# The Number of Overlapping Customers in Erlang-A Queues: An Asymptotic
Approach

Young Myoung Ko
Department of Industrial and Management Engineering
Pohang University of Science Technology
<EMAIL_ADDRESS>

Jamol Pender 111Corresponding Author
School of Operations Research and Information Engineering
Cornell University
<EMAIL_ADDRESS>

Jin Xu
School of Management
Huazhong University of Science and Technology
<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate the number of customers that overlap or coincide
with a virtual customer in an Erlang-A queue. Our study provides a novel
approach that exploits fluid and diffusion limits for the queue to approximate
the mean and variance of the number of overlapping customers. We conduct a
detailed analysis of the fluid and diffusion limit differential equations to
derive these approximations. We also construct new accurate approximations for
the mean and variance of the waiting time in the Erlang-A queue by combining
fluid limits with the polygamma function. Our findings have important
implications for queueing theory and evaluating the overlap risk of more
complicated service systems.

##  1 Introduction

Have you ever wondered how many people you come in contact with when you visit
a store or a retail shop? While this is an interesting question on its own, it
has taken on greater significance in light of the COVID-19 pandemic. Most
research on COVID-19 has utilized deterministic compartmentalized models to
estimate infection rates and spread dynamics, as demonstrated in works such as
Dandekar et al. [6], Nguemdjo et al. [33], Kaplan [21]. However, we must
acknowledge the importance of stochastic effects in determining the spread, as
evidenced in works like Drakopoulos et al. [12], Palomo et al. [37], Pang and
Pardoux [39], Forien et al. [16], Moein et al. [32]. As shoppers crowd stores
to purchase essential goods such as water and non-perishable items, the risk
of virus transmission increases. Service facilities and systems have
implemented several measures, including the installation of air filters,
transparent barriers, and mandatory mask-wearing for patrons. In addition,
these systems have adopted various forms of social and physical distancing to
minimize customer proximity, as outlined in Bove and Benoit [4].

Nonetheless, there are certain places where maintaining distance between
people is not always feasible, such as in workplaces or crowded retail
environments. In these situations, it becomes crucial to understand the extent
to which customers overlap with one another. Recent research by Kang et al.
[20], Palomo and Pender [35, 36, 38] has shed light on how to calculate
customer overlap times in both single-server and multi-server queues.
Specifically, Kang et al. [20] demonstrates how overlaps can be used to
calculate a new $R_{0}$ value for analyzing infection rates in
compartmentalized epidemic models. Meanwhile, Palomo and Pender [35] proves
that the overlap distribution is exponential for the M/M/1 queue and depends
explicitly on the service distribution, and demonstrates via simulation that a
similar result holds true in the non-exponential service setting as well.
Recent studies by Palomo and Pender [36, 38] have brought fresh insights into
the analysis of overlap times. Palomo and Pender [36] analyzes overlap times
in the infinite server queue, while Palomo and Pender [38] extends the
analysis to the case of batch arrivals, highlighting the practical
applications of this analysis for transportation systems like trains and
buses.

Unfortunately, the existing literature on the number of overlaps has only
focused on infinite server systems, as demonstrated by Palomo and Pender [35].
In an infinite server queue, the number of overlaps is equivalent to the queue
length upon arrival and the number of additional arrivals during service, as
there is no waiting. However, in the context of a finite server system,
customer waiting can increase the number of overlaps. Consequently, the number
of overlaps in an infinite server system can serve as a lower bound for any
finite server system without abandonment. Given the prevalence of abandonment
in real-world queueing systems, it is crucial to explore how abandonment and a
finite number of servers impact the number of overlaps experienced by
customers. While recent research by Ko and Xu [22] has analyzed the total
amount of overlapping time in a time-varying multi-server queue, their work
has not investigated the number of overlaps or the impact of abandonment on
the system. Thus, our work aims to address this gap in the literature.

Our paper presents a new analysis that provides critical insights into finite
server systems with customer abandonment. Specifically, our analysis enables
us to determine the number of overlaps that occur accurately and sheds light
on the distribution of potential overlaps, allowing us to establish prediction
intervals. This analysis has practical applications in preventing large
overlaps, and can also serve as a valuable design tool for constructing
appropriate overlaps by regulating arrival rates, service distributions, and
the number of servers. Overall, our analysis offers a valuable contribution to
the field of queueing systems and has the potential to inform decision-making
in a variety of service system applications.

###  1.1 Main Contributions of the Paper

The main contributions of this work can be summarized as follows:

  * •

We derive the exact expressions for the Erlang-A fluid and diffusion
differential equations.

  * •

We provide new approximations for the mean and variance of the number of
overlapping customers.

  * •

We provide new approximations for the mean and variance of the waiting time in
the Erlang-A queue using the digamma and trigamma functions.

###  1.2 Organization of the Paper

The remainder of this paper is organized as follows. Section 2 introduces the
Erlang-A queueing model and its importance in stochastic network theory. We
also derive exact expressions for the fluid and diffusion variance
differential equations. In Section 3, we introduce novel approximations for
the mean and variance of the number of overlaps for a virtual customer. Our
work also includes new approximations for the mean and variance of the wait
time for the Erlang-A queue. We also validate the effectiveness of these
approximations through simulation experiments. Lastly, in Section 4, we
summarize our findings and suggest promising areas for future research in this
area.

##  2 The Erlang-A Queueing Model

The Erlang-A queueing model is a fundamental queueing model in the applied
probability literature. The seminal work of Mandelbaum et al. [29] shows that
the queue length process for an $M(t)/M/c+M$ queueing system
$Q\equiv\\{Q(t)|t\geq 0\\}$ is represented by the following stochastic, time
changed integral equation:

$\displaystyle Q(t)$ $\displaystyle=$ $\displaystyle
Q(0)+\mathit{\Pi}_{1}\left(\int^{t}_{0}\lambda(s)ds\right)-\mathit{\Pi}_{2}\left(\int^{t}_{0}\mu\cdot(Q(s)\wedge
c)ds\right)$
$\displaystyle-\mathit{\Pi}_{3}\left(\int^{t}_{0}\theta\cdot(Q(s)-c)^{+}ds\right),$

where $\mathit{\Pi}_{i}\equiv\\{\mathit{\Pi}_{i}(t)|t\geq 0\\}$ for
$i={1,2,3}$ are i.i.d. standard (rate 1) Poisson processes and $(x\wedge
y)=\min\\{x,y\\}$. Thus, the sample path of the Erlang-A queue can be written
in terms of three independent unit rate Poisson processes. A deterministic
time change for $\mathit{\Pi}_{1}$ transforms it into a non-homogeneous
Poisson arrival process with rate $\lambda(t)$ that counts the customer
arrivals that occurred in the time interval [0,t). A random time change for
the Poisson process $\mathit{\Pi}_{2}$, gives us a departure process that
counts the number of customers that have received service from any one of the
$c$ servers. We implicitly assume that the number of servers is
$c\in\mathbb{Z}^{+}$ and that each server works at rate $\mu$. Finally, a
random time change of $\mathit{\Pi}_{3}$ counts the number of customers that
abandon the system before beginning service. We also assume that the
abandonment distribution is exponential and the abandonment rate is equal to
$\theta$.

The Erlang-A queueing model and its variants are very well studied in the
queueing literature, see for example Zeltyn and Mandelbaum [49], Whitt [47],
Mandelbaum and Zeltyn [30], Gurvich et al. [17], Engblom and Pender [15],
Pender [42], Niyirora and Pender [34], Aktekin and Ekin [1], Braverman et al.
[5], Pender [43], Pender and Massey [44], Bitton et al. [3], Azriel et al.
[2], van Leeuwaarden et al. [46]. The Erlang-A queue has been so extensively
analyzed since several important queueing models are special cases of it. One
special case is the infinite server queue. The infinite server queue can be
constructed from the Erlang-A queue in two ways. The first way is to set the
number of servers to infinity i.e., $c=\infty$. The second way to construct
the infinite server queue is to set the service rate $\mu$ equal to the
abandonment rate $\theta$. A second special case of the Erlang-A is when one
sets the abandonment parameter $\theta=0$. In this case, we shut off the
abandonment process and we obtain the Erlang-C queue. Finally, when we let
$\theta$ get large i.e., $\theta\to\infty$, we obtain an Erlang-B queue, see
for example Hampshire et al. [19]. Thus, the Erlang-A queueing model
encapsulates three special cases of queueing models and this reflects its
importance as a model in the applied probability and queueing literature.

