},\bar{\eta}^{(N)}_{*})\\\
\\\ \qquad\qquad\,\big{\lVert}&&\big{\rVert}\\\ \qquad\mbox{\tiny{Thm 3.6 of
\cite[cite]{[\@@bibref{}{KanRam10}{}{}]}}}&&\mbox{\tiny{Thm 6.2 of
\cite[cite]{[\@@bibref{}{KanRam12}{}{}]} and {\bf Thm 3.2}}}\\\
\qquad\qquad\big{\Downarrow}&&\big{\Downarrow}\\\ \\\
Z(t)=(X(t),\nu_{t},\eta_{t})&\stackrel{{\scriptstyle\mbox{\tiny{{\bf Thm
3.2}}}}}{{\longrightarrow}}&\qquad z_{*}=(x_{*},(\lambda\wedge
1)\nu_{*},\lambda\eta_{*})\end{array}$ Figure 1. Interchange of Limits Diagram

Along with the tightness of $(\bar{Z}^{(N)}_{*})_{N\in{\mathbb{N}}}$
established in [24], Theorem 3.2 of the present article completes the diagram
by establishing (for a class of service distributions) the right vertical
arrow (as explained in Remark 3.3) as well as the bottom horizontal arrow,
though the latter only when $\lambda\neq 1$ (i.e., in the subcritical and
supercritical regimes). It would be worthwhile in the future to investigate
whether this result can be extended further, in particular to establish
converence even in the critical regime $\lambda=1$, possibly under additional
conditions such as a finite second moment condition, like that imposed in
Theorem 3.9 of [25] (to study large-time behavior of fluid limits in the
absence of reneging).

##  4\. Proof of Theorem 3.2

We assume throughout this section that Assumption 2.1 holds. We then have the
following elementary lemma.

######  Lemma 4.1.

Fix $\lambda\geq 0$ and, given any $\eta_{0}\in{\mathcal{M}}_{F}[0,H^{r})$,
let $\eta=(\eta_{t})_{t\geq 0}$ be the solution to (2.3). Then
$\eta_{t}\Rightarrow\lambda\eta_{*}$ as $t\rightarrow\infty$.

###### Proof.

Fix $\psi\in{\mathcal{C}}_{b}({\mathbb{R}}_{+})$. In view of (2.18), the
boundedness of $\psi$, the finiteness of the measure $\eta_{0}$ the dominated
convergence theorem and the fact that
$\bar{G}^{r}(x+t)/\bar{G}^{r}(x)\rightarrow 0$ for every $x\in[0,H^{r})$ as
$t\rightarrow\infty$, together imply that the first term on the right-hand
side of (2.18) vanishes. On the other hand, since the mean patience time
$\int_{0}^{\infty}\bar{G}^{r}(s)ds$ is finite, the dominated convergence
theorem shows that the last term on the right-hand side of (2.18) converges to
$\langle\psi,\lambda\eta_{*}\rangle$. This concludes the proof that
$\eta_{t}\Rightarrow\lambda\eta_{*}$ as $t\rightarrow\infty$. ∎

###  4.1. Proof in the Subcritical Regime

In this section we prove part (1) of Theorem 3.2. Fix $\lambda\in(0,1)$ and
$(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$. Suppose $(X,\nu,\eta)$ is a
solution to the fluid equations, and let $(D,K,R,S,Q,B)$ be the corresponding
auxiliary processes.

The weak convergence of $\eta_{t}$ to $\lambda\eta_{*}$ as $t\to\infty$
follows from Lemma 4.1. We now analyze the remaining components of the
solution. Using the definition of $D$ from (2.6), setting $\psi=h^{s}$ in
(2.19), interchanging the order of integration, using integration by parts and
the fact $G^{s}(0+)=0$, we obtain

$\displaystyle D(t)=\displaystyle\int_{0}^{t}\langle h^{s},\nu_{s}\rangle\,ds$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\left(\int_{[0,H^{s})}\frac{g^{s}(x+s)}{\bar{G}^{s}(x)}\nu_{0}(dx)+\int_{[0,s]}g^{s}(s-u)dK(u)\right)ds$
$\displaystyle=$
$\displaystyle\int_{[0,H^{s})}\frac{G^{s}(t+x)-G^{s}(x)}{\bar{G}^{s}(x)}\nu_{0}(dx)+\int_{[0,t]}G^{s}(t-u)dK(u)$
$\displaystyle=$
$\displaystyle\int_{[0,H^{s})}\frac{G^{s}(t+x)-G^{s}(x)}{\bar{G}^{s}(x)}\nu_{0}(dx)+\int_{0}^{t}K(s)g^{s}(t-s)ds.$

Substituting this in (2.11), using (2.16) and (2.12), and performing repeated
integration by parts, we obtain

$\displaystyle X(t)$ $\displaystyle=$ $\displaystyle X(0)+\lambda
t-\int_{[0,H^{s})}\frac{G^{s}(t+x)-G^{s}(x)}{\bar{G}^{s}(x)}\nu_{0}(dx)-\int_{0}^{t}K(s)g^{s}(t-s)ds-R(t)$
$\displaystyle=$ $\displaystyle X(0)+\lambda
t-\int_{[0,H^{s})}\frac{G^{s}(t+x)-G^{s}(x)}{\bar{G}^{s}(x)}\nu_{0}(dx)$
$\displaystyle\qquad-\int_{0}^{t}(Q(0)+\lambda s-Q(s)-R(s))g^{s}(t-s)ds-R(t)$
$\displaystyle=$ $\displaystyle
X(0)-Q(0)G^{s}(t)-\int_{[0,H^{s})}\frac{G^{s}(t+x)-G^{s}(x)}{\bar{G}^{s}(x)}\nu_{0}(dx)+\int_{0}^{t}Q(s)g^{s}(t-s)ds$
$\displaystyle\qquad+\lambda\int_{0}^{t}\bar{G}^{s}(t-s)ds+\int_{0}^{t}R(s)g^{s}(t-s)ds-R(t)$
$\displaystyle=$ $\displaystyle
X(0)-Q(0)G^{s}(t)-\int_{[0,H^{s})}\frac{G^{s}(t+x)-G^{s}(x)}{\bar{G}^{s}(x)}\nu_{0}(dx)+\int_{0}^{t}Q(s)g^{s}(t-s)ds$
$\displaystyle\qquad+\int_{0}^{t}\left(\lambda-\int_{0}^{Q(s)}h^{r}((F^{\eta_{s}})^{-1}(y))dy\right)\bar{G}^{s}(t-s)ds,$

which implies that for each $t\geq 0$,

(4.3) $X(t)\leq
X(0)-Q(0)G^{s}(t)-\int_{[0,H^{s})}\frac{G^{s}(t+x)-G^{s}(x)}{\bar{G}^{s}(x)}\nu_{0}(dx)+\lambda\int_{0}^{t}\bar{G}^{s}(u)du+\int_{0}^{t}Q(s)g^{s}(t-s)ds.$

We now make use of the following simple observation.

######  Lemma 4.2.

$\limsup_{t\rightarrow\infty}\int_{0}^{t}Q(s)g^{s}(t-s)ds\leq\limsup_{t\rightarrow\infty}Q(t).$

###### Proof.

Let $q:=\limsup_{t\rightarrow\infty}Q(t)$. Then for each $\epsilon>0$, there
exists $T_{\epsilon}<\infty$ such that $Q(t)\leq q+\epsilon$ for all $t\geq
T_{\epsilon}$. So for each $t>T_{\epsilon}$, it follows that

$\displaystyle\int_{0}^{t}Q(s)g^{s}(t-s)ds$ $\displaystyle=$
$\displaystyle\int_{0}^{T_{\epsilon}}Q(s)g^{s}(t-s)ds+\int_{T_{\epsilon}}^{t}Q(s)g^{s}(t-s)ds$
$\displaystyle\leq$ $\displaystyle\left(\sup_{0\leq s\leq
T_{\epsilon}}Q(s)\right)(G^{s}(t)-G^{s}(t-T_{\epsilon}))+(q+\epsilon)G^{s}(t-T_{\epsilon}).$

By taking the limit supremum as $t\rightarrow\infty$ of both sides, we have
$\limsup_{t\rightarrow\infty}\int_{0}^{t}Q(s)g^{s}(t-s)ds\leq q+\epsilon$. The
lemma follows on taking $\epsilon\rightarrow 0$. ∎

Continuing with the proof of Theorem 3.2(1), taking the limit supremum in
(4.3), and using Lemma 4.2, the identity $\int_{0}^{\infty}G^{s}(u)du=1$ from
Assumption 2.1, the fact that
$\lim_{t\rightarrow\infty}(G^{s}(t+x)-G^{s}(x))/\bar{G}^{s}(x)\rightarrow 1$
for every $x$, the bounded convergence theorem and the identity
$X(0)=Q(0)+\langle{\bf 1},\nu_{0}\rangle$ from (2.13), we obtain

(4.4)
$\limsup_{t\rightarrow\infty}X(t)\leq\lambda+\limsup_{t\rightarrow\infty}\int_{0}^{t}Q(s)g^{s}(t-s)ds\leq\lambda+\limsup_{t\rightarrow\infty}Q(t).$

We now claim that there exists $T^{\prime}<\infty$ such that $\langle
1,\nu_{t}\rangle<1$ for all $t\geq T^{\prime}$. We argue by contradiction to
prove the claim. If the claim is false, note that for any $T^{\prime}<\infty$,
there would exist $T>T^{\prime}$ such that $\langle 1,\nu_{T}\rangle=1$. Then,
due to (2.15), we would have
$\limsup_{t\rightarrow\infty}X(t)=\limsup_{t\rightarrow\infty}Q(t)+1$, which
contradicts (4.4) since $\lambda<1$. Thus, fix $T^{\prime}<\infty$ as in the
claim. Then, by Lemma 2.7,
$(X^{[T^{\prime}]},\nu^{[T^{\prime}]},\eta^{[T^{\prime}]})$ solves the fluid
equations with arrival rate $\lambda$ and initial condition
$(X(T^{\prime}),\nu_{T^{\prime}},\eta_{T^{\prime}})$ and hence, (2.19) holds
with $\nu$ and $K$ replaced with $\nu^{[T^{\prime}]}$ and $K^{[T^{\prime}]}$,
respectively. Since $\nu^{T^{\prime}}(t)=\nu_{T^{\prime}+t}$ and by (2.15),
(2.12) and (2.16), $Q(T^{\prime}+\cdot)\equiv 0$,
$R^{[T^{\prime}]}(\cdot)\equiv 0$ and
$K^{T^{\prime}}(t)=K(T^{\prime}+t)-K(T^{\prime})=\lambda t,$ $t\geq 0$, this
implies that for every $\psi\in{\mathcal{C}}_{b}[0,H^{s})$,

$\int_{[0,H^{s})}\psi(x)\nu_{T^{\prime}+t}(dx)=\int_{[0,H^{s})}\psi(x+t)\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{T^{\prime}}(dx)+\int_{0}^{t}\psi(t-s)\bar{G}^{s}(t-s)\lambda
ds.$

Then, arguing as in the proof of Lemma 4.1, sending $t\rightarrow\infty$, and
invoking the bounded convergence theorem, the first integral on the right-hand
side vanishes, and the second integral converges to
$\lambda\int_{[0,H^{s})}\psi(x)\bar{G}^{s}(x)dx$. Recalling that
$\nu_{*}(dx)=\bar{G}^{s}(x)dx$ and $\int_{0}^{\infty}\bar{G}^{s}ds=1$ from
Assumption 2.1, it follows that $\nu_{t}\Rightarrow\lambda\nu_{*}$. In turn,
by the continuous mapping theorem this implies $\langle{\bf
1},\nu_{t}\rangle\Rightarrow\lambda$ as $t\rightarrow\infty$. When combined
with (2.10) and the fact that $\lambda<1$, this implies that as
$t\rightarrow\infty$, the weak limits of $X(t)$ and $\langle{\bf
1},\nu_{t}\rangle$ coincide and are equal to $\lambda$. This concludes the
proof of the first assertion of Theorem 3.2(1).

Now, if the initial condition $(X(0),\eta_{0},\nu_{0})$ had the law $\mu$ of a
random fixed point with arrival rate $\lambda<1$, then the convergence just
established would imply that ${\mathbb{P}}(\eta_{0}=\lambda\eta_{*})=1$ and
${\mathbb{P}}(\nu_{0}=\lambda\nu_{*})=1$. By the continuous mapping theorem,
the latter implies that almost surely $\langle{\bf
1},\nu_{0}\rangle=\langle{\bf 1},\lambda\nu_{*}\rangle=\lambda$. Since
$\lambda<1$, it then follows from (2.10) that $X(0)=\lambda$ almost surely,
thus proving that $\mu=\delta_{z_{*}^{\lambda}}$ with
$z_{*}^{\lambda}=(\lambda,\lambda\eta_{*},\lambda\nu_{*})$. This completes the
proof of Theorem 3.2(1).

