Uncertainty in modeling overdispersed queues

-An analysis of scaling limits for infinite-server queues in a random environment.

Mariska Heemskerk

Supervisors: prof. dr. Michel Mandjes & prof. dr. Johan van Leeuwaarden

June 25, 2015

Korteweg-de Vries Institute for Mathematics

Faculty of Science

University of Amsterdam

Contents

1 Introduction 4

2 Overdispersion in an infinite-server context 9

2.1 Pre-limit results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Limit results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Model with correlated arrival streams: functional central limit theorem 17

3.1 Qualitative behavior of first two moments . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Functional central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Large deviations 26

4.1 Univariate large deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Large deviations for the coupled model . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Conclusions and outlook 33

5.1 Alternative models using mixed Poisson arrivals . . . . . . . . . . . . . . . . . . . 33

5.2 Correlated queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3 General service times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Abstract

Large-scale service systems are typically modeled using a multi-server system with a nonho-

mogeneous Poisson arrival process (NHPP). Empirical studies provide statistical evidence

that actual arrival processes in call centers and healthcare often fluctuate more than the

NHPP-assumption can capture. This calls for arrival processes and hence queueing systems

that model overdispersion. To this end, we introduce a mixed Poisson arrival process in a

random environment. Studying this arrival process in the context of an infinite-server queue,

we establish multiple stochastic-process limits through asymptotic analysis. This gives rise

to a trichotomy in system behavior, in which different regimes arise depending on a delicate

interplay between the overdispersion caused by the mixed Poisson distribution and the rate

at which the random environment changes.

1 Introduction

Queueing theory was developed in order to solve capacity-sizing problems for service systems [7],

a type of stochastic networks consisting of a combination of installed work-processing resources

and typically a large number, say N , of work-generating sources. Each source generates arrivals

at a certain arrival rate. Examples of service system usage include modeling service operations

in call centers and healthcare. The capacity-sizing problem is to determine the optimal capacity

to guarantee a certain desired system performance (here the capacity is defined as the number

of servers). The required capacity typically increases with the level of uncertainty in the system.

In service system modeling, as in stochastic modeling in general, one can distinguish three types

of uncertainty [14]. First, there is uncertainty about the choice of the probabilistic model that

is used to approximate reality. This is referred to as model uncertainty. Given that the model

is appropriate, there can still be uncertainty concerning the parameters of the model: parameter

uncertainty. Finally, of course, once the model and parameters are set, intrinsic stochastic

variability provides for uncertainty of the outcomes of a process.

Taking into account the economies-of-scale principle, service systems are typically modeled using

many-server queues, where the canonical models are M/M/s and M/M/∞ systems with Poisson

arrivals, exponentially distributed service times and s or infinitely many servers, respectively.

In this thesis, we consider infinite-server queues, as this will make the analysis of our rather

complicated setting tractable. As observed by Whitt [13], the infinite-server assumption is natural

when the goal to achieve is to immediately serve all jobs. Besides, using techniques as PSA and

MOL, first-order insights in system behavior in a finite-server setting can be obtained afterwards

[15].

In the majority of studies, it is assumed that the arrival rate of the combined work-generating

sources, the so-called total arrival rate∑N

i=1 λi = Nλ, is a known parameter that can be extracted

from the data as the mean number of arrivals per time unit [2]. Consequently, the arrivals of the

N sources are considered as one single process, the arrival process, the arrival rate of which equals

the total arrival rate. Given an M/M/∞ system, the assumption of independent sources (the

individual arrival processes are independent processes) results in a Poisson arrival process with

mean and variance both equal to Nλ. However, recent studies suggest that this independence

assumption seems invalid: there is growing evidence that in many practical cases sources are

highly correlated [2]. For instance, it is observed that the number of arrivals fluctuate more than

we would expect from a Poisson process [12, 10, 4]. Given a non-negative random variable X,

the index of dispersion or the variance-to-mean ratio is

VMRX =VarX

EX.

Under the standard Poisson assumption the index of dispersion of the number of arrivals in a

time unit equals 1, whereas in other models it can be larger (smaller) than 1; this phenomenon

is referred to as overdispersion (underdispersion).

4

Overdispersion was observed in all kinds of data sets, for instance in call center data [4] and

hospital arrivals [9]. As this violates the Poisson assumption, it renders results as the widely

used square-root safety capacity prescription deficient for implementation. Indeed, Zeevi et al.

[2] showed that this capacity rule performs poorly when handling overdispersed data. However,

they observe that in practice as well as in literature, forecast errors due to inadequate capacity

prescriptions are often ascribed to stochastic variability. Of course, in the event of significant

overdispersion, such malfunctioning should be explained by another type of uncertainty instead.

Only recently, researchers have begun to recognize the urge to add extra built-in uncertainty

to service system models. As first suggested by Whitt [13], overdispersion can be explained by

parameter uncertainty, which actually could be viewed (but not necessarily) as a manifestation

of correlation between sources. The simplest way to implement parameter uncertainty is using

mixed Poisson arrivals; for related work, see e.g. [8, 10, 11] and references therein.

B Parameter uncertainty as a manifestation of correlated sources

When modeling an arrival process with overdispersed arrivals from a many-sources perspective

while holding on to the paradigm of an M/M/∞ system, it can be concluded from the above

that we should implement correlation between sources. This can be done via a random arrival

rate: instead of λ, the sources all have random arrival rates sampled from the distribution of

some random variable Λ, such that an underlying random mechanism determines the rates for all

sources. In more general language: parameter uncertainty causes extra variability in the arrival

process, rendering it a useful implementation when modeling dispersed data.

To understand the relation between parameter uncertainty and correlated sources, note that

correlation can be interpreted as the effect of an underlying random mechanism that influences

multiple processes. As before, suppose that there is one arrival rate for all sources (λi = λ for

i ∈ 1, . . . , N), which is called the state of the total arrival process. The idea behind a random

arrival rate is that λ is replaced by a random variable Λ such that EΛ = λ: an underlying random

mechanism determines the state.

In previous work, overdispersion is often modeled using so-called mixed Poisson arrivals. As we

will show, this construction leads to a level of overdispersion proportional to the size of the system

and the variance-to-mean ratio of Λ. Consider the following extension of a Poisson process.

Definition 1.1. Let Λ : Ω1 → R a non-negative random variable, absolutely continuous with

density fΛ(λ). Then XΛ = (XΛt )t∈T is a mixed Poisson process if conditional on the value of

Λ, XΛ is a Poisson process. Then XΛt ∼ mPoisson(Λt), i.e. the distribution of XΛ

t is mixed

Poisson and defined via:

P(XΛt = n) = EΛ

[(Λt)nn!

e−Λt]

=

∫ ∞0

(λt)n

n!e−λtfΛ(λ) dλ.

5

Let PΛi denote the mixed Poisson arrival process for source i, with random parameter Λ. When

the PΛi all respond to the same realization of Λ, the total arrival process P :=

∑Ni=1 P

Λi is mixed

Poisson with parameter NΛ, hence the process is dispersed. Note that the PΛi are correlated

through their parameter

From the system’s perspective, the mean number of mixed Poisson arrivals with random rate

NΛ in 1 time unit is Nλ and its variance is Nλ + N2Var(Λ). Hence for large values of N the

variance is huge compared the mean, and the VMR equals

1 +NVar(Λ)

λ. (1)

This VMR is tunable via Var(Λ), which is an important first step in fitting a model to actual

overdispersed data. We next enrich this mixed Poisson setting with a random environment in

order to establish a more flexible, and more realistic formalism for actual arrival processes.

B Channeling uncertainty via random environment

Mixed Poisson arrivals are overdispersed simply due to the arrival rate being randomly drawn

from the distribution of Λ, thus considerably generalizing the case where the rate is a deterministic

value EΛ = λ. In order to channel parameter uncertainty, we introduce a sampling frequency : not

only is the arrival rate a random unknown value, it also keeps changing over time at a certain pace.

The classical mixed Poisson setting is then enriched by extending the random rate to a random

environment, where a new arrival rate is independently drawn from Λ with frequency 1∆ , for some

∆ > 0. One can imagine that in a rapidly changing random environment, the randomness of Λ

does not play a role anymore, whereas in a slowly changing random environment, the variability

of Λ has a lot of influence on the variance of the arrival process. This trade-off between the

influence of the sampling frequency and parameter uncertainty is a convenient and interesting

device to have in a model. It could be used to tune the model in such a way that the resulting

level of overdispersion fits the data.

Num

bero

f arr

ival

s()

Time ( )Δ 2Δ 3Δ 4Δ0

Figure 1: Mixed Poisson arrivals in a random environment.

6

Figure 1 shows a realization of a mixed Poisson arrival process in a random environment. The

number of arrivals per time unit is locally Poisson and fluctuates proportionally to its local

parameter, independently drawn from the distribution of the random parameter Λ with EΛ = λ.

As the paramater changes every ∆ time units in this random environment, the number of arrivals

per time unit fluctuates more severely than normal Poisson data would (e.g. the sample variance

is larger than the sample mean, which is approximately EΛ = λ).

In order to explain how we extend the standard mixed Poisson construction with a random

environment, let us now define the precise arrival process to be used in our model. Consider a

Markovian arrival process with a random arrival rate that is newly sampled after each ∆ time

units, i.e. at deterministic sampling times reoccuring at frequency 1∆ . Assuming N sources, the

arrival process is an instance of a Cox process [5] with time-dependent arrival rate NΛ∆(s),

defined as

Λ∆(s) :=∑j

Λj1[j∆,(j+1)∆)(s), (2)

with i.i.d. Λj distributed as Λ, absolutely continuous with density fΛ(λ) and finite moments. The

number of arrivals after t time units, denoted by Pt, is mixed Poisson with random parameter

N∫ t

0 Λ∆(s) ds, hence if∫ t

0 Λ∆(s) ds =∑t/∆

j=1 λj∆ (suppose t/∆ ∈ N), then

P(Pt = n) =

∫ ∞0· · ·∫ ∞

0

(N∫ t

0 Λ∆(s) ds)n

n!e−N

∫ t0 Λ∆(s) ds

t/∆∏j=1

fΛ(λj) dλ1 . . . dλt/∆.

