A large deviations principle for infiniteserver queues in a random environment
Queueing Syst
A large deviations principle for infiniteserver queues in a random environment
H. M. Jansen 0 1 2
M. R. H. Mandjes 0 1 2
K. De Turck 0 1 2
S. Wittevrongel 0 1 2
Mathematics Subject Classification 0 1 2
B H. M. Jansen 0 1 2
M. R. H. Mandjes 0 1 2
K. De Turck 0 1 2
0 Laboratoire Signaux et Systèmes (L2S, CNRS UMR8506), École CentraleSupélec, Université Paris Saclay , 3 Rue Joliot Curie, Plateau de Moulon, 91190 GifsurYvette , France
1 TELIN, Ghent University , SintPietersnieuwstraat 41, 9000 Ghent , Belgium
2 Kortewegde Vries Institute for Mathematics, University of Amsterdam , Science Park 904, 1098 XH Amsterdam , The Netherlands
This paper studies an infiniteserver queue in a random environment, meaning that the arrival rate, the service requirements, and the server work rate are modulated by a general càdlàg stochastic background process. To prove a large deviations principle, the concept of attainable parameters is introduced. Scaling both the arrival rates and the background process, a large deviations principle for the number of jobs in the system is derived using attainable parameters. Finally, some known results about Markovmodulated infiniteserver queues are generalized and new results for several background processes and scalings are established in examples.
Infiniteserver queue; Random environment; Modulation; Large deviations principle

60K25 · 60F10
1 Introduction
The infiniteserver queue is one of the fundamental models in queueing theory. Its
distinguishing feature is the presence of an infinite number of servers, so that jobs are
served independently and there are no waiting times. This leads to explicit formulas for
many quantities of interest, especially for M/M/∞ queues, where jobs arrive according
to a Poisson process and the service requirements have an exponential distribution. In
practice, however, one often observes timevarying arrival intensities, service
requirement distributions, and server work rates. This calls for adequate modeling.
A natural way to incorporate timedependence is to consider an infiniteserver queue
in a random environment. In this case there is an independent background process that
modulates the arrival rate, the service requirement distribution, and the work rate of
the servers.
1.1 Model
In this paper, we study the case where the background process is a general stochastic
process J whose paths are rightcontinuous and have finite left limits, i.e., J has càdlàg
paths. The process J modulates the arrival rate, the service requirement distribution,
and the server work rate in the following way. When J is in state x , jobs arrive according
to a Poisson process with intensity λ (x ). Upon arrival, a job draws an independent
service requirement with distribution Fx if J is in state x when the job arrives. Then
the service requirement of the job is processed by a server, whose work rate is μ (x )
while J is in state x . Immediately after its service requirement has been processed, a
job leaves the system.
1.2 Main result
The main result of this paper is a full large deviations principle (LDP) for the transient
number of jobs in the system, under a scaling of the arrival rate and the background
process. To arrive at this result, we first show that the number of jobs in the system at
time t ≥ 0 has a Poisson distribution with random parameter φt ( J ). Then we scale
λ → nλ and we scale J → Jn such that the normalized random parameter φt ( Jn)
satisfies an LDP. Under this scaling, we derive the LDP for the transient number of
jobs in the system.
1.3 Literature
The amount of literature related to our main result is quite small. Moreover, almost
all papers on infiniteserver queues in a random environment (with notable exception
[
5
]) study Model I or Model II (cf. [
3
]). In both models, jobs arrive according to a
Poisson process with intensity λ (x ) when the background process is in state x . In
Model I, service requirements have a standard exponential distribution and servers
work at rate μ (x ) when the background process is in state x . This is equivalent to the
jobs being subject to a modulated hazard rate. In Model II, service requirements have
an exponential distribution with parameter κ (x ) when the background process is in
state x and servers work at constant rate 1.
An early reference is [
17
], which analyzes Model I when the background process
is a continuoustime Markov chain. Important results in [
17
] are a recursion for the
factorial moments of the number of jobs and the observation that the steadystate
distribution is not of some ‘matrixPoisson’ type.
Other important results can be found in [
8
], which studies Model I when the
background process is a semiMarkov process with finite state space. The crucial
observation in [
8
] is that the stationary number of jobs has a Poisson distribution with
a random parameter that is determined by the background process. Moreover, the
factorial moments of the number of jobs are computed via a recursion. These results are
generalized in [
14
].
The observation in [
8
] is used to obtain timescaling results in both the central limit
regime and the large deviations regime. In the central limit regime, [
2,4
] derive central
limit theorems for Markovmodulated infiniteserver queues for several models and
scalings. In this regime, the socalled deviation matrix (cf. [7]) plays an important role.
In the large deviations regime, [
3,5
] compute optimal paths to obtain rate functions
under a linear scaling of the arrival rates, given that the background process is an
irreducible continuoustime Markov chain. The former studies Model I, whereas the
latter studies Model II for a class of service requirement distributions that includes the
exponential distribution.
As mentioned, we show that the number of jobs in the system has a Poisson
distribution with a random parameter. This means that the probability distribution of the
number of jobs in the system is a mixture of Poisson distributions. This significantly
complicates a large deviations analysis: even in elementary cases a mixture may not
satisfy an LDP. Nevertheless, papers such as [
1,6,11
] have studied large deviations
of mixtures and identified conditions under which a mixture does satisfy an LDP.
However, our model does not fit into the framework of these publications. We will
elaborate on this in the next section.
1.4 Contributions
In more detail, the contributions of this paper are the following. We generalize known
models by considering a general càdlàg background process instead of a semiMarkov
background process with finite state space. We also generalize known models by
including general service requirement distributions. Moreover, in our model the
background process modulates both the service requirement distributions and the server
work rate, whereas previous papers considered models in which either the service
requirement distributions or the server work rate was modulated. In particular, our
model generalizes Model I and Model II.
Using elementary arguments, we show that in this model the transient number of
jobs has a Poisson distribution with random parameter. We scale the arrival rate
linearly and we scale the background process such that the normalized random parameter
satisfies an LDP. Under this scaling, we obtain a full LDP for the number of jobs in the
system. To the best of our knowledge, this is the first time that a full LDP is presented
for modulated infiniteserver queues. To prove the LDP, we exploit properties of the
queueing system, introduce the concept of attainable parameters, and use a
generalization of Varadhan’s Lemma. These tools enable us to avoid the assumptions in [
1,6,11
].
The theory is illustrated by examples that show rate functions that cannot be obtained
via background processes with finite state space. Additionally, we show that
completely different background processes may lead to the same LDP, even in highly
nontrivial cases. We also show examples that do not fit into the framework of [
1,6,11
].
1.5 Organization
The rest of this paper is organized as follows. In Sect. 2, we give a more practical
motivation to study our model and discuss some of the technical subtleties. In Sect. 3,
we describe the model and provide some of its basic properties. Additionally, we
fix some notation. In Sect. 4, we introduce the concept of attainable parameters and
prove an LDP for the number of jobs in the system. In Sect. 5, we show that the rate
function corresponding to this LDP has a simple description when we do not scale the
background process. As an illustration, we work out some examples. In Sect. 6, we give
examples in which we do scale the background process. In Sect. 7, we briefly discuss
the results and point out some topics for future research. The appendices provide some
technical details about the number of jobs in the system (Appendix 1), continuity and
convergence in Skorokhod space (Appendix 2), and properties of Poisson random
variables (Appendix 3).