A key insight from [18] is that for multi-server queueing systems, it is
natural to scale up the arrival rate and the number of servers simultaneously.
This scaling – known as the _Halfin-Whitt_ scaling – has been an important
technique for modeling call centers in the queueing literature, see for
example Pang et al. [41]. Since the $M(t)/M/c+M$ queueing process is a special
case of a single node _Markovian service network_ , we can also construct an
associated, _uniformly accelerated_ queueing process where both the new
arrival rate $\eta\cdot\lambda(t)$ and the new number of servers $\eta\cdot c$
are scaled by the same factor $\eta>0$. Thus, using the _Halfin-Whitt_ scaling
for the Erlang-A model, we arrive at the following sample path representation
for the queue length process as

$\displaystyle Q^{\eta}(t)$ $\displaystyle=$ $\displaystyle
Q^{\eta}(0)+\mathit{\Pi}_{1}\left(\int^{t}_{0}\eta\cdot\lambda(s)ds\right)-\mathit{\Pi}_{2}\left(\int^{t}_{0}\mu\cdot(Q^{\eta}(s)\wedge\eta\cdot
c)ds\right)$
$\displaystyle-\mathit{\Pi}_{3}\left(\int^{t}_{0}\theta\cdot(Q^{\eta}(s)-\eta\cdot
c)^{+}ds\right)$ $\displaystyle=$ $\displaystyle
Q^{\eta}(0)+\mathit{\Pi}_{1}\left(\int^{t}_{0}\eta\cdot\lambda(s)ds\right)-\mathit{\Pi}_{2}\left(\int^{t}_{0}\eta\cdot\mu\cdot\left(\frac{Q^{\eta}(s)}{\eta}\wedge
c\right)ds\right)$
$\displaystyle-\mathit{\Pi}_{3}\left(\int^{t}_{0}\eta\cdot\theta\cdot\left(\frac{Q^{\eta}(s)}{\eta}-c\right)^{+}ds\right).$

Applying the Halfin-Whitt scaling and taking the following limits gives us the
_fluid_ models of [29], i.e.,

$\lim_{\eta\to\infty}\frac{1}{\eta}Q^{\eta}(t)=q(t)\hskip
8.53581pt\mathrm{a.s.}$

where the deterministic process $q(t)$, the _fluid mean_ , is governed by the
one dimensional ordinary differential equation (ODE),

$\raisebox{-0.4pt}{$\stackrel{{\scriptstyle\bullet}}{{q}}$}(t)=\lambda(t)-\mu\cdot(q(t)\wedge
c)-\theta\cdot(q(t)-c)^{+}.$ (1)

Moreover, if one takes a diffusion limit i.e.,

$\lim_{\eta\to\infty}\sqrt{\eta}\left(\frac{1}{\eta}Q^{\eta}(t)-q(t)\right)\Rightarrow\tilde{Q}(t),\hskip
8.53581pt$

one obtains a diffusion process where the variance of the diffusion is given
by the following ODE,

$\displaystyle\raisebox{0.9pt}{$\stackrel{{\scriptstyle\bullet}}{{\mathrm{Var}}}$}\left[\tilde{Q}(t)\right]$
$\displaystyle=$ $\displaystyle\lambda(t)+\mu\cdot(q(t)\wedge
c)+\theta\cdot(q(t)-c)^{+}$ (2)
$\displaystyle-2\cdot\mathrm{Var}\left[\tilde{Q}(t)\right]\cdot\left(\mu\cdot\\{q(t)<c\\}+\theta\cdot\\{q(t)\geq
c\\}\right).$

In what follows, we provide an extensive analysis of the differential
equations that are obtained from the fluid and diffusion limits. Despite the
equations being ubiquitous in the literature, this paper provides the first
detailed analysis of the fluid and diffusion differential equation dynamics in
the constant arrival rate case. We begin with the analysis of the differential
equations of the fluid limits.

###  2.1 Fluid Analysis

In this section, we provide results regarding the fluid limit dynamics when
the arrival rate is constant. Before we describe our main results regarding
the fluid analysis, we provide a standard result for linear differential
equations.

######  Lemma 2.1.

Let $q(t)$ be the solution to the following differential equation

$\raisebox{-0.4pt}{$\stackrel{{\scriptstyle\bullet}}{{q}}$}=\lambda(t)-\mu(t)q(t)$

where $q(0)=q_{0}$. Then the solution for any value of $t$ is given by

$\displaystyle q(t)$ $\displaystyle=$ $\displaystyle
q_{0}\exp\left\\{-\int^{t}_{0}\mu(s)ds\right\\}$
$\displaystyle+\left(\exp\left\\{-\int^{t}_{0}\mu(s)ds\right\\}\cdot\left(\int^{t}_{0}\lambda(s)\exp\left\\{\int^{s}_{0}\mu(r)dr\right\\}ds\right)\right).$

###### Proof.

This follows from standard results on ordinary differential equations by
varying parameters (see [45]). ∎

Now that we characterize the dynamics of $q(t)$ with time-varying arrival rate
$\lambda(t)$ and service rate $\mu(t)$, our next result provides the explicit
solution to $q(t)$ in the steady state, i.e., when the arrival and service
rates are both constant.

######  Proposition 2.2.

When $\lambda(t)$ and $\mu(t)$ are constant, the solution $q(t)$ to Equation
(1) is given as:

$\displaystyle q(t)$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{\lambda-\mu c+\theta
c}{\theta}+\left(q(0)-\frac{\lambda-\mu c+\theta c}{\theta}\right)e^{-\theta
t}&\mbox{if }q(0)>c,\lambda>\mu c\\\ \frac{\lambda-\mu c+\theta
c}{\theta}+\left(q(0)-\frac{\lambda-\mu c+\theta c}{\theta}\right)e^{-\theta
t},&\mbox{if }q(0)>c,\lambda\leq\mu c,t\leq t_{1}^{*}\\\
\frac{\lambda}{\mu}+\left(c-\frac{\lambda}{\mu}\right)e^{-\mu(t-t^{*})},&\mbox{if
}q(0)>c,\lambda\leq\mu c,t>t_{1}^{*}\\\
\frac{\lambda}{\mu}+\left(q(0)-\frac{\lambda}{\mu}\right)e^{-\mu t}&\mbox{if
}q(0)\leq c,\lambda\leq\mu c\\\
\frac{\lambda}{\mu}+\left(q(0)-\frac{\lambda}{\mu}\right)e^{-\mu t},&\mbox{if
}q(0)\leq c,\lambda>\mu c,t\leq t_{2}^{*}\\\ \frac{\lambda-\mu c+\theta
c}{\theta}+\left(\frac{-\lambda+\mu
c}{\theta}\right)e^{-\theta(t-t^{*})},&\mbox{if }q(0)\leq c,\lambda>\mu
c,t>t_{2}^{*}\end{cases}$

where $t_{1}^{*}=\frac{\log\left(\frac{\theta q(0)-\lambda+\mu c-\theta c}{\mu
c-\lambda}\right)}{\theta}$ and
$t_{2}^{*}=\frac{\log\left(\frac{q(0)-\frac{\lambda}{\mu}}{c-\frac{\lambda}{\mu}}\right)}{\mu}.$

###### Proof.

When $q(0)>c$ and $\lambda>\mu c$, we apply Lemma 2.1 to the differential
equation given in Equation (1). When $q(0)>c$ and $\lambda\leq\mu c$, from
time 0 to time $t_{1}^{*}$ (which is the time that the differential equation
hits the value $c$), we can apply Lemma 2.1 to the differential equation to
obtain the solution. After $t_{1}^{*}$, we know the solution is larger than
$c$ so that it follows a new differential equation and the solution is also
given by Lemma 2.1. The discussion of the other cases are similar, so we omit
the details here. This completes the proof. ∎

Now that we have completely described the dynamics of the fluid queue length
as a function of time in the previous proposition. We can easily obtain the
next corollary following Proposition 2.2.