###  4.2. Proof of Theorem 3.2(2) when the hazard rate function is bounded
away from zero and infinity.

In this section we prove Theorem 3.2(2a) under Assumption 3.1(1). Fix
$\lambda>1$, suppose Assumption 2.1 is satisfied, Assumption 2.8 holds (with
$z_{*}^{\lambda}=(x_{*}^{\lambda},\nu_{*},\lambda\nu_{*})$ denoting the unique
element of ${\mathcal{I}}_{\lambda}$) and Assumption 3.1(1) holds (with
associated positive constants $\varepsilon_{h}>0$, $c_{h}<\infty$). For
notational convenience, we shall denote by $f^{*}(x)=\bar{G}^{s}(x)$ the
density of $\nu^{*}$. Note that the lower bound on $h^{s}$ implies that
$g^{s}$, and thus $f^{*}=\bar{G}^{s}$, is strictly positive on $(0,\infty)$.

Now, fix the initial condition $(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$, and
suppose $(X,\nu,\eta)$ is the associated solution to the fluid equations. We
will establish convergence, as $t\rightarrow\infty$, of the fluid age measure
$\nu_{t}$ described by (2.3) using an extended relative entropy functional in
a manner reminiscent of a Lyapunov function. Recall that ${\mathcal{P}}(E)$
denotes the space of probability measures on a measurable space $E$, and for a
finite measure $P$ on $E$, define the functional
$R:(P\|\cdot):{\mathcal{P}}(E)\mapsto(-\infty,\infty]$ by

(4.5)
$R(P\|Q):=\left\\{\begin{array}[]{ll}\displaystyle\int_{E}\log\frac{dP}{dQ}(x)dP(x)&\mbox{
if }P\ll Q,\\\ \infty&\mbox{ otherwise, }\end{array}\right.$

where $P\ll Q$ means $P$ is absolutely continuous with respect to $Q$ and we
use the convention $0\log 0=0$. We emphasize that we do not require $P$ to be
a probability measure, as we will often have to deal with sub-probability
measures, but when both $P$ and $Q$ are probability measures, this is simply
the relative entropy functional.

######  Remark 4.3.

If $c_{P}=P(E)>0$ denotes the total mass of $P$, then writing the above
integral as $\int_{E}\frac{dP}{dQ}\log\frac{dP}{dQ}dQ$ and using the convexity
of $x\mapsto x\log x$ on $(0,\infty)$ gives the lower bound

(4.6) $R(P\|Q)\geq c_{P}\log c_{P},$

which is attained by $P$ that is a constant multiple of the probability
measure $Q$. In particular, $R(P\|Q)$ may assume negative values. However,
when $P$ is a probability measure, $R(P\|Q)$ is always nonnegative and
$R(P\|Q)=0$ holds if and only if $P=Q$.

The proof of Theorem 3.2(2) will make use of the following properties of the
extended relative entropy functional.

######  Lemma 4.4.

Suppose $P$ and $Q$ are finite measures on ${\mathbb{R}}_{+}$, equipped with
the Borel $\sigma$-algebra, with $c_{P}:=P({\mathbb{R}}_{+})>0$ and
$Q({\mathbb{R}}_{+})=1$. If $P$ and $Q$, respectively, have densities $p$ and
$q$ (with respect to Lebesgue measure), then

(4.7)
$\int_{0}^{\infty}|p(x)-q(x)|dx\leq|c_{P}-1|+\Big{(}2c_{P}^{-1}|R(P\|Q)|+2|\log
c_{P}|\Big{)}^{1/2}.$

###### Proof.

First note that $c_{P}^{-1}P$ and $Q$ are probability measures, and so,
invoking Pinsker’s inequality (see, e.g., [15], p. 44) in the second
inequality below, we obtain

$\displaystyle\int_{0}^{\infty}|p(x)-q(x)|dx$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}|p(x)-c_{P}^{-1}p(x)|dx+\int_{0}^{\infty}|c_{P}^{-1}p(x)-q(x)|dx$
$\displaystyle\leq$
$\displaystyle|c_{P}-1|+\left(2R(c_{P}^{-1}P\|Q)\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle|c_{P}-1|+\left(2c_{P}^{-1}R(P\|Q)-2\log
c_{P}\right)^{1/2}$

which is clearly dominated by the right-hand side of (4.7). ∎

The second property is encapsulated in the following lemma, which crucially
relies on the lower bound on the hazard rate $h^{s}$, and whose proof is
relegated to Appendix A.

######  Lemma 4.5.

Let $f:[0,\infty)\mapsto[0,\infty)$ be a measurable function that satisfies
$\int_{0}^{\infty}fdx\leq 1$, suppose
$z_{f}:=\int_{0}^{\infty}h^{s}fdx<\infty$ and $\mu^{f}$ is the measure with
density $f$. Then

(4.8) $\int_{0}^{\infty}h^{s}(x)f(x)\log\frac{f(x)}{f^{*}(x)}dx-z_{f}\log
z_{f}\geq\varepsilon_{h}\int_{0}^{\infty}f(x)\log\frac{f(x)}{f^{*}(x)}dx=\varepsilon_{h}R(\mu^{f}\|\nu_{*}).$

The proof of Theorem 3.2(2) is somewhat involved and given in Section 4.2.2.
To help make some of those calculations more transparent, first in Section
4.2.1 we carry out some formal calculations (under more stringent conditions)
to provide intuition into why the extended relative entropy functional
$R(\cdot\|\nu^{*})$ may be a good candidate Lyapunov function for the problem
at hand (see also Remark 4.6).

####  4.2.1. A Formal Calculation

Observe that equation (2.3) characterizes $(\nu_{t})_{t\geq 0}$ as a weak
solution to a transport equation. Now, for the purposes of this formal
calculation only, suppose that $\nu_{0}$ has a density, denoted by $f_{0}$,
and for each $t>0$, suppose the measure $\nu_{t}$ has a sufficiently smooth
density, denoted by $f(x,t)$, $x\geq 0$. For conciseness, below we will use
$f(\cdot,t)$ to denote the function $x\mapsto f(x,t)$. Then by (2.17) and
(2.6), $\langle{\bf 1},\nu_{t}\rangle=\int_{0}^{\infty}f(\cdot,t)dx$ and
$\langle h,\nu_{t}\rangle=\int_{0}^{\infty}h^{s}f(\cdot,t)dx$, the transport
equation could be formally rewritten as the following partial differential
equation (PDE):

(4.9) $\partial_{t}f(x,t)=-\partial_{x}f(x,t)-h^{s}(x)f(x,t),\quad x>0,t>0,$

with the boundary condition $f(0,t)=K^{\prime}(t)$, which by (2.20), takes the
form

(4.10) $f(0,t)=\begin{cases}\lambda&\text{if
}\displaystyle\int_{0}^{\infty}f(\cdot,t)dx<1,\\\\[8.99994pt]
\displaystyle\int_{0}^{\infty}h^{s}f(\cdot,t)dx&\text{if
}\displaystyle\int_{0}^{\infty}f(\cdot,t)dx=1,\end{cases}$

and the initial condition

(4.11) $f(x,0)=f_{0}(x),\quad x>0.$

Proceeding with purely formal calculations to gain intuition, note that
$f^{*}=e^{-J}$, where $J(x):=\int_{0}^{x}h^{s}(y)dy<\infty$ for every $x>0$.
For $t>0$, define

$r_{t}:=R(\nu_{t}\|\nu^{*})=\int_{0}^{\infty}f(\cdot,t)\log\frac{f(\cdot,t)}{f^{*}}dx=\int_{0}^{\infty}f(\cdot,t)(\log
f(\cdot,t)+J)dx.$

Taking derivatives of both sides of the last equation with respect to $t$, and
using (4.9), we see that

$\displaystyle\frac{d}{dt}r_{t}$
$\displaystyle=\int_{0}^{\infty}\partial_{t}f(\cdot,t)(\log f(\cdot,t)+J+1)dx$
$\displaystyle=-\int_{0}^{\infty}(\partial_{x}f(\cdot,t)+h^{s}f(\cdot,t))(\log
f(\cdot,t)+J+1)dx.$

Since $f(\cdot,t)$ is integrable and $H^{s}=\infty$, it follows that
$\liminf_{x\rightarrow\infty}f(x,t)=0$. Using integration by parts, and
assuming (without justification) that $\lim_{x\rightarrow\infty}f(x,t)(\log
f(x,t)+J(x))=0$, we conclude that

$\displaystyle\int_{0}^{\infty}(\partial_{x}f(\cdot,t)(\log f(\cdot,t)+J+1)dx$
$\displaystyle=-f(0,t)(\log
f(0,t)+1)-\int_{0}^{\infty}f(\cdot,t)\left(\frac{\partial_{x}f(\cdot,t)}{f(\cdot,t)}+h^{s}\right)dx$
$\displaystyle=-f(0,t)\log f(0,t)-\int_{0}^{\infty}h^{s}f(\cdot,t)dx,$

On combining the last two equations, and recalling that $J=-\log f^{*}$, we
obtain

$\displaystyle\frac{d}{dt}r_{t}$ $\displaystyle=f(0,t)\log
f(0,t)-\int_{0}^{\infty}h^{s}f(\cdot,t)(\log f(\cdot,t)+J)dx$
$\displaystyle=f(0,t)\log
f(0,t)-\int_{0}^{\infty}h^{s}f(\cdot,t)\log\frac{f(\cdot,t)}{f^{*}}dx.$

Since $\int_{0}^{\infty}f(\cdot,t)dx=\langle{\bf 1},\nu_{t}\rangle\leq 1$ and
for almost every $t\in[0,\infty)$, (2.6) implies that
$\int_{0}^{\infty}hf(\cdot,t)dx<\infty$ for such $t$. we can apply the
estimate (4.8) from Lemma 4.5 with $f=f(\cdot,t)$ to obtain

$\int_{0}^{\infty}h^{s}f(\cdot,t)\log\frac{f(\cdot,t)}{f^{*}}dx\geq\left(\int_{0}^{\infty}h^{s}f(\cdot,t)dx\right)\log\left(\int_{0}^{\infty}h^{s}f(\cdot,t)dx\right)+\varepsilon_{h}\int_{0}^{\infty}f(\cdot,t)\log\frac{f(\cdot,t)}{f^{*}}dx.$

Substituting this into the previous display and using the boundary condition
(4.10), we have

(4.12)
$\frac{d}{dt}r_{t}\leq\begin{cases}-\varepsilon_{h}r_{t}+\lambda\log\lambda-\left(\int_{0}^{\infty}h^{s}f(\cdot,t)dx\right)\log\left(\int_{0}^{\infty}h^{s}f(\cdot,t)dx\right)&\text{if
}\int_{0}^{\infty}f(\cdot,t)dx<1,\\\ -\varepsilon_{h}r_{t}&\text{if
}\int_{0}^{\infty}f(\cdot,t)dx=1.\end{cases}$

This estimate does not directly imply the convergence of $r_{t}$ to zero.
However, the fact that it takes the form
$\frac{dr_{t}}{dt}\leq-\varepsilon_{h}r_{t}$ in the case
$\int_{0}^{\infty}f(\cdot,t)dx=1$ is a sign that the approach might be useful,
especially in the supercritical case ($\lambda>1$), where one might expect
that for sufficiently large $t$, $\int_{0}^{\infty}f(\cdot,t)dx=1$. However,
translating this intuition into a proof is not straightforward. The rigorous
argument provided in the next section indeed derives a version of (4.12) (with
some extra error terms), and copes with the more complicated structure of the
estimate in the case $\int_{0}^{\infty}f(\cdot,t)dx<1$, as well as the fact
that $r_{t}$ can go negative.

######  Remark 4.6.

Note that the PDE (4.9)-(4.10) has some similarities with the age-structured
model in equation (3) of [32], with $\nu=0$ and $d=b=h^{s}$, except that the
boundary condition (4.10) is more complicated. In particular, it is
discontinuous due to the appearance of the term $\lambda$ when
$\int_{0}^{\infty}f(\cdot,t)dx<1$. Furthermore, although $f^{*}$ can indeed be
seen as an eigenfunction corresponding to the eigenvalue $0$ of the stationary
equation (which corresponds to equation (7) of [32], again with $\nu=0$ and
$d=b=h^{s}$), since the hazard rate function $h^{s}$ is never integrable on
$[0,\infty)$, the solution to the dual equation (see (8) of [32]) appears not
to be well-defined. Thus, the results of [32] are not applicable to this
setting. Furthermore, a rigorous proof cannot in any case rely on an analysis
of the PDE because for general initial condition
$\nu_{0}\in{\mathcal{M}}_{F}[0,\infty)$, the measures $\nu_{t},t>0,$ need not
have densities, and even when they do, their densities have discontinuities in
both variables (these discontinuities will be apparent in the rigorous proof
in the next section). Nevertheless, along with the calculations given above,
this loose analogy further suggests that the extended relative entropy
functional may still serve as a Lyapunov function for the dynamics. That
verification of this property is non-trivial will be apparent on noting that
it requires additional conditions on $h^{s}$ and also a restriction to the
supercritical regime $\lambda>1$. In particular, it would be interesting to
see if the argument presented in the next section, or a modification thereof,
could relax conditions on $h^{s}$ to address a larger class of service
distributions, and also address the critical regime $\lambda=1$, which
currently we only address when the hazard rate function is decreasing (see
Proposition 4.12).