2 Motivation
Modulated infiniteserver queues are used to model various phenomena in
communications systems, road traffic, and hospital capacity planning, for example. For us, the
main practical motivation to study this model stems from biology. It is well known
that the production of molecules in a cell may be ‘bursty,’ meaning that periods of
high production activity are followed by periods of low production activity. In [
12
],
this phenomenon is modeled using an interrupted Poisson process, i.e., a Poisson
process that is modulated by a stochastic ON/OFF switch. After a molecule is
produced, it degrades during a random time interval. The resulting model for the number
of molecules (mRNA in the case of [
12
]) still present in the cell is, essentially, a
simple modulated infiniteserver queue. However, other publications (cf. [18, p. 22])
indicate that for more complicated production processes one may need a more general
background process with a larger state space.
Another important observation (cf. [19, pp. 605–606]) is that the production process
and the switching process may be on different time scales. This gives rise to different
regimes (cf. [19, Fig. 28.4]). Mathematically, this phenomenon is captured by the
modulated infiniteserver queue via a linear scaling of the arrival rate and a scaling of
the switching process.
Motivated by these observations, we study the modulated infiniteserver queue
using very few assumptions on the background process and the service requirement
distributions. In particular, we are interested in the large deviations behavior of the
number of jobs (molecules) in the system under a linear scaling of the arrival rate and
a very general scaling of the background process. Our goal is to prove a full large
deviations principle for the number of jobs in the system.
From a more mathematical point of view, this leads to some interesting problems.
First of all, the model that we consider includes general service requirements and
generalizes two wellknown models. Next to that, we impose very few assumptions
on the background process and its scaling, whereas other studies use semiMarkov
processes with finite state space and specific scalings. The scaled background processes
considered here induce rate functions that do not necessarily have compact level sets.
This leads to a large deviations problem that seems not to have been discussed before.
We will explain this in more detail.
In the next section, we introduce a linear scaling of the arrival rate λ → nλ and a
general scaling of the background process J → Jn. We denote the number of jobs in
the system at time t by Mn (t ) and we would like to prove an LDP for n1 Mn (t ). It turns
out that n1 Mn (t ) is a mixture of Poisson distributions with a certain mixing measure
νn. Large deviations of mixtures have been investigated in papers such as [
1,6,11
].
However, our model does not fit into the framework of these publications. We will
indicate why.
A probability measure Qn (·) is called a mixture if there exist a family of probability
measures {Qn (θ ; ·) : θ ∈ } (where the Qn (θ ; ·) are the ‘conditional’
probabilities) and a probability measure νn on (the mixing measure) such that Qn (F ) =
Qn (θ ; F ) dνn (θ ), where n ∈ N. To prove an LDP for a mixture, one clearly needs
to assume that each conditional probability satisfies an LDP and that the mixing
measure also satisfies an LDP (ignoring some trivial cases). In general, however, these
assumptions are not sufficient for a mixture to satisfy an LDP. Indeed, in [11, Ex. 4.2]
it is shown that a mixture may fail to satisfy an LDP under these assumptions, even
when νn = ν and thus νn automatically satisfies an LDP.
These problems may be circumvented by imposing additional assumptions. One
way is to take = R and νn = ν (cf. [11, Th. 4.2]). Another way is to assume
that the sequence of probability measures νn is exponentially tight (cf. [1, Th. 1]
and [6, Th. 2.3]). This assumption implies that the rate function corresponding to
νn is good, meaning that it has compact sublevel sets (cf. [10, Lem. 1.2.18] and
[6, Lem. 2.1]).
Although we do have = R in our case, we do not assume that νn = ν nor that
the rate function corresponding to νn has compact sublevel sets. Consequently, we
cannot use the known results about LDPs for mixtures. As mentioned in Sect. 1, we
solve this relying on special properties of the queueing model and a generalization
of Varadhan’s Lemma as presented in [
15
]. This approach allows us to work with a
much larger class of background processes and also shows that common assumptions
about good rate functions in large deviations theory may sometimes be unnecessarily
restrictive when dealing with queueing systems.
3 Model and problem description
We study an infiniteserver queue with modulated arrival rates, service requirements,
and server work rates. The precise mathematical setup of the model and some of its
basic properties are provided in Appendix 1. Heuristically, the model may be described
as follows.
Let ( J (t ))t≥0 be a càdlàg stochastic process with state space E , which is assumed
to be a metric space. We will refer to the process J as the background process or
modulating process. For each j ∈ E , let Z (1, j ) , Z (2, j ) , . . . be a sequence of
independent, nonnegative, identically distributed random variables with cumulative
distribution function Fj . We assume that the map (ω, j ) → Z (k, j ) (ω) is measurable.
Let λ and μ be continuous functions defined on E and taking values in [0, ∞).
While the background process is in state x ∈ E , jobs enter the system following a
Poisson process with intensity λ (x ) ≥ 0. When job k enters the system, its service
requirement is given by Z (k, y) if the background process is in state y ∈ E upon its
arrival. Server k processes this service requirement at rate μ (z) while the background
process is in state z ∈ E . Job k leaves the system when its service requirement has
been processed.
We denote a modulated infiniteserver queue (under the conditions detailed in
Appendix 1) by the quadruple ( J, Z , λ, μ). Additionally, we denote the number of
jobs in this system at time t by M (t ).
Given a modulated infiniteserver queue ( J, Z , λ, μ), we associate the map φt with
it, where φt : D ([0, ∞) ; E ) → [0, ∞) is given by
0
t
The map φt will be called the parameter map of ( J, Z , λ, μ). In Appendix 1 it is shown
that M (t ) has a Poisson distribution with random parameter φt ( J ). This will turn out
to be a crucial property in this paper.
We are interested in events with an unusual number of jobs in the system. More
precisely, we would like to prove an LDP for the number of jobs in the system. A
sequence of probability measures {τn}n∈N is said to satisfy an LDP with rate function
ρ if there exists a lower semicontinuous function ρ : X → [0, ∞] such that
for all closed sets F and
1
n
1
lim sup
n→∞
log τn (F ) ≤ − inf ρ (a)
a∈F
for all open sets G, where each τn is defined on the Borel σ algebra of the topological
space X . A sequence of random variables is said to satisfy an LDP with rate function
ρ if the sequence of measures induced by the random variables satisfies an LDP with
rate function ρ. Importantly, we do not assume that ρ is a good rate function, i.e., we
do not assume that ρ has compact level sets.
As mentioned, we would like to prove an LDP for the number of jobs in the system.
To analyze this problem, we will scale the arrival rates via λ (x ) → nλ (x ), i.e., we
linearly speed up the arrivals. In addition, we will scale the background process via
J → Jn. Formally, scaling λ (x ) → nλ (x ) and J → Jn means that we start with
an infiniteserver queue ( J, Z , λ, μ) and then consider the sequence of infiniteserver
queues {( Jn, Z , nλ, μ)}n∈N.