######  Corollary 2.3.

Suppose that $\lambda(t)$ and $\mu(t)$ are constant, and
$q(\infty)=\lim_{t\to\infty}q(t)$. Then we have

$\displaystyle q(\infty)$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{\lambda-\mu c+\theta c}{\theta}&\mbox{if
}\lambda>\mu c\\\ \frac{\lambda}{\mu}&\mbox{if }\lambda\leq\mu c\end{cases}$

###### Proof.

The results follows from Proposition 2.2 by letting $t\to\infty$. ∎

We then turn our attention to analyzing the diffusion variance differential
equations. Analyzing the fluid equations first is essential since the
diffusion variance differential equations depend on the fluid dynamics in an
explicit way.

###  2.2 Diffusion Variance Analysis

In this section, we provide an analysis of the diffusion variance differential
equations given in Equation (2). We start with explicit solutions to the
diffusion variance equations.

######  Proposition 2.4.

When $\lambda(t)$ and $\mu(t)$ are constant, the solution $v(t)$ to Equation
(2), is given as:

$\displaystyle v(t)=\begin{cases}\left(v(0)-q(0)-\frac{\mu c-\theta
c}{\theta}\right)e^{-2\theta
t}+\frac{\lambda}{\theta}+\left(q(0)-\frac{\lambda-\mu c+\theta
c}{\theta}\right)e^{-\theta t},&\mathrm{if}\ q(0)>c,\lambda>\mu c\\\
\left(v(0)-q(0)-\frac{\mu c-\theta c}{\theta}\right)e^{-2\theta
t}+\frac{\lambda}{\theta}+\left(q(0)-\frac{\lambda-\mu c+\theta
c}{\theta}\right)e^{-\theta t},&\mathrm{if}\ q(0)>c,\lambda\leq\mu c,t\leq
t_{1}^{*}\\\
(v(t_{1}^{*})-c)e^{-2\mu(t-t_{1}^{*})}+\frac{\lambda}{\mu}+\left(c-\frac{\lambda}{\mu}\right)e^{-\mu(t-t_{1}^{*})},&\mathrm{if}\
q(0)>c,\lambda\leq\mu c,t>t_{1}^{*}\\\ \left(v(0)-q(0)\right)e^{-2\mu
t}+\left(1-e^{-\mu t}\right)\cdot\frac{\lambda}{\mu}+q(0)e^{-\mu
t},&\mathrm{if}\ q(0)\leq c,\lambda\leq\mu c\\\ \left(v(0)-q(0)\right)e^{-2\mu
t}+\left(1-e^{-\mu t}\right)\cdot\frac{\lambda}{\mu}+q(0)e^{-\mu
t},&\mathrm{if}\ q(0)\leq c,\lambda>\mu c,t\leq t_{2}^{*}\\\
(v(t_{2}^{*})-c)e^{-2\theta(t-t_{2}^{*})}+\frac{\lambda}{\theta}+\left(\frac{-\lambda+\mu
c}{\theta}\right)e^{-\theta(t-t_{2}^{*})},&\mathrm{if}\ q(0)\leq c,\lambda>\mu
c,t>t_{2}^{*}\end{cases}$ (3)

where

$t^{*}_{1}=\frac{\log\left(\frac{\theta q(0)-\lambda+\mu c-\theta c}{\mu
c-\lambda}\right)}{\theta}\quad\mathrm{and}\quad
t^{*}_{2}=\frac{\log\left(\frac{q(0)-\frac{\lambda}{\mu}}{c-\frac{\lambda}{\mu}}\right)}{\mu}.$

###### Proof.

We prove the results in Equation (3) by discussing the following four cases.

CASE 1: For the first case with $q(0)>c$ and $\lambda>\mu c$, we have

$\stackrel{{\scriptstyle\bullet}}{{v}}$ $\displaystyle=$
$\displaystyle\lambda+\mu\cdot c+\theta\cdot(q(t)-c)-2\cdot\theta\cdot v(t).$

Using the theory of linear odes, this implies that

$\displaystyle v(t)$ $\displaystyle=$ $\displaystyle v(0)e^{-2\theta
t}+e^{-2\theta t}\int^{t}_{0}e^{2\theta s}\left(\lambda+\mu c-\theta c+\theta
q(s)\right)ds$ $\displaystyle=$ $\displaystyle v(0)e^{-2\theta
t}+(1-e^{-2\theta t})\cdot\left(\frac{\lambda+\mu c-\theta
c}{2\theta}\right)+\theta e^{-2\theta t}\int^{t}_{0}e^{2\theta s}q(s)ds$
$\displaystyle=$ $\displaystyle v(0)e^{-2\theta t}+(1-e^{-2\theta
t})\cdot\left(\frac{\lambda+\mu c-\theta c}{2\theta}\right)$ $\displaystyle+$
$\displaystyle\theta e^{-2\theta t}\int^{t}_{0}e^{2\theta
s}\left(\frac{\lambda-\mu c+\theta c}{\theta}+e^{-\theta
s}\left(q(0)-\frac{\lambda-\mu c+\theta c}{\theta}\right)\right)ds$
$\displaystyle=$ $\displaystyle v(0)e^{-2\theta t}+(1-e^{-2\theta
t})\cdot\left(\frac{\lambda+\mu c-\theta c}{2\theta}\right)$ $\displaystyle+$
$\displaystyle\left(\frac{\lambda-\mu c+\theta c}{2\theta}(1-e^{-2\theta
t})+\left(q(0)-\frac{\lambda-\mu c+\theta
c}{\theta}\right)\cdot\left(e^{-\theta t}-e^{-2\theta t}\right)\right)$
$\displaystyle=$ $\displaystyle\left(v(0)-q(0)+\frac{-\mu c+\theta
c}{\theta}\right)e^{-2\theta
t}+\frac{\lambda}{\theta}+\left(q(0)-\frac{\lambda-\mu c+\theta
c}{\theta}\right)e^{-\theta t}.$

We thus showed the first item of Equation (3).

CASE 2: For the second case where $q(0)>c$ and $\lambda<\mu c$, we have

$\displaystyle\raisebox{0.9pt}{$\stackrel{{\scriptstyle\bullet}}{{v}}$}=\begin{cases}\lambda+\mu
c+\theta q(t)-\theta c-2\theta v(t),&\mathrm{if}\ t\leq t^{*}\\\ \lambda+\mu
q(t)-2\mu v(t),&\mathrm{if}\ t>t^{*}\end{cases}$

where $t^{*}$ is equal to

$t^{*}_{1}=\frac{\log\left(\frac{\theta q(0)+\lambda-\mu c+\theta c}{\mu
c-\lambda}\right)}{\theta}.$

Using the theory of linear odes, this implies for $t\leq t^{*}_{1}$ that

$\displaystyle v(t)=\left(v(0)-q(0)+\frac{-\mu c+\theta
c}{\theta}\right)e^{-2\theta
t}+\frac{\lambda}{\theta}+\left(q(0)-\frac{\lambda-\mu c+\theta
c}{\theta}\right)e^{-\theta t}.$

Lastly for $t>t^{*}_{1}$ we have that

$\displaystyle v(t)$ $\displaystyle=$
$\displaystyle(v(t^{*}_{1})-c)e^{-2\mu(t-t^{*}_{1})}+\frac{\lambda}{\mu}+\left(c-\frac{\lambda}{\mu}\right)e^{-\mu(t-t^{*}_{1})}.$

We hence proved the second and third items in Equation (3).

