# Asymptotic product-form steady-state for generalized Jackson networks in
multi-scale heavy traffic

J.G. Dai
Cornell University
Peter Glynn
Stanford University
Yaosheng Xu
Amazon

###### Abstract

We prove that under a multi-scale heavy traffic condition, the stationary
distribution of the scaled queue length vector process in any generalized
Jackson network has a product-form limit. Each component in the product form
has an exponential distribution, corresponding to the Brownian approximation
of a single station queue. The “single station” can be constructed precisely
and its parameters have a good intuitive interpretation.

> Keywords: single-class queueing networks, product-form stationary
> distributions, reflected Brownian motions, interchange of limits, heavy
> traffic approximation, performance analysis
>
> Mathematics Subject Classification: 60K25, 60J27, 60K37

##  1 Introduction

Jackson (1957) introduced a class of first-come–first-serve (FCFS) queueing
networks that are known today as open Jackson networks. The defining
characteristics of a Jackson network are (a) all customers visiting a service
station are homogeneous in terms of the service time distribution and the
routing probabilities, and (b) all interarrival and service time distributions
are exponential. For a Jackson network, the queue length vector process is a
continuous time Markov chain (CTMC). Jackson’s pioneering contribution was
that when the traffic intensity at each station is less than one, the CTMC has
a unique stationary distribution that is of product form, meaning that the
steady-state queue length at the various stations of the network are
independent. This product-form result makes the computation of steady-state
performance measures highly tractable.

When the interarrival and service time distributions are general, the
corresponding network is known as a generalized Jackson network. For a
generalized Jackson network, the product-form result no longer holds in
general. Creating and justifying approximations for generalized Jackson
networks has been an active research area for more than 40 years; see, for
example, Whitt (1983) and Whitt and You (2020). For such a network, one can
compute a heavy traffic normalization factor for each station that is equal to

$\displaystyle\frac{1}{1-\rho_{j}},$

where $\rho_{j}$ is the traffic intensity at station $j$ (see (2.3) in Section
2 for the definition). When all the stations have roughly equal heavy traffic
normalization factors, Reiman (1984) and Johnson (1983) developed an
approximation to the finite-time dynamic behavior of such a network based on a
functional central limit theorem. The theorem applies one common scaling
(based on the heavy traffic normalization factor for an arbitrarily chosen
station) across all the different stations, and establishes that the resulting
scaled queue length vector process then converges to a multi-dimensional
reflecting Brownian motion (RBM) in heavy traffic (i.e. as the normalization
factor tends to infinity). This class of RBMs was first introduced in Harrison
and Reiman (1981). The stationary distribution of the RBM was shown to exist
in Harrison and Williams (1987), and they proved that the stationary
distribution of the RBM was characterized by the so-called basic adjoint
relationship (BAR). As with the pre-limit, the RBM stationary distribution is
not of product form in general; see Harrison (1978). By developing an
elaborate limit-interchange procedure, Gamarnik and Zeevi (2006) proved that
the stationary distribution of a generalized Jackson network is well
approximated by that of the corresponding RBM in the same “common scaling”
heavy traffic environment. Numerical algorithms have been developed for
computing the stationary distribution of such an RBM in low dimensions; see
Dai and Harrison (1992); Shen et al. (2002). These algorithms have a
complexity that does not scale well in the number of stations. A heuristic
sequential bottleneck decomposition (SBD) was proposed in Dai et al. (1994).

In this paper, we consider networks with a single server at each station. We
consider a network in which the heavy traffic normalization factors for the
different stations are all large, but _of widely separated magnitudes_. In
such a setting, it is natural to separately scale each station’s queue length
process by that station’s corresponding heavy-traffic normalization factor.
With this “multi-scaling” heavy traffic condition in place, we prove that the
stationary distribution of the separately scaled queue length vector process
in any generalized Jackson network converges to a tractable product-form
limit. Each component in the product form has an exponential distribution,
corresponding to the Brownian approximation of a single station queue. The
“single station” can be constructed precisely and has a good intuitive
interpretation, consistent with the bottleneck analysis advanced in Chen and
Mandelbaum (1991). Our proof critically relies on the BAR-approach recently
developed in Braverman et al. (2017) for queueing networks with general
interarrival and service time distributions. The BAR-approach provides an
alternative to the limit interchange procedure in Gamarnik and Zeevi (2006)
and Budhiraja and Lee (2009). Braverman et al. (2023) further demonstrates the
superiority of employing the BAR-approach for multiclass queueing networks
under priority service disciplines. Here, we demonstrate that the BAR-approach
is a natural approach to discover and justify results of the type discussed in
this paper. Our proof relies heavily on a steady-state moment bound for the
scaled queue length vector. This bound is proved in a companion paper Guang et
al. (2023).

We note that when the normalization factors $(1-\rho_{j})^{-1}$ are widely
separated, this implies that in heavy traffic, the diffusion temporal scalings
$(1-\rho_{j})^{2}$ that dictate the time scales governing the dynamics at each
station are also widely separated. It is this “time scale separation” that
leads to the independence across stations, as argued intuitively in Section 3.

##  2 Generalized Jackson network

We first define an open generalized Jackson network, following closely Section
2.1 of Braverman et al. (2017) both in terms of terminology and notation.
There are $J$ single-server stations in the network. Each station has an
infinite-size buffer that holds jobs waiting for service. Each station has an
external job arrival process. An arriving job is processed at various stations
in the network, one station at a time, before eventually exiting the network.
When a job arrives at a station and finds the server busy processing another
job, the arriving job waits in the buffer until its turn of service. After
being processed at one station, the job either is routed to another station
for further processing or exits the network, independent of all previous
history. Jobs at each station, either waiting or in service, are homogeneous
in terms of service times and routing probabilities. Jobs at each station are
processed following the first-come-first-serve (FCFS) discipline.

Let ${\cal J}=\\{1,\ldots,J\\}$ be the set of stations. The following
notational system follows Braverman et al. (2023) closely. For each station
$j\in\cal{J}$, we are given two nonnegative numbers
$\alpha_{j}\in\mathbb{R}_{+}$ and $\mu_{j}>0$, two sequences of i.i.d. random
variables $T_{e,j}=\\{T_{e,j}(i),i\in\mathbb{N}\\}$ and
$T_{s,j}=\\{T_{s,j}(i),i\in\mathbb{N}\\}$, and one sequence of i.i.d. random
vectors $\phi_{j}=\\{\phi_{j}(i),i\in\mathbb{N}\\}$. We assume $T_{e,1}$, …,
$T_{e,J}$, $T_{s,1}$, …, $T_{s,J}$, $\phi_{1}$, $\ldots$, $\phi_{J}$ are
independent. Following Braverman et al. (2023), we assume
$\mathbb{E}[T_{e,j}(1)]=1$, $\mathbb{E}[T_{s,j}(1)]=1$ such that $T_{e,j}(1)$
and $T_{s,j}(1)$ are unitized. Let

$\mathcal{E}=\\{j\in\mathcal{J}:\alpha_{j}>0\\}.$

For $j\in\mathcal{E}$, we use $T_{e,j}(i)/\alpha_{j}$ to represent the
interarrival time between the $(i-1)$th and $i$th external arriving jobs at
station $j$ and for, $j\in\mathcal{J}$, we use $T_{s,j}(i)/\mu_{j}$ to denote
the $i$th job service time at station $j$. Therefore, $\alpha_{j}$ is
interpreted as the external arrival rate and $\mu_{j}$ as the service rate at
station $j$. The random vector $\phi_{j}(1)$ takes vector $e^{(k)}$ with
probability $P_{jk}$ for $k\in\mathcal{J}$ and takes vector $e^{(0)}$ with
probability $P_{j0}\equiv 1-\sum_{k\in\mathcal{J}}P_{jk}$, where $e^{(k)}$ is
the $J$-vector with component $k$ being 1 and all other components being $0$,
and $e^{(0)}$ is the $J$-vector of zeros. When $\phi_{j}(i)=e^{(k)}$, the
$i$th departing job from station $j$ goes next to station $k\in\mathcal{J}$.
When $\phi_{j}(i)=e^{(0)}$, the job exits the network. We assume the $J\times
J$ routing matrix $P=(P_{ij})$ is transient, which is equivalent to requiring
that $(I-P)$ is invertible. This assumption ensures that each arriving job
will eventually exit the network, and thus the network is known as an open
generalized Jackson network.

We assume that there exists a $\delta_{0}>0$ such that

$\displaystyle\mathbb{E}\big{[}\big{(}T_{e,j}(1)\big{)}^{J+1+\delta_{0}}\big{]}<\infty,\quad\mathbb{E}\big{[}\big{(}T_{s,j}(1)\big{)}^{J+1+\delta_{0}}\big{]}<\infty,\quad
j\in\mathcal{J}.$ (2.1)

Denote by $c^{2}_{e,j}=\operatorname{Var}\left(T_{e,j}(1)\right)$ and
$c^{2}_{s,j}=\operatorname{Var}\left(T_{s,j}(1)\right)$. It is known that
$c^{2}_{e,j}$ and $c^{2}_{s,j}$ are the squared coefficients of variation for
interarrival time and service time, respectively.

Traffic Equation. Given the queueing network, let $\lambda$ be the unique
solution to the traffic equation

$\lambda=\alpha+P^{\prime}\lambda,$ (2.2)

where $P^{\prime}$ is the transpose of $P$. Here $\lambda_{j}$ is referred to
as the nominal total arrival rate to station $j$, including the external
arrival and the arrival from other stations. For each station
$j\in\mathcal{J}$, define the traffic intensity at station $j$ by

$\displaystyle\rho_{j}=\lambda_{j}/\mu_{j}.$ (2.3)

We assume that

$\displaystyle\rho_{j}<1,\quad j\in\mathcal{J}.$

Markov process. For $t\geq 0$ and $j\in\cal{J}$, let $Z_{j}(t)$ be the number
of jobs at station $j$, including possibly one being in service. Let
$R_{e,j}(t)$ be the residual time until the next external arrival to station
$j$. Let $R_{s,j}(t)$ be the residual service time for the job being processed
in the station $j$. If $Z_{j}(t)=0$, the residual service time is the service
time of next station $j$ job, meaning $R_{s,j}(t)=T_{s,j}(i)$ for an
appropriate $i\in\mathbb{N}$. We write $Z(t),R_{e}(t),R_{s}(t)$ as the vectors
of $Z_{j}(t),R_{e,j}(t)$, and $R_{s,j}(t)$, respectively. Define

$X(t)=(Z(t),R_{e}(t),R_{s}(t))$

for each $t\geq 0$. Then $\\{X(t),t\geq 0\\}$ is a Markov process with respect
to the filtration $\mathbb{F}^{X}\equiv\left\\{\mathcal{F}_{t}^{X};t\geq
0\right\\}$ defined on the state space
$\mathbb{Z}_{+}^{J}\times\mathbb{R}_{+}^{J}\times\mathbb{R}_{+}^{J}$, where
$\mathcal{F}_{t}^{X}=\sigma(\\{X(u);0\leq u\leq t\\})$.

##  3 Multi-scale heavy traffic and the main result

We consider a family of generalized Jackson networks indexed by $r\in(0,1)$.
We denote by $\mu^{(r)}_{j}$ the service rate at station $j$ in the $r$th
network. To keep the presentation clean, the service rate vector $\mu^{(r)}$
is the only network parameter that depends on $r\in(0,1)$. In particular,
arrival rate $\alpha$, unitized interarrival and service times, and routing
vectors do not depend on $r$. As a result, the nominal total arrival rate
$\lambda$ does not depend on $r$.

######  Assumption 3.1 (Multi-scale heavy traffic).

We assume for $r\in(0,1)$

$\displaystyle\mu^{(r)}_{j}-\lambda_{j}=r^{j},\quad j\in\mathcal{J}.$ (3.4)

Under condition (3.4), $\mu^{(r)}_{j}\to\lambda_{j}$ and

$\displaystyle\rho^{(r)}_{j}=\lambda_{j}/\mu_{j}^{(r)}=1-r^{j}/\mu^{(r)}_{j}\to
1$

as $r\downarrow 0$. Therefore, each station is in heavy traffic when $r\to 0$.
We call condition (3.4) the multi-scale heavy traffic condition. Dai (1995)
proves that the following assumption holds under some mild distributional
assumption on interarrival times.

######  Assumption 3.2 (Stability).

For each $r\in(0,1)$, the Markov process $\\{X^{(r)}(t),t\geq 0\\}$ is
positive Harris recurrent and it has a unique stationary distribution.

Under Assumption 3.2, we use

$\displaystyle X^{(r)}=\big{(}Z^{(r)},R_{e}^{(r)},R_{s}^{(r)}\big{)}$

to denote the steady-state random vector that has the stationary distribution.
It is well known that

$\displaystyle\mathbb{P}\\{Z^{(r)}_{j}=0\\}=1-\rho^{(r)}_{j}=r^{j}/\mu^{(r)}_{j},\quad
j\in\mathcal{J}.$ (3.5)

See, for example, (A.11) and (A.13) of Braverman et al. (2017) for a proof.

Throughout of this paper, we adopt the notation that $f(r)=O(g(r))$ means

$\displaystyle C_{1}\lvert g(r)\rvert\leq\lvert f(r)\rvert\leq C_{2}\lvert
g(r)\rvert\quad\text{ as }r\to 0$

for some $C_{1},C_{2}>0$, and $f(r)=o(g(r))$ means $f(r)/g(r)\to 0$ as $r\to
0$.

To state the main theorem, we introduce the $J\times J$ matrix $w=(w_{ij})$.
For each $j\in\mathcal{J}$,

$\displaystyle(w_{1j},\ldots,w_{j-1,j})^{\prime}=(I-P_{j-1})^{-1}P_{[1:j-1],j},$
(3.6)
$\displaystyle(w_{jj},\ldots,w_{J,j})^{\prime}=P_{[j:J],j}+P_{[j:J],[1:j-1]}(I-P_{j-1})^{-1}P_{[1:j-1],j},$
(3.7)

where prime denotes transpose, and, for $A,B\subseteq\mathcal{J}$, $P_{A,B}$
is the submatrix of $P$ whose entries are $(P_{k\ell})$ with $k\in A$ and
$\ell\in B$ with the following conventions: for $\ell\leq k$,

$\displaystyle[\ell:k]=\\{\ell,\ell+1,\ldots,k\\},\quad
P_{A,k}=P_{A,\\{k\\}},\quad P_{\ell,B}=P_{\\{\ell\\},B},\quad
P_{k}=P_{[1:k],[1:k]},$

and any expression involving $P_{A,B}$ is interpreted to be zero when either
$A$ or $B$ is the empty set. The expressions in (3.6) and (3.7) may look
complicated, but $(w_{ij})$ has the following probabilistic interpretation.
Recall that the routing matrix $P$ can be extended into a transition matrix of
a DTMC on state space $\\{0\\}\cup\mathcal{J}$ with $0$ being the absorbing
state. For each $i\in\mathcal{J}$ and $j\in\mathcal{J}$, $w_{ij}$ is the
probability that starting from state $i$, the DTMC will eventually visit state
$j$ while avoiding visiting states $\\{0,j+1,\ldots,J\\}$. See more discussion
of $(w_{ij})$ in Lemma 7.3 in Section 7.