Given the scalings λ (x ) → nλ (x ) and J → Jn, we denote the corresponding
number of jobs in the system by Mn (t ). It follows immediately from Eq. (1) that
Mn (t ) has a Poisson distribution with random parameter
0
t
nφt ( Jn) =
1 − FJn(s)
s
t
μ ( Jn (r )) dr
nλ ( Jn (s)) ds,
where φt is the parameter map associated with ( J, Z , λ, μ). The normalized
random parameter φt ( Jn) induces a sequence of probability measures {νn}n∈N on R via
νn (B) = P (φt ( Jn) ∈ B) for Borel sets B ⊂ R.
We will assume that the sequence of probability measures {νn}n∈N satisfies an LDP
with rate function ψ . The sequence {νn}n∈N trivially satisfies an LDP when νn = νn+1
for all n ∈ N, so this assumption covers the case in which the background process is
not scaled.
Given the scaling, we denote the number of jobs in the system at time t by Mn (t )
and consider the normalized random variable n1 Mn (t ). Our goal is to prove an LDP
for n1 Mn (t ) and to describe the corresponding rate function.
Throughout this paper, we will also use the following notation. We denote the
closure of a set A by cl A. We write B (x , ) for the open ball with center x ∈ Rd and
radius > 0 and B [x , ] for its closure. The Borel σ algebra of a topological space E
will be denoted by B (E ). For notational convenience, we will sometimes write R+ for
[0, ∞), B+ (x , ) for B (x , ) ∩ R+ and B+ [x , ] for B [x , ] ∩ R+. As is customary,
we define exp (−∞) = 0 and log (0) = −∞.
4 A large deviations principle
Let ( J, Z , λ, μ) be a modulated infiniteserver queue with associated parameter map
φt . In this section, we will prove an LDP for the number of jobs in the system under
a scaling of the arrival rates and the background process, i.e., we will prove an LDP
for n1 Mn (t ). It will turn out that socalled attainable parameters determine the rate
function corresponding to the LDP.
Definition 4.1 Given a scaling J → Jn, a real number γ ∈ [0, ∞) is called an
attainable parameter at time t ≥ 0 if for all > 0 there exists N ∈ N such that
P (φt ( Jn) ∈ B (γ , )) = νn (B (γ , )) > 0 for all n ≥ N . The set of all attainable
parameters at time t is denoted by R (t ).
The intuition behind attainable parameters is as follows. The number of jobs in
the system has a Poisson distribution with a random parameter that is completely
determined by the background process. Basically, the background process samples
the Poisson parameter. A real number γ is an attainable parameter if, for all n large
enough, the scaled background process samples parameters close to γ with positive
probability.
As mentioned before, we will prove an LDP for n1 Mn (t ) by scaling λ (x ) → nλ (x )
and J → Jn such that the sequence of probability measures {νn}n∈N induced by the
sequence of random parameters {φt ( Jn)}n∈N satisfies an LDP with rate function ψ .
The rate function I : R → [0, ∞] governing the LDP for n1 Mn (t ) is given by
where (γ ; ·) is the Fenchel–Legendre transform of the Poisson cumulant generating
function with parameter γ . It will turn out (cf. Lemma 4.2) that
However, we will take the infimum over R (t ) rather than over {ψ < ∞} to stress that
attainability of parameters is the crucial property for proving the LDP.
Before we can give the proof, we have to settle some technical details. First, it is not
immediately clear whether the function I is indeed a rate function or even whether I is
well defined. In particular, it is not clear whether R (t ) is a nonempty set. However, the
assumption that the sequence {νn}n∈N satisfies an LDP implies that R (t ) is nonempty,
as the following lemma shows.
Lemma 4.2 Let the scaling J → Jn be such that {νn}n∈N satisfies an LDP with rate
function ψ . Then R (t ) is a nonempty closed subset of [0, ∞) and {ψ < ∞} ⊂ R (t ).
Proof Suppose that γ ∈ R \ R (t ). Then there exists > 0 such that for all n ∈ N
there exists kn ∈ N such that kn ≥ n and νkn (B (γ , )) = 0. This implies that
B (γ , ) ⊂ R \ R (t ), so R (t ) is closed. Moreover, we must have
(2)
(3)
log νn (B (γ , )) = −∞ = −
so ψ (a) = ∞ for all a ∈ B (γ , ). Then R\R (t ) ⊂ {ψ = ∞} and {ψ < ∞} ⊂ R (t ).
The fact that ψ is a rate function implies that {ψ < ∞} is nonempty. The statement
of the lemma follows immediately.
From the previous lemma it follows that I is a well defined function. The fact that I
is a rate function is implied by Proposition 10.5 and the functions and ψ being rate
functions.
The next lemma is a generalization of Varadhan’s Lemma. Contrary to Varadhan’s
Lemma, it does not require that a given function f is continuous. Instead, it requires
that a weaker condition is fulfilled. We will use this lemma to obtain the large deviations
upper bound, by applying it to functions f of the form described in Proposition 10.4.
Lemma 4.3 Let X be a topological space and let {ξn}n∈N be a sequence of measures
defined on its Borel σ algebra. Suppose that {ξn}n∈N satisfies an LDP with rate
function . Let f : X → [−∞, 0] be a Borel measurable function such that f −1 ([a, b])
is a closed set for all a, b ∈ (−∞, 0] satisfying
Then it holds that
sup f (x ) −
x∈X
(x ) ≤ a ≤ b ≤ 0.
log
X
en f (x)ξn (dx ) ≤ sup f (x ) −
x∈X
(x ) .
Proof This follows immediately from [15, Cor. 2.3].
(4)
(5)
1
n
1
1
n
1
n
With these technical details settled, we can prove the following LDP for the number
of jobs in the system.
Theorem 4.4 Consider a modulated infiniteserver queue ( J, Z , λ, μ) as described
in Sect. 3. Scale λ (x ) → nλ (x ) and J → Jn such that {νn}n∈N satisfies an LDP with
rate function ψ . Then the rescaled number of jobs in the system n1 Mn (t ) satisfies an
LDP with rate function I as defined in Eq. (2), so
for any closed set F ⊂ R and
for any open set G ⊂ R.
Proof For γ ≥ 0, let P0 (γ ) , P1 (γ ) , P2 (γ ) , . . . denote a sequence of i.i.d. random
variables that have a Poisson distribution with parameter γ . Let F ⊂ R be a closed
set and let G ⊂ R be an open set.
To prove the upper bound (4), recall that Mn (t ) has a Poisson distribution with
random parameter nφt ( Jn). Then we may write
1
lim sup log P
n→∞ n
1
n
The inequality above follows from [11, Lem. 4.1].
According to Proposition 10.4, the function γ → − infa∈F (γ ; a) satisfies the
assumptions of Lemma 4.3. Moreover, {νn }n∈N satisfies an LDP both in R and in
[0, ∞) with rate function ψ (cf. [10, Lem. 4.1.5]). Hence, we may apply Lemma 4.3
to obtain
1
n
log
en[− infa∈F (γ ;a)] νn (dγ )
The fact that we only have to consider the infimum over R (t ) follows from Lemma 4.2.
This proves the upper bound.