CASE 3: For the third case where $q(0)\leq c$ and $\lambda<\mu c$, we have

$\stackrel{{\scriptstyle\bullet}}{{v}}$ $\displaystyle=$
$\displaystyle\lambda+\mu\cdot q(t)-2\cdot\mu\cdot v(t).$

Using the theory of linear odes, this implies that

$\displaystyle v(t)$ $\displaystyle=$ $\displaystyle v(0)e^{-2\mu t}+e^{-2\mu
t}\int^{t}_{0}e^{2\mu s}\left(\lambda+\mu q(s)\right)ds$ $\displaystyle=$
$\displaystyle v(0)e^{-2\mu t}+\left(1-e^{-2\mu
t}\right)\cdot\frac{\lambda}{2\mu}+\mu e^{-2\mu t}\int^{t}_{0}e^{2\mu
s}q(s)ds$ $\displaystyle=$ $\displaystyle v(0)e^{-2\mu t}+\left(1-e^{-2\mu
t}\right)\cdot\frac{\lambda}{2\mu}+\mu e^{-2\mu t}\int^{t}_{0}e^{2\mu
s}\left(\frac{\lambda}{\mu}+\left(q(0)-\frac{\lambda}{\mu}\right)e^{-\mu
s}\right)ds$ $\displaystyle=$ $\displaystyle v(0)e^{-2\mu t}+\left(1-e^{-2\mu
t}\right)\cdot\frac{\lambda}{\mu}+\mu e^{-2\mu
t}\int^{t}_{0}\left(q(0)-\frac{\lambda}{\mu}\right)e^{\mu s}ds$
$\displaystyle=$ $\displaystyle v(0)e^{-2\mu t}+\left(1-e^{-2\mu
t}\right)\cdot\frac{\lambda}{\mu}+\left(q(0)-\frac{\lambda}{\mu}\right)\left(e^{-\mu
t}-e^{-2\mu t}\right)$ $\displaystyle=$
$\displaystyle\left(v(0)-q(0)\right)e^{-2\mu t}+\left(1-e^{-\mu
t}\right)\cdot\frac{\lambda}{\mu}+q(0)e^{-\mu t}.$

We hence have the fourth item of Equation (3).

CASE 4: For the fourth case where $\lambda>\mu c$ and $q(0)\leq c$, we have

$\stackrel{{\scriptstyle\bullet}}{{v}}$ $\displaystyle=$
$\displaystyle\begin{cases}\lambda+\mu q(t)-2\mu v(t),&\mathrm{if}\ t\leq
t^{*}_{2}\\\ \lambda+\mu c+\theta q(t)-\theta c-2\theta v(t),&\mathrm{if}\
t>t^{*}_{2}\end{cases}$

where $t^{*}$ is equal to

$t^{*}_{2}=\frac{\log\left(\frac{q(0)-\frac{\lambda}{\mu}}{c-\frac{\lambda}{\mu}}\right)}{\mu}.$

Using the theory of linear odes, this implies for $t\leq t^{*}_{2}$ that

$\displaystyle v(t)$ $\displaystyle=$ $\displaystyle v(0)e^{-2\mu t}+e^{-2\mu
t}\int^{t}_{0}e^{2\mu s}\left(\lambda+\mu q(s)\right)ds$ $\displaystyle=$
$\displaystyle v(0)e^{-2\mu t}+\frac{\lambda}{2\mu}\left(1-e^{-2\mu t}\right)$
$\displaystyle+$ $\displaystyle\mu e^{-2\mu t}\int^{t}_{0}e^{2\mu
s}\left(\frac{\lambda}{\mu}+\left(q(0)-\frac{\lambda}{\mu}\right)e^{-\mu
s}\right)ds$ $\displaystyle=$ $\displaystyle v(0)e^{-2\mu
t}+\left(\frac{\lambda}{2\mu}+\frac{\lambda}{2\mu}\right)\left(1-e^{-2\mu
t}\right)+\mu e^{-2\mu t}\int^{t}_{0}e^{\mu
s}\left(q(0)-\frac{\lambda}{\mu}\right)ds$ $\displaystyle=$ $\displaystyle
v(0)e^{-2\mu
t}+\left(\frac{\lambda}{2\mu}+\frac{\lambda}{2\mu}\right)\left(1-e^{-2\mu
t}\right)+\left(q(0)-\frac{\lambda}{\mu}\right)\left(e^{-\mu t}-e^{-2\mu
t}\right)$ $\displaystyle=$ $\displaystyle(v(0)-q(0))e^{-2\mu
t}+\frac{\lambda}{\mu}+\left(q(0)-\frac{\lambda}{\mu}\right)e^{-\mu t}.$

Lastly for $t>t^{*}_{2}$ we have that

$\displaystyle
v(t)=(v(t^{*}_{2})-c)e^{-2\theta(t-t^{*}_{2})}+\frac{\lambda}{\theta}+\left(\frac{-\lambda+\mu
c}{\theta}\right)e^{-\theta(t-t^{*}_{2})}.$

We thus proved the fifth and sixth items of Equation (3). ∎

Now that we have provided an analysis of the transient dynamics of the
diffusion variance differential equations, we analyze the steady state
behavior of the diffusion variance differential equation in the following
result.

######  Corollary 2.5.

Let $v(t)$ be the solution to Equation (2), then in steady state we have

$\displaystyle v(\infty)$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{\lambda}{\mu},\mathrm{if}\ \lambda\leq\mu
c\\\ \frac{\lambda}{\theta},\mathrm{if}\ \lambda>\mu c.\end{cases}$

###### Proof.

The result follows directly from Proposition 2.4 by letting $v\to\infty$ in
Equation (3). ∎

What is apparent from the steady state variance is that, it is the same as the
steady state mean queue length when $\lambda\leq\mu c$. This is because the
system is effectively an infinite server queue in this regime. We also see
that the when the arrival rate $\lambda$ is larger than the maximum service
rate $\mu c$, then the steady state variance is $\lambda/\theta$. This is
interesting because when $\theta<\mu$, the mean is smaller than the variance
and when $\theta\geq\mu$, the steady state variance is smaller than the steady
state mean queue length. This describes over-dispersion and under-dispersion
respectively. For more on the over-dispersion and under-dispersion
relationships between the mean and variance in the Erlang-A queue see Daw and
Pender [9].

Now that we have a good understanding of the fluid mean and diffusion variance
of the Erlang-A queue, we will show how to use these results in the context of
studying overlaps in the Erlang-A queue.

##  3 The Number of Overlaps

In this section, we introduce the virtual overlap process, which counts the
number of customers that the virtual customer will overlap with during their
time in the queue. This process is important from a epidemiology perspective
since it would be the number of people that would need to be contact traced
for an exposure if the virtual customer had an infectious disease, see for
example Kang et al. [20], Palomo et al. [37].

Like the infinite server setting, the virtual customer will also overlap with
the number present in the queue and the number of customers that arrive during
the service time of the virtual customer. However, unlike the infinite server
setting, the virtual customer, who we assume does not abandon from the queue,
must also overlap with customers that also arrive during their wait for
service. Thus, the overlap process can be written as the following expression
in terms of the queue length $Q(t)$, the virtual waiting time $W(t)$, and the
service time of the virtual customer $\mathcal{S}$ i.e.

$\displaystyle O(t)$ $\displaystyle=$
$\displaystyle\underbrace{N\left(t+\mathcal{S}+W(t)\right)-N(t)}_{\text{Arrivals
During Service and Wait}}+\underbrace{Q(t)}_{\text{Queue Upon Arrival}}.$ (4)

Using the above representation, we can compute the mean number of overlaps at
time $t$ by taking the expectation of the overlap process. Thus, the mean
number of overlaps can be written as

$\displaystyle\mathbb{E}\left[O(t)\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[N\left(t+\mathcal{S}+W(t)\right)-N(t)+Q(t)\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[N\left(t+\mathcal{S}+W(t)\right)-N(t)\right]+\mathbb{E}\left[Q(t)\right]$
$\displaystyle=$
$\displaystyle\lambda\mathbb{E}\left[\mathcal{S}\right]+\lambda\mathbb{E}\left[W(t)\right]+\mathbb{E}\left[Q(t)\right].$

Unfortunately, the transient mean queue length and the transient mean wait
time are not known in closed form for the Erlang-A queue, except in the case
where $\mu=\theta$. This is a major difference between the Erlang-A and the
infinite server queue. In the infinite server queue, the mean wait time is
zero and mean queue can be written as an explicit integral with respect to the
service time distribution, see for example Eick et al. [13, 14]. Consequently,
we will use limit theory to approximate the mean number of overlaps.