######  Theorem 3.1.

Assume condition (2.1) and Assumptions 3.1-3.2 all hold. Then

$\displaystyle\Big{(}rZ_{1}^{(r)},\ldots,r^{J}Z_{J}^{(r)}\Big{)}\Rightarrow(Z^{*}_{1},\ldots,Z^{*}_{J}),\quad\text{
as }r\to 0,$ (3.8)

where “$\Rightarrow$” stands for convergence in distribution. Furthermore,
$Z^{*}_{1}$, $\ldots$, $Z^{*}_{J}$ are independent, and for each station
$j\in\mathcal{J}$, $Z^{*}_{j}$ is an exponential random variable with mean
$d_{j}$ given by

$\displaystyle d_{j}=$
$\displaystyle\frac{\sigma^{2}_{j}}{2(1-w_{jj})},\quad\text{ and }$ (3.9)
$\displaystyle\sigma_{j}^{2}=$
$\displaystyle\sum_{i<j}\alpha_{i}\bigl{(}w_{ij}^{2}c_{e,i}^{2}+w_{ij}(1-w_{ij})\bigr{)}+\alpha_{j}c_{e,j}^{2}+\sum_{i>j}\lambda_{i}\bigl{(}w_{ij}^{2}c_{s,i}^{2}+w_{ij}(1-w_{ij}))$
$\displaystyle+\lambda_{j}\bigl{(}c_{s,j}^{2}(1-w_{jj})^{2}+w_{jj}(1-w_{jj})\bigr{)}.$
(3.10)

In light of the results in literature, see for example Whitt and You (2020)),
the convergence of marginal distribution $r^{j}Z_{j}^{(r)}\Rightarrow
Z^{*}_{j}$ is not surprising. What is surprising is that $Z^{*}_{1}$,
$\ldots$, $Z^{*}_{J}$ are independent. In particular, while the distribution
of $(Z^{*}_{1},\ldots,Z^{*}_{J})$ is of product form, the pre-limit stationary
distribution is _not_ of product form, nor is the stationary distribution of
conventional heavy traffic limit, which is a multi-dimensional reflected
Brownian motion Reiman (1984).

The variance parameter $\sigma_{j}^{2}$ in (3.10) has the following appealing
interpretation. Consider a renewal arrival process $\\{E(t),t\geq 0\\}$ with
arrival rate $\nu$ and squared coefficient of variation $c^{2}$ of the
interarrival time. Each arrival has probability $p_{j}$ of going to station
$j$ instantly, independent of all previous history. Let $\\{A_{j}(t),t\geq
0\\}$ be the corresponding arrival process to station $j$. It is known that

$\displaystyle\lim_{t\to\infty}\frac{\text{Var}(A_{j}(t))}{t}=\bigl{(}c^{2}p_{j}^{2}+p_{j}(1-p_{j})\bigr{)}\nu.$
(3.11)

Thus, each of the first $J$ terms in the right side of (3.10) corresponds to
the variance parameter in (3.11) of $J$ arrival sources to station $j$. The
first $j-1$ terms correspond to arrival sources from lightly-loaded stations,
the $j$th term corresponds to the external arrival process to station $j$, and
the last $J-j$ terms correspond to arrival sources from service completions at
heavily-loaded stations. For a lightly-loaded station $\ell<j$, its external
arrival process serves as the renewal arrival process. For a heavily-loaded
station $k>j$, its service process serves as the renewal arrival process (as
if the server is always busy). The resulting formula $d_{j}$ in (3.9) is
consistent with the diffusion approximation of a $\sum_{i}G_{i}/GI/1$ queue
with $J$ external arrival processes and immediate feedback probability
$w_{jj}$ after the service at station $j$.

Theorem 3.1 suggests an obvious approximation to the stationary distribution
of the queue length vector process for a generalized Jackson network with
widely separated values of $(\mu_{j}-\lambda_{j})^{-1}$, $j\in\mathcal{J}$.
Our approximation is

$\displaystyle\big{(}Z_{1}(\infty),Z_{2}(\infty),\ldots,Z_{J}(\infty)\big{)}\stackrel{{\scriptstyle
d}}{{\approx}}\Big{(}\frac{Z^{*}_{1}}{\mu_{1}-\lambda_{1}},\frac{Z^{*}_{2}}{\mu_{2}-\lambda_{2}},\ldots,\frac{Z^{*}_{J}}{\mu_{J}-\lambda_{J}}\Big{)},$

where $(Z_{1}^{*},\ldots,Z_{J}^{*})$ is described in Theorem 3.1, and
$\stackrel{{\scriptstyle d}}{{\approx}}$ denotes “has approximately the same
distribution as” (and carries no rigorous meaning beyond that imparted by
Theorem 3.1).

Theorem 3.1 will be proved in Section 7. The proof relies on the BAR in
Section 4, the asymptotic BAR in Section 5, and state space collapse in
Section 6. In addition, the proof relies critically on a uniform moment bound
that is proved in the companion paper (Guang et al., 2023, Theorem 2). For
easy reference, we state that result here.

######  Proposition 3.1 (Theorem 2 of Guang et al. (2023)).

Assume (2.1), Assumption 3.1, and Assumption 3.2. There exists $r_{0}\in(0,1)$
and $\epsilon_{0}>0$ such that

$\sup_{r\in(0,r_{0})}\mathbb{E}\Big{[}\big{(}r^{j}Z_{j}^{(r)}\big{)}^{J+\epsilon_{0}}\Big{]}<\infty,\quad
j\in\mathcal{J}.$ (3.12)

######  Corollary 3.1.

Assume the assumptions in Proposition 3.1. Then

$\displaystyle\mathbb{E}\Big{[}\sum_{\ell<j}r^{\ell+1}Z^{(r)}_{\ell}\mid
Z_{j}^{(r)}=0\Big{]}=o(1)\quad\text{ as }r\to 0,\quad\text{ for
}j\in{\cal{J}}.$ (3.13)

###### Proof.

For each $j\in\mathcal{J}$ and $\ell<j$, by choosing $p=j+\epsilon_{0}$,
$q=(j+\epsilon_{0})/(j+\epsilon_{0}-1)$ and employing the Holder’s inequality

$\displaystyle\mathbb{E}\Big{[}r^{\ell}Z^{(r)}_{\ell}1_{\\{Z_{j}^{(r)}=0\\}}\Big{]}\leq\Big{(}\mathbb{E}\big{[}(r^{\ell}Z^{(r)}_{\ell})^{p}\big{]}\Big{)}^{1/p}\bigl{(}\mathbb{P}\\{Z_{j}^{(r)}=0\\}\bigr{)}^{1/q},$

one can show that

$\displaystyle r\mathbb{E}\Big{[}r^{\ell}Z^{(r)}_{\ell}\mid
Z_{j}^{(r)}=0\Big{]}$ $\displaystyle\leq
r\Big{(}\mathbb{E}\big{[}(r^{\ell}Z^{(r)}_{\ell})^{p}\big{]}\Big{)}^{1/p}\bigl{(}\mathbb{P}\\{Z_{j}^{(r)}=0\\}\bigr{)}^{1/q-1}$
$\displaystyle=\Big{(}\mathbb{E}\big{[}(r^{\ell}Z^{(r)}_{\ell})^{p}\big{]}\Big{)}^{1/p}(\mu^{(r)}_{j})^{1/(j+\epsilon_{0})}r^{\frac{\epsilon_{0}}{j+\epsilon_{0}}}=o(1),$

which proves (3.13). ∎

The moment bound (3.12) holds of course when interarrival and service time
distributions are exponential because $Z_{j}^{(r)}$ is geometrically
distributed with mean $\rho^{(r)}_{j}/(1-\rho^{(r)}_{j})$. In the exponential
case, Lemma 23 of Xu (2022) provides an alternative proof via Lyapunov
functions.

##  4 Basic adjoint relationship (BAR)

In this section, we present the basic adjoint relationship (BAR) that is the
starting point for the proof of Theorem 3.1. Following Braverman et al.
(2023), we present the BAR progressively, first when all distributions are
exponential, next when all distributions are bounded and otherwise general,
and finally when all distributions are general satisfying the moment condition
(2.1).

For each $\theta\in\mathbb{R}_{-}^{J}\equiv\\{x\in\mathbb{R}^{J}:x\leq 0\\}$,
define the exponential function $f_{\theta}:\mathbb{Z}^{J}_{+}\to\mathbb{R}$
via

$\displaystyle f_{\theta}(z)=\exp(\langle\theta,z\rangle),\quad
z\in\mathbb{R}_{+}^{J},$ (4.14)

where $\langle\theta,z\rangle=\sum_{j\in\mathcal{J}}\theta_{j}z_{j}$, and
$\mathbb{R}_{+}^{J}$ is the nonnegative orthant in $\mathbb{R}^{J}$. Define
the moment generating function (MGF) or Laplace transform $\phi^{(r)}$ of
$Z^{(r)}$

$\displaystyle\phi^{(r)}(\theta)=\mathbb{E}[f_{\theta}(Z^{(r)})],\quad\theta\in\mathbb{R}^{J}_{-},$
$\displaystyle\phi^{(r)}_{i}(\theta)=\mathbb{E}[f_{\theta}(Z^{(r)})\mid
Z^{(r)}_{i}=0],\quad i\in\mathcal{J}.$

It is well known that (3.8) is equivalent to

$\displaystyle\lim_{r\to\infty}\phi^{r}(r\eta_{1},\ldots,r^{J}\eta_{J})=\prod_{j\in\mathcal{J}}\frac{1}{1-d_{j}\eta_{j}},\quad\eta=(\eta_{1},\ldots,\eta_{J})^{\prime}\in\mathbb{R}^{J}_{-}.$
(4.15)

To prove (4.15), we will start with the case when interarrival and service
time distributions are exponential. In this case, $\left\\{Z^{(r)}(t),t\geq
0\right\\}$ is a continuous time Markov chain (CTMC) with discrete state space
$\mathbb{Z}_{+}^{J}$ for each $r\in(0,1)$. The generator $G^{(r)}$ for the
CTMC, applying to test function $f:\mathbb{Z}^{J}_{+}\to\mathbb{R}$, is given
by

$\displaystyle G^{(r)}f(z)$
$\displaystyle=\sum_{i\in\cal{J}}\alpha_{i}\Big{[}f(z+e^{(i)})-f(z)\Big{]}$
$\displaystyle+\sum_{i\in\cal{J}}\mu_{i}^{(r)}\mathbbm{1}(z_{i}>0)\sum_{j\in\\{0\\}\cup\cal{J}}P_{ij}\Big{[}f(z-e^{(i)}+e^{(j)})-f(z)\Big{]},\quad
z\in\mathbb{Z}^{J}_{+},$ (4.16)

where, again, $e^{(j)}$ is the $J$-vector with $j$th component being 1 and 0
for all other components, $e^{(0)}$ is the $J$-vector of zero, and

$\displaystyle P_{i0}=1-\sum_{j\in\mathcal{J}}P_{ij},\quad i\in\mathcal{J}.$
(4.17)

It follows from Proposition 3 of Glynn and Zeevi (2008) that the following
basic adjoint relationship (BAR) holds:

$\displaystyle\mathbb{E}\bigg{[}Gf\Big{(}{Z}^{(r)}\Big{)}\bigg{]}=0,\quad
f:\mathbb{Z}^{J}_{+}\to\mathbb{R}\text{ bounded}.$ (4.18)

For the exponential test function $f_{\theta}$ in (4.14), it is bounded when
$\theta\in\mathbb{R}^{J}_{-}$.

######  Lemma 4.1.

For exponential test functions $f_{\theta}$ in (4.14), BAR (4.18) becomes

$\displaystyle
Q(\theta)\phi^{(r)}(\theta)+\sum_{i\in\mathcal{J}}(\mu^{(r)}_{i}-\lambda_{i})\xi_{i}(\theta)\big{(}\phi^{(r)}(\theta)-\phi^{(r)}_{i}(\theta)\big{)}=0,\quad\theta\in\mathbb{R}^{J}_{-},$
(4.19)

where

$\displaystyle\gamma_{i}(\theta_{i})=e^{\theta_{i}}-1,$ (4.20)
$\displaystyle\xi_{i}(\theta)=P_{i0}\big{(}e^{-\theta_{i}}-1\big{)}+\sum_{j\in\cal{J}}P_{ij}\big{(}e^{\theta_{j}-\theta_{i}}-1\big{)},$
(4.21) $\displaystyle
Q(\theta)=\sum_{i\in\mathcal{J}}\bigl{(}\alpha_{i}\gamma_{i}(\theta_{i})+\lambda_{i}\xi_{i}(\theta)\bigr{)}.$
(4.22)

###### Proof.

One can easily verify that (4.16) becomes

$\displaystyle G^{(r)}f_{\theta}(z)=$
$\displaystyle\sum_{i\in\mathcal{J}}\alpha_{i}\gamma_{i}(\theta_{i})f_{\theta}(z)+\sum_{i\in\cal{J}}\mu_{i}^{(r)}\xi_{i}(\theta)f_{\theta}(z)\mathbbm{1}(z_{i}>0).$

It follows from (4.18) that

$\displaystyle\sum_{i\in\mathcal{J}}\alpha_{i}\gamma_{i}(\theta_{i})\mathbb{E}[f_{\theta}(Z^{(r)})]+\sum_{i\in\cal{J}}\mu_{i}^{(r)}\xi_{i}(\theta)\mathbb{E}[f_{\theta}(Z^{(r)})\mathbbm{1}(Z^{(r)}_{i}>0)]=0.$
(4.23)

Utilizing (3.5), one can verify that (4.23) is equivalent to (4.19). ∎

Now we assume interarrival and service time distributions are general so that
$\\{Z^{(r)}(t),t\geq 0\\}$ is not Markov. Instead, we focus on Markov process
$\\{X^{(r)}(t),t\geq 0\\}$. Recall that
$X^{(r)}=(Z^{(r)},R_{e}^{(r)},R^{(r)}_{s})$ denotes the random vector that is
distributed according to the stationary distribution. In the following, we
first assume interarrival and service time distributions are _bounded_. In
addition,

$\displaystyle\mathbb{P}\\{T_{e,j}(1)>0\\}=1,\quad\mathbb{P}\\{T_{s,j}(1)>0\\}=1,\quad
j\in\mathcal{J}.$ (4.24)

For $\theta\in\mathbb{R}^{J}$, define $\gamma_{i}(\theta)$ and
$\xi_{i}(\theta)$ via

$\displaystyle
e^{\theta_{i}}\mathbb{E}[e^{-\gamma_{i}(\theta_{i})T_{e,i}(1)}]=1,$ (4.25)
$\displaystyle\bigl{(}P_{i0}e^{-\theta_{i}}+\sum_{j\in\mathcal{J}}P_{ij}e^{\theta_{j}-\theta_{i}}\bigr{)}\mathbb{E}[e^{-\xi_{i}(\theta)T_{s,i}(1)}]=1,$
(4.26)

As explained in Braverman et al. (2017), under condition (4.24),
$\gamma_{i}(\theta_{i})$ and $\xi_{i}(\theta)$ are defined for _all_
$\theta\in\mathbb{R}^{J}$. Otherwise, they are defined only in a neighborhood
of $\theta=0$. Since we will choose

$\displaystyle\theta=(r\eta_{1},\ldots,r^{J}\eta_{J})\to 0\quad\text{as}\quad
r\to 0$

to be used in (4.15) for each fixed $\eta\in\mathbb{R}^{J}_{-}$, condition
(4.24) is _not_ needed for the proof of our theorem. In the rest of this
paper, we keep condition (4.24) for notational simplicity.