To prove the lower bound (5), let γ ∈ R (t ) and > 0. Define γ − = max {0, γ − }
and γ + = γ + . By definition of the set R (t ) there exists N such that
P (φt ( Jn) ∈ B (γ , )) > 0 for all n ≥ N .
Fix x ∈ G. Because G is open, there exists δ > 0 such that B (x , δ) ⊂ G. Observe
that
1
n
for all n ≥ N , where the equality follows from the fact that P (φt ( Jn) ∈ B (γ , )) > 0
for all n ≥ N . Then we get
Recall that φt ( Jn) satisfies an LDP with rate function ψ , so
lim inf
n→∞ n
1
log P (φt ( Jn) ∈ B (γ , )) ≥ −
by assumption. Moreover, it holds that
In the display above, the inequality follows from Lemma 8.2 and the second equality
is established in Proposition 10.3. Combining the results, we obtain that
1
n
− a∈iBn(fx,δ) (ξ ; a) − a∈Bin(fγ, ) ψ (a) .
This holds for all
> 0 and small enough δ > 0. Taking limits, we get
lim min
↓0 ξ∈ γ −,γ +
because ψ is lower semicontinuous. Similarly, we get limδ↓0 infa∈B(x,δ) (γ ; a) =
(γ ; x ). Hence, it follows that
1
n
≥ lδi↓m0 li↓m0 ξ∈ mγ−in,γ +
= −
(γ ; x ) + ψ (γ ) .
− a∈iBn(fx,δ) (ξ ; a) − a∈Bin(fγ, ) ψ (a)
Since x ∈ G and γ ∈ R (t ) were arbitrary, we obtain
1
n
P
which completes the proof.
The proof of Theorem 4.4 contains familiar elements. First, the upper bound is proved
using a Chernoff bound combined with a generalization of Varadhan’s Lemma. Second,
the lower bound is proved by considering ‘the most likely of all unlikely scenarios,’
which is similar to the method used in [
3,5
]. However, the proofs there relied on
properties of irreducible continuoustime Markov chains and the computation of
optimal paths, whereas we consider general càdlàg background processes via attainable
parameters.
5 Examples: unscaled background processes
Given a modulated infiniteserver queue ( J, Z , λ, μ) and a scaling λ → nλ and J →
Jn, Theorem 4.4 provides a full LDP for n1 Mn (t ) and describes the corresponding rate
function. In the upcoming examples, we will consider cases in which the background
process is not scaled and we will use Theorem 4.4 to verify or extend known results
and to obtain new results.
Throughout this section, we will assume that the background process is not scaled,
i.e., Jn = J for all n ∈ N for some càdlàg stochastic process J . This is similar to the
situation shown in the bottom figures in [19, Fig. 28.4], where the arrival process is
very fast relative to the background process. We will show how to obtain an LDP in
this case.
The following lemma is trivial, but plays a central role in this section.
Lemma 5.1 If Jn = J for all n ∈ N, then the sequence {φt ( Jn)}n∈N satisfies an LDP
with some rate function ψ . In this case R (t ) coincides with the support of φt ( J ) and
R (t ) = {ψ < ∞} = {ψ = 0}.
Proof Suppose that Jn = J for all n ∈ N. Let ν denote the law of φt ( J ). Clearly,
for each x ∈ R either ν (B (x , )) > 0 for all > 0 or there exists δ > 0 such that
ν (B (x , δ)) = 0. The set of all x ∈ R with ν (B (x , )) > 0 for all > 0 is the support
of φt ( J ). Hence, if Jn = J , then R (t ) equals the support of φt ( J ).
The rate function ψ : R → [0, ∞] is defined by taking ψ (a) = 0 for a ∈ R (t )
and ψ (a) = ∞ if a ∈/ R (t ).
Suppose that G ⊂ R is open. Then ν (G) > 0 if and only if G ∩ R (t ) = ∅. Hence,
lim infn→∞ n1 log ν (G) = − infa∈G ψ (a).
Suppose that F ⊂ R is closed. If F ∩ R (t ) = ∅, then trivially lim supn→∞ n1 log
ν (F ) ≤ 0 = − infa∈F ψ (a). If F ∩ R (t ) = ∅, then there exists an open set
F ∗ ⊃ F such that F ∗ ∩ R (t ) = ∅, because R (t ) is closed (cf. Lemma 4.2). Then
ν (F ∗) = 0 (see the argument for open sets G) and we have lim supn→∞ n1 log ν (F ) ≤
lim supn→∞ n1 log ν (F ∗) = −∞ = − infa∈F ψ (a).
Hence, when the background process is not scaled, we have the special property
that R (t ) = {ψ = 0}. This will enable us to compute explicit rate functions in the
examples. In these computations, we will extensively use the following properties of
the rate function I and properties of step functions in Skorokhod space.
Fig. 1 Graphs of the functions
(γ1; ·) and (γ2; ·) for
0 < γ1 < γ2 < ∞
∞
0
γ1
γ2
Recall that the rate function I is given by
and that R (t ) = {ψ = 0} (cf. Lemma 5.1). Hence, we get
In this case, we can give a simpler and more explicit description of I , using the
following properties of the function .
For γ ≥ 0, the function (γ ; ·) is the Fenchel–Legendre transform of the Poisson
cumulant generating function with parameter γ and is given by
(γ ; a) = ⎪⎨⎧ γ∞
a < 0;
a = 0;
⎪⎩ γ − a + a log (a/γ ) a > 0.
(γ1; a) ≤
(γ1; a) ≥
(γ2; a)
(γ2; a)
∀a ∈ 0, γ1 ;
∀a ∈ γ2, ∞) .
For γ = 0 and a > 0, we understand that γ − a + a log (a/γ ) = ∞. An important
observation is that the following inequalities hold for 0 ≤ γ1 ≤ γ2 < ∞:
See Fig. 1 for an illustration. Because in the present case I is just an infimum of Poisson rate functions, these inequalities imply that I has some special properties. They are described in the following proposition.
Proposition 5.2 In the present case, I (a) = 0 if and only if a ∈ R (t ). If I (a) > 0
for some a ∈ R, then exactly one of the following three scenarios is true:
1. a < c− = inf R (t ) and I (b) = (c−; b) for all b ∈ (−∞, c− ;
2. a > c+ = sup R (t ) and I (b) = (c+; b) for all b ∈ c+, ∞);
3. the previous two cases do not hold and I (b) = min { (c−; b) , (c+; b)} for all
b ∈ c−, c+ , where c− = sup (R (t ) ∩ (−∞, a)) and c+ = inf (R (t ) ∩ (a, ∞)).
α
β
γ
δ
Proof It follows immediately from Eqs. (6) and (7) that I (a) = 0 if and only if
a ∈ R (t ). Hence, I (a) > 0 implies that the distance of a to R (t ) is strictly positive,
since R (t ) is closed. The three scenarios now follow from the inequalities (8) and (9).
The previous proposition may seem rather abstract. To get some intuition, the following
example describes a typical rate function.