####  4.2.2. Proof of Theorem 3.2(2)

Fix $\lambda>1$ and recall the initial condition and associated solution
$(X,\nu,\eta)$ to the fluid equations. We start with the proof of part (a).
First, note that the limit $\eta_{t}\Rightarrow\lambda\eta_{*}$ in (3.1)
follows from Lemma 4.1. To establish the remaining limits, we begin with the
representation for the age measure $\nu_{t}$ given in (2.19), which shows that
$\nu_{t}=\theta_{t}+\mu_{t}$, where
$\theta_{t},\mu_{t}\in{\mathcal{M}}_{F}[0,\infty)$, are defined by

(4.13)
$\langle\psi,\theta_{t}\rangle:=\int_{[0,\infty)}\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\psi(x+t)\nu_{0}(dx)\quad\mbox{
and
}\quad\langle\psi,\mu_{t}\rangle:=\int_{0}^{\infty}\psi(x)\tilde{f}(x,t)dx,$

for every $\psi\in{\mathcal{C}}_{b}[0,\infty)$ and $\psi=h^{s}$, where for all
$t\geq 0$,

(4.14) $\tilde{f}(x,t):=\begin{cases}\bar{G}^{s}(x)k_{t-x}&x\in[0,t],\\\
0&x\in(t,\infty),\end{cases}$

where we recall that $k$, defined in (2.20), is a.e. equal to the derivative
$K^{\prime}$ of $K$.

Now, to estimate $d_{\text{TV}}(\mu_{t},\nu_{*})$, recall that
$f^{*}=\bar{G}^{s}$ is the density of $\nu^{*}$, and so both $\mu_{t}$ and
$\nu_{*}$ are absolutely continuous with respect to Lebesgue measure. Thus,
Lemma 4.4 shows that

(4.15) $d_{{\rm
TV}}(\mu_{t},\nu_{*})=\int_{0}^{\infty}\left|\tilde{f}(x,t)-f^{*}(x)\right|dx\leq|\langle{\bf
1},\mu_{t}\rangle-1|+\left(2\langle{\bf
1},\mu_{t}\rangle^{-1}|r_{t}|+2|\log\langle{\bf
1},\mu_{t}\rangle|\right)^{1/2},$

where, for $t\geq 0$,

(4.16) $\displaystyle r_{t}:=R(\mu_{t}\|\nu^{*})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\tilde{f}(x,t)\log\frac{\tilde{f}(x,t)}{f^{*}(x)}dx$
(4.17) $\displaystyle=$ $\displaystyle\int_{0}^{t}\bar{G}^{s}(t-x)k_{x}\log
k_{x}dx,$

with the last equality using the fact that $k_{t-x}=\tilde{f}(x,t)/f^{*}(x)$
due to (4.14). Since the expression in (2.20) and Assumption 3.1(1) show that
$k$ is strictly positive and bounded above by $\lambda\vee
c_{h}=\lambda\vee\sup_{x\in[0,\infty)}h(x)$, $r_{t}$ is well defined and
finite.

######  Remark 4.7.

Due to the pointwise convergence $\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\to
0$ as $t\to\infty$, the dominated convergence theorem shows that $\langle{\bf
1},\theta_{t}\rangle$, the total mass of $\theta_{t}$, converges to zero as
$t\to\infty$. Hence, $\theta_{t}$ converges to the zero measure in total
variation. Together with (4.15), it follows that in order to show
$B_{t}=\langle{\bf 1},\nu_{t}\rangle\to 1$ and
$d_{\text{TV}}(\nu_{t},\nu_{*})\to 0$ (and hence, $\nu_{t}\Rightarrow\nu_{*}$)
as $t\to\infty$, it suffices to prove that $\langle{\bf 1},\mu_{t}\rangle\to
1$ and $r_{t}\to 0$ as $t\rightarrow\infty$.

Our main goal in this section is to establish these limits.

######  Proposition 4.8.

Suppose Assumptions 2.1 and 3.1(1) hold, and $\lambda>1$. Then there exists
$T\in(0,\infty)$ such that $B(t)=1$ for all $t\geq T$. In addition,

(4.18) $\langle{\bf 1},\mu_{t}\rangle\to 1\quad\text{ and }\quad r_{t}\to
0,\quad\mbox{ as }t\to\infty,$

and also $\langle h,\nu_{t}\rangle\rightarrow 1$ as $t\rightarrow\infty$.

To establish this proposition, we proceed in several steps, establishing
various intermediate results in Steps 1–3, culminating in the proof of
Proposition 4.8 in Step 4.

##### Step 1.

We start with simple bounds on the measure $\theta_{t}$ defined in (4.13).
Recall the convention $0\log 0=0$.

######  Lemma 4.9.

We have $\sup_{t}\langle h^{s},\theta_{t}\rangle\leq c_{h}$,
$\int_{0}^{\infty}\langle h^{s},\theta_{t}\rangle dt<\infty$, and
$\int_{0}^{\infty}|\langle h^{s},\theta_{t}\rangle\log\langle
h^{s},\theta_{t}\rangle|dt<\infty$.

###### Proof.

Recall that $\theta_{t}=\nu_{t}-\mu_{t}$ is a nonnegative measure. Moreover,
substituting $\psi=h^{s}$ in (4.13), we have for each $t>0$,

(4.19) $\displaystyle\langle h^{s},\theta_{t}\rangle$
$\displaystyle=\int_{[0,\infty)}\frac{\bar{G}^{s}(x+t)h^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx).$

For the first assertion, note that for all $t\geq 0$, $\langle
h^{s},\theta_{t}\rangle\leq c_{h}\langle{\bf 1},\nu_{t}\rangle\leq c_{h}$. The
remaining claims will follow once we prove the following refinement of this
bound, namely, for all $t\geq 0$,

(4.20) $\langle h^{s},\theta_{t}\rangle\leq c_{h}e^{-\varepsilon_{h}t}.$

To see why this bound holds, first use the easily verifiable relation
$\bar{G}^{s}(y)=e^{-\int_{0}^{y}h^{s}(u)du}$ and the definition of
$\varepsilon_{h}$ to conclude that for all $x\geq 0$ and $t\geq 0$,
$\bar{G}^{s}(x+t)\leq\bar{G}^{s}(x)e^{-\varepsilon_{h}t}$. When substituted
into (4.19), this yields

$\langle h^{s},\theta_{t}\rangle\leq
e^{-\varepsilon_{h}t}\int_{[0,\infty)}h^{s}(x+t)\nu_{0}(dx)\leq
c_{h}e^{-\varepsilon_{h}t}\langle{\bf 1},\nu_{0}\rangle\leq
c_{h}e^{-\varepsilon_{h}t}.$

This proves (4.20) and completes the proof. ∎

##### Step 2.

We now obtain our main estimate on $r_{t}$ in Corollary 4.11, building off
preliminary estimates obtained in Lemma 4.10. In what follows, we will say
$(t_{1},t_{2})\subset[0,\infty)$ is a busy interval if $B_{t}=1$ for
$t\in(t_{1},t_{2})$, and say it is an excursion interval if $B_{t}<1$ for
$t\in(t_{1},t_{2})$ and $B_{t_{1}}=B_{t_{2}}=1$.

Let $m(\cdot)$ denote the modulus of continuity of the continuous function
$x\mapsto x\log x$ on the compact interval $[0,c_{h}]$. On $[0,e^{-1}]$ this
function is decreasing. Now, for $0\leq x<y\leq e^{-1}$, applying the
inequality $p\log p+(1-p)\log(1-p)\leq 0$ with $p=x/y$, we see that

$\displaystyle 0$
$\displaystyle\geq\frac{x}{y}\log\frac{x}{y}+\left(1-\frac{x}{y}\right)\log\left(1-\frac{x}{y}\right)=\frac{1}{y}[x\log
x+(y-x)\log(y-x)-y\log y].$

Hence, it follows that for $0\leq x<y\leq e^{-1}$,

$|x\log x-y\log y|=x\log x-y\log y\leq(x-y)\log(y-x)=|(x-y)\log(y-x)|.$

Moreover, in case $c_{h}>e^{-1}$, the function $x\mapsto x\log x$ is Lipschitz
on $[e^{-1},c_{h}]$. As a result, there is a constant $c_{1}$ (depending only
on $c_{h}$) such that

(4.21) $m(x)\leq|x\log x|+c_{1}x,\qquad x\in[0,c_{h}].$

######  Lemma 4.10.

For $t\geq 0$, define $\Upsilon_{t}:=m(\langle h^{s},\theta_{t}\rangle),$
where $\theta_{t}$ is defined by (4.13), and $m$ is the modulus of continuity
of $x\mapsto x\log x$, as defined above. If $(t_{1},t_{2})$ is a busy
interval, then

(4.22) $r_{t}\leq
r_{t_{1}}e^{-\varepsilon_{h}(t-t_{1})}+\int_{t_{1}}^{t}\Upsilon_{s}ds,\quad
t\in(t_{1},t_{2}).$

On the other hand, if $(t_{1},t_{2})$ is an excursion, then

(4.23) $r_{t_{2}}\leq r_{t_{1}}+\int_{t_{1}}^{t_{2}}\Upsilon_{s}ds,$

and

(4.24) $B^{\prime}(t)=\lambda-\langle h^{s},\nu_{t}\rangle,\qquad
t\in(t_{1},t_{2}).$

Furthermore, there exist finite positive constants $c_{r}$ and $c_{\rm lip}$
such that $\sup_{t}|r_{t}|\leq c_{r}$ and for any $0\leq s<t<\infty$,
$|r_{t}-r_{s}|\leq c_{\rm lip}|t-s|$, showing that the function $t\to r_{t}$
is globally Lipschitz on $[0,\infty)$.

###### Proof.

Note that although the function $\tilde{f}$ defined in (4.14) is discontinuous
in $t$ and in $x$, since $\bar{G}^{s}$ has a density, the relation (4.17)
shows that $r_{t}$ is differentiable (although not continuously
differentiable) with derivative

(4.25) $\displaystyle\frac{dr_{t}}{dt}$ $\displaystyle=$ $\displaystyle
k_{t}\log k_{t}-\int_{0}^{t}g^{s}(t-x)k_{x}\log k_{x}dx=k_{t}\log
k_{t}-\int_{0}^{t}g^{s}(x)k_{t-x}\log k_{t-x}dx.$

Substituting the identities $g^{s}=h^{s}\bar{G}^{s}=h^{s}f^{*}$ and
$k_{t-x}=\tilde{f}(x,t)/f^{*}(x)$ into (4.25), recalling the definition of
$\tilde{f}$ from (4.14) recalling the convention that $0\log 0=0$, and then
applying Lemma 4.5, with $f$ replaced with $\tilde{f}(\cdot,t)$, we obtain

$\displaystyle\frac{dr_{t}}{dt}$ $\displaystyle=k_{t}\log
k_{t}-\int_{0}^{\infty}h^{s}(x)\tilde{f}(x,t)\log\frac{\tilde{f}(x,t)}{f^{*}(x)}dx$
(4.26) $\displaystyle\leq k_{t}\log k_{t}-z_{\tilde{f}(\cdot,t)}\log
z_{\tilde{f}(\cdot,t)}-\varepsilon_{h}r_{t}^{+},$

where, as in Lemma 4.5,
$z_{\tilde{f}(\cdot,t)}=\int_{0}^{\infty}h^{s}(x)\tilde{f}(x,t)dx$, which is
equal to $\langle h^{s},\mu_{t}\rangle$ by (4.13).