For $\theta\in\mathbb{R}^{J}_{-}$, define

$\displaystyle\tilde{f}_{\theta}(x)=\exp\Big{(}\langle\theta,z\rangle-\sum_{i}\alpha_{i}\gamma_{i}(\theta_{i})u_{i}-\sum_{i}\mu^{(r)}_{i}\xi_{i}(\theta)v_{i}\Big{)},$

where
$x=(z,u,v)\in\mathbb{Z}^{J}_{+}\times\mathbb{R}^{J}_{+}\times\mathbb{R}^{J}_{+}$
and

$\displaystyle\tilde{\psi}^{(r)}(\theta)=\mathbb{E}[\tilde{f}_{\theta}(X^{(r)})],\quad\tilde{\psi}^{(r)}_{i}(\theta)=\mathbb{E}[\tilde{f}_{\theta}(X^{(r)})\mid
Z^{(r)}_{i}=0],\quad i\in\mathcal{J}.$

The following lemma states that BAR (4.19) for $\tilde{\psi}^{(r)}$ and
$\tilde{\psi}^{(r)}_{i}$ continues to hold. Namely,

######  Lemma 4.2.

Assume Assumption 3.2 holds and that interarrival and service times are
bounded, satisfying condition (4.24). Then,

$\displaystyle
Q(\theta){\tilde{\psi}}^{(r)}(\theta)+\sum_{i\in\mathcal{J}}(\mu^{(r)}_{i}-\lambda_{i})\xi_{i}(\theta)\bigl{(}\tilde{\psi}^{(r)}(\theta)-\tilde{\psi}_{i}^{(r)}(\theta)\bigr{)}=0,\quad\theta\in\mathbb{R}^{J}_{-},$
(4.27)

where $\gamma_{i}(\theta_{i})$, $\xi_{i}(\theta)$, and $Q(\theta)$ are defined
in (4.25), (4.26), and (4.22), respectively.

###### Proof.

It follows from Braverman et al. (2017) that

$\displaystyle\sum_{i\in\mathcal{J}}\alpha_{i}\gamma_{i}(\theta_{i})\mathbb{E}[\tilde{f}_{\theta}(Z^{(r)})]+\sum_{i\in\cal{J}}\mu_{i}^{(r)}\xi_{i}(\theta)\mathbb{E}[\tilde{f}_{\theta}(Z^{(r)})\mathbbm{1}(Z^{(r)}_{i}>0)]=0.$
(4.28)

Utilizing (3.5), one can verify that (4.28) is equivalent to (4.27). ∎

Even under the assumption that interarrival and service times are bounded,
Theorem 3.1 is novel and interesting. A first time reader may want to focus
their reading on this case.

In Section 5, we will develop an asymptotic BAR when interarrival and service
times are general, satisfying the moment condition (2.1). For that, we
consider a sequence of truncated interarrival and service times.

For $\theta\in\mathbb{R}^{J}$ and $t\in(0,1)$, we define
$\gamma_{j}(\theta_{j},t)$ and $\xi_{j}(\theta,t)$ via

$\displaystyle
e^{\theta_{j}}\mathbb{E}[e^{-\gamma_{j}(\theta_{j},t)(T_{e,j}(1)\wedge(1/t))}]=1,$
(4.29)
$\displaystyle\bigl{(}\sum_{k\in\mathcal{J}}P_{jk}e^{\theta_{k}-\theta_{j}}+P_{j0}e^{-\theta_{j}}\bigr{)}\mathbb{E}[e^{-\xi_{j}(\theta,t)(T_{s,j}(1)\wedge(1/t))}]=1.$
(4.30)

These definitions are identical to the ones in (4.25) and (4.26) except that
the interarrival and service times are truncated at level $1/t$. When
interarrival and service times are bounded by $1/t$,
$\gamma_{j}(\theta_{j},t)=\gamma_{j}(\theta_{j})$ and
$\xi_{j}(\theta,t)=\xi_{j}(\theta)$.

For each $\theta\in\mathbb{R}^{J}_{-}$ and each $r\in(0,1)$, define the
function $f_{(\theta,r)}(x)$ for
$x=(z,u,v)\in\mathbb{Z}^{J}_{+}\times\mathbb{R}^{J}_{+}\times\mathbb{R}^{J}_{+}$
via

$\displaystyle
f_{(\theta,r)}(x)=\exp\Bigl{(}\langle\theta,z\rangle-\sum_{j}\gamma_{j}(\theta_{j},r)(\alpha_{j}u_{j}\wedge
r^{-1})-\sum_{j}\xi_{j}(\theta,r)(\mu_{j}^{(r)}v_{j}\wedge r^{-1})\Bigr{)}.$
(4.31)

For each $\theta\in\mathbb{R}^{J}_{-}$, define

$\displaystyle{\psi}^{(r)}(\theta)=\mathbb{E}[f_{(\theta,r)}(X^{(r)})],\quad{\psi}^{(r)}_{j}(\theta)=\mathbb{E}\big{[}f_{(\theta,r)}(X^{(r)})\mid
Z^{(r)}_{j}=0\big{]}.$

######  Lemma 4.3.

Assume Assumption 3.2 holds and that interarrival and service times are
bounded, satisfying condition (4.24). Then,

$\displaystyle
Q^{(r)}(\theta){{\psi}}^{(r)}(\theta)+\sum_{i\in\mathcal{J}}(\mu^{(r)}_{i}-\lambda_{i})\xi_{i}(\theta,r)\bigl{(}{\psi}^{(r)}(\theta)-{\psi}_{i}^{(r)}(\theta)\bigr{)}=\epsilon(\theta,r),\quad\theta\in\mathbb{R}^{J}_{-},$
(4.32)

where $\gamma_{i}(\theta_{i},r)$ and $\xi_{i}(\theta,r)$ are defined in (4.29)
and (4.30), respectively, and

$\displaystyle{Q}^{(r)}(\theta)$
$\displaystyle=\sum_{i\in\mathcal{J}}\Big{(}\alpha_{i}\gamma_{i}(\theta_{i},r)+\lambda_{i}\xi_{i}(\theta,r)\Big{)},$
(4.33) $\displaystyle\epsilon(\theta,r)$
$\displaystyle=\sum_{i\in\mathcal{J}}\alpha_{i}\gamma_{i}(\theta_{i},r)\mathbb{E}\big{[}1(\alpha_{i}R^{(r)}_{e,i}>1/r)f_{(\theta,r)}(X^{(r)})\big{]}$
$\displaystyle+\sum_{i\in\mathcal{J}}\mu^{(r)}_{i}\xi_{i}(\theta,r)\mathbb{E}\big{[}1(\mu_{i}^{(r)}R^{(r)}_{s,i}>1/r,Z^{(r)}_{i}>0)f_{(\theta,r)}(X^{(r)})\big{]}.$
(4.34)

###### Proof.

It follows from Lemma 3.3 of Braverman et al. (2017) that

$\displaystyle\sum_{i\in\mathcal{J}}\alpha_{i}\gamma_{i}(\theta_{i},r)\mathbb{E}[f_{(\theta,r)}(Z^{(r)})\mathbbm{1}(\alpha_{i}R^{(r)}_{e,i}\leq
1/r)]$
$\displaystyle\qquad+\sum_{i\in\cal{J}}\mu_{i}^{(r)}\xi_{i}(\theta,r)\mathbb{E}[f_{(\theta,r)}(Z^{(r)})\mathbbm{1}(Z^{(r)}_{i}>0,\mu^{(r)}_{i}R^{(r)}_{s,i}\leq
1/r)]=0.$ (4.35)

Utilizing (3.5), one can verify that (4.35) is equivalent to (4.32). ∎

##  5 Asymptotic BAR

In this section, we state and prove the following asymptotic BAR.

######  Lemma 5.1.

Assume that Assumption 3.2 and moment condition (2.1) hold. Assume
$\theta=\theta(r)\in\mathbb{R}^{J}_{-}$ for each $r\in(0,1)$, satisfying

$\displaystyle\lvert\theta(r)\rvert\leq cr\quad r\in(0,1)$ (5.36)

for some constant $c>0$. Then, the following asymptotic BAR holds: as $r\to
0$,

$\displaystyle
Q^{*}(\theta)\psi^{(r)}(\theta)+\sum_{i\in\mathcal{J}}(\mu^{r}_{i}-\lambda_{i})\bar{\xi}_{i}(\theta)\bigl{(}\psi^{(r)}(\theta)-\psi^{(r)}_{i}(\theta)\bigr{)}$
$\displaystyle=-\frac{1}{2}\sum_{i\in\mathcal{J}}(\mu^{(r)}_{i}-\lambda_{i}){\xi}^{*}_{i}(\theta)\bigl{(}\psi^{(r)}(\theta)-\psi^{(r)}_{i}(\theta)\bigr{)}+o(r^{J}\lvert\theta\rvert)+o(\lvert\theta\rvert^{2}),$
(5.37)

where $\theta=\theta(r)\to 0$, and

$\displaystyle\bar{\xi}_{i}(\theta)=\sum_{j\in\mathcal{J}}P_{ij}\theta_{j}-\theta_{i},$
(5.38)
$\displaystyle\xi^{*}_{i}(\theta)=c_{s,i}^{2}\bigl{(}\sum_{j\in\mathcal{J}}P_{ij}\theta_{j}-\theta_{i}\bigr{)}^{2}+\sum_{j\in\mathcal{J}}P_{ij}\theta_{j}^{2}-\bigl{(}\sum_{j\in\mathcal{J}}P_{ij}\theta_{j}\bigr{)}^{2},$
(5.39) $\displaystyle
Q^{*}(\theta)=\frac{1}{2}\sum_{i\in\mathcal{J}}\big{(}\alpha_{i}\gamma_{i}^{*}(\theta_{i})+\lambda_{i}\xi_{i}^{*}(\theta)\big{)},\text{
with }$ (5.40)
$\displaystyle\gamma_{i}^{*}(\theta_{i})=c_{e,i}^{2}\theta_{i}^{2}.$ (5.41)

To prove the asymptotic BAR, we first establish a series of lemmas. The
following lemma follows the identical proof of Lemma 5.6 in Braverman et al.
(2023).

######  Lemma 5.2.

For each fixed $a>0$, there is a constant $d_{a}>0$ such that

$\displaystyle\lvert\gamma_{i}(\theta_{i},r)\rvert\leq
d_{a}\lvert\theta_{i}\rvert,\quad\lvert\xi_{i}(\theta,r)\rvert\leq
d_{a}\lvert\theta\rvert,\quad i\in\mathcal{J}$ (5.42)

for $r\in(0,1)$ and $\theta\in\mathbb{R}^{J}$ with $\lvert\theta\rvert\leq a$.

The following lemma provides Taylor expansions for $\gamma_{i}(\theta_{i},r)$
and $\xi_{i}(\theta,r)$, analogous to Lemma 5.7 in Braverman et al. (2023).
For completeness, we provide a full proof here.