###  3.1 The Transient Mean Number of Overlaps

In this section, we show how to approximate the transient mean number of
overlap using asymptotic analysis. We first leverage the results of Mandelbaum
et al. [29], which proves almost sure limit theorems for the queue length
process in the Halfin-Whitt regime. However, we also need results for the
virtual waiting time in order to fully analyze the overlap process. To do
this, we will exploit a recent result by Massey and Pender [31], which proves
the following theorem.

######  Theorem 3.1.

Let $W^{\eta}(t)$ be the virtual wait time of a customer at time t who is not
going to abandon, in the scaled process. Then we have

$\displaystyle\lim_{\eta\to\infty}W^{\eta}(t)$ $\displaystyle{\buildrel
a.s\over{=}}$ $\displaystyle w(t),$

where $w(t)$ satisfies the following equation

$\displaystyle w(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\theta}\log\left(1+\frac{\theta\cdot(q(t)-c)^{+}}{\mu
c}\right).$ (5)

Moreover, as $t\to\infty$ we have that when $\lambda>\mu c$

$\displaystyle\lim_{t\to\infty}w(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\theta}\log\left(\frac{\lambda}{\mu c}\right).$

###### Proof.

See Theorem 6 of Massey and Pender [31]. ∎

This result shows that the limiting virtual waiting time is a function of the
fluid queue length function $q(t)$. When the fluid queue length is less than
the number of servers $c$, then the virtual waiting time is equal to zero.
Moreover, when the fluid queue length is greater than the number of servers
$c$, then the virtual waiting time is positive. What is more important is that
the fluid limit for the virtual waiting time yields a deterministic function
of time. Thus, we are able to approximate the virtual waiting time with a non-
random function of time. Now that we have a limiting expression for the
virtual waiting time, we can define the scaled overlap process as follows

$\displaystyle
O^{\eta}(t)=N^{\eta}\left(t+\mathcal{S}+W^{\eta}(t)\right)-N^{\eta}(t)+Q^{\eta}(t).$
(6)

We will use the scaled overlap process to prove our main result of the paper,
which gives us a deterministic approximation for the transient mean number of
overlaps in the Erlang-A queue.

######  Theorem 3.2.

Let $O^{\eta}(t)$ be the scaled number of people that a customer who arrives
at time $t$ will overlap with. Then, we have

$\displaystyle\lim_{\eta\to\infty}\frac{1}{\eta}\mathbb{E}\left[O^{\eta}(t)\right]$
$\displaystyle=$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda}{\theta}\log\left(1+\frac{\theta\cdot(q(t)-c)^{+}}{\mu
c}\right)+q(t).$

where $q(t)$ is the solution to Equation (1). Moreover, when $t\to\infty$ we
have that

$\displaystyle\lim_{t\to\infty}\lim_{\eta\to\infty}\frac{1}{\eta}\mathbb{E}\left[O^{\eta}(t)\right]$
$\displaystyle=$ $\displaystyle\begin{cases}\frac{2\lambda}{\mu},\mathrm{if}\
\lambda\leq\mu c\\\
\frac{\lambda}{\mu}+\frac{\lambda}{\theta}\log\left(\frac{\lambda}{\mu
c}\right)+c+\frac{\lambda-\mu c}{\theta},\mathrm{if}\ \lambda>\mu
c.\end{cases}$ (7)

###### Proof.

Based on Equation (6) and Theorem 3.1, we have

$\displaystyle\lim_{\eta\to\infty}\frac{1}{\eta}\mathbb{E}\left[O^{\eta}(t)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\eta}\mathbb{E}\left[\left(N^{\eta}\left(t+\mathcal{S}+W^{\eta}(t)\right)-N^{\eta}(t)\right)\right]+\frac{1}{\eta}\mathbb{E}\left[Q^{\eta}(t)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\eta}\left(\eta\lambda\mathbb{E}\left[\mathcal{S}\right]+\eta\lambda\mathbb{E}\left[W^{\eta}(t)\right]\right)+q(t)$
$\displaystyle=$ $\displaystyle\frac{\lambda}{\mu}+\lambda w(t)+q(t)$
$\displaystyle=$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda}{\theta}\log\left(1+\frac{\theta\cdot(q(t)-c)^{+}}{\mu
c}\right)+q(t).$

In particular, when we look at steady state, we have that when $\lambda\leq\mu
c$,

$\displaystyle\lim_{t\to\infty}\lim_{\eta\to\infty}\frac{1}{\eta}\mathbb{E}\left[O^{\eta}(t)\right]$
$\displaystyle=$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda}{\theta}\log\left(1+\frac{\theta\cdot(q(\infty)-c)^{+}}{\mu
c}\right)+q(\infty)$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda}{\theta}\log\left(1+\frac{\theta\cdot\left(\frac{\lambda}{\mu}-c\right)^{+}}{\mu
c}\right)+\frac{\lambda}{\mu}$ $\displaystyle=$
$\displaystyle\frac{2\lambda}{\mu}.$

When $\lambda>\mu c$, we have

$\displaystyle\lim_{t\to\infty}\lim_{\eta\to\infty}\frac{1}{\eta}\mathbb{E}\left[O^{\eta}(t)\right]$
$\displaystyle=$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda}{\theta}\log\left(1+\frac{\theta\cdot(q(\infty)-c)^{+}}{\mu
c}\right)+q(\infty)$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda}{\theta}\log\left(\frac{\lambda}{\mu
c}\right)+c+\frac{\lambda-\mu c}{\theta}.$

This completes the proof. ∎

It is important to note that the expectation for Theorem 3.2 is necessary.
Without the expectation the overlap process would explicitly depend on
$\mathcal{S}$ and would be random. It is also worth noting that in the case
$\lambda\leq\mu c$, the steady state mean number of overlaps is equal to the
steady state mean number of overlaps in the infinite server queue as well. It
also does not depend on the staffing level $c$ since it behaves like an
infinite server in the steady state setting.

As we stated earlier, the Erlang-A model is a generalization of multiple
queueing system. We now discuss several special cases of the Erlang-A model in
the following based on our results in Theorem 3.2.

######  Remark 1.

(M/M/$\infty$ system): When $c\rightarrow\infty$, all the customers will enter
the service directly upon arrival. The system with constant arrival and
service rates becomes the M/M/$\infty$ system. We find from Equation (7) that
the number of overlapped customers will converge to $\frac{2\lambda}{\mu}$.
This result matches our analysis in [48]. It is important to distinguish
between letting $c\to\infty$ and letting $\theta\to\mu$. Notably, as $c$
approaches infinity, the waiting time converges to zero. Conversely, when
$\theta$ tends towards $\mu$, the waiting time does not approach zero.
Consequently, although the queue lengths are identical and share the same
sample path structure, the expected number of overlaps differs between the
systems. This observation of the two different systems emphasizes that the
number of overlaps is dependent on the customer’s experience instead of only
the queue length process. In particular, the customers in the case where
$c\to\infty$ all have the same experience in the system, however, in the case
where $\theta\to\mu$, abandoning customers have a different overlap experience
than those who wait in the queue and then get served. Finally, it is also
worth noting that the diffusion limits of the two systems are different, see
for example page 167 of Mandelbaum et al. [29]. This further highlights the
difference in the number of overlaps experienced by customers in the two
systems.

######  Remark 2.

(Erlang-B system): When letting the abandonment rate
$\theta\rightarrow\infty$, we have the Erlang-B system. According to Theorem
3.2, the expected number of people that the virtual customer overlaps in the
steady state is $\frac{2\lambda}{\mu}$ when $\lambda\leq\mu c$, and is
$\frac{\lambda}{\mu}+c$ when $\lambda>\mu c$. The reason is that no customer
will wait in the queue, so the virtual customer will always enter the service
directly. According to Corollary 2.3, the virtual customer will overlap with
$\frac{\lambda}{\mu}$ customers in the system when $\lambda\leq\mu c$, and $c$
customers when $\lambda>\mu c$. Moreover, during the service time, the virtual
customer is expected to overlap with $\frac{\lambda}{\mu}$ newly arrived
customers. Note that although the abandonment rate is infinity, we still
assume that every customer will arrive at the system first before abandon,
which explains why the virtual customer will overlap with new arrivals.