Now, suppose that $(t_{1},t_{2})$ is a busy interval for some $0\leq
t_{1}<t_{2}\leq\infty$. Then by (2.20) and Assumption 3.1(1), for
$t\in(t_{1},t_{2})$, $c_{h}\geq k_{t}=\langle h^{s},\nu_{t}\rangle=\langle
h^{s},\mu_{t}\rangle+\langle h^{s},\theta_{t}\rangle$, which implies
$k_{t}-z_{\tilde{f}(\cdot,t)}=\langle h^{s},\theta_{t}\rangle\geq 0$. Since
$m$ is the modulus of continuity of $x\mapsto x\log x$ on the interval
$[0,c_{h}]$, it follows that

$|k_{t}\log k_{t}-z_{\tilde{f}(\cdot,t)}\log z_{\tilde{f}(\cdot,t)}|\leq
m(\langle h^{s},\theta_{t}\rangle)=\Upsilon_{t}.$

When combined with (4.26), this shows that for any busy interval
$(t_{1},t_{2})$,

(4.27)
$\frac{dr_{t}}{dt}\leq\Upsilon_{t}-\varepsilon_{h}r_{t}^{+}\leq\Upsilon_{t}-\varepsilon_{h}r_{t},\quad
t\in(t_{1},t_{2}).$

Now, let $\tilde{r}$ denote the solution to the differential equation
$d\tilde{r}_{t}/dt=\Upsilon_{t}-\varepsilon_{h}\tilde{r}_{t}$ with the same
initial condition as $r$, namely $\tilde{r}_{t_{1}}=r_{t_{1}}$. Then
$\tilde{r}$ can be solved explicitly:

$\tilde{r}_{t}=\tilde{r}_{t_{1}}^{-\varepsilon_{h}(t-t_{1})}+\int_{t_{1}}^{t}e^{-\varepsilon_{h}(s-t_{1})}\Upsilon_{t-s}ds\leq
r_{t_{1}}e^{-\varepsilon_{h}(t-t_{1})}+\int_{t_{1}}^{t}\Upsilon_{s}ds,\quad
t\in(t_{1},t_{2}).$

A simple comparison theorem for ordinary differential equations then shows
that $r_{t}\leq\tilde{r}_{t}$ for $t\in(t_{1},t_{2})$. This proves (4.22).

Next, consider an excursion interval $(t_{1},t_{2})$. Then (2.20) implies that
$k_{t}=\lambda$ for $t\in(t_{1},t_{2})$. Moreover, it is not hard to see that
the fluid age equation (2.3) holds with the test function $\varphi\equiv{\bf
1}$, by approximating this function by compactly supported test functions
whose derivatives in $x$ are bounded. Since $\varphi_{x}=\varphi_{t}=0$,
differentiating the equation yields (4.24). Toward showing (4.23), recall that
$\langle{\bf 1},\nu_{t}\rangle=B(t)$ and $B(t_{1})=B(t_{2})=1$ by definition
of an excursion interval. Hence, it follows that

(4.28) $\frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}}\langle h^{s},\nu_{t}\rangle
dt=\lambda.$

Also, recalling $\theta_{t}=\nu_{t}-\mu_{t}$, we have

$|\langle h^{s},\mu_{t}\rangle\log\langle h^{s},\mu_{t}\rangle-\langle
h^{s},\nu_{t}\rangle\log\langle h^{s},\nu_{t}\rangle|\leq m(\langle
h^{s},\nu_{t}\rangle-\langle h^{s},\mu_{t}\rangle)=\Upsilon_{t}.$

As a result, for $t\in(t_{1},t_{2})$, the right-hand side of (4.26) is bounded
above by $\lambda\log\lambda-\langle h^{s},\nu_{t}\rangle\log\langle
h^{s},\nu_{t}\rangle+\Upsilon_{t}$. Integrating both sides of (4.26) and using
(4.28) and the convexity of $x\log x$, we have

$r_{t_{2}}-r_{t_{1}}\leq(t_{2}-t_{1})\lambda\log\lambda-\int_{t_{1}}^{t_{2}}\left(\langle
h^{s},\nu_{t}\rangle\log\langle
h^{s},\nu_{t}\rangle-\Upsilon_{t}\right)dt\leq\int_{t_{1}}^{t_{2}}\Upsilon_{t}dt.$

We now turn to the last assertion of the lemma. The bound $0\leq k_{t}\leq
c_{h}\vee\lambda$ implies that $|k_{t}\log k_{t}|\leq c_{2}$ for some finite
constant $c_{2}$. The boundedness of $t\to r_{t}$ thus follows from (4.17) and
the fact that $\int_{0}^{\infty}\bar{G}^{s}(x)dx=1$ (see Assumption 2.1(1)).
By (4.25), the bound on $|k_{t}\log k_{t}|$ also implies that $dr_{t}/dt$ is
bounded, and hence, that $t\mapsto r_{t}$ is globally Lipschitz on
$[0,\infty)$. ∎

As a corollary, we obtain our main estimate on $r_{t}$. For $t>0$, define

(4.29) $L(t):=\int_{0}^{t}1_{\\{B(s)=1\\}}ds,\quad t>0,\quad\mbox{ and
}\quad{\mathcal{B}}:=\\{t>0:B(t)=1\\}.$

######  Corollary 4.11.

For every $s\geq 0$ and $t>s$, $t\in{\mathcal{B}}$,

(4.30) $r_{t}\leq
c_{r}e^{-\varepsilon_{h}(L(t)-L(s))}+\int_{s}^{t}\Upsilon_{\tau}d\tau,$

where $c_{r}$ is the constant from Lemma 4.10.

###### Proof.

Fix $s\geq 0$ and $t>s$, $t\in{\mathcal{B}}$. Denote $t_{0}:=\inf\\{u\geq
s:B_{u}=1\\}$. Fix a nonempty open interval $(s_{0},s_{1})\subset(t_{0},t)$.
Then $(s_{0},s_{1})$ is said to be a maximal busy interval if it is a busy
interval that is not a proper subset of any open busy interval contained in
$(t_{0},t)$. Further, $(s_{0},s_{1})\subset(t_{0},t)$ is referred to as
admissible if it is either an excursion or a maximal busy interval. Since $B$
is continuous it is clear that ${\mathcal{O}}:=\\{u\in(s,t):B_{u}<1\\}$ is an
open set, and hence, can be written as a countable union of open intervals.
Thus, there are at most a countable number of excursions. Since any maximal
busy interval must be contiguous to one of the intervals comprising
${\mathcal{O}}$, it follows that the collection of admissible intervals is
also countable. For $u>0$, define a $u$-admissible interval to be an
admissible interval whose length is at least $u$. Denote by
${\mathcal{T}}_{u}$ the complement in $(t_{0},t)$ of the union of all
$u$-admissible intervals. Then, as $u\to 0$, the Lebesgue measure
$|{\mathcal{T}}_{u}|$ of this set clearly converges to zero.

Let $u>0$ be given, and let $I_{u}$ be the number of $u$-admissible intervals.
Since there are only a finite number of such intervals, we can label the
intervals $(t_{n},t_{n}^{\prime})$, $n=1,\ldots,I_{u}$ in such a way that
$s\leq t_{0}\leq t_{1}<t_{2}<t_{3}<\cdots<t_{I_{u}}\leq t$. Let $c_{\rm lip}$
denote the (global) Lipschitz constant of $t\mapsto r_{t}$, which exists by
Lemma 4.10. We now show by induction that, for $n=1,2,\ldots,I_{u}$,

(4.31) $r_{t_{n}}\leq
r_{t_{1}}e^{-\varepsilon_{h}\sum_{i=1}^{n-1}(L(t_{i}^{\prime})-L(t_{i}))}+\sum_{i=1}^{n-1}\int_{t_{i}}^{t_{i}^{\prime}}\eta_{\tau}d\tau+c_{\rm
lip}\sum_{i=1}^{n-1}(t_{i+1}-t^{\prime}_{i}),$

where a sum with the upper limit less than the lower limit is taken to be
zero.

Base Case: For $n=1$, (4.31) reduces to the trivial inequality $r_{t_{1}}\leq
r_{t_{1}}$, and thus is satisfied.

Induction step: Assuming (4.31) holds for an arbitrary
$n\in\\{1,\ldots,I_{u}-1\\}$, we show it holds for $n+1$. From (4.22) and
(4.23) of Lemma 4.10, along with the fact that $L(t_{n}^{\prime})-L(t_{n})$ is
equal to zero if $(t_{n},t_{n}^{\prime})$ is an excursion, and is equal to
$t_{n}^{\prime}-t_{n}$ if it is a busy interval, we have

$r_{t_{n}^{\prime}}\leq
r_{t_{n}}e^{-\varepsilon_{h}(L(t_{n}^{\prime})-L(t_{n}))}+\int_{t_{n}}^{t_{n}^{\prime}}\Upsilon_{\tau}d\tau.$

Using this estimate, the Lipschitz continuity of $r_{t}$ established in Lemma
4.10 and the induction hypothesis, it follows that

$\displaystyle r_{t_{n+1}}$ $\displaystyle\leq r_{t_{n}^{\prime}}+c_{\rm
lip}(t_{n+1}-t_{n}^{\prime})$ $\displaystyle\leq
r_{t_{n}}e^{-\varepsilon_{h}(L(t_{n}^{\prime})-L(t_{n}))}+\int_{t_{n}}^{t^{\prime}_{n}}\Upsilon_{\tau}d\tau+c_{\rm
lip}(t_{n+1}-t^{\prime}_{n})$
$\displaystyle\leq\Big{(}r_{t_{1}}e^{-\varepsilon_{h}\sum_{i=1}^{n-1}(L(t_{i}^{\prime})-L(t_{i}))}+\sum_{i=1}^{n-1}\int_{t_{i}}^{t_{i}^{\prime}}\Upsilon_{\tau}d\tau+c_{\rm
lip}\sum_{i=1}^{n-1}(t_{i+1}-t^{\prime}_{i})\Big{)}e^{-\varepsilon_{h}(L(t_{n}^{\prime})-L(t_{n}))}$
$\displaystyle\qquad+\int_{t_{n}}^{t_{n}^{\prime}}\Upsilon_{\tau}d\tau+c_{\rm
lip}(t_{n+1}-t^{\prime}_{n})$ $\displaystyle\leq
r_{t_{1}}e^{-\varepsilon_{h}\sum_{i=1}^{n}(L(t_{i}^{\prime})-L(t_{i}))}+\sum_{i=1}^{n}\int_{t_{i}}^{t_{i}^{\prime}}\Upsilon_{\tau}d\tau+c_{\rm
lip}\sum_{i=1}^{n}(t_{i+1}-t^{\prime}_{i}).$

This proves (4.31) by induction.

Next, note that each of the intervals $(t_{0},t_{1})$,
$(t^{\prime}_{i},t_{i+1})$ for $i=1,\ldots,I_{u}-1$, and $(t_{I_{u}},t)$, is a
subset of ${\mathcal{T}}_{u}$. Hence, we have

$L(t)-L(t_{0})-\sum_{i=1}^{I_{u}-1}(L(t^{\prime}_{i})-L(t_{i}))\leq(L(t_{1})-L(t_{0}))+\sum_{i=1}^{I_{u}-1}(L(t_{i+1})-T(t_{i}^{\prime}))+(L(t)-L(t_{I_{u}}))\leq|{\mathcal{T}}_{u}|.$

Hence, on applying (4.31) with $n=I_{u}$, noting that the last term on the
right-hand side is bounded by $c_{\rm lip}|{\mathcal{T}}_{u}|$, we obtain on
taking $u\to 0$, $r_{t}\leq
c_{r}e^{-\varepsilon_{h}(L(t)-L(t_{0}))}+\int_{t_{0}}^{t}\Upsilon_{\tau}d\tau$.
Finally, since by definition $s\leq t_{0}$ and $L(t_{0})=L(s)$, the lemma
follows. ∎

##### Step 3.

We now prove that $\bar{L}:=\sup_{t}L(t)=\infty$. Note that this implies that
the “servers” become busy infinitely often, as one might expect in the
supercritical regime $\lambda>1$. (We will later use this to prove the stonger
condition that the complement of ${\mathcal{B}}$ is bounded.)

Arguing by contradiction, assume that $\bar{L}<\infty$. By (4.13), clearly
$\langle{\bf 1},\mu_{t}\rangle\leq\langle{\bf 1},\nu_{t}\rangle\leq 1$ and, by
Assumption 3.1(1), $\langle h,\nu_{t}\rangle\leq c_{h}$, for all $t>0$.
However, (4.13), (4.14) and (2.20) together implly

$\displaystyle\langle{\bf 1},\mu_{t}\rangle=\int_{0}^{\infty}\tilde{f}(x,t)dx$
$\displaystyle=\int_{0}^{t}\bar{G}^{s}(x)k_{t-x}dx$
$\displaystyle=\int_{0}^{t}\bar{G}^{s}(x)[\lambda 1_{\\{B_{t-x}<1\\}}+\langle
h^{s},\nu_{t-x}\rangle 1_{\\{B_{t-x}=1\\}}]dx$
$\displaystyle\geq\lambda\int_{0}^{t}\bar{G}^{s}(x)dx-\lambda\int_{0}^{t}\bar{G}^{s}(x)1_{\\{B_{t-x}=1\\}}dx.$

Moreover, it is also true that

$\displaystyle\int_{0}^{t}\bar{G}^{s}(x)1_{\\{B_{t-x}=1\\}}dx$
$\displaystyle\leq\int_{0}^{t/2}1_{\\{B_{t-x}=1\\}}dx+\int_{t/2}^{t}\bar{G}^{s}(x)dx$
$\displaystyle\leq(L(t)-L(t/2))+\int_{t/2}^{\infty}\bar{G}^{s}(x)dx.$

Recalling that $\int_{0}^{\infty}\bar{G}^{s}(x)dx=1$ (see Assumption 2.1), if
$\bar{L}<\infty$ the above expression converges to zero as
$t\rightarrow\infty$. Hence, $\liminf_{t\to\infty}\langle{\bf
1},\mu_{t}\rangle\geq\lambda>1$, which is a contradiction. This proves
$\bar{L}=\infty$.