Note that on itself, introducing a sampling frequency does not lead to interesting results. For

instance, the mean number of arrivals per time slot is still Nλ and the variance is again Nλ +

N2Var(Λ); even when ∆ < 1 (suppose 1∆ ∈ N),

Var(P1) =1

∆·Var(P∆) +

1

∆(

1

∆− 1)Cov(P 1

∆, P2∆) =

1

∆(Nλ∆ +N2∆2Var(Λ)) +

1

∆(

1

∆− 1)N2∆2Var(Λ)

= Nλ+N2∆Var(Λ) +N2Var(Λ)−N2∆Var(Λ)

= Nλ+N2Var(Λ),

where P 1∆ and P 2

∆ are i.i.d. mixed Poisson random variables with random parameter NΛ∆.

Therefore, the VMR of the number of arrivals per time slot is the same as in (1). Nevertheless,

imposing a scaling on the sampling frequency, the asymptotic behavior of the resulting system

is interesting.

B Asymptotic analysis results

We have thus arrived at a single-class Markovian infinite-server queue with mixed Poisson arrival

governed by a random environment, so that the random arrival rate is refreshed at fixed points

in time. Surprisingly, despite the rather complicated set-up, it turns out to be possible to

7

compute the moments and the probability generating function of the queue length process. Also,

asymptotic analysis of the queue length under space and time scaling is tractable and will expose

powerful results, e.g. (functional) central limit results and results concerning rare events.

The asymptotic analysis is not only tractable, it can also be motivated as follows. Following

the economies-of-scale principle, we consider large systems (N large), rendering the sampling

frequency 1∆ small compared to NVar(Λ), the variance of the random arrival rate. The battle of

the sampling frequency against the variance of Λ is won before it’s fought; the variance will have

a lot of influence on the system’s behavior. In the limit, as N tends to infinity, the frequency

will get infinitely small compared to the system size, which will of course not lead to interesting

behavior. On the contrary, in the event of a relatively high sampling frequency the variance

of Λ has no impact on the system’s behavior. Therefore, we want to control the rate at which

the frequency increases as the system size grows. This is accomplished by scaling 1∆ to Nα

∆ for

some α > 0, where for α < 1 the variance eventually has the upper hand, whereas for α > 1, the

sampling frequency grows faster than the variance of Λ. Choosing α = 1, hence letting the system

size and the sampling frequency grow at the same rate, we find a third regime that combines the

two others. This trichotomy plays an important role in the analysis and will continue to pop up

throughout the different sections.

B Related work

This work is inspired by analyses of standard mixed Poisson arrival processes used to model

overdispersion in e.g. [2, 8, 9, 10, 11]. Here, it should be noted that in these studies, dispersed

data over disjoint time slots are modeled with individual, independent mixed Poisson arrival

processes, whereas we use a continuous dispersed arrival process to model at once data from,

say, a day.

In a way, the random environment adaptation of the standard mixed Poisson process resembles

Markov-modulation in arrival processes, where the process has rate λi whenever the background

process, a continuous-time Markov chain, is in state i. In this thesis, we will follow joint work

of Anderson, Blom and de Turck on Markov-modulated infinite-server queues [1, 3, 6]. Whereas

we scale the sampling frequency, in this setting the jump rate corresponding to the background

Markov process is scaled (sub/super-)linearly as the system size tends to infinity. This leads to

interesting asymptotic behavior; see [1] for the derivation of a functional central limit theorem

related to a generalized OU process, and [3, 6] for a rare-event analysis.

8

2 Overdispersion in an infinite-server context

In this section we present a transient and stationary analysis of a single-class Markovian infinite-

server queue with random environment of sample frequency 1∆ given by Λ(s) = Λ∆(s) as in (2).

Remember that we assumed that the sampled arrival rates are i.i.d. and distributed as a random

variable Λ > 0 with finite first two moments and density fΛ(·). The corresponding service times

are assumed i.i.d. (and in addition independent of the arrival process), exponentially distributed

random variables with mean 1/µ; some results carry over to the case of general i.i.d. service

times, as we indicate below. We particularly focus on asymptotic normality under a specific

parameter scaling.

2.1 Pre-limit results

In this first subsection we analyze the queue’s stationary behavior, in terms of the Laplace trans-

form and the corresponding moments. We then extend these findings to the associated transient

behavior. This subsection presents ‘pre-limit results’; later we impose parameter scalings which

facilitate the derivation of a central limit theorem.

B Transform of stationary behavior

Let M be the random variable associated with the stationary number of customers (also some-

times referred to as ‘jobs’) in the system. We describe this random variable in terms of its pgf

φ(z) := EzM .In the sequel we write pt := e−µt for the probability that a job present at kt is still present at

(k + 1)t and qt := (1 − e−µt)/(µt) for the probability that a job arriving at a uniform epoch in

[kt, (k + 1)t) is still present at (k + 1)t. Denote pt := 1− pt and qt := 1− qt.It is not hard to see that, owing to the fact that we observe the system in stationarity,

φ(z) =

( ∞∑k=0

P(M = k)

k∑`=0

z`(k

`

)p`∆(1− p∆)k−`

)

×

( ∞∑k=0

∫ ∞0

fΛ(λ)e−λ∆ (λ∆)k

k!

k∑`=0

z`(k

`

)q`∆(1− q∆)k−`dλ

)= φ(zp∆ + p∆)gΛ,∆(z), (3)

where gΛ,t(z) is defined as follows, with rt := tqt:

gΛ,t(z) :=

∫ ∞0

exp (−λrt(1− z)) fΛ(λ) dλ = E exp (−Λrt(1− z)) .

Observe that gΛ,t(z) is a pgf of ‘mixed Poisson’ type: conditional on Λ = λ the pgf corresponds

with that of a Poisson random variable with mean λrt.

9

An iteration argument yields

φ(z) =∞∏k=0

gΛ,∆

zpk∆ +k−1∑j=0

p∆pj∆

=∞∏k=0

gΛ,∆

(1− (1− z)pk∆

). (4)

This expression has an insightful interpretation. Note that the stationary number in the system,

M , can be written as the sum of M0,M1,M2, . . ., where Mk represents the number of jobs

that arrived in [−(k + 1)∆,−k∆) that are still present at time 0. Furthermore, observe that

the Mk are independent. It is easy to see that the probability of an arbitrary arriving job in

[−(k + 1)∆,−k∆) (i.e., having arrived at a uniform epoch in this interval) is still in the system

at time 0 with probability∫ ∆

0

1

∆e−µ(k∆+s)ds =

1

µ∆

(e−µk∆ − e−µ(k+1)∆

)=pk∆ − p

k+1∆

µ∆=pk∆p∆

µ∆. (5)

As a consequence,

EzMk =

∞∑`=0

∫ ∞0

fΛ(λ)e−λ∆ (λ∆)`

`!

∑m=0

zm(`

m

)(pk∆p∆

µ∆

)m(1−

pk∆p∆

µ∆

)`−mdλ

=

∫ ∞0

fΛ(λ) exp

(−λµpk∆p∆(1− z)

)dλ,

which, after an easy verification, turns out to equal φk(z) := gΛ,∆(1 − (1 − z)pk∆). The factor

φk(z) can thus be seen as the contribution of those jobs present at time 0 that arrived in the

interval [−(k + 1)∆,−k∆), for k = 0, 1, . . .. Therefore, the pgf φk(z) corresponds to that of a

mixed Poisson random variable, i.e., a Poisson random variable with random parameter

Vk :=Λkµpk∆p∆ = Λkp

k∆r∆,

with Λk the value of the arrival rate in the interval [−(k + 1)∆,−k∆). We conclude that M is

equal in distribution to the sum of a countably infinite number of independent mixed Poisson

random variables with random parameters Vk (and is consequently mixed Poisson itself).

Interestingly, the above argumentation carries over to the infinite-server model with i.i.d. general

service times; say that the jobs have the same distribution as a random variable B. Write Befor the random variable that corresponds to the residual lifetime of a job in the system in

stationarity from an arbitrary time in equilibrium. Then it is readily verified that we have the

following generalization of the probability in (5):∫ ∆

0

1

∆P(B > k∆ + s)ds =

EB∆

ξ(k,∆) with ξ(k,∆) := P(Be ∈ [k∆, (k + 1)∆]).

10

From this, (5) can be recovered for the case that jobs are exponentially distributed (then Beand B are identically distributed). The probability generating function of Mk for general service

times is

EzMk = E exp (−Λk · EB · ξ(k,∆) (1− z)) ,

hence Mk is mixed Poisson with random parameter Vk = Λk ·EB · ξ(k,∆), the expected fraction

of jobs with a residual time long enough to reach zero starting from a uniform epoch in the

interval [−(k + 1)∆,−k∆), for k ∈ 0, 1, . . .. We thus obtain that M is mixed Poisson, too:

EzM = E exp

(−∞∑k=0

Λk · EB · ξ(k,∆)(1− z)

).

Hence M is Poisson with random parameter∑∞

k=0 Vk =∑∞

k=0 Λk · EB · ξ(k,∆), with i.i.d.

Λk. Considering the special case Λk = λ a.s., we see (as expected) that M is Poisson with

(deterministic) parameter λEB.