######  Lemma 5.3.

Let $\theta(r)\in\mathbb{R}^{J}$ be given for each $r\in(0,1)$, satisfying
(5.36). Denoting $\theta=\theta(r)$,

$\displaystyle\gamma_{i}(\theta_{i},r)=\theta_{i}+\frac{1}{2}\gamma_{i}^{*}(\theta_{i})+o(r^{J}\theta_{i})+o(\theta_{i}^{2}),\quad
i\in\mathcal{E}$ (5.43)
$\displaystyle\xi_{i}(\theta,r)=\bar{\xi}_{i}(\theta)+\frac{1}{2}\xi_{i}^{*}(\theta)+o(r^{J}\lvert\theta\rvert)+o(\lvert\theta\rvert^{2})\quad
i\in\mathcal{J}.$ (5.44)

###### Proof of Lemma 5.3.

Fix $i\in\mathcal{J}$. For $r\in(0,1)$, define $g^{(r)}(x)=x\wedge r^{-1}$ for
$x\in\mathbb{R}_{+}$. Let

$\displaystyle
a^{(r)}=1/\mathbb{E}\big{[}g^{(r)}\big{(}T_{e,i}(1)\big{)}\big{]},\quad
b^{(r)}=\mathbb{E}\big{[}\big{(}g^{(r)}(T_{e,i}(1))\big{)}^{2}\big{]},\quad\sigma^{2}_{(r)}=b^{(r)}-(1/a^{(r)})^{2}.$

For $y\in\mathbb{R}$, define

$\displaystyle
c^{(r)}_{e,i}(y)=\gamma_{i}(y,r)-ya^{(r)}-\frac{1}{2}y^{2}(a^{(r)})^{3}\sigma^{2}_{(r)}.$

Lemma 4.2 of Braverman et al. (2017) proved that there exists $K>0$,

$\displaystyle\lim_{r\to 0}\sup_{0<\lvert y\rvert\leq
K}\frac{c^{(r)}_{e,i}(ry)}{r^{2}y^{2}}=0,$

which implies that for any $\theta_{i}(r)\in\mathbb{R}$ satisfying (5.36),

$\displaystyle c^{(r)}_{e,i}(\theta_{i}(r))=o((\theta_{i}(r))^{2})\quad\text{
as }r\to 0.$ (5.45)

Note that

$\displaystyle\mathbb{E}[T_{e,i}(1)]-\mathbb{E}[g^{(r)}(T_{e,i}(1))]=\mathbb{E}[(T_{e,i}(1)-r^{-1})1(T_{e,i}(1)>r^{-1})]$
$\displaystyle\leq
r^{J+\delta_{0}}\mathbb{E}([T_{e,i}(1))^{J+1+\delta_{0}}1(T_{e,i}(1)>r^{-1})]=o(r^{J+\delta_{0}})=o(r^{J}),$
$\displaystyle\mathbb{E}[(T_{e,i}(1))^{2}]-\mathbb{E}[(g^{(r)}(T_{e,i}(1)))^{2}]=\mathbb{E}\big{[}(T^{2}_{e,i}(1)-r^{-2})1(T_{e,i}(1)>r^{-1})\big{]}$
$\displaystyle\leq
r^{J-1+\delta_{0}}\mathbb{E}([T_{e,i}(1))^{J+1+\delta_{0}}1(T_{e,i}(1)>r^{-1})]=o(r^{J-1+\delta_{0}})=o(r^{J-1}).$

Therefore, using the facts that $\mathbb{E}[T_{e,i}(1)]=1$ and
$c^{2}_{e,i}={\rm Var}(T_{e,i}(1))$,

$\displaystyle
a^{(r)}=1/\mathbb{E}[g^{(r)}(T_{e,i}(1))]=1/(1+o(r^{J}))=1+o(r^{J}),$
$\displaystyle b^{(r)}=\mathbb{E}[T_{e,i}(1)]^{2}+o(r^{J-1}),$
$\displaystyle\sigma^{2}_{(r)}=c^{2}_{e,i}+o(r^{J-1}),$
$\displaystyle(a^{(r)})^{3}\sigma^{2}_{(r)}=c^{2}_{e,i}+o(r^{J-1}).$

Finally, (5.43) follows from (5.45) and

$\displaystyle
a^{(r)}\theta_{i}-\theta_{i}=\theta_{i}o(r^{J})=o(r^{J}\theta_{i}),$
$\displaystyle(a^{(r)})^{3}\sigma^{2}_{(r)}\theta^{2}_{i}-c^{2}_{e,i}\theta^{2}_{i}=\theta^{2}_{i}o(r^{J-1})=o(r^{J-1}\theta^{2}_{i})=o(\theta_{i}^{2}).$

Expansion (5.44) can be proved similarly, again using Lemma 4.2 of Braverman
et al. (2017).

∎

In the next lemma, recall $f_{(\theta,r)}(x)$ as defined in (4.31).

######  Lemma 5.4.

Assume that $\theta(r)\in\mathbb{R}^{J}_{-}$ for each $r\in(0,1)$ satisfies
(5.36). There exists an $M>0$ such that

$\displaystyle\sup_{r\in(0,1)}\sup_{x=(z,u,v)}f_{(\theta(r),r)}(x)\leq M.$
(5.46)

###### Proof.

It follows from (4.31) that for any $\theta(r)\in\mathbb{R}^{J}_{-}$
satisfying (5.36)

$\displaystyle
f_{(\theta(r),r)}(x)\leq\exp\Big{(}r^{-1}\sum_{j}\lvert\gamma_{j}(\theta_{j}(r),r)\rvert+r^{-1}\sum_{j}\lvert\xi_{j}(\theta(r),r)\rvert\Big{)}\leq
e^{d_{c}(\lvert\mathcal{E}\rvert+J)},$

where $d_{c}$ is a constant in Lemma 5.2, $c$ is a constant in (5.36), and
$\lvert\mathcal{E}\rvert$ is the cardinality of $\mathcal{E}$. ∎

######  Lemma 5.5.

For each $j\in\mathcal{J}$, as $r\to 0$,

$\displaystyle\mathbb{P}\big{\\{}\alpha_{j}R^{(r)}_{e,j}>1/r\big{\\}}=o(r^{J}),$
(5.47)
$\displaystyle\mathbb{P}\big{\\{}\mu^{(r)}_{j}R^{(r)}_{s,j}>1/r\big{\\}}=o(r^{J}).$
(5.48)

###### Proof.

To prove (5.47), we follow

$\displaystyle\mathbb{P}\\{\alpha_{j}R^{(r)}_{e,j}>1/r\\}$ $\displaystyle\leq
r^{J}\mathbb{E}[(\alpha_{j}R^{(r)}_{e,j})^{J}1(\alpha_{j}R^{(r)}_{e,j}>1/r)]$
$\displaystyle=\frac{r^{J}}{J+1}\mathbb{E}\big{[}\big{(}T_{e,j}(1)\big{)}^{J+1}1(T_{e,j}(1)>r^{-1})\big{]}=o(r^{J}),$

where first equality follows from Lemma 6.6 of Braverman et al. (2023), and
the second equality follows from condition (2.1).

Equation (5.48) can be proved similarly. ∎

###### Proof of Lemma 5.1.

It follows from (5.47), (5.48), and Lemma 5.4 that (5.46) that

$\displaystyle\mathbb{E}\big{[}1_{\\{\alpha_{i}R^{(r)}_{e,i}>1/r\\}}f_{(\theta(r),r)}(X^{(r)})\big{]}=o(r^{J}),\quad\mathbb{E}\big{[}1_{\\{\mu^{(r)}_{i}R^{(r)}_{s,i}>1/r\\}}f_{(\theta(r),r)}(X^{(r)})\big{]}=o(r^{J}).$

By Lemma 5.2,

$\displaystyle\gamma_{i}(\theta_{i}(r),r)=O(\theta_{i}(r)),\quad\xi_{i}(\theta(r),r)=O(\lvert\theta(r)\rvert)\quad
r\to 0.$

Therefore,

$\displaystyle\epsilon(\theta(r),r)=o(\lvert\theta(r)\rvert r^{J}),$ (5.49)

where $\epsilon(\theta,r)$ are defined in (4.34). The Taylor expansion

$\displaystyle Q(\theta(r))=Q^{*}(\theta(r))+o(\lvert\theta(r)\rvert^{2}))$
(5.50)

follows from (5.43), (5.44), and the fact that

$\displaystyle\sum_{i\in\mathcal{J}}\bigl{(}\alpha_{i}\bar{\gamma}_{i}(\theta_{i})+\lambda_{i}\bar{\xi}_{i}(\theta)\bigr{)}=0\quad\text{
for each }\theta\in\mathbb{R}^{J},$

where the last equality follows from traffic equation (2.2). Finally, the
lemma follows from BAR (4.32), the Taylor expansions (5.44) and (5.50), the
estimate (5.49), and the fact that $\psi^{(r)}(\theta(r))\leq 1$ and
$\psi^{(r)}_{i}(\theta(r))\leq 1$ for each $r\in(0,1)$ and $i\in\mathcal{J}$.

∎

We remark that when all distributions are exponential, one can verify that
$\gamma_{i}(\theta_{i})$ and $\xi_{i}(\theta)$ defined in (4.25) and (4.26)
are identical to the ones in (4.20) and (4.21). Therefore, Taylor expansions
for $\eta_{i}(\theta_{i})$ and $\xi_{i}(\theta)$ can be developed trivially,
which are still given by (5.43) and (5.44) with $c_{e,i}^{2}=1$ and
$c_{s,i}^{2}=1$, omitting $o(r^{J}\lvert\theta_{i}\rvert)$ and
$o(r^{J}\lvert\theta\rvert)$ terms because the lack of truncation in this
case.

##  6 State space collapse

Recall $(w_{ij})$ defined in (3.6) and (3.7). To prove Theorem 3.1, fixing a
$\eta\in\mathbb{R}^{J}_{-}$ and a $j\in\mathcal{J}$, we define
$\theta=(\theta_{1},\ldots,\theta_{J})^{\prime}$ with the components being
given as the following:

$\displaystyle\theta_{k}=\eta_{k}r^{k},\quad k=j+1,\ldots,J,$ (6.51)
$\displaystyle\theta_{j}=\eta_{j}r^{j}+\frac{1}{1-w_{jj}}\sum_{i=j+1}^{J}w_{ji}\theta_{i},$
(6.52) $\displaystyle\theta_{\ell}=w_{\ell
j}\theta_{j}+\sum_{i=j+1}^{J}w_{\ell i}\theta_{i},\quad\ell=1,\ldots,j-1.$
(6.53)

Similarly, we define
$\tilde{\theta}=(\tilde{\theta}_{,}\ldots,\tilde{\theta}_{J})^{\prime}$ with
the components being given as the following:

$\displaystyle\tilde{\theta}_{k}=\eta_{k}r^{k},\quad k=j+1,\ldots,J,$ (6.54)
$\displaystyle\tilde{\theta}_{j}=\eta_{j}r^{j+1/2}+\frac{1}{1-w_{jj}}\sum_{i=j+1}^{J}w_{ji}\theta_{i},$
(6.55) $\displaystyle\tilde{\theta}_{\ell}=w_{\ell
j}\tilde{\theta}_{j}+\sum_{i=j+1}^{J}w_{\ell
i}\theta_{i},\quad\ell=1,\ldots,j-1.$ (6.56)

We emphasize that the vectors $\theta$ and $\tilde{\theta}$ defined by
(6.51)-(6.56) depend on index $j$. We avoid writing $\theta^{(j)}$ for
notational simplicity. Whenever the index $j$ is clear from the context, this
simplification should not cause confusion. The reason for this particular form
of $\theta$ and $\tilde{\theta}$ will be clear from the proof of Theorem 3.1
in Section 7. Although $\tilde{\theta}$ is well defined for $j=J$ as well, it
is used later only for $j\in\mathcal{J}\setminus\\{J\\}$.

The next lemma is a consequence of Proposition 3.1.

######  Lemma 6.1 (State space collapse for $\phi^{(r)}$ and
$\phi^{(r)}_{j}$).

Fix $\eta\in\mathbb{R}^{J}_{+}$ and $j\in\mathcal{J}$. Let
$\theta=\theta(r,\eta)$ and $\tilde{\theta}=\tilde{\theta}(r,\eta)$ be defined
in (6.51)-(6.56). As $r\to 0$,

$\displaystyle\phi^{(r)}(\theta)-\phi^{(r)}(0,\ldots,0,r^{j}\eta_{j},\ldots,r^{J}\eta_{J})=o(1),$
(6.57)
$\displaystyle\phi^{(r)}(\tilde{\theta})-\phi^{(r)}(0,\ldots,0,0,r^{j+1}\eta_{j+1},\ldots,r^{J}\eta_{J})=o(1),$
(6.58)
$\displaystyle\phi^{(r)}_{j}(\theta)-\phi^{(r)}_{j}(0,\ldots,0,0,r^{j+1}\eta_{j+1},\ldots,r^{J}\eta_{J})=o(1),$
(6.59)
$\displaystyle\phi^{(r)}_{j}(\tilde{\theta})-\phi^{(r)}_{j}(0,\ldots,0,0,r^{j+1}\eta_{j+1},\ldots,r^{J}\eta_{J})=o(1).$
(6.60)

###### Proof.

The proof of (6.57) follows from the following relationships

$\displaystyle\phi^{(r)}(0,\ldots,0,r^{j}\eta_{j},\ldots,r^{J}\eta_{J})-\phi^{(r)}(\theta)=\mathbb{E}\Big{[}\Big{(}1-e^{\sum_{k<j}\theta_{k}Z^{(r)}_{k}+(\theta_{j}-r^{j}\eta_{j})Z^{(r)}_{j}}\Big{)}e^{\sum_{k\geq
j}\eta_{k}r^{k}Z^{(r)}_{k}}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}1-e^{\sum_{k<j}\theta_{k}Z^{(r)}_{k}+(\theta_{j}-r^{j}\eta_{j})Z^{(r)}_{j}}\Big{]}\leq\mathbb{E}\Big{[}\sum_{k<j}\lvert\theta_{k}\rvert
Z^{(r)}_{k}+\lvert\theta_{j}-r^{j}\eta_{j}\rvert Z^{(r)}_{j}\Big{]}$
$\displaystyle=\sum_{k<j}\frac{\lvert\theta_{k}\rvert}{r^{k}}\mathbb{E}\Big{[}r^{k}Z^{(r)}_{k}\Big{]}+\frac{\lvert\theta_{j}-r^{j}\eta_{j}\rvert}{r^{j}}\mathbb{E}\Big{[}r^{j}Z^{(r)}_{j}\Big{]}=o(1),$

where the last equality is due to $\theta_{k}=O(r^{j})=o(r^{k})$ for $k<j$,
$\theta_{j}-r^{j}\eta_{j}=O(r^{j+1})=o(r^{j})$, and (3.12).

The proof of (6.58) follows from the following relationships

$\displaystyle\phi^{(r)}(0,\ldots,0,r^{j+1}\eta_{j+1},\ldots,r^{J}\eta_{J})-\phi^{(r)}(\tilde{\theta})=\mathbb{E}\Big{[}\Big{(}1-e^{\sum_{k\leq
j}\tilde{\theta}_{k}Z^{(r)}_{k}}\Big{)}e^{\sum_{k>j}\eta_{k}r^{k}Z^{(r)}_{k}}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}1-e^{\sum_{k\leq
j}\tilde{\theta}_{k}Z^{(r)}_{k}}\Big{]}\leq\mathbb{E}\Big{[}\sum_{k\leq
j}\lvert\tilde{\theta}_{k}\rvert Z^{(r)}_{k}\Big{]}=\sum_{k\leq
j}\frac{\lvert\tilde{\theta}_{k}\rvert}{r^{k}}\mathbb{E}\Big{[}r^{k}Z^{(r)}_{k}\Big{]}=o(1),$

where the last equality is due to $\tilde{\theta}_{k}=O(r^{j+1/2})=o(r^{k})$
for $k\leq j$ and (3.12).