Example 5.3 Suppose that R (t ) = [α, β] ∪ γ , δ for some 0 < α < β < γ <
δ < ∞. Then the function I looks like the graph shown in Fig. 2: it equals 0 on the
intervals [α, β] and [γ , δ], whereas it equals the minimum of (β; ·) and (γ ; ·) on
the interval (β, γ ) in between. On the interval (−∞, α] the function I equals (α; ·)
and on the interval [δ, ∞) the function I equals (δ; ·).
In the remainder of this section, we focus on the modulated M/M/∞ queue
( J, Z , λ, μ) as described in Example 8.3 under a linear scaling of the arrival rates.
The associated parameter map φt is given by Eq. (16) (cf. Example 8.3), which is
continuous (cf. Lemma 9.4).
To compute R (t ) in this case, it is often convenient to use the following properties of
step functions in D ([0, ∞) ; E ). (For the definition of a step function, see Appendix 2.)
Recall that the set of all step functions in D ([0, ∞) ; E ) is denoted by S ([0, ∞) ; E ).
Lemma 5.4 If {φt ( f ) f ∈ S ([0, ∞) ; E )} ⊂ R (t ), then
R (t ) = cl {φt ( f ) f ∈ S ([0, ∞) ; E )} = {φt ( f ) f ∈ D ([0, ∞) ; E )} .
Proof This follows from Lemma 4.2 and Corollary 9.3 and the fact that φt is continuous
under the present assumptions.
Lemma 5.5 If {φt ( f ) f ∈ S ([0, ∞) ; E )} ⊂ R (t ), then R (t ) is a closed interval.
Proof It suffices to show that R (t ) is convex. Let fc1, fc2 ∈ S ([0, ∞) ; E ). We may
assume that φt f 1
c ≤ φt fc2 . For x ∈ [0, t ] we define the function gx via
gx (s) = 1{s<x} fc1 (s) + 1{s≥x} fc2 (s)
for s ∈ [0, ∞). Clearly, gx ∈ S ([0, ∞) ; E ) for each x ∈ [0, t ].
Since fc1 and fc2 are step functions, there exists a finite set E ∗ ⊂ E such that
gx ∈ S ([0, ∞) ; E ∗) for each x ∈ [0, t ]. Suppose that x1, x2 ∈ [0, t ] with x1 < x2
and x2 − x1 = . Then gx1 (s) = gx2 (s) for all s ∈ [0, t ] \ [x1, x2). Since the interval
[x1, x2) has length , Lemma 9.6 implies that
,
where λ+ = maxx∈E∗ λ (x ), μ+ = maxx∈E∗ μ (x ) and κ+ = maxx∈E∗ κ (x ). This
shows that the function x → φt (gx ) is a continuous function from [0, t ] to R.
Observe that φt (g0) = φt fc2 and φt (gt ) = φt fc1 . Now applying the
Intermediate Value Theorem to the continuous function x → φt (gx ), it follows that
φt fc1 , φt fc2
= [φt (gt ) , φt (g0)] ⊂ {φt (gx )  x ∈ [0, t ]} ⊂ R (t ) .
Combined with Lemma 5.4, this implies the statement of the lemma.
Let fc ∈ S ([0, ∞) ; E ) be a step function. Clearly, fc has a unique minimal
representation {(ti , αi )}ik=0, where k ∈ N, 0 = t0 < t1 < · · · < tk < ∞ and α0, . . . , αk ∈ E
are such that fc (t ) = αi for t ∈ ti , ti+1 and i = 0, . . . , k − 1 and fc (t ) = αk for
t ∈ [tk , ∞). Given this minimal representation, we define its truncated minimal step
size by
fc = 1 ∧ i=m1,i.n..,k {ti − ti−1} .
Additionally, we define tk+1 = tk ∨ t . The truncated minimal step size and tk+1 will
be used for computing attainable parameters.
In the upcoming examples, we would like to compute rate functions via attainable
parameters. To compute attainable parameters, we use the following strategy. We fix
a certain path f , often a step function. This gives us a parameter value φt ( f ). Then
we would like to show that, with positive probability, the background process stays
‘close’ to f , which will imply that φt ( f ) is an attainable parameter.
Staying ‘close’ to f depends on properties of E and the background process. In most
cases, the background process needs a little bit of room (both in time and in space) to
jump near a discontinuity of f . This is where the truncated minimal step size comes
in: it is an upper bound on the time we give the background process for jumping near
a discontinuity of a step function. The precise meaning of this will become clearer in
the examples.
The first example treats the familiar case of a Markovmodulated M/M/∞ queue,
i.e., the case in which the background process is an irreducible Markov chain. This
case is partly studied in [
3
] (Model I) and [
5
] (Model II). In the example, we recover
[3, Th. 2] and [5, Th. 1]. Additionally, we generalize these results to our model and
extend them to a full LDP.
Example 5.6 Let J be an irreducible, continuoustime Markov process with finite
state space E = {1, . . . , d}. We consider the modulated infiniteserver queue
( J, Z , λ, μ) under the scaling λ → nλ. Theorem 4.4 (combined with Lemma 5.1)
shows that n1 Mn (t ) satisfies an LDP with rate function I . This rate function may be
computed as follows.
1 1
f (t ) = αi−1 ∀t ∈ ti−1 + 2 k g, ti − 2 k g
∀i ∈ {1, . . . , k} ,
Intuitively speaking, the set W (g; ) consists of all paths f ∈ D ([0, ∞) ; E ) that
coincide with g on the intervals described above. These intervals cover [0, t ], except
around 0 and around time points at which g jumps.
Observe that the set W (g; ) is constructed such that each f ∈ W (g; ) coincides
with g on [0, t ], except possibly on a subset with Lebesgue measure at most . Since E
is finite and the parameter map φt is given by Eq. (16) under the present assumptions,
Lemma 9.6 then implies that
where 0 ≤ a− ≤ a+ < ∞ with a− = inf g∈D([0,∞);E) φt (g) and a+ =
supg∈D([0,∞);E) φt (g). Now applying Proposition 5.2, it follows that the rate
function I is given by
The result of the previous example depends neither on the initial distribution nor on
the transition rate matrix of the irreducible Markov chain. Moreover, the analysis in
the previous example implies the following lemma. It shows that we always obtain a
good rate function when the background process has a finite state space.
Lemma 5.7 Let J (1) be a background process with finite state space E and let J (2) be
an irreducible Markov chain with the same state space. Consider the two modulated
M/M/∞ queues J (1), Z , λ, μ and J (2), Z , λ, μ . Scaling λ → nλ, we obtain in
both cases an LDP for the number of jobs in the system with corresponding rate
functions I (1) and I (2). Then it holds that I (1) (a) ≥ I (2) (a) for all a ∈ R. In particular,
both I (1) and I (2) are good rate functions.
In the next example we will modulate an M/M/∞ queue by another Markovmodulated
infiniteserver queue. This setup differs from the setup considered in [
3,5
]. In
particular, the state space of the background process is countably infinite, so that we may
obtain a rate function that is not good.
Example 5.8 Consider a Markovmodulated infiniteserver queue as described in
[
17
], i.e., a Markovmodulated infiniteserver queue under the assumptions of Model I.
Assume that neither the arrival rates nor the server work rates are identically equal to
0 and that the system starts empty. Let J (t ) be the number of jobs in this
Markovmodulated infiniteserver queue at time t ≥ 0. Then J is a càdlàg stochastic process
and its state space is E = Z 0.