##### Step 4.

We now combine the above results to prove Proposition 4.8.

###### Proof of Proposition 4.8..

We first claim that to establish (4.18), it suffices to show that $B(t)=1$ for
all sufficiently large $t$. Recalling that $\Upsilon=m(\langle
h^{s},\theta\rangle)$ is integrable on $[0,\infty)$ by Lemma 4.9 and the bound
(4.21) on $m$, and that $0\leq L(t)\to\infty$ as $t\to\infty$ by Step 3, which
implies ${\mathcal{B}}$ is unbounded, we can send first $t\to\infty$ along
${\mathcal{B}}$ and then $s\to\infty$ in (4.30) of Corollary 4.11, to obtain
$\limsup_{t\to\infty,\ t\in{\mathcal{B}}}r_{t}\leq 0$. We cannot directly
deduce from this that the limit of $r_{t}$ along ${\mathcal{B}}$ is zero,
since $r_{t}=R(\mu_{t}\|\nu_{*})$ could be negative. However, for
$t\in{\mathcal{B}}$, $B(t)=\langle{\bf 1},\nu_{t}\rangle=1$ and hence,
$\langle{\bf 1},\mu_{t}\rangle=1-\langle{\bf 1},\theta_{t}\rangle$. Since
$r_{t}=R(\mu_{t}\|\nu_{*})$, and $\langle{\bf 1},\theta_{t}\rangle\to 0$ by
Remark 4.7, when combined with (4.6) this implies
$\limsup_{t\to\infty,t\in{\mathcal{B}}}r_{t}\geq\limsup_{t\to\infty,t\in{\mathcal{B}}}\langle{\bf
1},\mu_{t}\rangle\ln\langle{\bf 1},\mu_{t}\rangle=0$. Hence,

(4.32) $\lim_{t\to\infty,\ t\in{\mathcal{B}}}\langle{\bf
1},\mu_{t}\rangle=1\quad\text{and}\quad\lim_{t\to\infty,\
t\in{\mathcal{B}}}r_{t}=0,$

If ${\mathcal{B}}\supseteq[t_{0},\infty)$ for some finite $t_{0}$, this
clearly proves (4.18), and the claim follows.

We now turn to the proof of the fact that $B(t)=1$ outside a finite interval.
First note that, (4.32) and the Pinsker-type inequality (4.15) together show
that

$\lim_{t\to\infty,\
t\in{\mathcal{B}}}\int_{0}^{\infty}|\tilde{f}(x,t)-f^{*}(x)|dx=0.$

Thus, given $\varepsilon_{0}:=\frac{\lambda-1}{4}$ there exists
$T\in{\mathcal{B}}$ such that

(4.33) $c_{h}\langle{\bf 1},\theta_{t}\rangle<\varepsilon_{0}\quad\text{ and
}\quad
c_{h}\int_{0}^{\infty}|\tilde{f}(x,t)-f^{*}(x)|dx<\varepsilon_{0}\qquad\text{for
all }t\geq T,\ t\in{\mathcal{B}}.$

We claim that $[T,\infty)\subset{\mathcal{B}}$. Arguing by contradiction,
assume there exists $T^{\prime}>T$ for which $T^{\prime}\not\in{\mathcal{B}}$,
that is, such that $B({T^{\prime}})<1$. Let
$\tau:=\sup\\{t<T^{\prime}:B_{t}=1\\}$. By the continuity of $B$,
$T\leq\tau<T^{\prime}$ and $\tau\in{\mathcal{B}}$; in particullar, the
estimates in (4.33) are valid for $t=\tau$. Find $t^{*}>0$ so small that
$G^{s}(t^{*})<\frac{1}{4}$ and $0<t^{*}<T^{\prime}-\tau$. For all $0\leq t\leq
t^{*}$, applying Lemma 2.7 with $T=\tau$ and (2.19) with $\psi=h^{s}$, and
using the fact that (2.20) implies $K(\tau+t)-K(\tau)=\lambda t$ for
$t\in(0,t^{*})$, the identity $g^{s}=h^{s}\bar{G}^{s}$, the upper bound on
$h^{s}$ from Assumption 3.1(1) and (4.33), we obtain

$\displaystyle\langle h^{s},\nu_{\tau+t}\rangle$
$\displaystyle=\int_{0}^{\infty}\frac{g^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{\tau}(dx)+\lambda\int_{0}^{t}g^{s}(t-s)ds$
$\displaystyle=\int_{0}^{\infty}\frac{g^{s}(x+t)}{\bar{G}^{s}(x)}\theta_{\tau}(dx)+\int_{0}^{\infty}\frac{g^{s}(x+t)}{\bar{G}^{s}(x)}\nu^{*}(dx)+\int_{0}^{\infty}\frac{g^{s}(x+t)}{\bar{G}(x)}(\mu_{\tau}(dx)-\nu^{*}(dx))+\lambda\int_{0}^{t}g^{s}(t-s)ds$
$\displaystyle\leq c_{h}\langle{\bf
1},\theta_{\tau}\rangle+\int_{0}^{\infty}g^{s}(x+t)dx+c_{h}\int_{0}^{\infty}|f(\tau,x)-f^{*}(x)|dx+\lambda
G^{s}(t)$ $\displaystyle\leq\varepsilon_{0}+1-G^{s}(t)+\varepsilon_{0}+\lambda
G^{s}(t)$ $\displaystyle=1+(\lambda-1)G^{s}(t)+2\varepsilon_{0}\leq
1+3\varepsilon_{0}=\lambda-\varepsilon_{0}.$

Thus for all $\tau<s<\tau+t^{*}$, $\langle
h,\nu_{s}\rangle\leq\lambda-\varepsilon_{0}$. Next, since the interval
$(\tau,\tau+t^{*})$ is a subset of an excursion, equation (4.24) for $B$ is
valid for $t$ in that interval, and it follows that
$B^{\prime}(s)\geq\varepsilon_{0}$ for $s\in(\tau,\tau+t^{*})$. Bsy the
continuity of $B$,

$B(s)\geq 1+(t-\tau)\varepsilon_{0}>1,\qquad s\in(\tau,\tau+t^{*}),$

which is a contradiction. We have thus shown that $B(t)=1$ for all
sufficiently large $t$. Together with (4.32), this proves (4.18).

To conclude the proof of the proposition, it only remains to show that
$\langle h^{s},\nu_{t}\rangle\to 1$ as $t\to\infty$. Fix $T\in(0,\infty)$ such
that $B(t)=1$ for all $t\geq T$. Then using Lemma 2.7 and equation (2.19) with
$\psi(x)=h^{s}(x)$, and noting from (2.20) that $K^{\prime}(T+s)=\langle
h,\nu_{T+s}\rangle$ for all $s>0$, and recalling again that
$g^{s}=\bar{G}^{s}h^{s}$, we have

$\langle h^{s},\nu_{T+s}\rangle=z(s)+\int_{[0,s]}g^{s}(T+s-w)\langle
h^{s},\nu_{T+w}\rangle dw,$

where $z(s):=\int_{[0,\infty)}\frac{g^{s}(x+s)}{\bar{G}^{s}(x)}\nu_{T}(dx).$
Next, note that $g^{s}(s)=h^{s}(s)\bar{G}^{s}(s)\leq c_{h}\bar{G}^{s}(s)$.
Since $\bar{G}^{s}$ is decreasing and integrable over $[0,\infty)$, it is also
directly Riemann integrable (see Prop. 2.16(c), Ch. 9 of [14]), and thus, so
is $g^{s}$. Hence, by the key renewal theorem (Theor. 2.8, Ch. 9 of [14]),
$\langle h^{s},\nu_{T+s}\rangle$ converges as $s\to\infty$ to
$\int_{0}^{\infty}z(s)ds/\int_{0}^{\infty}xg^{s}(x)dx=\int_{0}^{\infty}z(s)ds$,
since by Assumption 2.1, $\int_{0}^{\infty}xg^{s}(x)dx=1$. Thus,

$\displaystyle\lim_{s\to\infty}\langle h^{s},\nu_{T+s}\rangle$
$\displaystyle=\int_{0}^{\infty}\int_{[0,\infty)}\frac{g^{s}(x+s)}{\bar{G}^{s}(x)}\nu_{T}(dx)ds$
$\displaystyle=\int_{[0,\infty)}\frac{1}{\bar{G}^{s}(x)}\Big{(}\int_{0}^{\infty}g^{s}(x+s)ds\Big{)}\nu_{T}(dx)=\int_{[0,\infty)}\nu_{T}(dx),$

which is equal to $1$ by our choice of $T$. This completes the proof of the
proposition. ∎

###  4.3. Proof of Convergence when the Hazard Rate Function is Decreasing

In this section, we assume throughout that Assumption 2.1 and Assumption
3.1(2) hold, and we establish Theorem 3.2(2) in this case, as well as Theorem
3.2(3). In addition, fix $\lambda\geq 1$, and suppose that $(X,\nu,\eta)$ is
the solution to the fluid equations with arrival rate $\lambda$ and some
initial condition $(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$. Also, recall
from (2.17) that $B(t)=\langle{\bf 1},\nu_{t}\rangle$, and define

(4.34)
$W(t):=B(t)-\int_{[0,H^{s})}\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx),\quad
t\geq 0.$

Note that $W(t)$ represents the fluid mass of jobs that arrived after time $0$
and are still in service at time $t$.

We will first establish the following key result.

######  Proposition 4.12.

Suppose Assumption 2.1 and Assumption 3.1(2) hold, and $\lambda\geq 1$. Then
we have

(4.35) $\lim_{t\rightarrow\infty}W(t)=\lim_{t\rightarrow\infty}B(t)=1,$

Further, if $\lambda>1$, there exists $T\in[0,\infty)$ such that $B(t)=1$ for
all $t\geq T$.

Before launching into the proof, we derive some useful relations. Setting
$\psi\equiv{\bf 1}$ in (2.19) and using integration by parts, it follows that

$\displaystyle B(t)=\langle{\bf 1},\nu_{t}\rangle$ $\displaystyle=$
$\displaystyle\int_{[0,H^{s})}\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)+\int_{0}^{t}\bar{G}^{s}(t-s)dK(s)$
$\displaystyle=$
$\displaystyle\int_{[0,H^{s})}\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)+K(t)-\int_{0}^{t}K(s)g^{s}(t-s)ds,$

which when rearranged yields

(4.37) $K(t)=W(t)+\int_{0}^{t}K(t-s)g^{s}(s)ds.$

Then, (4.3), (4.34) and the fact that $\nu_{t}$ is a sub-probability measure,
together imply that for each $t\geq 0$,

(4.38) $W(t)=\int_{0}^{t}\bar{G}^{s}(t-s)dK(s)\geq 0\quad\mbox{ and }\quad
W(t)\leq B(t)\leq 1.$

Together with (4.37) and the renewal theorem (see Chapter V of [6]), this
implies

(4.39) $K(t)=W(t)+Z(t),\qquad\mbox{ with }\quad Z(t):=\int_{0}^{t}W(t-s)\
dU_{s}(s),$

and $U_{s}$ is equal to the renewal function of the distribution with density
$g^{s}$. Now, (2.8) implies that

$D(t):=\int_{0}^{t}\langle h,\nu_{s}\rangle ds=B(0)-B(t)+K(t),\qquad t\geq 0.$

Then by (4.39), (4.34) and (4.3), we obtain

(4.40) $\displaystyle D(t)$ $\displaystyle=$ $\displaystyle\langle{\bf
1},\nu_{0}\rangle-\int_{[0,H^{s})}\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)+Z(t)$
$\displaystyle=$
$\displaystyle\int_{[0,H^{s})}\frac{G^{s}(x+t)-G^{s}(x)}{\bar{G}^{s}(x)}\nu_{0}(dx)+Z(t).$

Under Assumption 3.1(2), the hazard rate function $h^{s}$ is decreasing and
hence, by Theorem 3 of [12], the renewal function $U_{s}$ is concave. Since
$G^{s}$ has density $g^{s}$, the density $u_{s}:=U_{s}^{\prime}$ exists by
Proposition 2.7 of [6] and $u_{s}(x)=\sum_{n=1}^{\infty}(g^{s})^{\star n}(x),\
x\geq 0$, which in particular implies that $u_{s}(0)=g^{s}(0)$. Moreover, by
Alexandrov’s Theorem (cf. page 172 of [34]), the concavity of $U_{s}$ implies
that $u_{s}$ is non-increasing, that is,

(4.41) $u_{s}^{\prime}(t)\leq 0,\quad\mbox{ for a.e.\ }t\geq 0.$

Now, differentiation of both sides of the defining equation for $Z$ in (4.39)
yields

(4.42)
$Z^{\prime}(t)=W(t)u_{s}(0)+\int_{0}^{t}W(t-s)u_{s}^{\prime}(s)ds,\quad\mbox{
for a.e.\ }t\geq 0.$

On the other hand, differentiating the equation for $K$ in (4.39) and using
(2.20), one obtains, for a.e. $t\geq 0$,

(4.43) $\displaystyle W^{\prime}(t)$ $\displaystyle=$ $\displaystyle
K^{\prime}(t)-Z^{\prime}(t)$ (4.47) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\lambda-Z^{\prime}(t)&\mbox{ if
}B(t)<1,\\\ D^{\prime}(t)-Z^{\prime}(t)&\mbox{ if }B(t)=1\mbox{ and
}Q(t)>0,\\\ \lambda\wedge D^{\prime}(t)-Z^{\prime}(t)&\mbox{ if }B(t)=1\mbox{
and }Q(t)=0.\end{array}\right.$

Next, differentiating both sides of (4.40), we obtain for a.e. $t\geq 0$,

$D^{\prime}(t)=\int_{[0,H^{s})}\frac{g^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)+Z^{\prime}(t)\geq
Z^{\prime}(t).$

Therefore, by (4.43), for a.e. $t\geq 0$,

(4.48) $Z^{\prime}(t)\leq\lambda\quad\Rightarrow\quad W^{\prime}(t)\geq 0.$

We now establish some auxiliary results that will be used in the proof of
Proposition 4.12.