The proof of (6.59) follows from the following relationships

$\displaystyle\phi^{(r)}_{j}(0,\ldots,0,r^{j}\eta_{j},\ldots,r^{J}\eta_{J})-\phi^{(r)}_{j}(\theta)=\mathbb{E}\Big{[}\Big{(}1-e^{\sum_{k<j}\theta_{k}Z^{(r)}_{k}}\Big{)}e^{\sum_{k\geq
j}\eta_{k}r^{k}Z^{(r)}_{k}}\mid Z^{(r)}_{j}=0\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\Big{(}1-e^{\sum_{k<j}\theta_{k}Z^{(r)}_{k}}\Big{)}\mid
Z^{(r)}_{j}=0\Big{]}\leq\mathbb{E}\Big{[}\sum_{k<j}\lvert\theta_{k}\rvert
Z^{(r)}_{k}\mid Z^{(r)}_{j}=0\Big{]}$
$\displaystyle=\sum_{k<j}\frac{\lvert\theta_{k}\rvert}{r^{k+1}}\mathbb{E}\Big{[}r^{k+1}Z^{(r)}_{k}\mid
Z^{(r)}_{j}=0\Big{]}=o(1),$

where the last equality is due to $\theta_{k}=O(r^{j})$ for $k<j$ and (3.13).

The proof of (6.60) follows from the following relationships

$\displaystyle\phi^{(r)}_{j}(0,\ldots,0,r^{j+1}\eta_{j+1},\ldots,r^{J}\eta_{J})-\phi^{(r)}_{j}(\tilde{\theta})=\mathbb{E}\Big{[}\Big{(}1-e^{\sum_{k<j}\tilde{\theta}_{k}Z^{(r)}_{k}}\Big{)}e^{\sum_{k>j}\eta_{k}r^{k}Z^{(r)}_{k}}\mid
Z^{(r)}_{j}=0\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}\Big{(}1-e^{\sum_{k<j}\tilde{\theta}_{k}Z^{(r)}_{k}}\Big{)}\mid
Z^{(r)}_{j}=0\Big{]}\leq\mathbb{E}\Big{[}\sum_{k<j}\lvert\tilde{\theta}_{k}\rvert
Z^{(r)}_{k}\mid Z^{(r)}_{j}=0\Big{]}$
$\displaystyle=\sum_{k<j}\frac{\lvert\tilde{\theta}_{k}\rvert}{r^{k+1}}\mathbb{E}\Big{[}r^{k+1}Z^{(r)}_{k}\mid
Z^{(r)}_{j}=0\Big{]}=o(1),$

where the last equality is due to $\tilde{\theta}_{k}=O(r^{j+1/2})=o(r^{k+1})$
for $k<j$ and (3.13).

∎

######  Lemma 6.2 (State space collapse for $\psi^{(r)}$ and
$\psi^{(r)}_{j}$).

Assume that $\theta=\theta(r)\in\mathbb{R}^{J}_{-}$ for each $r\in(0,1)$,
satisfying (5.36). Then, as $r\to 0$,

$\displaystyle\phi^{(r)}(\theta)-\psi^{r}(\theta)=o(1),\quad$ (6.61)
$\displaystyle\phi_{j}^{(r)}(\theta)-\psi^{r}_{j}(\theta)=o(1)\quad\text{ for
}j\in\mathcal{J}.$ (6.62)

###### Proof.

We first prove (6.61). By (5.36) and Lemma 5.2,

$\displaystyle\lvert\gamma_{i}(\theta_{i},r)\rvert\leq
d_{c}\lvert\theta_{i}\rvert\leq c\,d_{c}r\quad\text{ and
}\quad\lvert\xi_{i}(\theta_{i},r)\rvert\leq d_{c}\lvert\theta_{i}\rvert\leq
c\,d_{c}r,\quad r\in(0,1),$

where $c>0$ is the constant in (5.36). It follows that for $r\in(0,1)$

$\displaystyle\lvert\phi^{(r)}(\theta)-\tilde{\psi}^{(r)}(\theta)\rvert$
$\displaystyle\leq\mathbb{E}\Big{\lvert}1-\exp\Big{(}-\sum_{i}\gamma_{i}(\theta_{i},r)(\alpha_{i}R^{(r)}_{e,i}\wedge
r^{-1})-\sum_{i}\xi_{i}(\theta,r)(\mu^{(r)}_{i}R^{(r)}_{s,i}\wedge
r^{-1})\Big{)}\Big{\rvert}$ $\displaystyle\leq
e^{c\,d_{c}}c\,d_{c}\Big{(}\sum_{i}\alpha_{i}r\mathbb{E}[R^{(r)}_{e,i}]+\sum_{i}\mu^{(r)}_{i}r\mathbb{E}[R^{(r)}_{s,i}]\Big{)}=o(1),$

where the second inequality follows from

$\displaystyle\lvert 1-e^{-x}\rvert\leq e^{\lvert x\rvert}\lvert x\rvert,\quad
x\in\mathbb{R},$

and the last equality follows from Lemma 4.5 of Braverman et al. (2017).
Therefore, (6.61) is proved.

We next prove (6.62). It follows that, for $r\in(0,1)$,

$\displaystyle\lvert\phi^{(r)}_{j}(\theta)-\tilde{\psi}^{(r)}_{j}(\theta)\rvert$
$\displaystyle\leq\mathbb{E}\Big{[}\,\big{\lvert}1-\exp\big{(}-\sum_{i}\gamma_{i}(\theta,r)(\alpha_{i}R^{(r)}_{e,i}\wedge(1/r))-\sum_{i}\xi_{i}(\theta,r)(\mu^{(r)}_{i}R^{(r)}_{s,i}\wedge(1/r))\big{)}\big{\rvert}\mid
Z^{(r)}_{j}=0\Big{]}$ $\displaystyle\leq
e^{c\,d_{c}}c\,d_{c}\Big{(}\sum_{i}\alpha_{i}r\mathbb{E}\big{[}R^{(r)}_{e,i}\mid
Z^{(r)}_{j}=0\big{]}+\sum_{i}\mu^{(r)}_{i}r\mathbb{E}\big{[}R^{(r)}_{s,i}\mid
Z^{(r)}_{j}=0\big{]}\Big{)},$

Now,

$\displaystyle r\mathbb{E}\big{[}R^{(r)}_{e,i}\mid
Z^{(r)}_{j}=0\big{]}=\frac{r\mathbb{E}\big{[}R^{(r)}_{e,i}1_{\\{Z^{(r)}_{j}=0\\}}\big{]}}{\mathbb{P}\\{Z^{(r)}_{j}=0\\}}\leq\frac{r\big{(}\mathbb{E}\big{[}\big{(}R^{(r)}_{e,i}\big{)}^{p}\big{]}\big{)}^{1/p}\big{(}\mathbb{P}\\{Z^{(r)}_{j}=0\\}\big{)}^{1/q}}{\mathbb{P}\\{Z^{(r)}_{j}=0\\}}$
$\displaystyle=r^{1-j/p}\Big{(}\mathbb{E}\big{[}\big{(}R^{(r)}_{e,i}\big{)}^{(J+\delta_{0})}]\Big{)}^{1/(J+\delta_{0})}(\mu_{j}^{(r)})^{1/p}=o(1)\quad\text{
for }i\in\mathcal{E},j\in\mathcal{J},$

where in obtaining the inequality, we have used Holder’s inequality with
$p=J+\delta_{0}$ ($\delta_{0}$ is given in (2.1)) and $1/p+1/q=1$, and the
last equality follows from Lemma 6.6 of Braverman et al. (2023) and moment
condition (2.1). Similarly, one can prove

$\displaystyle r\mathbb{E}\big{[}R^{(r)}_{s,i}\mid
Z^{(r)}_{j}=0\big{]}=o(1)\quad\text{ for }i,j\in\mathcal{J}.$

∎

##  7 Proof of Theorem 3.1

Fix $\eta\in\mathbb{R}^{J}_{+}$. With $\theta=\theta(r,\eta)$ and
$\tilde{\theta}=\tilde{\theta}(r,\eta)$ defined in (6.51)-(6.56), we state the
following lemma, which will be proved at the end of this section.

######  Lemma 7.1.

Assume (2.1), Assumption 3.1 and Assumption 3.2. Fix
$\eta\in\mathbb{R}^{J}_{+}$. For each $j\in\mathcal{J}$, define
$\theta=\theta(r,\eta)$ through (6.51)-(6.53). As $r\to 0$,

$\displaystyle\psi^{(r)}(\theta)-\frac{1}{1-d_{j}\eta_{j}}\psi^{(r)}_{j}(\theta)=o(1),$
(7.63)

For each $j\in\mathcal{J}\setminus\\{J\\}$, let
$\tilde{\theta}=\tilde{\theta}(r,\eta)$ be defined in (6.54)-(6.56). As $r\to
0$,

$\displaystyle\psi^{(r)}(\tilde{\theta})-\psi^{(r)}_{j}(\tilde{\theta})=o(1).$
(7.64)

We next state and prove the following lemma.

######  Lemma 7.2.

Fix $\eta\in\mathbb{R}^{J}_{-}$. For each $j\in\mathcal{J}$, as $r\to 0$,

$\displaystyle\phi^{(r)}(0,\ldots,0,r^{j}\eta_{j},\ldots,r^{J}\eta_{J})-\frac{1}{1-d_{j}\eta_{j}}\phi^{(r)}(0,\ldots,0,r^{j+1}\eta_{j+1},\ldots,r^{J}\eta_{J})=o(1)$
(7.65)

with the convention that
$\phi^{(r)}(0,\ldots,0,r^{j+1}\eta_{j+1},\ldots,r^{J}\eta_{J})=1$ when $j=J$.

###### Proof.

We first prove (7.65) for $j=J$. The proof follows the the following
equalities.

$\displaystyle\phi^{(r)}(0,\ldots,0,\ldots,0,r^{J}\eta_{J})-\frac{1}{1-d_{J}\eta_{J}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:ssc1})}}{{=}}\phi^{(r)}(\theta)-\frac{1}{1-d_{J}\eta_{J}}+o(1)\stackrel{{\scriptstyle(\ref{eq:ssc-
phi})}}{{=}}\psi^{(r)}(\theta)-\frac{1}{1-d_{J}\eta_{J}}+o(1)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:bar1})}}{{=}}\frac{1}{1-d_{J}\eta_{J}}\big{[}\psi^{(r)}_{J}(\theta)-1\big{]}+o(1)\stackrel{{\scriptstyle(\ref{eq:ssc-
phii})}}{{=}}\frac{1}{1-d_{J}\eta_{J}}\big{[}\phi^{(r)}_{J}(\theta)-1\big{]}+o(1)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:ssc2})}}{{=}}o(1),$

where $\theta=\theta(r,\eta)$ is defined through (6.51)-(6.53), and the
equation numbers above each equality justify that equality.

We now prove (7.65) for $j\in\mathcal{J}\setminus\\{J\\}$. The proof follows
the the following equalities.

$\displaystyle\phi^{(r)}(0,\ldots,0,r^{j}\eta_{j},\ldots,r^{J}\eta_{J})-\frac{1}{1-d_{j}\eta_{j}}\phi^{(r)}(0,\ldots,0,r^{j+1}\eta_{j+1},\ldots,r^{J}\eta_{J})$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:ssc1}),(\ref{eq:ssc3})}}{{=}}\phi^{(r)}(\theta)-\frac{1}{1-d_{j}\eta_{j}}\phi^{(r)}(\tilde{\theta})+o(1)\stackrel{{\scriptstyle(\ref{eq:ssc-
phi})}}{{=}}\psi^{(r)}(\theta)-\frac{1}{1-d_{j}\eta_{j}}\psi^{(r)}(\tilde{\theta})+o(1)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:bar1}),(\ref{eq:bar2})}}{{=}}\frac{1}{1-d_{j}\eta_{j}}\big{[}\psi^{(r)}_{j}(\theta)-\psi_{j}^{(r)}(\tilde{\theta})\big{]}+o(1)\stackrel{{\scriptstyle(\ref{eq:ssc-
phii})}}{{=}}\frac{1}{1-d_{j}\eta_{j}}\big{[}\phi^{(r)}_{j}(\theta)-\phi_{j}^{(r)}(\tilde{\theta})\big{]}+o(1)$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:ssc2}),(\ref{eq:ssc4})}}{{=}}o(1),$

where $\theta=\theta(r,\eta)$ is defined through (6.51)-(6.53),
$\tilde{\theta}=\tilde{\theta}(r,\eta)$ is defined through (6.54)-(6.56), and
the equation numbers above each equality justify that equality. ∎

###### Proof of Theorem 3.1.

Lemma 7.2 implies (4.15), which is equivalent to (3.8). Therefore, Lemma 7.2
implies Theorem 3.1. ∎

In the remainder of this section, we prove Lemma 7.1. To prove Lemma 7.1, we
first state and prove the following three lemmas.

######  Lemma 7.3.

Fix $j\in\mathcal{J}$. The following equations

$w_{ij}=P_{ij}+\sum_{k<j}P_{ik}w_{kj},\quad i\in\cal{J}$ (7.66)

has a unique solution $(w_{1j},\ldots,w_{J,j})$ that is given by (3.6) and
(3.7).

###### Proof.

The first $j-1$ equations in (7.66) uniquely solve
$(w_{1j},\ldots,w_{j-1,j})^{\prime}$, which is given by (3.6). (3.7) is simply
the vector version of the last $J-j+1$ equations with
$(w_{1j},\ldots,w_{j-1,j})^{\prime}$ being substituted by the right side of
(3.6). ∎

The quantity $w_{ij}$ has the following probabilistic interpretations.
Consider the DTMC on state space $\\{0\\}\cup\mathcal{J}$ corresponding to the
routing matrix $P$. Let $w_{ij}$ be the probability that starting from state
$i$, the DTMC will eventually visit state $j$, avoiding visiting states in
$\\{0,j+1,\ldots,J\\}$. By the “first-step” method, $w_{ij}$ satisfies (7.66).

######  Lemma 7.4 (Solution to a linear system).

Fix $\eta\in\mathbb{R}^{J}_{-}$ and $j\in\mathcal{J}$. Recall
$\theta=\theta(r,\eta)$ and $\tilde{\theta}=\tilde{\theta}(r,\eta)$ defined in
(6.51)-(6.56). Then, $\theta$ and $\tilde{\theta}$ satisfy the following
equations.