######  Remark 3.

(Erlang-C system): When shutting off the abandonment process, we have the
Erlang-C system. Based on Equation (7), the number of overlapped customers in
the fluid limit becomes $2\frac{\lambda}{\mu}$ for $\lambda\leq\mu c$. The
reason is that the fluid limit of the virtual waiting time becomes 0 in the
steady state, according to Theorem 3.1. In the fluid limit, the virtual
customer will overlap with $\frac{\lambda}{\mu}$ customers upon arrival, and
other $\frac{\lambda}{\mu}$ customers during service.

The transient mean queue length and wait time are not known in closed form.
However, if we condition on the queue length, the waiting time distribution is
known in closed form for the Erlang-A model. By conditioning on the number of
customers ahead of you given that you are waiting, it is easily seen that the
waiting time has a hypoexponential distribution i.e.,

$\displaystyle W_{k}=\sum^{k}_{j=0}Y_{j}$

where $Y_{j}\sim\mathrm{Exp}(\mu c+\theta\cdot j)$ and $k$ is the number of
customers that are ahead of the customer upon the arrival. Note that $j$ is
allowed to be zero when the queue length is identical to the number of servers
since in this case, the customer needs to wait an exponential amount of time
with rate $\mu c$. With this hypoexponential representation, we can compute
the conditional mean and variance of the waiting time. The conditional mean
waiting time is

$\displaystyle\mathbb{E}\left[W_{k}\right]$ $\displaystyle=$
$\displaystyle\sum^{k}_{j=0}\frac{1}{\mu c+\theta\cdot j}$ (8)
$\displaystyle=$
$\displaystyle\frac{1}{\theta}\sum^{k}_{j=0}\frac{1}{\frac{\mu c}{\theta}+j}$
$\displaystyle=$ $\displaystyle\frac{1}{\theta}\left(\psi\left(\frac{\mu
c}{\theta}+k+1\right)-\psi\left(\frac{\mu c}{\theta}\right)\right)$

where $\psi(x)$ is the digamma function. Moreover, the variance is

$\displaystyle\mathrm{Var}\left[W_{k}\right]$ $\displaystyle=$
$\displaystyle\sum^{k}_{j=0}\frac{1}{(\mu c+\theta\cdot j)^{2}}$ (9)
$\displaystyle=$
$\displaystyle\frac{1}{\theta^{2}}\sum^{k}_{j=0}\frac{1}{\left(\frac{\mu
c}{\theta}+j\right)^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\theta^{2}}\left(\psi^{(1)}\left(\frac{\mu
c}{\theta}\right)-\psi^{(1)}\left(\frac{\mu c}{\theta}+k+1\right)\right)$

where $\psi^{(1)}(x)$ is the trigamma function. One should note that both the
digamma and the trigamma functions are special cases of the the Hurwitz-
Riemann zeta function defined as

$\displaystyle\zeta(s,\alpha)=\sum^{\infty}_{j=0}\frac{1}{(j+\alpha)^{s}}.$

Now that we have explicit expressions for the mean and variance of the waiting
time given that there are $k$ customers in front of the current arrival, we
should be able to leverage the fluid limits for the queue length process to
approximate the mean and variance and of the waiting time at any time $t$.

We know that a first order Taylor expansion around the mean is given by the
following expression

$\displaystyle f(Q(t))$ $\displaystyle\approx$ $\displaystyle
f\left(\mathbb{E}\left[Q(t)\right]\right)+f^{\prime}\left(\mathbb{E}\left[Q(t)\right]\right)\cdot\left(Q(t)-\mathbb{E}\left[Q(t)\right]\right).$

Thus, the mean of the function can be approximated by

$\displaystyle\mathbb{E}\left[f(Q(t))\right]$ $\displaystyle\approx$
$\displaystyle f\left(\mathbb{E}\left[Q(t)\right]\right)\approx
f\left(q(t)\right)$

and the variance of the function can be approximated by

$\displaystyle\mathrm{Var}\left[f(Q(t))\right]$ $\displaystyle\approx$
$\displaystyle
f^{\prime}\left(\mathbb{E}\left[Q(t)\right]\right)^{2}\cdot\mathrm{Var}\left[Q(t)\right]\approx
f^{\prime}\left(q(t)\right)^{2}\cdot v(t).$

Finally, it is also important to compute an approximation for the covariance
as well since we will need it later for computing the variance of the number
of overlaps in the Erlang-A queue.

$\displaystyle\mathrm{Cov}\left[f(Q(t)),Q(t)\right]$ $\displaystyle\approx$
$\displaystyle\mathrm{Cov}\left[f(\mathbb{E}\left[Q(t)\right])+f^{\prime}(\mathbb{E}\left[Q(t)\right])(Q(t)-\mathbb{E}\left[Q(t)\right]),Q(t)\right]$
$\displaystyle=$ $\displaystyle
f^{\prime}(\mathbb{E}\left[Q(t)\right])\cdot\mathrm{Var}\left[Q(t)\right]$
$\displaystyle\approx$ $\displaystyle f^{\prime}(q(t))\cdot v(t).$

It is worth mentioning that the covariance result is well known in physics as
the linear noise approximation. We will show in the sequel how to use these
approximations for estimating the mean and variance of the waiting time for
the Erlang-A queue. We will also find that these approximations of the waiting
time are essential for computing the mean and variance of the number of
overlaps in the Erlang-A queue as well.

By using a first order Taylor expansion around the mean queue we obtain the
following expression as an approximation for the mean waiting time

$\displaystyle\mathbb{E}\left[W(t)\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\frac{1}{\theta}\left(\psi\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right)-\psi\left(\frac{\mu
c}{\theta}\right)\right)\right]$ (10) $\displaystyle=$
$\displaystyle\frac{1}{\theta}\mathbb{E}\left[\left(\psi\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right)\right)\right]-\frac{1}{\theta}\psi\left(\frac{\mu
c}{\theta}\right)$ $\displaystyle\approx$
$\displaystyle\frac{1}{\theta}\left(\psi\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)-\psi\left(\frac{\mu c}{\theta}\right)\right).$

Moreover, a first order Taylor expansion also yields the following
approximation for the variance of the waiting time

$\displaystyle\mathrm{Var}\left[W(t)\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\mathrm{Var}\left[W(t)\mid
Q(t)\right]\right]+\mathrm{Var}\left[\mathbb{E}\left[W(t)\mid
Q(t)\right]\right]$ (11) $\displaystyle=$
$\displaystyle\mathbb{E}\left[\frac{1}{\theta^{2}}\left(\psi^{(1)}\left(\frac{\mu
c}{\theta}\right)-\psi^{(1)}\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right)\right)\right]$ $\displaystyle+$
$\displaystyle\mathrm{Var}\left[\frac{1}{\theta}\left(\psi\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right)-\psi\left(\frac{\mu
c}{\theta}\right)\right)\right]$ $\displaystyle=$
$\displaystyle\frac{1}{\theta^{2}}\left(\psi^{(1)}\left(\frac{\mu
c}{\theta}\right)-\mathbb{E}\left[\psi^{(1)}\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right)\right]\right)$ $\displaystyle+$
$\displaystyle\frac{1}{\theta^{2}}\mathrm{Var}\left[\psi\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right)\right]$ $\displaystyle\approx$
$\displaystyle\frac{1}{\theta^{2}}\left(\psi^{(1)}\left(\frac{\mu
c}{\theta}\right)-\mathbb{E}\left[\psi^{(1)}\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right)\right]\right)$ $\displaystyle+$
$\displaystyle\frac{1}{\theta^{2}}\cdot\psi^{(1)}\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)^{2}\cdot\mathrm{Var}\left[Q(t)\right]$
$\displaystyle\approx$
$\displaystyle\frac{1}{\theta^{2}}\left(\psi^{(1)}\left(\frac{\mu
c}{\theta}\right)-\psi^{(1)}\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)\right)$ $\displaystyle+$
$\displaystyle\frac{1}{\theta^{2}}\cdot\psi^{(1)}\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)^{2}\cdot\\{q(t)>c\\}\cdot v(t).$

Note that we have replaced the value $k$ in the conditional mean and variance
formulas with the fluid limit queue length at time $t$.