Consider the modulated M/M/∞ queue ( J, Z , λ, μ) and impose the scaling λ →
nλ. Then n1 Mn (t ) satisfies an LDP with rate function I , according to Theorem 4.4
and Lemma 5.1. This rate function may be computed as follows.
Recall that J stays in state m ∈ E during [t, t + t ] with positive probability
for arbitrarily large t . Moreover, because neither the arrival rates nor the server
work rates are identically equal to 0, the process J also has the following property.
If J (t ) = m1 at time t ≥ 0, then it jumps to state m2 ∈ E during [t, t + t ] with
positive probability for arbitrarily small t .
Roughly speaking, these two properties mean that the background process is
irreducible, in the sense that it can jump to or stay in any state during any time interval
we would like. Of course, this is very similar to the Markov chain being irreducible
in the previous example. Consequently, our strategy for determining the attainable
parameters will be very similar, although there are some subtleties related to the state
space being infinite.
Fix any g ∈ S ([0, ∞) ; E ) with minimal representation {(ti , αi )}ik=0 and take any
∈ (0, 1). Let W (g; ) denote the set of all f ∈ D ([0, ∞) ; E ) with
1 1
f (t ) = αi−1 ∀t ∈ ti−1 + 2 k g, ti − 2 k g
∀i ∈ {1, . . . , k} ,
Observe that each f ∈ W (g; ) coincides with g, except possibly on a subset with
Lebesgue measure at most . Moreover, each f ∈ W (g; ) takes values in the finite
set E ∗ = {0, . . . , α+}, where α+ = max {αi i ∈ {0, . . . , k}}. Since the parameter map
φt is given by Eq. (16) under the present assumptions, Lemma 9.6 implies that
as
→ 0.
The two properties of the background process described above imply that
P ( J ∈ W (g; )) > 0. It follows that {φt (g)  g ∈ S ([0, ∞) ; E )} ⊂ R (t ). Write
a− = inf g∈D([0,∞);E) φt (g) and a+ = supg∈D([0,∞);E) φt (g). Lemmas 5.4 and 5.5
imply that R (t ) = a−, a+ if a+ < ∞ and R (t ) = a−, ∞ if a+ = ∞. Hence,
⎪⎩⎪⎪
if a+ = ∞. Note that I is not a good rate function if a+ = ∞.
The previous example only depends on the state space being countable and discrete
and on the background process being irreducible in the sense described above.
Consequently, the same result holds if the background process is an irreducible Markov
process with a countable, discrete state space.
In the last example of this section we compare rate functions that are obtained
using two different background processes. One background process is a Markov chain,
whereas the other background process is a reflected Brownian motion, which has an
uncountable state space. It turns out that both background processes lead to the same
LDP, even though the background processes are completely different. Apparently, two
very different modulating processes may lead to the same rate function for the LDP,
even if the arrival rates, service requirements and server work rates are nontrivial.
Example 5.9 Let E = [
0, 1
] be equipped with the Euclidean metric. Recall that,
under the present assumptions, the Z (k, j ) depend on the function κ. Assume that
λ : [
0, 1
] → [
0, 1
] is given by λ (x ) = x , κ : [
0, 1
] → [
0, 1
] is given by κ (x ) = 1
and μ : [
0, 1
] → [
0, 1
] is given by μ (x ) = 1 − x .
Let J MC be an irreducible, continuoustime Markov chain with state space {0, 1}.
Let J rBM be a reflected Brownian motion with reflecting barriers 0 and 1. For
simplicity, assume that J rBM starts in x0 ∈ (0, 1), so
J rBM (t ) = x0 + W (t ) + L (t ) − U (t )
for some standard Brownian motion W , lowerregulator process L and upperregulator
process U (cf. [
9
]).
Consider the two modulated M/M/∞ queues J MC, Z , λ, μ and J rBM, Z , λ, μ .
Under the scaling λ → nλ, both n1 MnrBM (t ) and n1 MnMC (t ) satisfy an LDP with the
same good rate function I , which is given by
I (a) = ⎪⎨⎧ 0∞
⎪⎩
(13)
Then we get and Now observe that sup
P J rBM ∈ W (g; ) ≥ P (x0 + W ∈ W (g; )) > 0,
due to the definition of J rBM and W being a Brownian motion.
It follows that {φt (g)  g ∈ S ([0, ∞) ; E )} ⊂ RrBM (t ), so RrBM (t ) = [0, t ] and
the corresponding rate function is given by the function I above.
In this section, we considered examples in which the background process was not
scaled. As shown, this implies some special properties, which we can use to explicitly
compute rate functions. In the next section, we will scale the background process, too.
Although explicit computations are not possible in general, there are still cases for
which we may derive rate functions.
6 Examples: scaled background processes
In this section, we will give two examples in which the background process is scaled.
In the first example, we will consider the Markovmodulated M/M/∞ queue and derive
an explicit rate function under a superlinear timescaling. This scaling corresponds to
the top figures in [19, Fig. 28.4], where the arrival process is very slow relative to the
background process.
In the second example, we will consider a new model: we take the service
requirements from Example 8.4 and let the background process be a Brownian motion. Besides
being useful for modeling purposes, Brownian motion also induces mixing measures
that are not exponentially tight. We will show this and derive an LDP using
Theorem 4.4. In this case, the rate function will be given as the solution of a variational
problem.
Example 6.1 Consider the modulated M/M/∞ queue ( J, Z , λ, μ) with parameter
map φt , as described in Example 8.3. Assume that J is an irreducible continuoustime
Markov chain with finite state space {1, . . . , d} and generator matrix Q.
Denote the stationary distribution corresponding to Q by π = (π1, . . . , πd ) and
define μ∞ = dj=1 π j μ j and
d
j=1
t =
π j λ j
0
t
e−κ j μ∞(t−s) ds =
1 − e−κ j μ∞t .
d
j=1
π j
λ j
κ j μ∞
Scale λ → nλ and J → Jn, where Jn (t ) = J n1+ t . It is easy to see that scaling
J → Jn is equivalent to scaling Q → n1+ Q.
The sequence of random parameters {φt ( Jn)}n∈N satisfies an LDP with rate function
ψ , where
Indeed, this follows from the fact that
and
ψ (a) =
0 a = t ;
∞ a = t .
for all η > 0. These equalities are an immediate result from the proof of [3, Th. 3].
Given this LDP for {φt ( Jn)}n∈N, Theorem 4.4 implies that n1 Mn (t ) satisfies an
LDP with rate function I , where
Hence, under this superlinear timescaling of the background Markov chain, the LDP
for n1 Mn (t ) is governed by a Poisson rate function with parameter t .
Example 6.2 Consider the (nonexponential) modulated infiniteserver queue
( J, Z , λ, μ) with parameter map φt , as described in Example 8.4. Assume that the
background process J is a standard Brownian motion W on [0, ∞). By W we denote
its restriction to the interval [0, t ]. The sample paths of W are elements of C0 [0, t ],
the space of continuous functions f : [0, t ] → R with f (0) = 0.