B First two moments

We now evaluate the first two moments of M . This is an interesting computation in its own

right, but it also provides useful results that we can use when considering this system under a

specific parameter scaling (as we do in the next subsection).

Differentiating (3) and letting z ↑ 1, we obtain a fixed-point equation,

φ′(1) = φ′(1)e−µ∆ + g′Λ,∆(1) = φ′(1)e−µ∆ + r∆ EΛ.

Hence EM = φ′(1) = r∆ EΛ/(1 − e−µ∆) = EΛ/µ. This mean could have been computed more

directly as well, using a standard identity for conditional means:

EM =∞∑k=0

EMk =∞∑k=0

E [E[Mk |Λk]] . (6)

Then observe that (Mk |Λk) is Poisson, and hence its mean equals Vk, its parameter. In the

sequel, we denote vk := EVk. As a result, (6) equals

EM =∞∑k=0

vk =EΛ

µ

∞∑k=0

pk∆p∆ =EΛ

µ.

Note that for general service times, this generalizes to EM = EΛ · EB.

For the variance we use that

φ′′(1) = φ′′(1)p2∆ + 2φ′(1)p∆g

′Λ,∆(1) + g′′Λ,∆(1),

11

and hence

φ′′(1) =2φ′(1) p∆ g

′Λ,∆(1)

1− p2∆

+g′′Λ,∆(1)

1− p2∆

= 2

(EΛ

µ

)2 p∆

1 + p∆+

EΛ2

µ2

1− p∆

1 + p∆.

It thus follows that, after some algebra,

VarM = φ′′(1) + φ′(1)− (φ′(1))2

=EΛ

µ+ C

Var Λ

µ2, (7)

where C := (1− p∆)/(1 + p∆).

Alternatively, one could use the ‘law of total variance’ to identify VarM :

VarM =∞∑k=0

VarMk =∞∑k=0

E[Var(Mk |Λk)] +∞∑k=0

Var(E[Mk |Λk]). (8)

Observe that, because of the ‘mixed Poisson property’, E[Var(Mk |Λk)] = EVk =: vk, and as a

result the first term at the right-hand side of (8) equals EM. The second term, which is inherently

non-negative, gives rise to ‘overdispersion’, i.e., the effect that the variance of the stationary

number in the system exceeds the corresponding mean. This is a distinguishing feature compared

to the analogous system in which the Poissonian arrival rate is deterministic: the stationary

number of jobs in an M/G/∞ queue is Poisson, and cannot accommodate any overdispersion. In

order to evaluate the second term in the right-hand side of (8), we note that Var(E(Mk |Λk)) =

VarVk, which equals

VarVk =1

µ2p2k

∆ p2∆ · Var Λ. (9)

Substituting (9) in the second term in the right-hand side of (8), we find that VarM equals (7),

as desired.

From the above reasoning we conclude that formula (7) lends itself to a nice interpretation. The

first term, EΛ/µ that is, is the contribution to the variance that one would have if the arrival

rate would have had the deterministic value EΛ, whereas the second term needs to be added to

deal with the intrinsic variability of the arrival rate.

For the model with general service times the counterpart of (7) is

VarM = EB EΛ + C (EB)2 Var Λ, with C :=

∞∑k=0

(ξ(k,∆)

)2.

B Transient behavior

As the system’s transient behavior can be dealt with in the same way, we restrict ourselves to

a short account of this. We let the system start empty (for ease of presentation; a non-empty

12

initial condition can be analyzed without any additional difficulty). Denote by M(t) the number

of jobs present at time t. Then it follows that, for n ∈ N,

M(n∆) =n∑j=1

Mj ,

where (for given n) Mj represents the number of jobs that have arrived between (j − 1)∆ and

j∆ and are still around at time n∆, with pgf

EzMj =∞∑k=0

∫ ∞0

fΛ(λ)e−λ∆ (λ∆)k

k!

k∑`=0

z`(k

`

)(pn−j∆ p∆

µ∆

)`(1−

pn−j∆ p∆

µ∆

)k−`dλ

=

∫ ∞0

fΛ(λ) exp

(−λµpn−j∆ p∆(1− z)

)dλ,

as before. As the individual random variables M1, M2, . . . are independent, M(n∆) is mixed

Poisson:

EzM(n∆) =n∏j=1

E exp

(−Λ

µpn−j∆ p∆(1− z)

).

Again the model with general service times can be dealt with as well. Following the steps that

are standard by now, we obtain that M(n∆) has a mixed Poisson distribution, as

EzMj =

∫ ∞0

fΛ(λ) exp (−λEB ξ(j,∆)(1− z)) dλ = E exp (−ΛEB ξ(j,∆)(1− z)) .

2.2 Limit results

This section focuses on the central limit regime that results from simultaneously scaling both the

sampling frequency 1∆ and the arrival rate Λ to infinity, in a controlled way. More specifically,

we consider the following regime:

Λ 7→ NΛ 1∆ 7→

Nα

∆ , (10)

where the parameter N ∈ N is sent to ∞. The scaled counterpart of Λ(s) is N · Λ(N)(s), where

Λ(N)(s) :=∞∑j=0

Λj1[j∆N−α,(j+1)∆N−α)(s).

In words: the sampling frequency and the arrival rates are both inflated, but, importantly, this

occurs at rates that are not necessarily identical. An important conclusion from the computations

in the remainder of this subsection, is that the behavior of the number of jobs in the system for

α > 1 (relatively high rate-renewal frequency) crucially differs from that for α < 1 (relatively

low rate-renewal frequency); the case α = 1 will be treated as a special case.

13

We consider a sequence of systems indexed by N , where the N -th system uses a sampling fre-

quency of Nα

∆ and i.i.d. mixed Poisson arrival rates with random parameter NΛ. Let M (N)

denote the stationary number of jobs in the system. We start our exposition by a preliminary

calculation, in which we compute the mean and variance of M (N) and study their behavior for

large N , and indeed observe the above-mentioned trichotomy. Then, after centering and nor-

malizing M (N), we derive a central limit theorem. This result will serve as an illustration for

the reader, to create intuition as for why the scaled stationary number of customers is asymp-

totically normal. However, the goal of the next section is to derive a functional central limit

theorem for the (scaled) transient process M (N)(·) (for a system that is more general than the

one under consideration now). Given the behavior of the transient system, asymptotic normality

in stationarity could be derived by sending t→∞.

B Qualitative behavior of first two moments: trichotomy in the variance

First, we identify the steady-state mean and variance in our scaling regime, using (6) and (7).

We find that

EM (N) = NEΛ

µ(11)

VarM (N) = NEΛ

µ+N2 1− e−µ∆N−α

1 + e−µ∆N−αVar Λ

µ2∼ N EΛ

µ+N2−α∆Var Λ

2µ, (12)

where the ‘∼’ in the last display indicates that the ratio of the left-hand side and right-hand side

converges to 1 as N →∞. We thus observe that

VarM (N) ∼

N

EΛ

µif α > 1;

N2−α∆Var Λ

2µif α < 1;

N

(EΛ

µ+

∆Var Λ

2µ

)if α = 1.

(13)

These three cases allow an appealing interpretation, that was already mentioned in the intro-

duction. For α > 1, the sampling frequency will overrule the variability of Λ and the variance

of M (N) is linear in N . Consequently, the model behaves essentially as an M/M/∞. For α < 1,

we find a superlinear relation between N and Var Λ, and both the sampling frequency and the

variance of Λ play a role. We indeed find that the asymptotic variance grows faster than the

asymptotic mean for α < 1; in this regime the system is overdispersed. For α = 1, the variance

is ‘slightly larger’ than for α > 1, but it is still linear in N . In this case the sampling frequency

and the variance of Λ diverge at the same rate, resulting in a balanced variance for M (N) that

combines the two former cases.

14

Due to the above computation, VarM (N) is essentially proportional to Nγ , with γ := max1, 2−α. As a consequence, one may expect that, under (10),

M (N) :=M (N) − EM (N)

Nγ/2(14)

converges to a (zero-mean) normally distributed random variable. It is this property that we

verify now.

B Asymptotic normality

We show how to establish asymptotic normality for M (N), i.e., the centered and normalized

version of the random variable M (N). We use a direct proof, in which we evaluate the Laplace

transform of M (N) and show that it converges to that of a (zero-mean) Gaussian random variable

as N → ∞; appealing to Levy’s convergence theorem, we establish the desired convergence in

distribution.

The proof is included to provide further motivation and intuition for the scaling proposed. In

the next section, the setting is considerably generalized and additionally, a functional result

is derived: we prove weak convergence of (a centered and normalized version of) the process

M (N)(·) to a generalized Ornstein-Uhlenbeck (OU) process.

Theorem 2.1 (clt). As N →∞, M (N) converges to a zero-mean normally distributed random

variable with variance

σ2 :=EΛ

µ1α≥1 +

∆Var Λ

2µ1α≤1.

Proof. Let φ(N)(z) be the counterpart of (4) under scaling as in (10) and g(N)Λ,∆(z) the counterpart

of gΛ,∆. Then

φ(N)(z) =∞∏k=0

g(N)Λ,∆

(1− (1− z)e−µkN−α∆

).

We are interested in the behavior of M (N) in the central limit regime, hence we need to analyze

the limiting distribution (as N → ∞, that is) of (14). To this end, we evaluate its Laplace

transform:

E exp

(−sM

(N) − EM (N)

Nγ/2

)= exp

(sN1−γ/2EΛ

µ

)φ(N)

(e−s/N

γ/2). (15)

One readily obtains that

log φ(N)(

e−s/Nγ/2)

=∞∑k=0

logE exp

(−NΛ

µ

(1− e−µN

−α∆)(

1− e−s/Nγ/2)

e−µkN−α∆

). (16)

15

The right-hand side of (16) is a sum of cumulants (which is a cumulant itself), all related to the

random variable Λ. Let m` denote the `-th cumulant moment of Λ (for ` ∈ N); in particular

m1 = EΛ and m2 = Var Λ. In addition, we define

ζ(N)k (s) := −N

µ

(1− e−µN

−α∆)(

1− e−s/Nγ/2)

e−µkN−α∆.