###  3.2 The Transient Variance of the Number of Overlaps

In addition to the mean, we are also interested in approximating the variance
of the number of overlaps. With the variance, we are able to understand the
variation around our approximations of the mean. This implies that we can
construct prediction intervals for the number of overlaps one might expect at
any time $t$. We know from Equation (4) that the overlap process satisfies the
following equation

$\displaystyle O(t)=N\left(t+\mathcal{S}+W(t)\right)-N(t)+Q(t).$

Thus, the variance of the number overlap is equal to

$\displaystyle\mathrm{Var}[O(t)]$ $\displaystyle=$
$\displaystyle\mathrm{Var}\left[N\left(t+\mathcal{S}+W(t)\right)-N(t)+Q(t)\right]$
$\displaystyle=$
$\displaystyle\mathrm{Var}\left[N\left(t+\mathcal{S}\right)-N(t)+N\left(t+\mathcal{S}+W(t)\right)-N\left(t+\mathcal{S}\right)+Q(t)\right]$
$\displaystyle=$
$\displaystyle\mathrm{Var}\left[N\left(t+\mathcal{S}\right)-N(t)\right]+\mathrm{Var}\left[N\left(t+\mathcal{S}+W(t)\right)-N\left(t+\mathcal{S}\right)+Q(t)\right]$
$\displaystyle=$
$\displaystyle\mathrm{Var}\left[N\left(t+\mathcal{S}\right)-N(t)\right]+\mathrm{Var}\left[N\left(W(t)\right)+Q(t)\right]$
$\displaystyle=$
$\displaystyle\mathrm{Var}\left[N\left(t+\mathcal{S}\right)-N(t)\right]+\mathrm{Var}\left[N\left(W(t)\right)\right]+\mathrm{Var}\left[Q(t)\right]$
$\displaystyle+$ $\displaystyle
2\mathrm{Cov}\left[N\left(W(t)\right),Q(t)\right]$ $\displaystyle=$
$\displaystyle\lambda\mathbb{E}\left[\mathcal{S}\right]+\lambda\mathbb{E}\left[W(t)\right]+\lambda^{2}\mathrm{Var}[\mathcal{S}]+\lambda^{2}\mathrm{Var}[W(t)]+\mathrm{Var}\left[Q(t)\right]$
$\displaystyle+$ $\displaystyle 2\lambda\mathrm{Cov}\left[W(t),Q(t)\right]$
$\displaystyle=$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda^{2}}{\mu^{2}}+\lambda\mathbb{E}\left[W(t)\right]+\lambda^{2}\mathrm{Var}[W(t)]+\mathrm{Var}\left[Q(t)\right]$
$\displaystyle+$ $\displaystyle 2\lambda\mathrm{Cov}\left[W(t),Q(t)\right].$

The exact variance of both $W(t)$ and $Q(t)$ and their covariance are unknown.
Fortunately, because of the fluid and diffusion limits, we have approximations
for the mean and variance of the queue length and waiting times. Thus, it
remains for us to derive a transient approximation for the covariance between
the waiting time and the queue length at time $t$. In order to compute the
covariance of the two processes, we use a conditioning argument based on the
queue length, which is natural given the conditional mean waiting time formula
of Equation (8). We outline this argument below.

$\displaystyle\mathrm{Cov}\left[W(t),Q(t)\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[W(t)\cdot
Q(t)\right]-\mathbb{E}\left[W(t)\right]\cdot\mathbb{E}\left[Q(t)\right]$
$\displaystyle=$
$\displaystyle\sum^{\infty}_{k=0}k\cdot\mathbb{E}\left[W(t)|Q(t)=k\right]\cdot\mathbb{P}\left(Q(t)=k\right)-\mathbb{E}\left[W(t)\right]\cdot\mathbb{E}\left[Q(t)\right]$
$\displaystyle=$
$\displaystyle\sum^{\infty}_{k=c}k\cdot\left(\sum^{k-c}_{j=0}\frac{1}{\mu
c+\theta\cdot
j}\right)\cdot\mathbb{P}\left(Q(t)=k\right)-\mathbb{E}\left[W(t)\right]\cdot\mathbb{E}\left[Q(t)\right]$
$\displaystyle=$
$\displaystyle\sum^{\infty}_{k=c}k\cdot\left(\psi\left(\frac{\mu
c}{\theta}+k-c\right)-\psi\left(\frac{\mu
c}{\theta}\right)\right)\cdot\mathbb{P}\left(Q(t)=k\right)-\mathbb{E}\left[W(t)\right]\cdot\mathbb{E}\left[Q(t)\right]$
$\displaystyle=$ $\displaystyle\mathbb{E}\left[Q(t)\left(\psi\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right)-\psi\left(\frac{\mu
c}{\theta}\right)\right)\right]-\mathbb{E}\left[W(t)\right]\cdot\mathbb{E}\left[Q(t)\right]$
$\displaystyle=$ $\displaystyle\mathrm{Cov}\left[\psi\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right)-\psi\left(\frac{\mu
c}{\theta}\right),Q(t)\right]$ $\displaystyle=$
$\displaystyle\mathrm{Cov}\left[\psi\left(\frac{\mu
c}{\theta}+(Q(t)-c)^{+}\right),Q(t)\right]$ $\displaystyle\approx$
$\displaystyle\mathrm{Var}\left[Q(t)\right]\cdot\psi^{(1)}\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)\cdot\\{q(t)>c\\}$ $\displaystyle\approx$
$\displaystyle v(t)\cdot\psi^{(1)}\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)\cdot\\{q(t)>c\\}.$

Using this result, we can approximate the covariance between the waiting time
and the queue length as the product of the variance and the trigamma function,
which is evaluated at the fluid limit. By using this approximation, we arrive
at the following formula for approximating the transient variance of the
number of overlaps.

$\displaystyle\mathrm{Var}[O(t)]$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda^{2}}{\mu^{2}}+\lambda\mathbb{E}\left[W(t)\right]+\lambda^{2}\mathrm{Var}[W(t)]+\mathrm{Var}\left[Q(t)\right]+2\lambda\mathrm{Cov}\left[W(t),Q(t)\right]$
(12) $\displaystyle\approx$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda^{2}}{\mu^{2}}+\frac{\lambda}{\theta}\left(\psi\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)-\psi\left(\frac{\mu
c}{\theta}\right)\right)+v(t)$ $\displaystyle+$
$\displaystyle\frac{\lambda^{2}}{\theta^{2}}\left(\psi^{(1)}\left(\frac{\mu
c}{\theta}\right)-\psi^{(1)}\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)\right)$ $\displaystyle+$
$\displaystyle\frac{\lambda^{2}}{\theta^{2}}\cdot\psi^{(1)}\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)^{2}\cdot\\{q(t)>c\\}\cdot v(t)$
$\displaystyle+$ $\displaystyle 2\lambda v(t)\cdot\psi^{(1)}\left(\frac{\mu
c}{\theta}+(q(t)-c)^{+}\right)\cdot\\{q(t)>c\\}.$

We can approximate the steady variance by setting $t\to\infty$. Thus, when
$\lambda>\mu c$

$\displaystyle\lim_{t\to\infty}\mathrm{Var}[O(t)]$ $\displaystyle\approx$
$\displaystyle\frac{\lambda}{\mu}+\frac{\lambda^{2}}{\mu^{2}}+\frac{\lambda}{\theta}\left(\psi\left(\frac{\lambda}{\theta}\right)-\psi\left(\frac{\mu
c}{\theta}\right)\right)+\frac{\lambda}{\theta}+\frac{\lambda^{3}}{\theta^{3}}\psi^{(1)}\left(\frac{\lambda}{\theta}\right)^{2}$
$\displaystyle+$
$\displaystyle\frac{\lambda^{2}}{\theta^{2}}\left(\psi^{(1)}\left(\frac{\mu
c}{\theta}\right)-\psi^{(1)}\left(\frac{\lambda}{\theta}\right)\right)+2\frac{\lambda^{2}}{\theta}\cdot\psi^{(1)}\left(\frac{\lambda}{\theta}\right)$

and when $\lambda<\mu c$

$\displaystyle\lim_{t\to\infty}\mathrm{Var}[O(t)]$ $\displaystyle\approx$
$\displaystyle\frac{2\lambda}{\mu}+\frac{\lambda^{2}}{\mu^{2}}.$

It is important to note that the last equation is precisely the same variance
of the infinite server queue setting. Thus, when the queue is underloaded, the
number of overlaps behaves similarly to an infinite server queue.