######  Lemma 4.13.

Suppose $\lambda\geq 1$. Then there is no $T\in(0,\infty)$ and $c\in(0,1)$
such that $W(t)<c$ for all $t\geq T$. The same assertion also holds when $W$
is replaced with $B$.

###### Proof.

Suppose the statement of the lemma is not true, that is, suppose there exists
$T>0$ and $c\in(0,1)$ such that $W(t)<c$ for all $t\geq T$. Since
$\int_{[0,H^{s})}\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)\rightarrow
0$ as $t\rightarrow\infty$, by (4.34), there exists $T^{\prime}>T$ such that
$B(t)<1$ for all $t\geq T^{\prime}$. In turn, by (2.8), it follows that
$K^{\prime}(t)=\lambda$, for all $t\geq T^{\prime}$, and hence (4.3) and
(4.34) imply that

$W(t)=\int_{0}^{t}\bar{G}^{s}(t-s)dK(s)=\int_{0}^{T^{\prime}}\bar{G}^{s}(t-s)dK(s)+\lambda\int_{T^{\prime}}^{t}\bar{G}^{s}(t-s)ds.$

As $t\rightarrow\infty$, the first term converges to zero by the dominated
convergence theorem and the pointwise limit $\bar{G}^{s}(t-s)\rightarrow 0$.
For the same reason, the second term converges to
$\lim_{t\to\infty}\lambda\int_{0}^{t}\bar{G}^{s}(t-s)ds=\lambda\int_{0}^{\infty}\bar{G}^{s}(s)ds$,
which is equal to $\lambda$ by (2.1) of Assumption 2.1. Thus,
$\lim_{t\rightarrow\infty}W(t)\geq\lambda$, which is a contradiction, thus
proving the first assertion of the lemma. Since, by (4.34),
$B(t)-W(t)=\int_{[0,H^{s})}\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)\rightarrow
0$ as $t\rightarrow\infty$, the same assertion holds also for $B$. ∎

Next, substituting into (4.42) the inequality (4.41), the relation
$u_{s}(0)=g^{s}(0)$ and the fact that $W(t)\in[0,1]$ for each $t\geq 0$ due to
(4.38), we see that

(4.49) $Z^{\prime}(t)\leq W(t)u_{s}(0)=W(t)g^{s}(0)\leq g^{s}(0)\quad\mbox{for
a.e.\ }t\geq 0.$

We also observe that since the hazard rate function $h^{s}$ is decreasing by
Assumption 3.1(2), then $g^{s}(0)>0$. (Otherwise, if $g^{s}(0)=0$, then
$h^{s}(0)=0$, which implies that $0\leq h^{s}(t)\leq h^{s}(0)=0$ for each
$t\geq 0$ and thus, $g^{s}(t)=0$ for all $t\geq 0$, which would contradict the
fact that $g^{s}$ is the density of $G^{s}$.) Therefore, for
$n\in{\mathbb{N}}\cup\\{0\\}$ and $\varepsilon\in(0,\frac{1}{2})$, define

(4.50)
$\lambda_{n}:=\frac{\lambda-\varepsilon}{g^{s}(0)}\left(\sum_{i=0}^{n}\left(1-\frac{1}{g^{s}(0)}\right)^{i}\right)=\frac{\lambda-\varepsilon}{g^{s}(0)}\frac{1-\left(1-\frac{1}{g^{s}(0)}\right)^{n+1}}{1-\left(1-\frac{1}{g^{s}(0)}\right)}=(\lambda-\varepsilon)\left(1-\left(1-\frac{1}{g^{s}(0)}\right)^{n+1}\right),$

and

(4.51) $\tau_{n}:=\sup\\{t>0:W(t)<\lambda_{n}\\}.$

If $\tau_{n}<\infty$, then

(4.52) $W(\tau_{n}+t)\geq\lambda_{n}\quad\forall t\geq 0.$

######  Lemma 4.14.

Suppose $\lambda\geq 1$, $\varepsilon\in(0,\frac{1}{2})$ and
$g^{s}(0)>\lambda-\varepsilon$. Then $\tau_{n}<\infty$ and hence, (4.52) holds
for all $n\in{\mathbb{N}}$ with $n<n^{*}$, where
$n^{*}:=\sup\left\\{n\in{\mathbb{N}}_{0}:\lambda_{n}<1\right\\}$, and also for
$n=n^{*}$, if $n^{*}<\infty$.

###### Proof.

Since $g^{s}(0)>\lambda-\varepsilon>\frac{1}{2}$ by the assumptions of the
lemma, it follows that $|1-\frac{1}{g^{s}(0)}|<1$ and (4.50) then implies that

(4.53) $\lambda_{n}\rightarrow\lambda-\varepsilon\mbox{ as
}n\rightarrow\infty.$

We prove the lemma by induction. We first start with the base case $n=0$,
where $\lambda_{0}=(\lambda-\varepsilon)/g^{s}(0)$. Note that $\lambda_{0}<1$
by the assumptions of the lemma. We argue by contradiction to show that

(4.54) $W(t)<\frac{\lambda-\varepsilon}{g^{s}(0)}\quad\mbox{ for all
}t\in(0,\tau_{0}).$

Note that (4.54) holds trivially if $\tau_{0}=0$. So, suppose $\tau_{0}>0$ and
(4.54) does not hold. Then there must exist $0<t_{1}<\tau_{0}$ for which
$W(t_{1})\geq\frac{\lambda-\varepsilon}{g^{s}(0)}$. It follow from (4.49) and
(4.48), that for a.e. $t\in(t_{1},\tau_{0})$, the inequality
$W(t)<\frac{\lambda-\varepsilon}{g^{s}(0)}$ implies that $Z^{\prime}(t)\leq
W(t)g^{s}(0)<\lambda-\varepsilon<\lambda$ and hence $W^{\prime}(t)\geq 0$.
Since $W$ is absolutely continuous,
$W(t)\geq\frac{\lambda-\varepsilon}{g^{s}(0)}=\lambda_{0}$ for all
$t\in[t_{1},\tau_{0})$ (see Lemma B.1). This contradicts the definition of
$\tau_{0}$, and thus, (4.54) holds. If $\tau_{0}=\infty$, then (4.54) implies
$W(t)<\lambda_{0}<1$ for all $t>0$, which contradicts Lemma 4.13. Thus,
$\tau_{0}<\infty$. This completes the proof of the base case.

Now, suppose that $\tau_{k}<\infty$ for some $k\in{\mathbb{N}}\cup\\{0\\},$
with $k<n^{*}$ if $n^{*}<\infty$. It follows that $\lambda_{k+1}<1$ by the
choice of $k$ and the definition of $n^{*}$. By the definition of $\tau_{k}$
and the continuity of $W$,

(4.55) $W(\tau_{k}+t)\geq\lambda_{k}\mbox{ for all }t\in[0,\infty).$

Then for a.e. $t\geq 0$, by (4.42), (4.41), (4.55) and the relations $W(t)\geq
0$ and $u_{s}(0)=g^{s}(0)$, we have

(4.56) $\displaystyle Z^{\prime}(\tau_{k}+t)$ $\displaystyle=$ $\displaystyle
W(\tau_{k}+t)g^{s}(0)+\int_{0}^{t}W(\tau_{k}+t-s)u_{s}^{\prime}(s)ds+\int_{t}^{\tau_{k}+t}W(\tau_{k}+t-s)u_{s}^{\prime}(s)ds$
(4.57) $\displaystyle<$ $\displaystyle
W(\tau_{k}+t)g^{s}(0)+\lambda_{k}\left(u_{s}(t)-g^{s}(0)\right).$

Since Assumption 3.1(2) implies that the integrable function $g^{s}$ is also
bounded, it lies in $\mathbb{L}^{1+\varepsilon}(0,\infty)$ for any
$\varepsilon>0$, and satisfies $g^{s}(t)\rightarrow 0$ as
$t\rightarrow\infty$. Thus, by Theorem 12 of [39] we can conclude that
$\lim_{t\rightarrow\infty}u_{s}(t)=1$. Hence, there exists $\sigma_{k}>0$ such
that

(4.58)
$(\lambda-\varepsilon)+\lambda_{k}(u_{s}(t)-1)=(\lambda-\varepsilon)+\lambda_{k}\left(g^{s}(0)-1\right)+\lambda_{k}\left(u_{s}(t)-g^{s}(0)\right)<\lambda\mbox{
for all }t\geq\sigma_{k}.$

We now show that the following statement cannot hold:

(4.59)
$W(\tau_{k}+t)<\lambda_{k+1}=\frac{\lambda-\varepsilon}{g^{s}(0)}+\lambda_{k}\left(1-\frac{1}{g^{s}(0)}\right)\mbox{
for all }t>\sigma_{k},$

where the equality follows from (4.50). Indeed, if this were true, then this
would imply that $W(t)<\lambda_{k+1}<1$ for all
$t\geq\sigma_{k}^{\prime}:=\tau_{k}+\sigma_{k}$, which contradicts Lemma 4.13.
Thus, (4.59) does not hold or, in other words, there exists
$\tau_{k}^{\prime}\in(\sigma_{k}^{\prime},\infty)$ such that
$W(\tau_{k}^{\prime})\geq\lambda_{k+1}$. We now show that for a.e.
$t\in(0,\infty)$, if $W(\tau_{k}^{\prime}+t)<\lambda_{k+1}$ then
$W^{\prime}(\tau_{k}^{\prime}+t)\geq 0$. Indeed, if the first inequality is
true, then substituting this into (4.57) with $\tau_{k}^{\prime}$ in place of
$\tau_{k}$, and using (4.58), it follows that
$Z^{\prime}(\tau^{\prime}_{k}+t)<\lambda$. When combined with (4.48) the
latter implies $W^{\prime}(\tau_{k}^{\prime}+t)\geq 0$. Hence (applying Lemma
B.1 with $f=W$, $c=\lambda_{k+1}$, $T=\tau_{k}^{\prime}$, $S=\infty$), it
follows that $W(\tau_{k}^{\prime}+t)\geq\lambda_{k+1}$ for all $t\geq 0$, thus
showing that $\tau_{k+1}\leq\tau_{k}^{\prime}<\infty$. By induction, it
follows that for each $0\leq n<n^{*}$, $\tau_{n}<\infty$ and hence, (4.52)
holds, and if $n^{*}<\infty$ then also $\tau_{n^{*}}<\infty$ and (4.52) holds
with $n=n^{*}$. This completes the proof of the lemma.

∎

We are now in a position to present the proof of Proposition 4.12.