Then it follows that

log φ(N)(

e−s/Nγ/2)

=∞∑`=1

m`

`!

∞∑k=0

(ζ

(N)k (s)

)`. (17)

Let us first consider the contribution of the term corresponding to ` = 1. Observe that

m1

∞∑k=0

ζ(N)k (s) = −N EΛ

µ

(1− e−s/Nγ/2

)∼ −sN1−γ/2EΛ

µ+ s2N1−γEΛ

2µ(18)

as N → ∞. Note that the first term in the right-hand side of (18) cancels the first term in the

right-hand side of (15), so that we are left with the second term, i.e.

s2N1−γEΛ

2µ. (19)

The second term in (17), corresponding to ` = 2, gives

N2 Var Λ

2µ2

(1− e−µN−α∆

)2

1− e−2µN−α∆

(1− e−s/Nγ/2

)2∼ 1

2s2N2−α−γ ∆Var Λ

2µ. (20)

Now compare the asymptotic expansion identified in (19) and (20). In case α > 1, we have

that γ = 1, so that (19) equals 12 s

2 EΛ/µ, whereas (20) converges to zero. On the other hand,

for α < 1 we have γ = 2 − α, and hence (19) converges to zero, whereas (20) behaves as14s

2 ∆Var Λ/µ. Finally, if α = 1, we indeed find that both terms converge to a finite positive

limit.

We now check that the terms in (17) for ` ≥ 3 vanish as N →∞. It is readily verified that

∞∑k=0

(ζ

(N)k (s)

)`∼ N `

(1− e−µN−α∆

)`1− e−`µN−α∆

(1− e−s/Nγ/2

)`∼ N ` · µ

`N−α`∆`

`µN−α∆

s`

Nγ`/2,

which behaves like N δ, with δ = δ(α) := `(1− α− γ/2) + α. In case α ≥ 1, we have γ = 1 and

hence δ = `(12 − α) + α = 1

2`+ α(1− `) ≤ 12`+ 1− ` = 1− `

2 ; in case α < 1, we have γ = 2− αand hence δ = −`α/2 + α = α(1 − `

2). Hence δ < 0 for ` ≥ 3 and the corresponding terms in

(17) can indeed be neglected. We have therefore established that, as N →∞,

logE exp

(−sM

(N) − EM (N)

Nγ/2

)→ 1

2σ2 s2,

as claimed.

16

3 Model with correlated arrival streams: functional central limit

theorem

In this section we generalize the central limit result of Thm. 2.1 in two ways. Firstly, we establish

its functional version: the centered and normalized transient queue length process converges to

a limiting process of Ornstein-Uhlenbeck type with parameters that depend on the regime. Ad-

ditionally, we consider a multidimensional setting with correlated arrivals: every arrival triggers

jobs in multiple queues. The resulting functional central limit theorem is a multidimensional

Gaussian process. We explicitly identify the corresponding correlation structure.

Let us start by describing the mechanics of this setting. The system works as a fork-join model

in which d queues are fed by a single arrival process that is constructed as the one featuring in

the previous section. The service times in queue j are i.i.d. exponential random variables with

mean µ−1j ; the service processes of the individual queues are independent, and also independent

of the arrival process.

Clearly, both the pre-limit results and the asymptotic normality that were established in the

former section can be extended to this more general setting. Moreover, we perform the same

scaling as before: the sampling frequency is sped up by a factor Nα, while the (random) arrival

rate is blown up by a factor N . Let M(N)j (t) denote the number of customers at time t in the

j-th queue of the scaled system.

We now present an alternative way of writing M(N)j (t), which facilitates the use of the martingale

central limit theorem. We introduce the functional

Ψ[X](t) :=

∫ t

0X(s) ds,

mapping the stochastic process X(s) : s ∈ [0, t] to a real number; observe that µj Ψ[M(N)j ](t)

can be interpreted as the ‘cumulative service capacity’ in queue j in the interval [0, t]. In addition,

we write Λ(t) for 1t

∫ t0 Λ(s) ds, such that t · Λ(t) is the total input rate up to time t. The scaling

counterpart of Λ(t) is N Λ(N)(t), where

Λ(N)(t) :=1

t

∫ t

0Λ(N)(s) ds.

By the law of large numbers, Λ(t) and Λ(N)(t) converge a.s. to EΛ (as t → ∞ and N → ∞,

respectively).

With Y0(·) and Yj(·) denoting independent unit-rate Poisson processes, we can characterize

M(N)j (t) as follows:

M(N)j (t)

d= M

(N)j (0) + Y0

(Nt · Λ(N)(t)

)− Yj

(µj Ψ[M

(N)j ](t)

). (21)

17

The objective in this section is to derive a d-dimensional functional central limit theorem (fclt)

for the M(N)j (·). This result characterizes the time-dependent number of customers in the scaled

system and makes explicit how the correlated arrivals lead to correlation between the individual

queue length processes. It will be stated and proved in the second subsection; first we consider

the queues’ stationary behavior by presenting the corresponding first two moments of the system

in stationarity (including the covariance of individual queues).

3.1 Qualitative behavior of first two moments

Note that the individual queue lengths are only coupled through the arrival process, so under

scaling (10), the mean and variance of M(N)j are as in (11) and (12):

EM (N)j = N

EΛ

µj,

VarM(N)j = N

EΛ

µj+N2Var Λ

µ2j

1− pj(∆N−α)

1 + pj(∆N−α)∼ N EΛ

µj+N2−α∆Var Λ

2µj.

Hence, we find the same behavior as in (13), and interestingly, the same trichotomy is observed

for the covariance between M(N)i and M

(N)j , for i, j ∈ 1, . . . , d, as stated in the next lemma.

Lemma 3.1 (Covariance in M (N)). For i, j ∈ 1, . . . , d with i 6= j, and for large N ,

Cov(M(N)i ,M

(N)j ) ∼

NEΛ

µi + µjif α > 1;

N2−α∆Var(Λ)

µi + µjif α < 1;

N

(EΛ

µi + µj+

∆Var(Λ)

µi + µj

)if α = 1.

(22)

Proof. Without loss of generality, we take i = 1 and j = 2. We first consider the non-scaled

model, by studying the joint probability generating function,

EzM1(k∆)1 z

M2(k∆)2 =

k∏j=1

ξjk(z1, z2)

where ξjk(z1, z2) is the contribution due to the slot between (j − 1)∆ and j∆; as z1 and z2 are

held fixed for the moment, we suppress them. Now introduce the functions

fi(r, k) := e−µi(k∆−r), gj(µ, k) :=1− e−µ∆

µ∆e−µ(k−j)∆,

18

where it is noted that gj(µ, k) looks like e−µ(k−j)∆ for ∆ small. In addition, we define the

quantities

ζ++jk :=

∫ j∆

(j−1)∆

1

∆f1(r, k)f2(r, k)dr = gj(µ1 + µ2, k),

ζ+−jk :=

∫ j∆

(j−1)∆

1

∆f2(r, k)(1− f1(r, k))dr = gj(µ1, k)− gj(µ1 + µ2, k),

ζ−+jk :=

∫ j∆

(j−1)∆

1

∆(1− f1(r, k))f2(r, k)dr = gj(µ2, k)− gj(µ1 + µ2, k),

ζ−−jk :=

∫ j∆

(j−1)∆

1

∆(1− f1(r, k))(1− f2(r, k))dr = 1− gj(µ1, k)− gj(µ2, k) + gj(µ1 + µ2, k),

Using the arguments that are standard by now,

ξjk : = E

( ∞∑`=0

e−λ∆ (λ∆)`

`!

(ζ++jk z1z2 + ζ+−

jk z1 + ζ−+jk z2 + ζ−−jk

)`)= E exp

(Λ∆

(ζ++jk z1z2 + ζ+−

jk z1 + ζ−+jk z2 + ζ−−jk − 1

))∼ E exp

(Λ∆

(2∏i=1

((zi − 1)e−µi(k−j)∆ + 1

)− 1

))for small ∆.

Because the contributions to the Mi(k∆) resulting from different time intervals are independent,

we obtain that

Cov(M1(k∆),M2(k∆)

)=

k∑j=1

(∂2

∂z1∂z2ξjk(z1, z2)− ∂

∂z1ξjk(z1, z2)

∂

∂z2ξjk(z1, z2)

)∣∣∣∣z1↑1,z2↑1

.

Now imposing scaling (10) and considering the stationary behavior by letting k tend to infinity,

it is readily derived that

E zM(N)1

1 zM

(N)2

2 ∼∞∏j=1

E exp

(Λ∆N1−α

(2∏i=1

((zi − 1)e−µij∆/N

α+ 1)− 1

));

observe that, for reasons of symmetry, we replaced k − j by j in the definition of the gj(µ, k).

We thus arrive at

Cov(M

(N)1 ,M

(N)2

)∼

∞∑j=1

((EΛ2)∆2N2−2αe−(µ1+µ2)j∆/Nα

+ EΛ ∆N1−αe−(µ1+µ2)j∆/Nα)

−∞∑j=1

2∏i=1

(EΛ ∆N1−αe−µij∆/N

α)

=EΛ ∆N1−α + Var(Λ) ∆2N2−2α

1− e−(µ1+µ2)∆

Nα

,

19

which behaves for N large as (22).