$\displaystyle\bar{\xi}_{\ell}(\theta)=0,\quad\bar{\xi}_{\ell}(\tilde{\theta})=0,\quad\ell=1,\ldots,j-1,$
(7.67) $\displaystyle\bar{\xi}_{j}(\theta)=(1-w_{jj})r^{j}\eta_{j},$ (7.68)
$\displaystyle\bar{\xi}_{j}(\tilde{\theta})=(1-w_{jj})r^{j+1/2}\eta_{j}.$
(7.69)

###### Proof.

With $\theta_{k}=r^{k}\eta_{k}$ for $k=j+1,\ldots,J$ being fixed, the linear
system of equations

$\displaystyle\bar{\xi}_{\ell}(\theta)=0,\quad\ell=1,\ldots,j-1,$
$\displaystyle\bar{\xi}_{j}(\theta)=(1-w_{jj})r^{j}\eta_{j}$

has $j$ equations and $j$ variables $(\theta_{1},\ldots,\theta_{j})$. It has a
unique solution $(\theta_{1},\ldots,\theta_{j})^{\prime}$ given by (6.52) and
(6.53). Similarly, fixing $\tilde{\theta}_{k}=r^{k}\eta_{k}$ for
$k=j+1,\ldots,J$, the linear system of equations

$\displaystyle\bar{\xi}_{\ell}(\tilde{\theta})=0,\quad\ell=1,\ldots,j-1,$
$\displaystyle\bar{\xi}_{j}(\tilde{\theta})=(1-w_{jj})r^{j+1/2}\eta_{j}$

has a unique solution given by (6.55) and (6.56). ∎

######  Lemma 7.5.

Fix $j\in\mathcal{J}$. Recall $\sigma^{2}_{j}$ is defined in (3.10). Then,

$\displaystyle\sigma^{2}_{j}=2Q^{*}(w_{1j},\ldots,w_{j-1,j},1,0,\ldots,0),$
(7.70)

where $Q^{*}$ is defined in (5.40).

###### Proof.

Define

$\displaystyle
u=(w_{1j},\ldots,w_{j-1,j},1,0,\ldots,0)^{\prime}\in\mathbb{R}^{J},$ (7.71)

where prime denotes transpose. It follows from (7.66) that

$\displaystyle\sum_{k\in\mathcal{J}}P_{ik}u_{k}=w_{ij},\quad i\in\mathcal{J}.$

Recall the definition $\xi^{*}_{i}(u)$ in (5.39). It follows that

$\displaystyle\xi^{*}_{i}(u)$
$\displaystyle=c^{2}_{s,i}\Bigl{(}-u_{i}+\sum_{k\in\mathcal{J}}P_{i,k}u_{k}\Bigr{)}^{2}+\sum_{k\in\mathcal{J}}P_{ik}u_{k}^{2}-\big{(}\sum_{k\in\mathcal{J}}P_{ik}u_{k}\big{)}^{2}$
$\displaystyle=c_{s,i}^{2}(-u_{i}+w_{ij})^{2}+\sum_{k\in\mathcal{J}}P_{ik}(u_{k}^{2}-u_{k})+w_{ij}-w_{ij}^{2}.$

For $u$ in (7.71),

$\displaystyle u_{i}+w_{ij}=\begin{cases}0&i<j,\\\ -1+w_{jj},&i=j,\\\
w_{ij}&i>j.\end{cases}$

Denoting

$\displaystyle
b_{i}=\sum_{k\in\mathcal{J}}P_{ik}(u_{k}^{2}-u_{k})+w_{ij}-w_{ij}^{2},$

one has the following

$\displaystyle\sum_{i=1}^{J}\lambda_{i}\xi^{*}_{i}(u)=\lambda_{j}c_{s,j}^{2}(1-w_{jj})^{2}+\sum_{i>j}\lambda_{i}c_{s,i}^{2}w_{ij}^{2}+\sum_{i\in\mathcal{J}}\lambda_{i}b_{i},$

where the last term $\sum_{i\in\mathcal{J}}\lambda_{i}b_{i}$ is equal to

$\displaystyle\sum_{i\in\mathcal{J}}\lambda_{i}\sum_{k\in\mathcal{J}}P_{ik}(u_{k}^{2}-u_{k})+\sum_{i\in\mathcal{J}}\lambda_{i}w_{ij}(1-w_{ij})$
$\displaystyle=\sum_{k\in\mathcal{J}}(\lambda_{k}-\alpha_{k})(u_{k}^{2}-u_{k})+\sum_{i\in\mathcal{J}}\lambda_{i}w_{ij}(1-w_{ij})$
$\displaystyle=\sum_{k<j}(\alpha_{k}-\lambda_{k})(w_{kj}-w_{kj}^{2})+\sum_{i\in\mathcal{J}}\lambda_{i}w_{ij}(1-w_{ij})$
$\displaystyle=\sum_{k<j}\alpha_{k}(w_{kj}-w_{kj}^{2})+\sum_{i\geq
j}\lambda_{i}w_{ij}(1-w_{ij}).$

Following the definition of the quadratic function $Q^{*}(u)$ in (5.40) with
$\eta^{*}_{i}(u_{i})$ defined in (5.41), one has

$\displaystyle 2Q^{*}(u)$
$\displaystyle=\alpha_{j}c_{e,j}^{2}+\sum_{i<j}\alpha_{i}c_{e,i}^{2}w^{2}_{ij}+\lambda_{j}c_{s,j}^{2}(1-w_{jj})^{2}+\sum_{i>j}\lambda_{i}c_{s,i}^{2}w_{ij}^{2}$
$\displaystyle+\sum_{i<j}\alpha_{i}(w_{ij}-w_{ij}^{2})+\sum_{i\geq
j}\lambda_{i}w_{ij}(1-w_{ij})=\sigma^{2}_{j}.$

∎

######  Lemma 7.6.

Fix $\eta\in\mathbb{R}^{J}_{-}$ and fix $j\in\mathcal{J}$. Let
$\theta=\theta(r,\eta)$ and $\tilde{\theta}=\tilde{\theta}(r,\eta)$ be defined
in (6.51)-(6.56). Then,

$\displaystyle\lim_{r\to
0}\frac{Q^{*}(\theta)}{r^{2j}}=\eta_{j}^{2}Q^{*}(w_{1j},\ldots,w_{j-1,j},1,0,\ldots,0),$
(7.72) $\displaystyle\lim_{r\to
0}\frac{Q^{*}(\tilde{\theta})}{r^{2j+1}}=\eta_{j}^{2}Q^{*}(w_{1j},\ldots,w_{j-1,j},1,0,\ldots,0).$
(7.73)

###### Proof.

The limit (7.72) follows from

$\displaystyle\theta_{\ell}=w_{\ell
j}r^{j}\eta_{j}+o(r^{j}),\quad\ell=1,\ldots,j-1,$
$\displaystyle\theta_{j}=r^{j}\eta_{j}+o(r^{j}),$
$\displaystyle\theta_{k}=o(r^{j}),\quad k=j+1,\ldots,J.$

The limit (7.73) can be checked similarly using

$\displaystyle\tilde{\theta}_{\ell}=w_{\ell
j}r^{j+1/2}\eta_{j}+o(r^{j+1/2}),\quad\ell=1,\ldots,j-1,$
$\displaystyle\tilde{\theta}_{j}=r^{j+1/2}\eta_{j}+o(r^{j+1/2}),$
$\displaystyle\tilde{\theta}_{k}=o(r^{j+1/2}),\quad k=j+1,\ldots,J.$

∎

###### Proof of Lemma 7.1.

We first prove (7.63). Fix $\eta\in\mathbb{R}^{J}_{-}$ and $j\in\mathcal{J}$.
Recall $\theta=\theta(r,\eta)$ defined in (6.51)-(6.53). Using (7.67) and
multiscale heavy traffic condition (3.4), the asymptotic BAR (5.37) becomes

$\displaystyle Q^{*}(\theta)\psi^{(r)}(\theta)+\sum_{i\geq
j}r^{i}\bar{\xi}_{i}(\theta)\bigl{(}\psi^{(r)}(\theta)-\psi^{(r)}_{i}(\theta)\bigr{)}$
$\displaystyle=-\frac{1}{2}\sum_{i\in\mathcal{J}}r^{i}{\xi}^{*}_{i}(\theta)\bigl{(}\psi^{(r)}(\theta)-\psi^{(r)}_{i}(\theta)\bigr{)}+o(r^{J}\lvert\theta\rvert)+o(\lvert\theta\rvert^{2}).$
(7.74)

From definition (6.51)-(6.53), it is clear that $\theta_{i}=O(r^{j})$ for each
$i\in\mathcal{J}$. Therefore, $\bar{\xi}_{i}(\theta)=O(r^{j})$ and
${\xi}^{*}_{i}(\theta)=O(r^{2j})$ for $i\in\mathcal{J}$. Hence, (7.74) leads
to

$\displaystyle
Q^{*}(\theta)\psi^{(r)}(\theta)+r^{j}\bar{\xi}_{j}(\theta)\bigl{(}\psi^{(r)}(\theta)-\psi^{(r)}_{i}(\theta)\bigr{)}=o(r^{2j}).$

Using (7.68), the preceding equation becomes

$\displaystyle
Q^{*}(\theta)\psi^{(r)}(\theta)+(1-w_{jj})r^{2j}\eta_{j}\bigl{(}\psi^{(r)}(\theta)-\psi^{(r)}_{i}(\theta)\bigr{)}=o(r^{2j}).$

Dividing both sides by $r^{2j}$ and using (7.72) and (7.70), we have

$\displaystyle(\sigma^{2}_{j}/2)\eta_{j}^{2}\psi^{(r)}(\theta)+(1-w_{jj})\eta_{j}\bigl{(}\psi^{(r)}(\theta)-\psi^{(r)}_{i}(\theta)\bigr{)}=o(1),$

from which (7.63) follows.

We next prove (7.64). Fix $j\in\mathcal{J}\setminus\\{J\\}$. Recall
$\tilde{\theta}=\tilde{\theta}(r,\eta)$ defined defined in (6.54)-(6.56).
Using (7.67) and multiscale heavy traffic condition (3.4), the asymptotic BAR
(5.37) becomes

$\displaystyle Q^{*}(\tilde{\theta})\psi^{(r)}(\tilde{\theta})+\sum_{i\geq
j}r^{i}\bar{\xi}_{i}(\tilde{\theta})\bigl{(}\psi^{(r)}(\tilde{\theta})-\psi^{(r)}_{i}(\tilde{\theta})\bigr{)}$
$\displaystyle=-\frac{1}{2}\sum_{i\in\mathcal{J}}r^{i}{\xi}^{*}_{i}(\tilde{\theta})\bigl{(}\psi^{(r)}(\tilde{\theta})-\psi^{(r)}_{i}(\tilde{\theta})\bigr{)}+o(r^{J}\lvert\tilde{\theta}\rvert)+o(\lvert\tilde{\theta}\rvert^{2}).$

From definition (6.54)-(6.56), it is clear that
$\tilde{\theta}_{i}=O(r^{j+1/2})$ for each $i\in\mathcal{J}$. Therefore,
$\bar{\xi}_{i}(\tilde{\theta})=O(r^{j+1/2})$ and
${\xi}^{*}_{i}(\tilde{\theta})=O(r^{2j+1})$ for $i\in\mathcal{J}$. Hence,
(7.74) leads to

$\displaystyle
Q^{*}(\tilde{\theta})\psi^{(r)}(\tilde{\theta})+r^{j}\bar{\xi}_{j}(\tilde{\theta})\bigl{(}\psi^{(r)}(\tilde{\theta})-\psi^{(r)}_{i}(\tilde{\theta})\bigr{)}=o(r^{2j+1}).$

Using (7.69), the preceding equation becomes

$\displaystyle
Q^{*}(\tilde{\theta})\psi^{(r)}(\tilde{\theta})+(1-w_{jj})r^{2j+1/2}\eta_{j}\bigl{(}\psi^{(r)}(\tilde{\theta})-\psi^{(r)}_{i}(\tilde{\theta})\bigr{)}=o(r^{2j+1}).$

Dividing both sides by $r^{2j+1/2}$ and using (7.73) and (7.70), we have

$\displaystyle\sqrt{r}(\sigma^{2}_{j}/2)\eta_{j}^{2}\psi^{(r)}(\theta)+(1-w_{jj})\eta_{j}\bigl{(}\psi^{(r)}(\tilde{\theta})-\psi^{(r)}_{i}(\tilde{\theta})\bigr{)}=o(\sqrt{r}),$

from which (7.64) follows. ∎

## References

  * Braverman et al. (2017) Braverman, A., Dai, J. and Miyazawa, M. (2017).  Heavy traffic approximation for the stationary distribution of a generalized Jackson network: the BAR approach.  Stochastic Systems 7 143–196.
URL http://projecteuclid.org/euclid.ssy/1495785619

  * Braverman et al. (2023) Braverman, A., Dai, J. and Miyazawa, M. (2023).  The BAR-approach for multiclass queueing networks with SBP service policies.  arXiv preprint arXiv:2302.05791 .
URL https://arxiv.org/abs/2302.05791

  * Budhiraja and Lee (2009) Budhiraja, A. and Lee, C. (2009).  Stationary distribution convergence for generalized Jackson networks in heavy traffic.  Math. Oper. Res. 34 45–56.
URL http://dx.doi.org/10.1287/moor.1080.0353

  * Chen and Mandelbaum (1991) Chen, H. and Mandelbaum, A. (1991).  Discrete flow networks: bottleneck analysis and fluid approximations.  Math. Oper. Res. 16 408–446.
URL https://doi.org/10.1287/moor.16.2.408

  * Dai (1995) Dai, J. G. (1995).  On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models.  Ann. Appl. Probab. 5 49–77.
URL
http://links.jstor.org/sici?sici=1050-5164(199502)5:1<49:OPHROM>2.0.CO;2-4&origin=MSN

  * Dai and Harrison (1992) Dai, J. G. and Harrison, J. M. (1992).  Reflected Brownian motion in an orthant: numerical methods for steady-state analysis.  The Annals of Applied Probability 2 65–86.
  * Dai et al. (1994) Dai, J. G., Nguyen, V. and Reiman, M. I. (1994).  Sequential bottleneck decomposition: an approximation method for open queueing networks.  Operations Research 42 119–136.
  * Gamarnik and Zeevi (2006) Gamarnik, D. and Zeevi, A. (2006).  Validity of heavy traffic steady-state approximation in generalized Jackson networks.  Ann. Appl. Probab. 16 56–90.
URL http://dx.doi.org/10.1214/105051605000000638

  * Glynn and Zeevi (2008) Glynn, P. W. and Zeevi, A. (2008).  Bounding stationary expectations of Markov processes.  In Markov processes and related topics: a Festschrift for Thomas G. Kurtz, vol. 4 of Inst. Math. Stat. Collect. Inst. Math. Statist., Beachwood, OH, 195–214.
URL http://dx.doi.org/10.1214/074921708000000381