Equip C0 [0, t ] with the supremum metric. Of course, we may view the function
φt as a map from C0 [0, t ] to [0, ∞) and this map is continuous under the supremum
metric.
Scale λ → nλ and J → Jnα for some fixed α ∈ [0, ∞), where Jnα is given by a
timescaling: Jnα (s) = W n−αs for s ≥ 0. Under this scaling, the arrivals are sped
up linearly, whereas the time scale of the Brownian motion is slowed down sublinearly,
linearly, or superlinearly.
Since W is a Brownian motion, we have φt Jn1 =d φt √1n W = φt √1n W .
Schilder’s Theorem (cf. [10, Th. 5.2.3]) states that √1n W satisfies an LDP in C0 [0, t ]
with good rate function
ξ ( f ) =
∞
Here, H1 ([0, t ]) denotes the set of all absolutely continuous functions f ∈ C0 [0, t ]
that have square integrable derivative f˙.
Recall that the parameter map φt is given by Eq. (17) and that φt is continuous
under the supremum metric on C0 [0, t ]. The contraction principle (cf. [10, Th. 4.2.1])
now implies that φt Jn1 satisfies an LDP with good rate function ψ , where ψ is given
by
ψ (a) = inf {ξ ( f )  f ∈ H1 ([0, t ]) , φt ( f ) = a} .
Clearly, ψ (a) = 0 if and only if a = φt ( f0), where f0 (s) = 0 for all s ∈ [0, t ]. Now
writing
1
n
log P φt J α
n ∈ B
= nα−1 1
nα log P φt n−α/2W
∈ B
for Borel sets B, it is straightforward to verify that for each α ∈ [0, ∞) the random
variable φt Jnα satisfies an LDP with rate function ψ α, which takes the following
form. For α > 1, we have ψ α (a) = 0 if a = φt ( f0) and ψ α (a) = ∞ if a = φt ( f0).
For α = 1, we have ψ α = ψ . For α ∈ [0, 1), we have ψ α (a) = 0 if a ∈ {ψ < ∞}
and ψ α (a) = ∞ if a ∈ {ψ = ∞}.
Observe that for α ∈ [0, 1) the set {ψ < ∞} is not necessarily compact, for instance
when λ (x ) = 1 + x 2 and μ (x ) = κ (x ) = 1. Hence, the sequence of probability
measures induced by φt Jnα may not be exponentially tight for α ∈ [0, 1). For
α ∈ (0, 1), this scaling is not covered by the results in [
1,6,11
].
Nevertheless, it follows from Theorem 4.4 that n1 Mnα (t ) satisfies an LDP with rate
function I α, where n1 Mnα (t ) is the number of jobs in the system when the background
process is Jnα and I α is given by
I α (a) =
inf
γ ∈Rα(t)
(γ ; a) + ψ α (γ ) .
Now recall that {ψ α < ∞} ⊂ Rα (t ). Also observe that {ξ < ∞} = H1 ([0, t ]) and
that {ψ < ∞} = {φt ( f ) f ∈ H1 ([0, t ])}. Then we may rewrite I α as I α (a) =
(φt ( f0) ; a) if α > 1, I α (a) = inf f ∈H1([0,t]) (φt ( f ) ; a) if α ∈ [0, 1) and
I α (a) = f ∈Hi1n(f[0,t]) [ (φt ( f ) ; a) + ψ (φt ( f ))]
7 Discussion and concluding remarks
In this paper, we studied an infiniteserver queue in a random environment and proved
a full LDP for the transient number of jobs in the system. The proof of this LDP has
two essential ingredients, namely the result that the transient number of jobs in the
system has a Poisson distribution with a random parameter and the assumption that
the random parameter satisfies an LDP. Hence, the large deviations behavior of the
random parameter seems to be the crucial factor that determines the large deviations
behavior of the number of jobs in this specific queueing system.
The rate function corresponding to the LDP for the number of jobs is rather
abstract. Nevertheless, we showed in the examples how to compute the rate
function in certain specific cases. In particular, we recovered earlier obtained results for
Markovmodulated infiniteserver queues and strengthened these to a full LDP.
Additionally, we proved LDPs when the background process has an uncountable state space.
In all examples, knowledge about the behavior of the background process could be
exploited to describe the rate function.
The results in this paper also show that we do not have to restrict ourselves to
background processes with finite state space or service requirements with an exponential
distribution. Moreover, the proof of the LDP shows that assumptions about good rate
functions used to study large deviations of mixtures may be unnecessarily restrictive
when dealing with queueing systems.
There are several interesting topics for future research on the modulated
infiniteserver queue presented here. In this paper, we only looked at large deviations of
the number of jobs at a fixed time t ≥ 0. However, for certain applications it may be
desirable to know the deviations over the whole time interval [0, t ]. Therefore, it would
be interesting to consider sample path large deviations. Also moderate deviations could
be worth investigating, so as to bridge the gap between the central limit theorems and
the large deviations results for modulated infiniteserver queues. It is unlikely, though,
that we may obtain such results under as few assumptions as in this paper.
Furthermore, it would be interesting to see whether the large deviations results
for modulated infiniteserver queues carry over to modulated Ornstein–Uhlenbeck
processes. To the best of our knowledge, this has not been investigated so far.
Acknowledgments This research has been partly funded by the Interuniversity Attraction Poles
Programme initiated by the Belgian Science Policy Office.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
Appendix 1: Transient number of jobs in the system
In this section, we provide the precise mathematical description of the model and
determine the distribution of the number of jobs in the system at time t ≥ 0, which
is denoted by M (t ). We mentioned in Sect. 1 that the steadystate distribution of the
number of jobs in the system has already been determined for specific background
processes and service requirements in Model I and Model II. However, in this case we
would like to determine the transient distribution given a general càdlàg background
process for the model described below, which generalizes Model I and Model II and
includes general service times.
Throughout this section, we denote by D ([0, ∞) ; E ) the space of càdlàg functions
from [0, ∞) to E , where E is a metric space with metric ρ. Throughout, we assume
that E is equipped with the Borel σ algebra B (E ) induced by ρ. We define, in the
usual way, a metric d◦ on D ([0, ∞) ; E ) that generates the Skorokhod J1 topology.
(For more details, see Appendix 2 and references there.)
Let ( , F , P) be a probability space on which we have defined a standard Poisson
Y and
process Y and a càdlàg stochastic process J with state space E . Assume that F∞
J are independent.
F∞
Also assume that, for each j ∈ E , we have defined on ( , F , P) a sequence of
independent, identically distributed, nonnegative random variables Z (1, j ), Z (2, j ) , . . .
such that the map (ω, j ) → Z (k, j ) (ω) is Z ⊗ B (E ) /B ([0, ∞]) measurable,
where Z = σ (Z (k, j )  k ∈ N, j ∈ E ). We denote the cumulative distribution
function of Z (1, j ) by Fj . Note that F ∞Y, F∞
J and Z are independent and that the maps
(ω, j, t ) → 1{Z(k, j)(ω)≤t} and ( j, t ) → Fj (t ) are measurable with respect to the
obvious σ algebras.