Recall that γ = max1, 2−α; due to the above computation we have that the covariance matrix

of the d-dimensional vector M (N) is essentially proportional to Nγ . Therefore, we expect that

the centered and normalized version of the joint stationary number of customers converges to a

(zero-mean) d-dimensional normal random vector with covariance matrix C such that

Cij =

1α≤1

EΛ

µi+ 1α≥1

∆Var(Λ)

2µiif i = j,

1α≤1EΛ

µi + µj+ 1α≥1

∆Var(Λ)

µi + µjif i 6= j.

3.2 Functional central limit theorem

The objective of this section is to derive a functional limit theorem for M (N)(t), the vector

describing the numbers of customers in the scaled system at time t. To this end, we consider the

process M(N)j (·) := M

(N)j (·)/N , for which we have

M(N)j (t) = M

(N)j (0) +N−1Y0

(Nt · Λ(N)(t)

)−N−1Yj

(Nµj Ψ[M (N)](t)

).

Applying the law of large numbers to Poisson processes, the following useful lemma is obtained;

see e.g. [1].

Lemma 3.2. Let Y be a unit-rate Poisson process. Then for any U > 0, almost surely

limN→∞

sup0≤u≤U

∣∣∣∣Y (Nu)

N− u∣∣∣∣ = 0.

The uniform convergence in Lemma 3.2 entails that M(N)j (t) converges almost surely to the

solution of the functional equation

%j(t) = %j,0 + tEΛ− µj Ψ[%j ](t), (23)

as N → ∞, under the proviso that M(N)j (0) converges a.s. to some value %j,0 for j = 1, . . . , d.

The solution is given by a convex mixture of the initial position %j(0) = %j,0 and the limiting

value EΛ/µj :

%j(t) = %j,0e−µjt +EΛ

µj

(1− e−µjt

).

Having identified this ‘fluid limit’, the next objective is to establish an fclt for the centered and

normalized process

U(N)j (t) := Nβ

(M

(N)j (t)− %j(t)

),

with β := 1− γ/2 = min12 ,

α2 . Here we closely follow the proof in [1], where the idea is to use

a martingale central limit theorem (mclt) , so as to obtain weak convergence to a (generalized)

Ornstein-Uhlenbeck process. The version that we need in our setting is stated below.

20

Theorem 3.3 (mclt, [1]). Let R(N)N∈N be a sequence of Rd-valued martingales. Assume that

the following condition on the jump sizes is met:

limN→∞

E[sups≤t

∣∣∣R(N)(s)−R(N)(s−)∣∣∣] = 0; (24)

in addition, assume that, as N →∞,[R

(N)i , R

(N)j

]t→ Cij(t)

for a deterministic function Cij(t), continuous in t for all t > 0 and for i, j = 1, . . . , d.

Then the process R(N) weakly converges to a centered Gaussian process W with independent

increments whose covariance matrix is characterized by

E[Wi(t) ·Wj(t)

T]

= Cij(t).

Introducing compensated unit-rate Poisson processes Yj(t) := Yj(t)− t, we define

Y(N)0 (t) := Nβ−1

Y0

(Nt · Λ(N)(t)

)...

Y0

(Nt · Λ(N)(t)

) , Y

(N)(t) := Nβ−1

Y1

(Nµ1 Ψ[M

(N)1 ](t)

)...

Yd

(Nµd Ψ[M

(N)d ](t)

) (25)

both define sequences of d-dimensional processes amenable for this mclt.

Lemma 3.4. Consider the d-dimensional processes Y(N)0 (·) and Y

(N)(·). If α ≥ 1, then as

N → ∞ these processes converge weakly to d-dimensional zero-mean Brownian motions with

covariance matrices K0(t) := (tEΛ)11T and K(t) := diagµ1 Ψ[%1](t), . . . , µd Ψ[%d](t); if α < 1

the limiting covariance matrices equal 0.

This result is in correspondence with the fact that the first martingale consists of d identical

processes, whereas the other consists of d independent processes.

Proof. We start by checking the conditions of Thm. 3.3. First, observe that for each N , Y(N)0 (·)

and Y(N)

(·) are d-dimensional real-valued martingales; scaled and centered Poisson processes

satisfy the martingale property. Also, condition (24) is met, as both for R(N) = Y(N)0 and

R(N) = Y(N)

,

limN→∞

E[sups≤t

∣∣∣R(N) −R(N)(s−)∣∣∣] <∞,

whereas Nβ−1 ≤ 1/√N converges to zero.

21

Let us determine the entries of the corresponding quadratic covariation matrices:[Nβ−1Y0

(Nt · Λ(N)(t)

)]t

= N2β−2Y0

(Nt · Λ(N)(t)

),[

Nβ−1Yj

(Nµj Ψ[M

(N)j ](t)

)]t

= N2β−2Yj

(Nµj Ψ[M (N)](t)

),

whereas, for i 6= j,[Nβ−1Yi

(Nµ1 Ψ[M

(N)i ](t)

), Nβ−1Yj

(Nµj Ψ[M

(N)j ](t)

)]t

= 0

(using that, for i 6= j, Yi(·) and Yj(·) are independent, so that their quadratic covariation is zero).

Note that β = min12 ,

α2 , so that 2β − 2 = min−1, α − 2. Now observe that for α ≥ 1 (and

hence 2β − 2 = −1), by virtue of Lemma 3.2 and the fluid limit derived in (23),

limN→∞

N2β−2Y0

(Nt · Λ(N)(t)

)= tEΛ,

limN→∞

N2β−2Yj

(Nµj Ψ[M

(N)j ](t)

)= µj Ψ[%j ](t).

Thm. 3.3 yields that the processes converge weakly to d-dimensional Brownian motions with

covariance matrices K0(t) and K(t), respectively.

For α < 1, we have 2β − 2 = α − 2 < −1, and hence the entries of the covariance matrices all

vanish as N →∞. Therefore, both Y(N)0 (·) and Y

(N)(·) converge to a process identical to 0.

Stated below is the main theorem of this section: an fclt for the process U (N)(·). In line with

earlier findings, three regimes need to be distinguished: α > 1 (the fast regime), α < 1 (the slow

regime), and α = 1 (the intermediate regime).

Theorem 3.5 (fclt). As N → ∞, U (N)(·) converges weakly to a zero-mean d-dimensional

Gaussian process with covariance matrix

Cij(t) :=

1α≥1

(EΛ

µi+ %i,0e−µit

)(1− e−µit) + 1α≤1

∆Var Λ

2µi(1− e−2µit) if i = j,(

1α≥1EΛ

µi + µj+ 1α≤1

∆Var Λ

µi + µj

)· (1− e−(µi+µj)t) if i 6= j.

(26)

Proof. We first write, for j = 1, . . . , d, using the definition of U(N)j (t),

U(N)j (t) = Nβ

(M

(N)j (0) +

Y0

(Nt · Λ(N)(t)

)N

−Yj(Nµj Ψ[M

(N)j ](t))

)N

− %j(t)

).

Adding and subtracting %j,0, this expression is equivalent to

Nβ(M

(N)j (0)− %j,0

)−Nβ

(%j(t)− %j,0

)+Nβ

(Y0

(Nt · Λ(N)(t)

)N

−Yj(Nµj Ψ[M

(N)j ](t)

)N

),

22

which, by filling out the explicit form of %j(t), simplifies to

U(N)j (t) = U

(N)1,j (t) + U

(N)2,j (t) + U

(N)3,j (t),

with

U(N)1,j (t) := U

(N)j (0)− µj Ψ[U

(N)j ](t),

U(N)2,j (t) := Nβ

(t · Λ(N)(t)− tEΛ

),

U(N)3,j (t) := Nβ−1

(Y0

(Nt · Λ(N)(t)

)− Yj

(Nµj Ψ[M

(N)j ](t)

)).

We consider the three individual components separately.

Component U(N)1,j (t) consists of the starting value of the process, which is assumed to

converge to some value Uj(0), minus a reverting term. It is now straightforward that, as

N →∞, U (N)1,j (·) converges to the solution of U1,j(t) = Uj(0)− µj Ψ[Uj ](t).

Then consider U(N)2,j (·). For α ≤ 1 (and hence β = α

2 ), the standard fclt for partial sums

of i.i.d. random variables entails that, as N →∞,

U(N)2,j (·)→

√∆Var Λ · V (·),

with V (·) a standard Brownian motion. On the other hand, the limiting process equals 0

for α > 1, as we then have β = 12 <

α2 .

Finally, from Lemma 3.4, we conclude that U(N)3 (·) weakly converges to a d-dimensional

zero-mean Brownian motion with covariance matrix K0(t) +K(t) if α ≥ 1, and to 0 else.

Using the above observations, we can now complete the proof. Each of the three regimes will be

considered separately.

1. Fast regime (α > 1). We have obtained above that U (N)(·) converges weakly to the solution

U(·) of the d-dimensional stochastic integral equation given by

Uj(t) = Uj(0)− µj Ψ[Uj ](t) +W0(tEΛ) +Wj (µj Ψ[%j ](t)) for j = 1, . . . , d

with W0(·),W1(·), . . . ,Wd(·) independent standard Brownian motions, or, equivalently

Uj(t) = Uj(0)− µj Ψ[Uj ](t) + Wj(tEΛ + µj Ψ[%j ](t))

with W1(·), . . . , Wd(·) standard Brownian motions (but not independent). It takes a routine

calculation to find that

Uj(t) = e−µjt(Uj(0) +

∫ t

0

√EΛ + µj%j(s) e

µjs dWj(s)

).

23

All linear combinations of the Uj(t) are normally distributed, and we conclude that this d-

dimensional process is multivariate normal. It is readily seen that the expectations equal

Uj(0)e−µjt. For the variances we find, after an elementary computation,

VarUj(t) = e−2µjt

(∫ t

0(EΛ + µj%j(s)) e

2µjs ds

)=

(EΛ

µj+ %j,0e

−µjt)

(1− e−µjt).