###  3.3 Numerical Experiments

To understand how our approximations for the mean queue length and the mean
number of overlapping customers perform, we present eight different numerical
examples below. Before we provide the examples, we list the parameter values
for each of the examples in Table 1. In all simulation examples, we simulate
the sample paths 10,000 times to produce each curve.

Parameters | $\lambda$ | $\mu$ | $\theta$ | $C$ | $Q(0)$
---|---|---|---|---|---
Figure (a) | 10 | 1 | 0.5 | 30 | 10
Figure (b) | 10 | 1 | 0.5 | 30 | 50
Figure (c) | 10 | 1 | 2 | 30 | 10
Figure (d) | 10 | 1 | 2 | 30 | 50
Figure (e) | 40 | 1 | 0.5 | 30 | 10
Figure (f) | 40 | 1 | 0.5 | 30 | 50
Figure (g) | 40 | 1 | 2 | 30 | 10
Figure (h) | 40 | 1 | 2 | 30 | 50
Table 1: Parameters for Examples

In Figure 1, we present a plot of the simulated mean queue length, which has
been approximated by employing the fluid limit as expressed in Equation (1),
denoted as “analytical” in the figures. Our findings indicate that across
various parameter settings outlined in Table 1, the fluid approximation
consistently and accurately estimates the mean dynamics when juxtaposed with
the simulated values. This observation underscores the reliability and
effectiveness of the fluid approximation in accurately characterizing the
behavior of the mean queue length.

In Figure 2, we plot the standard deviation of the queue length for the
parameters outlined in Table 1. We observe that the approximation provided by
the diffusion variance, as given in Equation (2), consistently performs well
at approximating the corresponding simulated values, for all of the parameter
values. Thus, this illustrates the accuracy of the diffusion variance in
effectively approximating the dynamic behavior of the queue length standard
deviation. Moreover, with good approximations for the standard deviation, it
becomes feasible to construct prediction intervals for the queue length. Such
prediction intervals serve as valuable tools for assessing the range within
which the actual queue length is likely to fall, providing a measure of
confidence in our approximations.

In Figure 3, we plot the simulated mean virtual waiting time with two
different approximations. The first approximation, denoted as “analytical 1”
in the figures, is derived from the fluid limit, as outlined in Theorem 3.1
and is given in Equation (5). The second approximation, denoted as “analytical
2” in the figure, is given by Equation (10), which depends on the digamma
function and the fluid queue length. We observe that across all parameter
settings provided in Table 1, the approximation employing the digamma function
consistently outperforms the fluid-based version in estimating the virtual
waiting time.

In Figure 4, we plot the simulated standard deviation of the virtual waiting
time, utilizing an approximation derived from the trigamma function as given
in Equation (11). The derivation of this approximation involved a Taylor
expansion of the variance and the utilization of conditional mean and variance
formulas for the wait time, as outlined in Equations (8) and (9). Our
observations reveal that regardless of the parameter settings examined in
Table 1, the trigamma function consistently demonstrates excellent performance
in approximating the standard deviation of the virtual waiting time. This
consistency and accuracy highlight the reliability and effectiveness of the
trigamma-based approximation method.

Moving on to Figure 5, we explore the mean number of overlapping customers as
a function of time. It is noteworthy that the approximations presented in
Theorem 3.2, denoted as “analytical” in the figures, show remarkable accuracy
across all parameter values considered. The high level of precision maintained
by these approximations reinforces their reliability and usefulness in
practical settings.

Figure 6 plots the standard deviation of the number of overlapping customers
over time for the aforementioned eight examples in Table 1. Our investigation
reveals that the two variance approximations presented in Equation (2)
(“analytical 1” in the figures) and Equation (12) (“analytical 2” in the
figures) both exhibit strong performance across all parameter values. The
approximation based on the fluid and diffusion limits, as outlined in
Equations (1) and (2), consistently outperforms the approximation given by
Equation (12). Consequently, with these robust approximations at our disposal,
it becomes feasible to construct reliable prediction intervals for determining
the customers that a virtual customer may overlap with.

By extending and refining our analysis using advanced approximation techniques
derived from trigamma functions and fluid and diffusion limits, we have
established a solid foundation for predicting and understanding various
aspects of the virtual waiting time and customer overlap dynamics. These
findings have significant implications for enhancing the efficiency and
effectiveness of virtual customer service systems in a number of practical
applications.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 1: Fluid Mean Number in System vs. Simulation.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 2: Standard Deviation Number of Customers (Analytical vs. Simulation).

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 3: Mean Virtual Waiting Time (Analytical vs Simulation).

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 4: Standard Deviation of Virtual Waiting Time (Analytical vs.
Simulation).

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 5: Mean Number of Overlapping Customers (Analytical vs. Simulation).

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 6: Standard Deviation of Number of Overlapping Customers (Analytical
vs. Simulation).

##  4 Conclusion

In this paper, we present a novel analysis of the mean and variance of the
number of overlaps in the Erlang-A queue. Our contribution extends the current
literature by considering both abandonment and a finite number of servers,
thus providing a more realistic model. To achieve this, we employ a
methodology based on the fluid and diffusion differential equations introduced
by Mandelbaum et al. [29, 28], Massey and Pender [31]. Specifically, we derive
exact expressions for these equations using the theory of linear differential
equations. Moreover, we utilize these exact expressions to approximate the
number of overlaps for a virtual customer that will not abandon. Our results
show that our fluid and diffusion-based approximations offer reliable
estimates of the mean and variance of the number of overlapping customers in
the Erlang-A queue.

As a side result, we also present new approximations for the mean and variance
of the waiting time in the Erlang-A queue. Notably, our approximations are
functions of the digamma and trigamma functions, respectively. These
approximations offer a significant improvement over existing results and can
be used to enhance the performance of queueing systems in practical
applications. Overall, our work contributes to a better understanding of the
behavior of the Erlang-A queue and provides useful tools for its analysis in
real-world scenarios.

This work opens up several potential avenues for future research that could be
valuable to pursue. Firstly, we could explore a more general queueing model
with abandonment, such as the $G/G/C+G$ queue. Although some limit theorems
exist for this model Liu and Whitt [25, 26], the analysis of the virtual
waiting time and its relationship with the queue length is currently
unavailable. A thorough investigation of this relationship would provide
insights into how the generality of the arrival, service, and abandonment
processes could impact the number of overlaps.

Additionally, we could consider other types of queueing models, such as
multidimensional network queueing models like those of Liu and Whitt [24, 27],
Pender and Massey [44], batch queueing models like those explored in Pang and
Whitt [40], Daw and Pender [8], Daw et al. [11], and even models with self-
exciting arrivals Koops et al. [23], Daw and Pender [7], Daw et al. [10].
These extensions would offer further opportunities to investigate the impact
of various system parameters on the number of overlaps and the waiting time
and they represent promising avenues for future research that we intend to
pursue.

## Acknowledgements

Jamol Pender would like to acknowledge the gracious support of the National
Science Foundation DMS Award # 2206286. This work was also supported in part
by the National Research Foundation of Korea (NRF) grants (No.
2021R1A2C1094699 and 2021R1A4A1031019) funded by the Korea government
(Ministry of Science and ICT, MSIT).

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