###### Proof of Proposition 4.12.

We first prove the proposition when $\lambda=1$. For this, we consider two
cases.
Case 1a:  $g^{s}(0)\leq 1$. In this case, (4.49) shows that $Z^{\prime}(t)\leq
1$ for a.e. $t\geq 0$, then (4.48) implies that for a.e. $t\geq 0$,
$W^{\prime}(t)\geq 0$. Since $W$ is absolutely continuous by (4.38) this
implies that $W$ is increasing on $[0,\infty)$ and
$b:=\lim_{t\rightarrow\infty}W(t)$ exists. Furthermore, (4.34) and the fact
that $\bar{G}^{s}(x+t)\rightarrow 0$ as $t\rightarrow\infty$ for every
$x\in[0,H^{s})$, imply $b=\lim_{t\rightarrow\infty}B(t).$ We now argue by
contradiction to show that $b=1$. Suppose $b<1$, then for any $T<\infty$,
there exists $T_{1}<\infty$ such that for $t\geq 0$, $B(T_{1}+t)<1$ and thus,
by (2.20) $K^{\prime}(T_{1}+t)=1$. Now, recalling $B(\cdot)=\langle{\bf
1},\nu_{\cdot}\rangle$ from (4.3) and combining Lemma 2.7 and Theorem 2.6, it
follows that (2.19) holds with $\psi={\bf 1}$, and $\nu_{t}$ and $K_{t}$
replaced with $\nu_{T_{1}+t}$, and $K_{T_{1}+t}-K_{T_{1}}$, respectively, or
in other words, for each $t\geq 0$,

$\displaystyle B(T_{1}+t)$ $\displaystyle=$
$\displaystyle\int_{[0,H^{s})}\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{T_{1}}(dx)+\int_{0}^{t}\bar{G}^{s}(t-s)K^{\prime}(T_{1}+s)ds.$

When combined with the relation $K^{\prime}(T_{1}+\cdot)=\lambda=1$ a.e., this
implies that for each $t\geq 0$.

$\displaystyle B(T_{1}+t)$ $\displaystyle=$
$\displaystyle\int_{[0,H^{s})}\frac{\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{T_{1}}(dx)+\int_{0}^{t}\bar{G}^{s}(t-s)ds,$

Sending $t\rightarrow\infty$, using $\bar{G^{s}}(x+t)\to 0$ pointwise and the
dominated convergence theorem, as well as (2.2) of Assumption 2.1, this
implies $b=\lim_{t\rightarrow\infty}B(t)=1$. This contradicts the supposition
that $b<1$, and thus proves that $b=1$.

Case 1b:  $g^{s}(0)>1$. In this case, by (4.50),

$\lambda_{n}=(1-\varepsilon)\left(1-\left(1-\frac{1}{g^{s}(0)}\right)^{n+1}\right)<1\mbox{
for all }n\geq 1.$

Thus, by Lemma 4.14, for each $n\geq 1$, we have $\tau_{n}<\infty$ and so
(4.52) implies $\liminf_{t\rightarrow\infty}W(t)\geq\lambda_{n}\mbox{ for each
}n\geq 1$. By (4.53), we obtain $\liminf_{t\rightarrow\infty}W(t)\geq
1-\varepsilon$. Sending $\varepsilon\downarrow 0$, we obtain
$\liminf_{t\rightarrow\infty}W(t)\geq 1$. Since
$\limsup_{t\rightarrow\infty}W(t)\leq 1$ by (4.38) it follows that in fact
$\lim_{t\rightarrow\infty}W(t)=1$. When combined with (4.34) and the fact that
$\bar{G}^{s}(x+t)\rightarrow 0$ as $t\rightarrow\infty$ for every
$x\in[0,H^{s})$, it follows that $\lim_{t\rightarrow\infty}B(t)=1,$ thus
proving the proposition in this case.

We next prove the proposition for the case that $\lambda>1$. Let
$\varepsilon>0$ be small enough such that $\lambda-\varepsilon>1$. We now
consider two cases.
Case 2a:  $g^{s}(0)\leq\lambda-\varepsilon$. In this case, (4.49) shows that
$Z^{\prime}(t)\leq\lambda-\varepsilon<\lambda$ for a.e. $t\geq 0$, and hence,
(4.48) implies that for a.e. $t\geq 0$, $W^{\prime}(t)\geq 0$. Moreover, by
(4.43), we have $W^{\prime}(t)=\lambda-Z^{\prime}(t)\geq\varepsilon\mbox{ if
}B(t)<1$. By the definition of $W$ in (4.34), we obtain

$B^{\prime}(t)=W^{\prime}(t)+\int_{[0,H^{s})}\frac{g^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx).$

Since $h^{s}$ is decreasing, we have $h^{s}(x+t)\leq h^{s}(0)$ for each
$x\in[0,H^{s}-t)$, and an application of the dominated convergence theorem
shows that

$\int_{[0,H^{s})}\frac{g^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)\leq\int_{[0,H^{s})}\frac{h^{s}(0)\bar{G}^{s}(x+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)\rightarrow
0\mbox{ as }t\rightarrow\infty.$

The last three displays together imply that there exists $T\in(0,\infty)$ such
that $B^{\prime}(t)>\varepsilon/2$ whenever $B(t)<1$ for a.e.
$t\in[T,\infty)$. Since $B$ is bounded (by $1$), the inequality $B(t)<1$
cannot hold for all $t\geq T$. In other words, there must exist $T^{\prime}>T$
such that $B(T^{\prime})=1$. Since $B$ is absolutely continuous and bounded by
$1$ (applying Lemma B.1 with $f=B$, $c=1$, $T=T^{\prime}$ and $S=\infty$), we
conclude that $B(t)=1$ for all $t\in[T^{\prime},\infty)$.

Case 2b:  $g^{s}(0)>\lambda-\varepsilon$. Then $n^{*}<\infty$ since (4.50)
shows that $\lambda_{n}\uparrow(\lambda-\varepsilon)>1$ as
$n\rightarrow\infty$. Since by Lemma 4.14, $\tau_{n^{*}}<\infty$, then the
continuity of $W$ dictates that $W(\tau_{n^{*}})=\lambda_{n^{*}}$. Together
with (4.57) with $k=n^{*}$ and the fact that $W$ is bounded by $1$ due to
(4.38), this implies that for a.e. $t\geq 0$,

$\displaystyle Z^{\prime}(\tau_{n^{*}}+t)$ $\displaystyle\leq$ $\displaystyle
W(\tau_{n^{*}}+t)g^{s}(0)+\lambda_{n^{*}}\left(u_{s}(t)-g^{s}(0)\right)$
$\displaystyle\leq$
$\displaystyle(1-\lambda_{n^{*}})g^{s}(0)+\lambda_{n^{*}}u_{s}(t).$

By the definition of $n^{*}$, we have
$\lambda_{n^{*}}<1\leq\lambda_{n^{*}+1}$. Together with the definition of
$\lambda_{n}$ in (4.50), this implies that

$1-\lambda_{n^{*}}\leq\lambda_{n^{*}+1}-\lambda_{n^{*}}=\frac{\lambda-\varepsilon}{g^{s}(0)}\left(1-\frac{1}{g^{s}(0)}\right)^{n^{*}+1}.$

Combining the above two displays, we obtain

$Z^{\prime}(\tau_{n^{*}}+t)\leq(\lambda-\varepsilon)\left(1-\frac{1}{g^{s}(0)}\right)^{n^{*}+1}+\lambda_{n^{*}}u_{s}(t).$

Recalling that $\lim_{t\rightarrow\infty}u_{s}(t)=1$ and using the expression
for $\lambda_{n^{*}}$ from (4.50), it follows that as $t\rightarrow\infty$,

$(\lambda-\varepsilon)\left(1-\frac{1}{g^{s}(0)}\right)^{n^{*}+1}+\lambda_{n^{*}}u_{s}(t)\rightarrow(\lambda-\varepsilon)\left(1-\frac{1}{g^{s}(0)}\right)^{n^{*}+1}+\lambda_{n^{*}}=\lambda-\varepsilon.$

Thus, for all $t$ large enough,
$Z^{\prime}(\tau_{n^{*}}+t)<\lambda-\varepsilon/2$. However, note that by
(4.43), we have

$W^{\prime}(\tau_{n^{*}}+t)=\lambda-Z^{\prime}(\tau_{n^{*}}+t)>\varepsilon/2\mbox{
if }B(\tau_{n^{*}}+t)<1.$

By the definition of $W$ in (4.34), we obtain

$B^{\prime}(\tau_{n^{*}}+t)=W^{\prime}(\tau_{n^{*}}+t)+\int_{[0,H^{s})}\frac{g^{s}(x+\tau_{n^{*}}+t)}{\bar{G}^{s}(x)}\nu_{0}(dx).$

Since $h^{s}$ is decreasing, it follows that $h^{s}(x+\tau_{n^{*}}+t)\leq
h^{s}(0)$ for each $x\in[0,H^{s})$, and we obtain by the dominated convergence
theorem that

$\int_{[0,H^{s})}\frac{g^{s}(x+\tau_{n^{*}}+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)\leq\int_{[0,H^{s})}\frac{h^{s}(0)\bar{G}^{s}(x+\tau_{n^{*}}+t)}{\bar{G}^{s}(x)}\nu_{0}(dx)\rightarrow
0\mbox{ as }t\rightarrow\infty.$

The rest of the proof follows as in Case 1b. The last four displays imply that
there exists $T\in(0,\infty)$ such that $B^{\prime}(t)>\varepsilon/4$ whenever
$B(t)<1$ for a.e. $t\in[T,\infty)$. By the boundedness of $B$ it follows that
there exists $T^{\prime}>T$ such that $B(T^{\prime})=1$. Thus, for a.e. $t\geq
T^{\prime}$, we have $B^{\prime}(t)>\varepsilon/4$ whenever $B(t)<1$. In turn
(by Lemma B.1) this implies that $B(t)=1$ for all $t\in[T^{\prime},\infty)$.
Since all possible cases have been considered, this concludes the proof of the
proposition. ∎

We now consider convergence properties of the measure-valued age process.

######  Lemma 4.15.

For $\lambda\geq 1$, under the assumptions of Proposition 4.12, suppose there
exists $T<\infty$ such that $B(t)=1$ for all $t\geq T$. Then
$\nu_{t}\Rightarrow\nu_{*}$ and $\langle h^{s},\nu_{t}\rangle\rightarrow 1$ as
$t\rightarrow\infty$.

###### Proof.

By invoking Lemma 2.7, we can assume without loss of generality that $T=0$.
Then $B(t)=\langle{\bf 1},\nu_{t}\rangle=1$ for all $t\geq 0$, and so by (4.6)
of Corollary 4.4 of [25], $K$ has the representation

$K(t)=\int_{0}^{t}\left(\int_{[0,H^{s})}\frac{G^{s}(x+t-s)-G^{s}(x)}{\bar{G}^{s}(x)}\nu_{0}(dx)\right)dU_{s}(dx),\
t\geq 0.$

In view of the representation for the fluid age measure in (2.19), the
convergence $\nu_{t}\Rightarrow\nu_{*}$ is then a direct consequence of Lemma
6.2 of of [25] with $\pi=\nu$. Finally, since $h^{s}$ is bounded and monotone
by Assumption 3.1(2), the set of its discontinuities is countable and thus has
zero Lebesgue measure. Since $\nu_{*}$ is an absolutely continuous measure,
the continuous mapping theorem implies $\langle
h^{s},\nu_{t}\rangle\rightarrow\langle
h^{s},\nu_{*}\rangle=\int_{0}^{\infty}g^{s}(x)dx=1$, as $t\to\infty$. This
concludes the proof of the lemma. ∎

###  4.4. Uniqueness of Random Fixed Points

We now show how the convergence results of the last two sections can be
bootstrapped to conclude, under Assumption 2.8, the existence of a unique
random fixed point.

######  Proposition 4.16.

Suppose $\lambda\geq 1$, Assumptions 2.1 and 2.8 hold and suppose that for any
solution $(X,\nu,\eta)$ to the fluid equations with arrival rate $\lambda$ and
initial condition $(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$,

(4.60) $\eta_{t}\Rightarrow\lambda\eta_{*}\quad\mbox{ and }\quad
B_{t}\rightarrow 1.$

Then any random fixed point $\mu$ for the fluid equations with arrival rate
$\lambda$ satisfies $\mu=\delta_{z^{\lambda}_{*}}$, where
$z^{\lambda}_{*}=(x^{\lambda}_{*},\nu_{*},\lambda\eta_{*})$, with
$x^{\lambda}_{*}$ being the unique element of ${\mathcal{X}}_{\lambda}$ in
(2.24).