Likewise, for the covariance, with

Ui(t) :=√EΛ

∫ t

0eµis dW0(s) +

∫ t

0

õi%i(s)e

µis dWi(s),

it follows that, for i 6= j,

Cov(Ui(t), Uj(t)) = e−(µi+µj)t EUi(t) Uj(t)

= e−(µi+µj)t EΛ · E[∫ t

0e−µis dW0(s) ·

∫ t

0e−µjs dW0(s)

]= e−(µi+µj)t EΛ

∫ t

0e(µi+µj)s ds

=EΛ

µi + µj(1− e−(µi+µj)t).

This shows (26) for α > 1.

2. Slow regime (α < 1). In the slow regime, we have convergence to the solution of

Uj(t) = Uj(0)− µj Ψ[Uj ](t) +(√

∆Var Λ)V (t),

which can be written as

dUj(t) = −µjUj(t) dt+(√

∆Var Λ)

dV (t).

Therefore the Uj(·) are OU-processes, with marginal solutions, for j = 1, . . . , d:

Uj(t) = e−µjt(Uj(0) +

∫ t

0

√∆Var(Λ) eµjs dV (s)

).

As before, we can conclude that this d-dimensional process is multivariate normal with expecta-

tions Uj(0)e−µjt. Computations as above reveal that for α < 1, as claimed in (26),

Cov(Ui(t), Uj(t)) =∆Var(Λ)

µi + µj(1− e−(µi+µj)t).

3. Intermediate regime (α = 1). In this regime, we have a combination of the processes from the

other cases:

dUj(t) = −µjUj(t) dt+√EΛ dW0(t) +

õj%j(t) dWj(t) +

√∆Var(Λ) dV (t).

24

The marginal solutions are, for j = 1, . . . , d, equal to

Uj(t) = e−µjt(Uj(0) +

∫ t

0

√EΛ + ∆Var(Λ) eµjs dW0(s) +

∫ t

0

õj%j(s) e

µjs dWj(s)

).

Again, we conclude that this d-dimensional process is multivariate normal with expectations

Uj(0)e−µjt; routine computations yield the desired covariance matrix, as given in (26). This

completes the proof.

It is interesting to study the impact of the scaling parameter α on the correlation between the

individual queue lengths. Remarkably, it turns out that for α 6= 1 this correlation depends on

the service rates only, whereas for α = 1 also the first and second moment of Λ play a role; see

the following corollary for a result on the stationary regime.

Corollary 3.6 (Correlation coefficients). In stationarity, the correlation coefficient for i 6= j

satisfies

limN→∞

Corr(M(N)i ,M

(N)j ) = cij(α) ·

√µiµj

µi + µj∈ [0, 1], (27)

for some constant cij(α) ∈ [1, 2]; in addition, cij(α) = 1 for α > 1 and cij(α) = 2 for α < 1.

Proof. From Thm. 3.5, as t→∞,

Cij(t)→ 1α≥1EΛ

µi + µj1i 6=j+ 1α≤1

∆Var(Λ)

µi + µj.

We observe that (27) holds, with

cij(α) =EΛ 1α≥1 + ∆Var(Λ) 1α≤1

EΛ 1α≥1 + 12∆Var(Λ) 1α≤1

,

which equals 1 for α > 1, and 2 for α < 1.

For α = 1, we find interesting boundary behavior: when ∆ → 0 the random arrival rate Λ

becomes increasingly deterministic (i.e., Var Λ ↓ 0), cij(α) converges to 1, as was the case for

α > 1. If, on the other hand, EΛ ↓ 0, we find that cij(α) converges to 2, which equals the result

for α < 1. It is also observed that √µiµj

µi + µj≤ 1

2,

due to the inequality of arithmetic and geometric means.

25

4 Large deviations

Whereas the previous section focused on the behavior of the process M (N)(·) around the mean

(by establishing an fclt), this section analyzes rare events. As it will turn out, the observed

trichotomy applies here again. The results again relate to the setting with d coupled queues; for

ease we first present (and prove) the results for d = 1, to return to the coupled model at the end

of the section.

4.1 Univariate large deviations

As before, we study the system resulting from an arrival process with time-dependent scaled

arrival rate N · Λ(N)(s), where

Λ(N)(s) = Λi1[i∆N−α,(i+1)∆N−α)(s)

with i.i.d. Λi distributed as Λ > 0. We are interested in the tail asymptotics of M (N)(t); more

specifically, our objective is to evaluate the decay rate

limN→∞

1

NlogP

(MN (t)

N≥ a

).

We start this section with a number of preliminary results.

The following law of large numbers holds: almost surely, as N →∞, for any t > 0,

Λ(N)(t) :=1

t

∫ t

0Λ(N)(s) ds→ EΛ.

Hence, for all δ > 0, with Sδ := [EΛ− δ,EΛ + δ],

P(

Λ(N)t ∈ Sδ

)→ 1 as t→∞.

Furthermore, define

κt

(Λ(N)

):=

∫ t

0Λ(N)(s)e−µ(t−s) ds.

Then, as observed earlier, M (N)(t) is a mixed Poisson random variable, with random

parameter N · κt(Λ(N)). It is verified that (given that the queue starts empty), almost

surely, as N →∞, for any t > 0,

M (N)(t)

N→ %(t) :=

EΛ

µ(1− e−µt) =

∫ t

0EΛ e−µ(t−s) ds.

The following theorem enables us to compute the large deviations for our model, using the

moment generating function (mgf) of M (N)(t).

26

Theorem 4.1 (Gartner-Ellis). Let (ZN )N≥0 be a sequence of random variables such that the

mgf EeθZN exists and is finite for θ > 0 and let a > limN→∞EZNN . Define

I(θ) := θa− limN→∞

1

NlogE eθZN .

Then

limN→∞

1

NlogP

(ZNN≥ a

)= − sup

θI(θ).

Note that the maximizing value for θ is strictly positive, as we have that I(0) = 0 and I(θ) < 0

for θ < 0. Namely, the slope of limN→∞1N logE eθZN is smaller than a, as limN→∞

1N logE eθZN

is convex in θ and its slope equals limN→∞EZNN < a in zero. Therefore, from now on we will

only consider strictly positive values for θ, which implies for instance that eθ − 1 > 0.

In order to apply Gartner-Ellis, we first compute the mgf of M (N)(t). To this end, we use the

known expression for the pgf of a mixed Poisson random variable, and then plug in z = eθ:

EeθM(N)(t) = E exp

(Nκt(Λ

(N))(eθ − 1))

=

btNα

∆c−1∏

i=0

E exp

(NΛi(e

θ − 1)

∫ (i+1) ∆Nα

i ∆Nα

e−µ(t−s) ds

)

× E exp

(NΛbtNα

∆c(e

θ − 1)

∫ t

btNα∆c ∆Nα

e−µ(t−s) ds

).

After taking the logarithm and dividing by N , we observe that for large values of N it is justified

to replace t by (btNα

∆ c+ 1) ∆Nα , as an immediate consequence of (btNα

∆ c+ 1) ∆Nα → t. We rely on

this simplification in the rest of our analysis, i.e. we work with, as N →∞,

1

Nlog

btNα

∆c∏

i=0

E exp

(NΛ(eθ − 1)

∫ (i+1) ∆Nα

i ∆Nα

e−µ(t−s) ds

),

which simplifies to

1

N

btNα

∆c∑

i=0

logE exp

(NΛ(eθ − 1)e−µt

(eµ(i+1)∆/Nα − eµi∆/N

α

µ

)).

This in turn behaves as (N large)

1

N

btNα

∆c∑

i=0

logE exp(

∆N1−α Λ(eθ − 1)e−µ(t−i∆/Nα)),

27

which, for reasons of symmetry, can be replaced by

1

N

btNα

∆c∑

i=0

logE exp(

∆N1−α Λ(eθ − 1)e−µi∆/Nα). (28)

As before, we distinguish between the cases α > 1, α = 1, and α < 1. Informally, in the

first regime, the sampling frequency is so high that the Λi’s can replaced by EΛ; the system’s

M/M/∞ behavior is also inherited when considering large deviations. In the second regime, on

the contrary, rather detailed information on the distribution of Λ is required. The intermediate

regime is the least insightful, as it involves a rather complicated integral.

1. Fast regime (α > 1). Our analysis heavily relies on the following proposition.

Proposition 4.2. If α > 1, then

limN→∞

1

NlogEeθM

(N)(t) = %(t)(eθ − 1).

Proof. The proof consists of a lower bound (which is relatively straightforward) and an upper

bound (which is more involved).

Lower bound. Due to Jensen’s inequality, (28) majorizes

1

N

btNα

∆c−1∑

i=0

(∆N1−α EΛ (eθ − 1)e−µi∆/N

α).

The stated lower bound follows by recognizing a Riemann sum, and by letting N go to ∞, thus

obtaining

EΛ

∫ t

0e−µs ds(eθ − 1) =

EΛ

µ(1− e−µt)(eθ − 1).

Upper bound. We fix an ` ∈ (0, α). Let

KN :=∆

Nα· bN `c ∼ ∆N `−α, KN :=

⌈(btN

α

∆c+ 1) · bN `c−1

⌉∼ t

∆Nα−`.

Clearly, KN ∼ t/KN , and bearing in mind that α > `, KN → 0 and KN →∞.