  * Guang et al. (2023) Guang, J., Chen, X. and Dai, J. (2023).  Uniform moment bounds for generalized Jackson networks in multi-scale heavy traffic.  arXiv preprint arXiv .
  * Harrison (1978) Harrison, J. M. (1978).  The diffusion approximation for tandem queues in heavy traffic.  Advances in Applied Probability 10 886–905.
  * Harrison and Reiman (1981) Harrison, J. M. and Reiman, M. I. (1981).  Reflected Brownian motion on an orthant.  Ann. Probab. 9 302–308.
URL
http://links.jstor.org/sici?sici=0091-1798(198104)9:2<302:RBMOAO>2.0.CO;2-P&origin=MSN

  * Harrison and Williams (1987) Harrison, J. M. and Williams, R. J. (1987).  Brownian models of open queueing networks with homogeneous customer populations.  Stochastics 22 77–115.
URL http://dx.doi.org/10.1080/17442508708833469

  * Jackson (1957) Jackson, J. R. (1957).  Networks of waiting lines.  Operations Research 5 518–521.
  * Johnson (1983) Johnson, D. P. (1983).  Diffusion approximations for optimal filtering of jump processes and for queueing networks.  Ph.D. thesis, University of Wisconsin.
URL
http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:8316215

  * Reiman (1984) Reiman, M. I. (1984).  Open queueing networks in heavy traffic.  Mathematics of Operations Research 9 441–458.
URL http://dx.doi.org/10.1287/moor.9.3.441

  * Shen et al. (2002) Shen, X., Chen, H., Dai, J. G. and Dai, W. (2002).  The finite element method for computing the stationary distribution of an SRBM in a hypercube with applications to finite buffer queueing networks.  Queueing Systems 42 33–62.
  * Whitt (1983) Whitt, W. (1983).  The queueing network analyzer.  Bell System Technical Journal 62 2779–2815.
  * Whitt and You (2020) Whitt, W. and You, W. (2020).  Heavy-traffic limits for stationary network flows.  Queueing Syst. 95 53–68.
URL https://doi-org.proxy.library.cornell.edu/10.1007/s11134-019-09645-8

  * Xu (2022) Xu, Y. (2022).  WWTA load-balancing for parallel-server systems with heterogeneous servers and multi-scale heavy traffic limits for generalized jackson networks.  Ph.D. thesis, Cornell University.


# Please Don’t Go - A Comprehensive Approach to Increase Women’s Participation
in Open Source Software

Bianca Trinkenreich  Northern of Arizona University
Flagstaff, AZ, USA
<EMAIL_ADDRESS>
###### Abstract

Women represent less than 24% of employees in the software development
industry and experience various types of prejudice and bias. Despite various
efforts to increase diversity and multi-gendered participation, women are even
more underrepresented in Open Source Software (OSS) projects. In my PhD, I
investigate the following question: How can OSS communities increase women’s
participation in their projects? I will identify different OSS career pathways
and develop a holistic view of women’s motivations to join or leave OSS, as
well as their definitions of success. Based on this empirical investigation, I
will work together with the Linux Foundation to design attraction and
retention strategies focused on women. Before and after implementing the
strategies, I will conduct empirical studies to evaluate the state of the
practice and understand the implications of the strategies.

###### Index Terms:

open source software, women, gender, diversity, participation, success, career

##  I Problem and Research Statement

Open Source Software (OSS) development is a collaborative endeavor in which
expert developers distributed around the globe create software solutions [1,
2]. Many OSS projects count on a community of volunteers to succeed, and such
a community needs newcomers for their sustainability and growth. The lack of
gender diversity in OSS projects has gained increasing attention from
practitioners and researchers.

Diversity in software development teams can take many different forms,
including gender, experience, culture, and technical knowledge. Some teams are
more diverse in one attribute and less in others [3]. Gender diversity
positively affects productivity by bringing together different perspectives;
improving outcomes [3], innovation, and problem-solving capacity; and
fostering a healthier work environment [4]. A diverse development team is more
likely to properly comprehend users’ needs, contributing to a better alignment
between the delivered software and its intended customers [5].

Although organizations are taking actions to increase gender diversity, the
percentage of women in OSS projects are in average lower than 10%. Only 7.5%
of the contributions to public code from the last 50 years were authored by
women [6]. Women represent only 5.2% of the contributors to the Apache
Software Foundation [7], 9.9% in Linux kernel [8], and 10% of OpenStack
contributors [9], three of the largest and most well-known OSS communities.
Indeed, women represent only 9% of GitHub users [10].

Considering the benefits of having a more gender-diverse team, researchers are
also increasingly focusing on understanding the low representation of women in
OSS.

Research suggests that gender bias and sexist behavior pervade OSS [11, 12].
Women feel frustrated when they are the only woman on a development team or
when their input is under-valued or ignored, even on topics in which they have
expert knowledge [13]. Within OSS projects, the notion of meritocracy reigns,
following the logic that quality speaks for itself and will be rewarded [14].
Continually finding themselves on the bottom rung, it is no surprise that many
women report experiencing “imposter syndrome” [3]. Gender biases can represent
a a persistent barrier for women to join OSS [15].

Strategies suggested by previous work to attract and retain women include
issuing code of conduct statements [16, 17, 18]; adopting feminist and social
justice principles [18]; promoting women to leadership positions [19];
providing spaces for women to build their leadership capacity and engage in
developing the community with norms and values consistent with their own [20],
focusing on the first social experiences through programs such as mentorships
[21]; and reforming the systemic gender-bias and providing inclusive tools and
infrastructure [15]. Strategies discouraged by the literature include setting
quotas for women, since merely increasing the proportion of women can lead to
the flattening of the slope of the relationship between behavioral femaleness
and outcomes and activate questionable stereotypes [22].

Consistent with the general finding that women’s participation in OSS remains
very low, survey and anecdotal evidence have indicated that attracting and
retaining women as contributors in OSS projects has been particularly
challenging [23, 24, 25].

##  II Research Goals and Hypothesis

I will investigate women’s participation in OSS, including their motivations
to join, what attracts them to OSS, the pathways they follow as they progress
on a project (or to achieve their perceptions of success), the challenges they
face and their motivations to stay, take breaks, or leave, and what attracts
or repels them from OSS. Ultimately, the overall goal of this project is to
help OSS projects devise strategies to attract and retain women, while helping
these women contributors to attain their own goals. For this purpose, I will
create strategies to 1. attract and retain women based on the forces
compelling them to join and stay in OSS, and 2. diminish the forces compelling
them to drop out of OSS, as depicted in Fig. 1.

Figure 1: Contributors joining model presenting the stages and forces that act
during the joining process

The process of joining OSS projects can be understood into two stages:
Onboarding and Contributing [26]. Explaining the process in different stages
helps to visualize the different forces that compel contributors towards
staying with or leaving the project [27].

Motivations drive both the onboarding and contributing stages of joining OSS.
An individual can have one or a set of motivation(s) to join, and shift to (a)
different motivation(s) to stay (or not) [23]. Shifts in motivation might
occur due to changes in the OSS landscape, or they might reflect the journey
an individual makes and their growth since first joining [28, 26].

Individuals do not only behave to achieve immediate rewards; they might also
act to reach or maintain a consistency of action points beyond the attainment
of specific and immediate goals [28]. In addition to motivations, self-
perceptions of success affect choices they make for personal and professional
lives, including educational options and decisions of where to work [29].
Aligned with this, I argue that an OSS community or organization can better
attract and retain women contributors when they consider both their
motivations and the multitude of factors that underpin what means success to
them.

I will identify the challenges that women report when contributing to OSS, and
what would make them take breaks or even leave altogether. I will collect
their advice for other women and their suggestions about possible actions to
increase inclusivity, and also use as input for the strategies to attract and
retain them.

In this dissertation, I tackle the research question How can OSS communities
increase women’s participation in OSS projects?. To guide my exploration of an
answer to the research question, I defined specific questions:

  * •

RQ.1 What strategies can be employed to attract and retain more women
contributors to OSS projects?

  * •

RQ.2. How can these strategies help to increase the percentage of women in OSS
projects?

##  III Expected Contributions

The theoretical contribution of this dissertation is multi-fold, including
identification of: the pathways that can be followed by OSS contributors (from
any gender); women’s motivations to join (or not), stay, take breaks, and
leave OSS projects; women contributors’ multi-faceted definitions of success;
contributions to the state-of-the-art, including the current challenges women
face, women’s advice for other women, and suggestions to make OSS projects
more inclusive.

The practical contributions include guidelines for women seeking a career in
OSS, showing the different roles, activities, backgrounds, and necessary
skills. Women can use the guidelines to develop a training plan, and learn and
improve the skills necessary for their preferred pathway. I will also provide
actionable mechanisms for OSS projects to encourage women to join and keep
contributing. The strategies will be implemented in a large OSS project that
seeks to increase women’s participation (Linux Kernel) and evaluated in
practice, as I explain in Section IV-C.

To the best of my knowledge, there is no work that provides strategies to
increase women’s participation in OSS based on what I call “women’s desires
and beliefs about OSS;” that is, their motivations to join, stay, or leave,
their perceptions of success, and the challenges they encounter. Also there is
no work that offers a guideline of career pathways and roles in OSS that
highlights for women the different ways they can contribute, be successful,
and achieve their goals.

##  IV Research Methodology

My research has three stages and adopts mixed methods to accomplish its goal,
as depicted in Figure 2.

To help answer RQ.1. What strategies can be employed to attract and retain
more women as contributors to OSS projects? of this dissertation, I’ve
executed Stage 1 and planned Stage 2 to explore the career pathways, goals,
definitions of success, and motivations that influence a contributor’s
decision to join, stay, take breaks, or leave an OSS project. The studies
include all genders contributors as I use segmented analysis to compare
findings between genders.

Figure 2: Research Design

###  IV-A Stage 1 - Explore career pathways, motivations, and definitions of
success

I started this project with a study based on interviews to find the different
pathways followed by successful OSS contributors [30]. We interviewed 17
participants (12 of them identified their gender as women) that were invited
speakers at OSCON (Open Source Software Conference). I have qualitatively
analyzed the results to identify career paths, how they joined, which roles
and activities they perform, and how they arrived at their current position in
OSS. Understanding that participation in OSS projects includes more than
writing code [9], we found that people can build a career in OSS through
different roles and activities, and with different backgrounds, including
those not related to writing software. These activities, while crucial to the
survival of the OSS community, are currently performed by hidden or largely
unacknowledged figures in the community. They (or their contributions) are not
visible when navigating the project repository data. As a result they are not
seen as the central figures of the project. This not only is demotivating,
because of the lack of recognition, but it can also discourage individuals who
lack a Computer Science background or interest in coding-related activities.

Next, I conducted a second study based on interviews and a survey to
understand the multi-faceted definition of success in OSS. Success in OSS
encompasses more than code contributions alone; as evident in one
participant’s OSS journey, who shared in our first study that success in OSS
involves, who said it involves “contributing more than code, [including]
contributing documentation, processes …the governance of the project” (P7).
However, currently there is a misconception that success in OSS is only
achieved through activities related to source code [31, 32, 33, 34].

The disproportionate emphasis on code can make other types of contributions
seem less valuable. This may specifically disadvantage women, given that the
majority of code authors are men, and social ties may influence women’s
inclusion in technical activities [11, 35] resulting in women’s lower
engagement [36]. Affirming this, the majority (144 out of 165) of men who
answered our survey were coders. OSS contributors, however, comprise a
heterogeneous group, with diverse talents, skills, career goals, and
motivations [37, 38, 28, 39]. Also from the first study, we found that some
contributors perform a variety of non-code related activities (e.g., advocacy,
technical writing, translation, project management) [30] and follow different
pathways than the celebrated “onion model” [11, 40, 30].

Given the fact that OSS communities comprise more players than simply “code
warriors,” the community’s definition of success ought to broaden beyond the
quantity of code one produces. The way we define success has a remarkable
impact on the choices we make in our personal and professional lives. Without
such a broadened understanding, how would it possible support diverse
individuals whose background, career goals, and pathways do not fit the
typical onion model career mold? I have investigated the different career
pathways, goals, and the self-definitions of success through interviews with
27 OSS contributors who are recognized as successful in their communities, and
a follow-up open survey with 193 OSS contributors. This study provides nuanced
definitions of success perceptions in OSS, and show that OSS contributors have
a broader perspective on success than the narrow focus on code-related
activities—which is better supported by current tools and practices. Our
results include 26 categories of definitions of success. A segmented analysis
by gender showed that women consider recognition more than men as part of
their definition for success. The literature shows that men relate success to
tangible and objective outcomes, but, contrary to the research in other
domains [29, 41, 42], definitions of success that are considered subjective
were also cited by men. This study was also turned into a paper that is
currently under peer review.

Following the first two studies, I conducted a third study based on a survey
about the current motivations that drive OSS contributors to join and stay in
OSS. Our field’s understanding of what motivates people to contribute to OSS
is still fundamentally grounded in studies from the early 2000s, and much has
changed since the early days of OSS [43]. OSS today enjoys a place of
distinction in producing key technologies and providing learning; from the
first study we could see that OSS also offers different career opportunities.
With such drastic changes to the status of OSS, we considered it likely that
what motivates people to join OSS also has evolved since the early days. It is
time to revisit the fundamental question of what drives people to contribute
to OSS today. Shifts in motivation occur not only due to changes to the OSS
landscape, but also in reflection of the journey an individual makes and their
growth since they first joined [26]. Currently, there is a lack an
understanding of the differences in motivation for the early joiners compared
to those who are well-entrenched in OSS.

Aiming to support both the attraction of new members and the retention of
existing contributors, we ran this study to understand the current motivations
of OSS contributors, how they shifted from OSS’s infancy [31, 37] in response
to the changing landscape, and how motivation changes after members join OSS.
Through a survey with 242 OSS contributors, the results showed that intrinsic
and internalized motivations currently explain what drives most contributors
today. On the extrinsic end, _career_ is relevant to many contributors,
contrary to _pay_ , which only explains why less than one-third of the
respondents contributing to OSS. Contributing to OSS often transforms
extrinsic motivations when joining to intrinsic ones to stay. Whereas
ideology, own-use, or education-related programs can be an impetus to join
OSS, individuals stay for intrinsic reasons (fun, altruism, reputation, and
kinship). Results also showed that some motivations to contribute to OSS have
endured since the 2000s: learning, fun, knowledge sharing, and a belief that
source code should be open—all core tenets of OSS. Others have seen a marked
difference. Social aspects (e.g., altruism, kinship, and reputation) have
increased in the ranking, whereas participating in OSS to “scratch one’s own
itch” has dropped. Very few women from the respondents joined OSS because of
traditional motivations, but continued for _reciprocity_ , which was one of
the lowest factors for men. This study was accepted and will be presented at
ICSE 2021 in the Research Track.