###### Proof.

Fix $\lambda\geq 1$ and let $\mu$ be a random fixed point for the fluid
equations with arrival rate $\lambda$. Let $(X(0),\nu_{0},\eta_{0})$ be a
random element taking values in
${\mathbb{R}}_{+}\times{\mathcal{M}}_{F}[0,H^{s})\times{\mathcal{M}}_{F}[0,H^{r})$
with law $\mu$ and let $(X,\nu,\eta)$ be the solution to the fluid equations
with arrival rate $\lambda$ and initial condition
$(X(0),\nu_{0},\eta_{0})\in{\mathfrak{S}}$. Since
$\eta_{t}\Rightarrow\lambda\eta_{*}$ and $B_{t}\rightarrow 1$ by assumption
and the laws of $\eta_{t}$ and $\nu_{t}$ are invariant in $t$ since $\mu$ is a
random fixed point, we have ${\mathbb{P}}(\eta_{0}=\lambda\eta_{*})=1$ and
${\mathbb{P}}(B_{t}=\langle{\bf 1},\nu_{t}\rangle=1)=1$. Further, by
continuity of $B$, we have ${\mathbb{P}}$-almost surely, $B_{t}=1$ for all
$t\geq 0$. Then by Lemma 4.15, it follows that $\nu_{t}\Rightarrow\nu_{*}$ as
$t\rightarrow\infty$. Since the law of $\nu_{t}$ is invariant in $t$, it
follows that ${\mathbb{P}}(\nu_{0}=\nu_{*})=1$.

To complete the proof, it only remains to show that
${\mathbb{P}}(X(0)=x_{*}^{\lambda})=1$. Since almost surely, for all $t\geq
0$, $B(t)=1$ and $\eta_{t}=\lambda\eta_{*}$, the relations (2.13) and (2.12)
show that almost surely for all $t\geq 0$, $X(t)=Q(t)+1$ and

$R(t)=\int_{0}^{t}\left(\int_{0}^{Q(s)}h^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(y))dy\right)ds=\lambda\int_{0}^{t}G^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(Q(s)))ds.$

Moreover, using the fact that almost surely for each $t\geq 0$,
$\nu_{t}=\nu_{*}$, and hence, $D(t)=t\langle h^{s},\nu_{*}\rangle=t$, we have
from (2.13), (2.11) and the fact that $E=E^{\lambda}$ that almost surely for
each $t\geq 0$,

(4.61) $\displaystyle Q(t)$ $\displaystyle=$ $\displaystyle
Q(0)+(\lambda-1)t-\lambda\int_{0}^{t}G^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(Q(s)))ds$
(4.62) $\displaystyle=$ $\displaystyle
Q(0)+\int_{0}^{t}\left(\lambda{\bar{G}}^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(Q(s)))-1\right)ds.$

We now consider two cases.
Case 1:  $\lambda=1$. In this case, we have

$\int_{0}^{t}\left(\lambda{\bar{G}}^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(Q(s)))-1\right)ds=-\int_{0}^{t}G^{r}(\left(F^{\eta_{*}}\right)^{-1}(Q(s)))ds.$

It is clear from (4.62) that $Q$ is decreasing on $[0,\infty)$. By the non-
negativity of $Q$, $q_{*}:=\lim_{t\rightarrow\infty}Q(t)$ exists and the fact
that $X(t)=Q(t)+1$ implies $\lim_{t\rightarrow\infty}X(t)=x_{*}:=q_{*}+1$.
Note that $G^{r}(\left(F^{\eta_{*}}\right)^{-1}(q_{*}))=0$ since, otherwise,
$Q(t)\rightarrow-\infty$ as $t\rightarrow\infty$, which contradicts the non-
negativity of $Q$. Therefore, by Assumption 2.8, the definition of
${\mathcal{X}}_{\lambda}$ in (2.24) and the fact that $\lambda-1=0$, it
follows that $x_{*}$ is equal to the unique element $x_{*}^{\lambda}$ of
${\mathcal{X}}_{\lambda}$. As before, since $\mu$ is a random fixed point,
this implies that ${\mathbb{P}}(X(0)=x_{*}^{\lambda})=1$.
Case 2:  $\lambda>1$. In this case, it is clear from (4.62) that $Q$ is
differentiable on $(0,\infty)$, and

(4.63)
$Q^{\prime}(t)=\lambda{\bar{G}}^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(Q(s)))-1\mbox{
for each }t>0.$

First note that since by (2.14), $Q(t)\leq\langle{\bf
1},\eta_{t}\rangle=\lambda\langle{\bf 1},\eta_{*}\rangle$, $Q(t)$ is bounded.
We now argue by contradiction to show that
$q_{*}=\lim_{t\rightarrow\infty}Q(t)$ exists. Suppose this is not the case.
Then, since $Q$ is bounded on $[0,\infty)$, $Q$ must oscillate for the limit
not to exist. By the continuity of $Q$, this implies there must exist two
sequences of times $\\{t_{n},n\geq 1\\}$ and $\\{s_{n},n\geq 1\\}$ with
$t_{n}\rightarrow\infty$, $s_{n}\rightarrow\infty$ as $n\rightarrow\infty$,
and $\varepsilon>0$ such that $|Q(t_{n})-Q(s_{n})|>\varepsilon$ for all $n$
sufficiently large, and $Q^{\prime}(t_{n})=Q^{\prime}(s_{n})=0$ for each
$n\geq 1$. By (4.62), the latter relation implies

$G^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(Q(t_{n})))=G^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(Q(s_{n})))=\frac{\lambda-1}{\lambda}\mbox{
for all }n\geq 1.$

Since $Q$ is bounded, there exist $0\leq q_{i}\leq\lambda\langle{\bf
1},\eta^{r}\rangle$, $i=1,2$ and is a subsequence $\\{n_{k},k\geq 1\\}$ such
that $Q(t_{n_{k}})\rightarrow q_{1}$ and $Q(s_{n_{k}})\rightarrow q_{2}$ as
$k\rightarrow\infty$. It follows that $|q_{1}-q_{2}|\geq\varepsilon$ and
$G^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(q_{1}))=G^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(q_{2}))=\frac{\lambda-1}{\lambda}$,
where we have used the fact that $G^{r}$ and
$\left(F^{\lambda\eta_{*}}\right)^{-1}$ are continuous, with the latter
continuity holding because $\lambda\eta_{*}$ has a density
$\lambda\bar{G}^{r}$ that is strictly positive on its support. By Assumption
2.8, we have $q_{1}=q_{2}$ which contradicts $|q_{1}-q_{2}|\geq\varepsilon$.
Thus, $q_{*}=\lim_{t\rightarrow\infty}Q(t)$ exists and then
$\lambda{\bar{G}}^{r}(\left(F^{\lambda\eta_{*}}\right)^{-1}(Q(t)))-1=0$ since
otherwise by (4.62) $Q$ will not have a limit. We can then argue as in Case 1
that $X(t)\rightarrow q_{*}+1=x_{*}^{\lambda}$, and thus
${\mathbb{P}}(X(0)=x_{*}^{\lambda})=1$. This completes the proof of the
theorem. ∎

##  5\. Results Regarding the Multiclass Model

Here we consider the model with multiple classes operating under a fixed
priority discipline. This model, with general class-dependent service time and
patience time distributions, was analyzed in [9] and convergence at the fluid
scale, uniformly on compact time intervals, was established. Here, we study
the long-time behavior under the additional assumption that the service time
distribution does not depend on the job class, and its hazard rate function is
bounded away from zero and infinity, that is, satisfies Assumption 3.1(1). For
simplicity of exposition, we also assume that the reneging distributions are
exponential (but may depend on the job class) since the main motivation is to
deduce the optimality of a certain priority scheduling rule (known as the
$c\mu/\theta$ rule; see details below) discussed in [9], which is not expected
to hold beyond the exponential reneging case. As shown in [9], this optimality
result relies on the convergence of the invariant distributions of the fluid-
scaled process, as $N\to\infty$, to the unique element of the invariant
manifold of the fluid limit (under assumptions that ensure such uniqueness).
However, the convergence result in [9] (specifically Theorem 4.3 therein)
suffers from the same flaw as that described for the single-class case in
Remark 3.3; namely, from the proof in [9] one can only deduce that the
invariant distributions of the $N$-server systems exist, are tight and that
any subsequential limit of the sequence of invariant distributions must be a
random fixed point of the fluid equations (defined analogously to Definition
2.10). As explained in Remark 2.11 in the single-class setting, in order to
show that there is a unique random fixed point (which must then coincide with
the unique element of the invariant manifold) it suffices to establish the
long-time convergence of the solution of the fluid equations with any initial
condition to the unique element of the invariant manifold. Thus, the limit
interchange result that we prove here fixes the flaw in the main optimality
result of [9] under the additional assumptions on the service distribution
stated above. This leaves open the question of whether there is also a limit
interchange for class dependent service times and when hazard rates are not
necessarily bounded. We present the fluid equations in Section 5.1, and then
state and prove the theorem in Section 5.2.

###  5.1. Fluid Model Equations for the Multiclass System

Analogous to the single-class case, for each class $i\in\\{1,\ldots,J\\}$, we
denote by $B_{i}$, $X_{i}$ and $Q_{i}$ nonnegative functions that represent
the fluid analogs of the number in service, number in system and number in
queue, let the nonnegative, nondecreasing functions $D_{i}$, $K_{i}$ and
$R_{i}$ represent the fluid analogs of cumulative class $i$ departures from
service, cumulative entries to service and cumulative reneging, and let
$\nu_{i}$ represent the fluid analog of the measure-valued function that
encodes the ages of class $i$ jobs in service. Since we assume exponential
reneging times, we will not require the potential reneging measures
$\eta_{i}$, but only the reneging rate $\theta_{i}>0$. Also, let
$(X,\theta,\nu,B,Q,D,K,R)$ be the corresponding vector-valued processes whose
$i$th component is given by
$(X_{i},\theta_{i},\nu_{i},B_{i},Q_{i},D_{i},K_{i},R_{i})$. We describe the
fluid equations only for the special case when all service distributions are
identical, with common cumulative distribution function $G=G^{s}$, hazard rate
function $h=h^{s}$ and support $[0,H)=[0,H^{s})=[0,\infty)$, and arrival rates
$\lambda_{i}>0,i=1,\ldots,J$.

Before we present the fluid model equations, let us comment on the special
form that the single-server fluid model equations (of Definition 2.3) take
when the reneging is exponential. In this case, the reneging hazard rate is
constant, namely $h^{r}(t)=\theta$ for all $t$, and thus equation (2.12) takes
the form

$R(t)=\theta\int_{0}^{t}Q(s)ds,$

and thus there is no longer any need to keep track of the potential reneging
measure $\eta$. Accordingly, in the multiclass setting, our fluid model is an
extension of such a modified set of fluid equations where the equation of $R$
is similar to the above display, and from which $\eta$ is absent.

######  Definition 5.1.

Given arrival and reneging rate vectors $\lambda\in(0,\infty)^{J}$ and
$\theta\in(0,\infty)^{J}$, and initial condition
$(X(0),\nu_{0})\in[0,\infty)^{J}\times({\mathcal{M}}_{F}[0,\infty))^{J}$, a
tuple
$(B,X,Q,D,K,R,\nu)\in({\mathcal{D}}_{{\mathbb{R}}^{J}_{+}}({\mathbb{R}}_{+}))^{3}\times({\mathcal{D}}^{+}_{{\mathbb{R}}^{J}_{+}}({\mathbb{R}}_{+}))^{3}\times({\mathcal{D}}_{{\mathcal{M}}_{F}[0,\infty)}({\mathbb{R}}_{+}))^{J}$
is said to be a solution to the multiclass fluid equations with initial
condition $(X(0),\nu_{0})$ and arrival and reneging rate vectors $\lambda$ and
$\theta$ if equations (5.1)–(5.2) below are satisfied: For
$\varphi\in{\mathcal{C}}^{1}_{c}([0,\infty)\times{\mathbb{R}}_{+})$, and
$t\geq 0$,

$\displaystyle\langle\varphi(\cdot,t),\nu_{i,t}\rangle$
$\displaystyle=\langle\varphi(\cdot,0),\nu_{i,0}\rangle+\int_{0}^{t}\langle\varphi_{x}(\cdot,s)+\varphi_{t}(\cdot,s),\nu_{i,s}\rangle
ds$ (5.1) $\displaystyle\quad-\int_{0}^{t}\langle
h(\cdot)\varphi(\cdot,s),\nu_{i,s}\rangle
ds+\int_{0}^{t}\varphi(0,s)dK_{i}(s),$

where $B,D,R$ are the auxiliary processes given by

(5.2) $B_{i}(t)=\langle 1,\nu_{i,t}\rangle,\qquad D_{i}(t)=\int_{0}^{t}\langle
h,\nu_{i,s}\rangle ds,\qquad R_{i}(t)=\theta_{i}\int_{0}^{t}Q_{i}(s)ds.$

and for $t\geq 0$, $K,B,D$ satisfy the following balance equations and basic
relations:

(5.3) $\displaystyle B_{i}$ $\displaystyle=B_{i,0}-D_{i}+K_{i},$ (5.4)
$\displaystyle X_{i}(t)$
$\displaystyle=X_{i,0}-D_{i}(t)+\lambda_{i}t-R_{i}(t),$ (5.5) $\displaystyle
Q_{i}$ $\displaystyle=X_{i}-B_{i},$