The first step is to bound (28) from above by N−1∑KN−1

m=0 Im, where for independent Λn ∼ Λ

Im :=

bN`c−1∑n=0

logE exp(

∆N1−α Λ(eθ − 1)e−µ(mbN`c+n)∆/Nα)

= logE exp

∆N1−αbN`c−1∑n=0

Λne−µ(mbN`c+n)∆/Nα(eθ − 1)

≤ logE exp

∆N1−αbN`c−1∑n=0

Λne−µmKN (eθ − 1)

.

28

The main idea is to intersect the expectation with the event

EN :=

1

bN `c

bN`c∑n=1

Λn < EΛ + δ

and its complement, for an arbitrarily chosen δ > 0. To this end, we introduce JN,m :=

∆N1−α exp(−µmKN )(eθ − 1); realize that JN,m → 0 as N → ∞ (as a consequence of α > 1).

We thus have

Im ≤ log

E

exp(JN,m

bN`c−1∑n=0

Λn)∣∣∣∣∣∣EN)

+ E

exp(JN,m

bN`c−1∑n=0

Λn)1E cN

.

The idea of the remainder of the proof is to show that

lim supN→∞

1

N

KN−1∑m=0

logE

exp(JN,m

bN`c−1∑n=0

Λn)∣∣∣∣∣∣EN

= %(t)(eθ − 1) > 0, (29)

whereas

lim supN→∞

1

N

KN−1∑m=0

logE

exp(JN,m

bN`c−1∑n=0

Λn)1E cN

,≤ 0, (30)

which implies the claim.

Let us start by establishing (29). Because of the conditioning, the expression in the left-hand

side of (29) is bounded from above by

lim supN→∞

1

N

KN−1∑m=0

logE exp(

∆N1−α bN `c(EΛ + δ

)e−µmKN (eθ − 1)

)

=(EΛ + δ

)KN

KN−1∑m=0

e−µmKN (eθ − 1).

For ` ∈ (0, α), this Riemann sum converges to

(EΛ + δ)

∫ t

0e−µs ds(eθ − 1) =

EΛ + δ

µ(1− e−µt)(eθ − 1)

as N →∞. Letting δ ↓ 0, we derive (29).

We continue by proving (30) by a change-of-measure argument. We define the (exponentially

twisted) measure Q ≡ Qδ such that EQΛ = EΛ + δ, with mgf

EQ[eθΛ] =E[e(θ+θ)Λ]

E[eθΛ],

29

where θ ≡ θδ > 0 maximizes θ(EΛ + δ)− logEeθΛ. The change-of-measure machinery then yields

E

exp

JN,mbN`c−1∑n=0

Λn

1E cN

= EQ

exp

(JN,m − θ) bN`c−1∑n=0

Λn

1E cN

.Recalling that the Λn are nonnegative random variables, and realizing that JN,m − θ → −θ < 0

as α > 1, we conclude that for N large the expression in the previous display is majorized by

Q(E cN ), which converges to 1

2 as N → ∞ due to the central limit theorem. As a consequence,

the left-hand side of (30) is bounded from above by

lim supN→∞

1

N

KN∑m=0

log1

2= − log 2 · lim sup

N→∞

KN

N≤ 0.

This proves the claim.

An application of ‘Gartner-Ellis’ then yields the following result.

Proposition 4.3. If α > 1,

limN→∞

1

NlogP

(M (N)(t)

N≥ a

)= − sup

θ

(θa− %(t)(eθ − 1)

)= a log

(%(t)

a

)− %(t) + a.

2. Slow regime (α < 1). Here two cases need to be distinguished: Λ having either a bounded or

an unbounded support.

Let us start with the unbounded case. It means that for all x > 0 there is an ε = εx > 0 such

that P(Λ > x) > ε. As a consequence, (28) is bounded from below by

1

N

btNα

∆c∑

i=0

logE exp(

∆N1−α xe−µi∆/Nα(eθ − 1)

)+

1

N

btNα

∆c∑

i=0

log ε.

The first of these two terms converges to

x

µ(1− e−µt)(eθ − 1),

wheres the second converges to 0 (due to α < 1). Letting x→∞ leads to

limN→∞

1

NlogEeθM

(N)(t) =∞.

It means that the supremum over θ ≥ 0 in ‘Gartner-Ellis’ is attained at θ = 0, and hence we find

decay rate 0. Intuitively, Λ is re-sampled relatively infrequently, and as a consequence there is a

relatively high likelihood to attain a large value.

30

Now consider the bounded case. For ease we assume that Λ attains value in a discrete set of

positive values, of which x is the largest (with probability p > 0), and y < x the one-but-largest.

As above,

limN→∞

1

NlogEeθM

(N)(t) ≥ x(1− e−µt)(eθ − 1)

µ. (31)

In addition, (28) is majorized by

1

Nlog

btNα

∆c∑

i=0

(p exp

(∆N1−α x(eθ − 1)e−µi∆/N

α)

+ (1− p) exp(

∆N1−α y(eθ − 1)e−µi∆/Nα))

,

which converges to (use x > y and α < 1) to the right-hand side of (31). Applying ‘Gartner-Ellis’,

we thus find that

limN→∞

1

NlogP

(M (N)(t)

N≥ a

)= − sup

θ

(θa− x(1− e−µt)(eθ − 1)

µ

)= a log

(x(1− e−µt)

µa

)+ a− x(1− e−µt)

µ.

3. Intermediate regime (α = 1). In this case it is directly derived that

limN→∞

1

NlogEeθM

(N)(t) =1

∆

∫ t

0logE exp [Λ∆e−µs(eθ − 1)] ds,

and hence

limN→∞

1

NlogP

(M (N)(t)

N≥ a

)= − sup

θI(θ) = − sup

θ

(θa− 1

∆

∫ t

0logE exp [Λ∆e−µs(eθ − 1)] ds

).

In the deterministic case, Λ = λ that is, this is the same result as for α > 1, but in general

we cannot compute the integral. We will now consider a special case in which computation is

possible.

Suppose that Λ ∼ exp(ν). In order to find the value for θ maximizing I(θ), we differentiate with

respect to θ and solve

∂

∂θ

1

∆

∫ t

0logE exp [Λ∆e−µs(eθ − 1)] ds =

∫ t

0e−µs

EΛ exp [Λ∆e−µs(eθ − 1)]

E exp [Λ∆e−µs(eθ − 1)]ds = a, (32)

where, with z = ∆(eθ − 1),

E exp [Λ∆e−µs(eθ − 1)] =

∫ ∞0

ν exp [λ(∆e−µs(eθ − 1)− ν

)] dλ

=ν

ν − ze−µs

EΛ exp [Λ∆e−µs(eθ − 1)] =ν

(ν − ze−µs)2

31

Therefore, (32) equals

a =

∫ t

0

e−µs

ν − ze−µsds =

1

µν

∫ 1

e−µt

1

1− zνx

dx

=log (1− z

ν e−µt)− log (1− zν )

µz

=1

µz· log

(ν − ze−µtν − z

)Which comes down to solving µaz e−µt−eµaz

1−eµaz = µaν. Therefore, define f(x) := xC1−ex

1−ex − C2 and

note that there are certain conditions on C1 and C2 under which f(x) has multiple zeroes. These

could be computed numerically. Let x a zero of f(x) such that θ with x = µa∆(eθ−1) maximizes

I(θ). Then

limN→∞

1

NlogP

(M (N)(t)

N≥ a

)= −I(θ).

4.2 Large deviations for the coupled model

We conclude the section by pointing out how the large devations for the coupled model (where

each arrival generates work in d queues) can be dealt with. The multivariate version of the

Gartner-Ellis theorem entails that, modulo the validity of mild regularity conditions to be im-

posed on the set A ⊂ Rd+,

limN→∞

1

NlogP

(M (N)(t)

N∈ A

)= − inf

a∈Asupθ

(d∑i=1

θiai − limN→∞

1

NlogE exp

( d∑i=1

θiM(N)i (t)

)).

The problem therefore reduces to characterizing the limiting log moment generating function. It

takes standard computations to verify that for α > 1, relying on specific intermediate results in

the proof of Lemma 3.1,

limN→∞

1

NlogE exp

( d∑i=1

θiM(N)i (t)

)= EΛ

(∫ t

0

d∏i=1

(e−µis(eθi − 1) + 1

)ds− t

),

whereas for α = 1 it turns out to equal

1

∆

∫ t

0logE exp

(Λ∆

(d∏i=1

(e−µis(eθi − 1) + 1

)− 1

))ds.

For α < 1 we again find decay rate 0 in the case Λ is unbounded, while in the bounded case the

probability of interest is (as in the single-dimensional case) essentially determined by the largest

value it can attain. Note that these results indeed reduce to the univariate case if we plug in

d = 1.

32

5 Conclusions and outlook

In this thesis, we have introduced a new framework to model systems with overdispersed arrivals

in the form of an infinite-server system with mixed Poisson arrivals in a random environment.

For this model, we were able to derive stochastic-process limits and large-deviations results, and

in both cases, we revealed a trichotomy in system behavior depending on the imposed scaling

on system size and sampling frequency. In the process of finding these results, we encountered

various possible directions for future research. Having said this, we would like finish off by

exploring some extensions of the model, considering general service times and correlated queues.

In order to place the model in a broader perspective, we will start this final section with a

discussion on some alternative models for overdispersion.

5.1 Alternative models using mixed Poisson arrivals

The three alternative models that we will discuss have been introduced in order to deal with

overdispersion. Although they all involve mixed Poisson arrival processes, it is noted that they

also show crucial differences.

Jongbloed and Koole [8] considered the setting of an M/M/s system, but instead of Poisson

arrivals assume mixed Poisson arrivals, so that conditional on the outcome of some random

variable Λ, say Λ = λ, the number of arrivals in a unit length interval is Poisson with rate λ.

In [8] they show how the additional uncertainty in Λ leads to more uncertainty in the system’s

behavior and resulting dimensioning rules.