From the studies performed in Stage 1, I observed that although motivation and
success perceptions are interrelated and complement each other, they can play
different roles. As an example, one of our participants reported being
motivated to join by “reputation”, but perceived success as “getting paid to
contribute”. Individuals with diverse backgrounds and understandings of
success may need different engagement strategies [30]. By understanding that
success is polyvalent in OSS, communities can leverage different definitions
of success to support the growth of diverse individuals. For contributors who
consider success as ”Having contacts in several different communities”,
communities can promote meetups to help increase social capital.

###  IV-B Stage 2 - Understanding women’s participation in OSS

I am currently analyzing the results of a systematic literature review to
identify studies about women’s participation in OSS projects, including what
types of activities women perform in OSS projects, their motivations to join,
stay, not join, and leave, challenges they face, and strategies to increase
women’s participation in OSS projects. The search string was be executed in
ACM, Scopus, Compendex and IEEE.

In parallel with the literature review, I am running a case study based on
survey and interviews to understand women’s challenges, career pathways,
perceptions of success, and motivations in the Linux Kernel. I will start with
an on-line survey with all the current contributors Linux Kernel. Besides
demographics, the survey’s questionnaire will include open and closed
questions about motivations, success, challenges, advice for their women
peers, and suggestions to increase women’s participation and suggestions for
metrics to evaluate the sense of virtual community [44, 45]. After qualitative
and quantitative analysis of the data, I will corroborate the findings through
semi-structured interviews [46]. For the interviews I will first identify
survey respondents who had agreed to follow up conversations. From this set of
respondents, I will randomly select a set of interview participants to balance
the demographics distribution and include different genders, as well those who
had never taken breaks and those who contributed on a regular basis, then took
a break, and came back. In addition, I will mine Linux Kernel repositories to
identify contributors who contributed on a regular basis for a certain time
and left, and try to recruit them for an interview with the Linux Kernel
community managers. Following this, I will interview and qualitatively analyze
data using coding procedures and compare different genders on: their
motivations to join, stay, take breaks, and leave; the categories of
challenges they face; how they evolve and the pathways they take; how they
perceive success; and their suggestions for improving inclusivity. Added to
this data, I will collect information about the results of the actions this
project had already taken to increase inclusivity.

###  IV-C Stage 3 - Strategies to Increase Women in OSS

In this Stage I will design actionable strategies to increase women’s
participation. The interventions will focus on increasing their sense of
belonging and stickiness with the project. The community managers of Linux
Kernel will evaluate the feasibility of the strategies. The data collected in
both Stages 1 and 2 will be used as design input.

In order to answer the RQ.2. How can the strategies help increase the rates of
women in OSS projects?, and aligned with the Plan-Do-Check-Act (PDCA) approach
[47, 48], during the execution of the strategies I will interact and collect
feedback from the contributors and the community managers to evaluate, learn,
and improve the strategies. Additionally, I will run post-study debriefing
interview sessions with women who participated in the strategies to collect
their impressions, and the positive and negative points to be used as lessons
learned. I will track of the number of women before and after the strategies
were implemented to evaluate whether the strategies helped to increase the
number of women contributors.

##  V Conclusion

This proposal focuses on increasing women’s participation in OSS projects.
More specifically, I take an holistic approach to women’s motivations to join
(or not), stay, take breaks, or leave OSS, together with their definitions of
success, to design attraction and retention strategies focused on women’s
aspirations within OSS. Through preliminary findings, I show the many
different career pathways that women can follow find success in OSS, how their
motivations to start and to stay can shift, and that their definition of
success is multi-faceted and nuanced. Success can include both objective
perspectives (e.g., monetary rewards, amount of contribution) as well as
subjective perceptions (e.g., recognition in the community, satisfaction).
Although contributors’ goals may end up at the same destination, the routes
they take to arrive there can be many and divergent, rooted in their diverse
motivations and perceptions of success.

## References

  * [1] S. Oreg and O. Nov, “Exploring motivations for contributing to open source initiatives: The roles of contribution context and personal values,” _Computers in human behavior_ , vol. 24, no. 5, pp. 2055–2073, 2008.
  * [2] A. Forte and C. Lampe, “Defining, understanding, and supporting open collaboration: Lessons from the literature,” _American Behavioral Scientist_ , vol. 57, no. 5, pp. 535–547, 2013.
  * [3] B. Vasilescu, D. Posnett, B. Ray, M. G. van den Brand, A. Serebrenik, P. Devanbu, and V. Filkov, “Gender and tenure diversity in github teams,” in _Proceedings of the 33rd annual ACM conference on human factors in computing systems_ , 2015, pp. 3789–3798.
  * [4] C. P. Earley and E. Mosakowski, “Creating hybrid team cultures: An empirical test of transnational team functioning,” _Academy of Management journal_ , vol. 43, no. 1, pp. 26–49, 2000.
  * [5] M. J. Muller and S. Kuhn, “Participatory design,” _Communications of the ACM_ , vol. 36, no. 6, pp. 24–28, 1993.
  * [6] S. Zacchiroli, “Gender differences in public code contributions: a 50-year perspective,” _IEEE Software_ , 2020.
  * [7] F. Sharan, “Asf committer diversity survey. accessed: 2020-10-16,” https://cwiki.apache.org/confluence/display/COMDEV/ASF+Committer+Diversity+Survey+-+2016, 2016\.
  * [8] Bitergia, “Gender-diversity analysis of the linux kernel technical contributions. accessed: 2020-10-16,” https://blog.bitergia.com/2016/10/11/gender-diversity-analysis-of-the-linux-kernel-technical-contributions, 2016\.
  * [9] D. Izquierdo, N. Huesman, A. Serebrenik, and G. Robles, “Openstack gender diversity report,” _IEEE Software_ , vol. 36, no. 1, pp. 28–33, 2018.
  * [10] B. Vasilescu, A. Serebrenik, and V. Filkov, “A data set for social diversity studies of github teams,” in _2015 IEEE/ACM 12th working conference on mining software repositories_. IEEE, 2015, pp. 514–517.
  * [11] D. Nafus, “‘patches don’t have gender’: What is not open in open source software,” _New Media & Society_, vol. 14, no. 4, pp. 669–683, 2012.
  * [12] J. Terrell, A. Kofink, J. Middleton, C. Rainear, E. Murphy-Hill, C. Parnin, and J. Stallings, “Gender differences and bias in open source: Pull request acceptance of women versus men,” _PeerJ Computer Science_ , vol. 3, p. e111, 2017.
  * [13] B. Vasilescu, V. Filkov, and A. Serebrenik, “Perceptions of diversity on git hub: A user survey,” in _2015 IEEE/ACM 8th International Workshop on Cooperative and Human Aspects of Software Engineering_. IEEE, 2015, pp. 50–56.
  * [14] J. Feller and B. Fitzgerald, “A framework analysis of the open source software development paradigm,” in _ICIS 2000 proceedings of the twenty first international conference on information systems_. Association for Information Systems (AIS), 2000, pp. 58–69.
  * [15] C. Mendez, H. S. Padala, Z. Steine-Hanson, C. Hilderbrand, A. Horvath, C. Hill, L. Simpson, N. Patil, A. Sarma, and M. Burnett, “Open source barriers to entry, revisited: A sociotechnical perspective,” in _Proceedings of the 40th International Conference on Software Engineering_ , 2018, pp. 1004–1015.
  * [16] P. Tourani, B. Adams, and A. Serebrenik, “Code of conduct in open source projects,” in _2017 IEEE 24th international conference on software analysis, evolution and reengineering (SANER)_. IEEE, 2017, pp. 24–33.
  * [17] A. Lee and J. C. Carver, “Floss participants’ perceptions about gender and inclusiveness: a survey,” in _2019 IEEE/ACM 41st International Conference on Software Engineering (ICSE)_. IEEE, 2019, pp. 677–687.
  * [18] M. Fossatti, “Gender, diversity, and inclusion in open source communities,” _XRDS: Crossroads, The ACM Magazine for Students_ , vol. 26, no. 4, pp. 46–48, 2020.
  * [19] A. Bosu and K. Z. Sultana, “Diversity and inclusion in open source software (oss) projects: Where do we stand?” in _2019 ACM/IEEE International Symposium on Empirical Software Engineering and Measurement (ESEM)_. IEEE, 2019, pp. 1–11.
  * [20] Y. Qiu, K. J. Stewart, and K. M. Bartol, “Joining and socialization in open source women’s groups: an exploratory study of kde-women,” in _IFIP International Conference on Open Source Systems_. Springer, 2010, pp. 239–251.
  * [21] V. Kuechler, C. Gilbertson, and C. Jensen, “Gender differences in early free and open source software joining process,” in _IFIP International Conference on Open Source Systems_. Springer, 2012, pp. 78–93.
  * [22] B. Vedres and O. Vasarhelyi, “Gendered behavior as a disadvantage in open source software development,” _EPJ Data Science_ , vol. 8, no. 1, p. 25, 2019\.
  * [23] R. Ghosh, R. Glott, B. Krieger, and G. Robles, “Free/libre and open source software: Survey and study,” 2002.
  * [24] T. Weiss, “Panel: Open-source needs more women developers,” _Computerworld_ , 2005.
  * [25] P. Wurzelová, F. Palomba, and A. Bacchelli, “Characterizing women (not) contributing to open-source,” in _2019 IEEE/ACM 2nd International Workshop on Gender Equality in Software Engineering (GE)_. IEEE, 2019, pp. 5–8.
  * [26] I. Steinmacher, M. A. Gerosa, and D. Redmiles, “Attracting, onboarding, and retaining newcomer developers in open source software projects,” in _Workshop on GSD in a CSCW Perspective_ , 2014.
  * [27] I. Steinmacher, I. Wiese, A. P. Chaves, and M. A. Gerosa, “Why do newcomers abandon open source software projects?” in _2013 6th International Workshop on Cooperative and Human Aspects of Software Engineering (CHASE)_. IEEE, 2013, pp. 25–32.
  * [28] G. Von Krogh, S. Haefliger, S. Spaeth, and M. W. Wallin, “Carrots and rainbows: Motivation and social practice in open source software development,” _MIS quarterly_ , pp. 649–676, 2012.
  * [29] L. S. Dyke and S. A. Murphy, “How we define success: A qualitative study of what matters most to women and men,” _Sex Roles_ , vol. 55, no. 5-6, pp. 357–371, 2006.
  * [30] B. Trinkenreich, M. Guizani, I. Wiese, A. Sarma, and I. Steinmacher, “Hidden figures: Roles and pathways of successful oss contributors,” _Computer Supported Cooperative Work (CSCW)_ , vol. 4, no. 180, 2020.
  * [31] K. R. Lakhani and R. G. Wolf, “Why hackers do what they do: Understanding motivation and effort in free/open source software projects,” 2003.
  * [32] B. Fitzgerald, “The transformation of open source software,” _MIS quarterly_ , vol. 30, no. 3, pp. 587–598, 2006.
  * [33] G. Robles, I. Steinmacher, P. Adams, and C. Treude, “Twenty years of open source software: From skepticism to mainstream,” _IEEE Software_ , vol. 36, no. 6, pp. 12–15, 2019.
  * [34] I. Steinmacher, G. Robles, B. Fitzgerald, and A. Wasserman, “Free and open source software development: the end of the teenage years,” _Journal of Internet Services and Applications_ , vol. 8, no. 1, p. 17, Dec 2017. [Online]. Available: https://doi.org/10.1186/s13174-017-0069-9
  * [35] H. S. Qiu, A. Nolte, A. Brown, A. Serebrenik, and B. Vasilescu, “Going farther together: The impact of social capital on sustained participation in open source,” in _2019 IEEE/ACM 41st International Conference on Software Engineering (ICSE)_. IEEE, 2019, pp. 688–699.
  * [36] ——, “Going farther together: The impact of social capital on sustained participation in open source,” in _Proceedings of the 41st International Conference on Software Engineering_ , ser. ICSE ’19. Piscataway, NJ, USA: IEEE Press, 2019, pp. 688–699. [Online]. Available: https://doi.org/10.1109/ICSE.2019.00078
  * [37] A. Hars and S. Ou, “Working for free motivations of participating in open source software projects,” in _HICSS’04_ , 2004, pp. 25–31.
  * [38] G. Hertel, S. Niedner, and S. Herrmann, “Motivation of software developers in open source projects: an internet-based survey of contributors to the linux kernel,” _Research policy_ , vol. 32, no. 7, pp. 1159–1177, 2003.
  * [39] R. A. Ghosh, “Understanding free software developers: Findings from the floss study,” _Perspectives on free and open source software_ , vol. 28, pp. 23–47, 2005.
  * [40] E. H. Trainer, C. Chaihirunkarn, A. Kalyanasundaram, and J. D. Herbsleb, “From personal tool to community resource: What’s the extra work and who will do it?” in _18th ACM Conference on Computer Supported Cooperative Work & Social Computing_, 2015, pp. 417–430.
  * [41] Y. Cho, J. Park, S. Jeoung, B. Ju, J. You, A. Ju, C. K. Park, H. Y. Park _et al._ , “How do south korean female executives’ definitions of career success differ from those of male executives?” _European Journal of Training and Development_ , 2017.
  * [42] A. M. Porter, “Physics phds ten years later: Success factors and barriers in career paths. results from the phd plus 10 study.” _AIP Statistical Research Center_ , 2019.
  * [43] I. Steinmacher, G. Robles, B. Fitzgerald, and A. Wasserman, “Free and open source software development: the end of the teenage years,” _Journal of Internet Services and Applications_ , vol. 8, no. 17, 2017.
  * [44] A. L. Blanchard, “Developing a sense of virtual community measure,” _CyberPsychology & Behavior_, vol. 10, no. 6, pp. 827–830, 2007.
  * [45] D. Abfalter, M. E. Zaglia, and J. Mueller, “Sense of virtual community: A follow up on its measurement,” _Computers in Human Behavior_ , vol. 28, no. 2, pp. 400–404, 2012.
  * [46] P. Mihas, _Learn to Use an Exploratory Sequential Mixed Method Design for Instrument Development_. SAGE Publications, Limited, 2019.
  * [47] W. A. Shewhart and W. E. Deming, _Statistical method from the viewpoint of quality control_. Courier Corporation, 1986\.
  * [48] W. E. Deming, _Out of the Crisis_. MIT press, 2018.