Maman [10] again considers an M/M/s-setting, yet making a stronger assumption than Jongbloed

and Koole by restricting the random arrival rate to be of the form Λ = λ+ λcX for some zero-

mean random variable X with finite variance. Overdispersion due to parameter uncertainty thus

increases with c. Maman established scaling limits for the stationary queue length behavior in

the large-λ limit and a specific coupling between λ and s of the form (using unit mean exponential

service times) s ≈ λ+ βλc, where β is a Quality-of-Service parameter. Although the model and

scalings differ from our model, Maman also found a trichotomy in scaling limits, depending on

the value of c. Consequently, she identifies three levels of arrival rate variability: conventional

(c ≤ 12), moderate (c ∈ (1

2 , 1)) and extreme (c = 1). Comparing these levels with our regime, we

observe striking similarities. For c ≤ 12 , the situation is indeed conventional and the square-root

rule applies. This corresponds to the case α > 1 in our model, where the system indeed behaves

essentially like M/M/∞. Note that in our model, we considered the case where α = 1 (c = 12)

separately; although there is no (persistent) overdispersion in this regime, we observed slightly

different behaviour. For c ∈ (12 , 1), Maman proposes the so-called qed-c staffing rule to replace

the classical square-root rule; additional capacity is needed to hedge for substantial parameter

uncertainty. This situation corresponds to our slow regime (0 < α < 1), where overdispersion

is observed. For c = 1, Maman introduces a new regime in which qed performances can not

33

be achieved: the ed regime. The standard deviation of M is the same size as its mean. This

situation corresponds to setting α = 0 in our model; the sampling frequency is fixed while the

system size grows to ∞ and there is no scaling limit.

Mathijsen et al. [11] considered a discrete-time queue, with arrivals per slot following some mixed

Poisson distribution, and a capacity of serving a deterministic number of s jobs per slot. Like

in Maman [10], large-scale system limits are obtained, where A, the generic random variable

describing the arrivals per slot, and s are coupled: s = EA + βσA. Again, this gives rise to a

trichotomy, depending on how the variance of A grows with the system size.

The three models in [8, 10, 11] provided a source of inspiration for this work, and share at some

level the feature of the trichotomy due to the level or growth of overdispersion. The details of

the models being rather different, we will not pursue a further comparison among these models

here. However, this is an interesting topic for future research.

5.2 Correlated queues

Considering d queues using a single arrival process, we can think of an extension where each queue

has an additional, individual arrival process (all independent). Additionally, the arrivals from

the joint arrival process could be distributed over the queues according to a certain probability

distribution, in two ways. Denote such arrival processes as follows:

Pi(t) = Pi(t) + Pthinned by i(t),

with Pi(t) for i ∈ 1, . . . , N i.i.d. Poisson processes, and P (t) yet another independent Poisson

process that generates jobs for all queues. Using the pgf, it can be shown that the resulting

processes are again Poisson. The joint pgf for the Pthinned by i(t) is

E∏i∈I

zPthinned by i(t)i = e−λt · eλt·

∏Ni=1(pi(zi−1)+1),

which equals the pgf for∑N

i=1 Pthinned by i(t).

Obviously, the queue length processes of the d queues are positively correlated when the jobs of P

enter the queue i with probability pi, indepedently of what happens at the other sources. Defining

the Pi in such a way that the jobs of P are sent to exactly one of the i queues would provide

us with negatively correlated sources. Note that from a call center point of view, we would

not expect that arrivals enter at exactly the same time (only with probability zero), whereas

in a hospital setting arriving patients might generate jobs in multiple queues (e.g. doctors or

machines).

5.3 General service times

We mentioned several times that results could be generalized for the case of general service

times. In order to see that, we consider the stationary queue length process M with general

34

service times. The corresponding pgf is

φ(z) =∞∏k=0

E exp (−ΛkEBP(Be ∈ [k, k + 1])(1− z))

= E exp (−∞∑k=0

ΛkEBP(Be ∈ [k∆, (k + 1)∆])(1− z)).

Imposing the usual scaling,

Λ 7→ NΛ,1

∆7→ Nα

∆,

the pgf of the scaled stationary queue length M (N) will look like

φ(N)(z) = E exp (−∞∑k=0

NΛkEB · P(Be ∈ [k∆N−α, (k + 1)∆N−α))(1− z)).

We can now determine the first two moments of M (N):

EM (N) = NEΛEB.

VarM (N) = NEΛEB +N2C(N)(EB)2Var Λ.

where C(N) =∑∞

k=0 P(Be ∈ [kN−α, (k + 1)N−α))2. As N → ∞, we find that C(N) = O(N−α)

under the assumption of exponential service times. Namely, we see that

C(N) =∞∑k=0

(e−(k+1)N−α − e−kN−α

)2=

(e−N−α − 1)2

1− e−2N−α =1− e−N−α

1 + e−N−α =N−α +O(N−2α)

2 +O(N−α)=

1

2N−α.

This raises the question: What is the order of C(N) for general service times? If we find that it

is of order N−α, all results in this thesis could be generalized.

Let B denote a deterministic service time. We need to impose some extra conditions to obtain

that

C(N) =

∞∑k=0

P(Be ∈ [kN−α, (k + 1)N−α))2 = O(N−α).

35

Remembering that P(Be ∈ [k, k + 1)) = 1EB∫ 1

0 P(B > k + s) ds, we obtain

P(Be ∈ [kN−α, (k + 1)N−α)) =1

EB

∫ N−α

0P(B > kN−α + s) ds

=N−α

B

∫ 1

0P(NαB > k + t) dt

=N−α

B

∫ 1

01NαB>k+t) dt

=

N−α

B if k ≤ NαB − 1

1− N−α

B k if k ∈ (NαB − 1, NαB)

0 if k ≥ NαB.

For C(N) =∑∞

k=0 P(Be ∈ [kN−α, (k + 1)N−α))2, this means that

C(N) = bNαB − 1c ·(N−α

B

)2

+

(1− N−α

BbNαBc

)2

∼ N−α

B.

We now begin to see why C(N) = O(N−α) in general. In fact, it holds for general service times

that:

P(Be ∈ [kN−α, (k + 1)N−α)) =N−α

EB

∫ 1

0P(NαB > k + t) dt ≤ N−α

EBP(NαB > k).

This gives

C(N) ≤(N−α

EB

)2

·∞∑k=0

P(NαB > k)2 ≤ N−2α

(EB)2·Nα

∞∑k=0

P(B > k)2 =1

(EB)2

∞∑k=0

P(B > Nαk)2.

From this we conclude that the distribution of B should be such that the above series is of order

N−α, resulting in the desired order of magnitude for the variance of M (N). We could study large

deviations to determine whether this condition holds for certain distributions for B. Note that

for exponential service times, the above assumption indeed holds:

∞∑k=0

P(B > Nαk)2 =

∞∑k=0

e−2µNαk =1

1− e−2µNα ∼N−α

2µ<∞ for large N.

36

References

[1] D. Anderson, J. Blom, M. Mandjes, H. Thorsdottir, and K. de Turck. A functional central

limit theorem for a Markov-modulated infinite-server queue. Methodology and Computing

in Applied Probability, 2014.

[2] A. Bassamboo, S. R. Ramandeep, and A. J. Zeevi. Capacity sizing under parameter uncer-

tainty: Safety staffing principles revisited. Management Science, 56(10):1668–1686, 2010.

[3] J. Blom, K. Turck, O. Kella, and M. Mandjes. Tail asymptotics of a Markov-modulated

infinite-server queue. Queueing Syst. Theory Appl., 78(4):337–357, 2014.

[4] L. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn, and L. Zhao. Statistical

analysis of a telephone call center: A queueing-science perspective. Journal of the American

Statistical Association, 100, 2005.

[5] D.R. Cox. Some statistical methods connected with series of events journal of the royal

statistical society. Journal of the Royal Statistical Society, Series B (Methodological),

17(2):129–164, 1955.

[6] K. De Turck and M. Mandjes. Large deviations of an infinite-server system with a linearly

scaled background process. Performance Evaluation, 75-76:36–49, 2014.

[7] A. K. Erlang. Solution of some problems in the theory of probabilities of significance in

automatic telephone exchanges. Elektroteknikeren, 13, 1917.

[8] G. Jongbloed and G. Koole. Managing uncertainty in call centres using poisson mixtures.

Applied Stochastic Models in Business and Industry, 17(4):307–318, 2001.

[9] S. Kim and W. Whitt. Are call center and hospital arrivals well modeled by nonhomogeneous

poisson processes? Manufacturing & Service Operations Management, 16(3):464–480, 2014.

[10] S. Maman. Uncertainty in the demand of service: The case of call centers and emergency

departments. M. Sc. Thesis, Technion - Israel Institute of Technology, Haifa, Israel, 2009.

[11] B.M.J. Mathijsen, A.J.E.M. Janssen, J.S.H. van Leeuwaarden, A.P. Zwart, and A. Man-

delbaum. Robust heavy-traffic approximations for service systems facing overdispersed de-

mand.

[12] S.G. Steckley, S.G. Henderson, and V. Mehrotra. Forecast errors in service systems. Prob-

ability in the Engineering and Informational Sciences, 23, 2009.

[13] W. Whitt. Dynamic staffing in a telephone call center aiming to immediately answer all

calls. Oper. Res. Lett., 24(5):205–212, 1999.

37

[14] W. Whitt. Stochastic models for the design and management of customer contact centers:

some research directions. 2002.

[15] W. Whitt, L. V. Green, and P. J. Kolesar. Coping with time-varying demand when set-

ting staffing requirements for a service system. Production and Operations Management,

16(1):13–39, 2007.

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