Intuitively, Z (k, j ) describes the service requirement of job k if the background
process J is in state j upon arrival of job k. The measurability assumption means that
the background process should select the particular service requirement of job k ‘in
a measurable and independent way.’ This is, of course, a very reasonable assumption
and easily verifiable in many cases.
Now we know what the background process looks like and how the service
requirements are modulated. To modulate the arrival rate and the server work rate, we take
continuous functions λ : E → [0, ∞) and μ : E → [0, ∞). Then the arrival rate at
time s ≥ 0 will be given by λ ( J (s)) and the server work rate at time s ≥ 0 will be
given by μ ( J (s)).
Given a background process J , service requirements Z (k, j ) and functions λ and
μ under the conditions described above, we will denote a modulated infiniteserver
queue by the quadruple ( J, Z , λ, μ).
The modulated infiniteserver queue ( J, Z , λ, μ) is constructed as follows. We
define the modulated Poisson process Y via
Y (t ) = Y
0
t
The process Y will be the arrival process. We denote the jump times of Y by
τ1, τ2, . . . and the jump times of Y by τ 1, τ 2, . . .. For convenience, we set τ0 = τ 0 = 0.
The jump times τk and τ k are related via τk = − (τ k ) and τ k = (τk ), where
(t ) = 0t λ ( J (s)) ds and − (r ) = inf {t ≥ 0  (t ) ≥ r }.
Define the interarrival times σk = τk − τk−1 and σ k = τ k − τ k−1 for k ∈ N.
For later use, we note that σ 1, σ 2, . . . is a sequence of i.i.d. random variables with a
standard exponential distribution.
At time t = 0 there are no jobs in the system. At each jump time of Y exactly one
job arrives. Hence, the number of jobs that have entered the system during the time
interval [0, t ] is given by the (a.s. finite) random variable k∞=1 1{τk ≤t}.
When job k enters the system at time τk , its service requirement is given by
Z (k, J (τk )). In other words, job k draws an independent service requirement with
cumulative distribution function FJ (τk ). Job k leaves the system when its service
requirement has been processed by the server, whose work rate is modulated by the
background process J and is equal to μ ( J (s)) for s ≥ 0.
Hence, job k has both entered and left the system before time t ≥ 0 if and only if
τk ≤ t and Z (k, J (τk )) ≤ [τk ,t) μ ( J (r )) dr . We get
M (t ) =
∞
k=1
1{τk ≤t} − 1{τk ≤t}1 Z(k,J (τk ))≤ [τk ,t) μ(J (r)) dr
.
Note that M (t ) is a càdlàg stochastic process.
If J is deterministic, then it is relatively easy to determine the distribution of M (t ).
For instance, one may compute the characteristic function of M (t ) via the following
steps.
Suppose that J (ω, t ) = f (t ) for all ω ∈ and t ≥ 0 for some function f ∈
D ([0, ∞) ; E ). We may write the characteristic function of M (t ) as
E exp (iθ M (t )) = E exp iθ
1{τk ≤t} − 1{τk ≤t}1 Z(k,J (τk ))≤ tt∧τk μ(J (r)) dr
= E1{τ1>t} +
E1{τn≤t;τn+1>t}
exp iθ n −
Z(k, f (τk ))≤ tt∧τk μ( f (r)) dr
.
Clearly, E1{τ1>t} = e− 0t λ( f (s)) ds = e− (t). We are left with computing the infinite
sum above. Fix n ∈ N and note that
∞
k=1
1
n
k=1
1
E1{τn≤t;τn+1>t} exp iθ n −
= E 1{τn≤t;τn+1>t} exp (iθ n)
Summarizing, we get Next, observe that
t
t∧τk
t∧τk
n
k=1
because Y and the collection of service requirements are independent. For convenience,
we write
h (τk ) = exp (iθ ) + (1 + exp (iθ )) F f (τk )
μ ( f (r )) dr .
E exp (iθ M (t )) = E1{τ1>t} +
E exp
−iθ
= E1{τn≤t;τn+1>t}
n
τ1, τ2, . . .
exp (iθ ) + (1 + exp (iθ )) F f (τk )
,
h
− (yk ) dyn . . . dy1
[h (zk ) λ ( f (zk ))] dzn . . . dz1.
h
− xk+
dxn . . . dx1
We define xk+ = x1 + · · · + xk . Straightforward calculations give
∞
n=1
n
k=1
n
eτ n
k=1
n
k=1
E
1{τn≤t}
h (τk ) e (τn)
= E1{τ n≤ (t)}
n
h
− (τ k )
= 0
= 0
zn−1 k=1
· · · 0
(t) n
yn−1 k=1
n
Now note that for an integrable function g we have
t n t
0
g (s) ds
0 F f (s)
.
Now we may write
E exp (iθ M (t )) = E1{τ1>t} +
E1{τn≤t;τn+1>t}h (τk )
k =n0 k1! ne=x1p (iθ )
= e− (t) +
∞ e− (t)
n=1
= e(−1 −(t)enx∞=p0(kiθ=n)0)k1!0t eFxpf(s()iθ )
(1 − exp (iθ ))
= e− (t) ∞
k=0 n∞=0 k1! exp (iθ )
(1 − exp (iθ )) t
0 F f (s)
0 F f (s)
(t ) + exp (iθ )
t
= e0xpF f((esx)p (isθ )μ−(1f)(r )t) d1r −λF(ff(s()s)) dts
0
s
∞
t
(t ) k
μ ( f (r )) dr
λ ( f (s)) ds .
Hence, in this case M (t ) has a Poisson distribution with parameter φt ( f ), where
Given our modulated infiniteserver queue, we may view φt as a map from
D ([0, ∞) ; E ) to [0, ∞) and we will call φt the parameter map associated with the
modulated infiniteserver queue.
Note that s → F f (s) st μ ( f (r )) dr is B ([0, ∞)) /B ([
0, 1
]) measurable, as it
is a composition of measurable maps. Also note that
(14)
t
0 ≤
and that s → λ ( f (s)) is càdlàg. Hence, the integral in Eq. (14) is actually well defined
and finite.
Now suppose that J is not deterministic. In this case, we may use the independence
of J and standard arguments to obtain that
E exp (iθ M (t )) = EE exp (iθ M (t ))  J
= E exp
exp (iθ ) − 1
t
μ ( J (r )) dr
0
t
0
t
s
t
s
t
We summarize our findings in the following lemma.
Lemma 8.1 Under the stated conditions, M (t ) has a Poisson distribution with
random parameter φt ( J ), where φt is the parameter map associated with the modulated
infiniteserver queue as defined via Eq. (14) and thus
φt ( J ) =
1 − FJ (s)
μ ( J (r )) dr
λ ( J (s)) ds.
If we scale λ (x ) → nλ (x ) and J → Jn, then the number of jobs in the system Mn (t )
has a Poisson distribution with random parameter nφt ( Jn).
Lemma 8.1 states that M (t ) has a Poisson distribution with random parameter
φt ( J ), meaning that
E exp (iθ M (t )) = Eu (φt ( J ) , θ ) =
where u (γ , ·) : R → C is the characteristic function of a Poisson distribution with
parameter γ ∈ [0, ∞) and ν is the law of φt ( J ). We may also describe this as M (t )
having a mixed Poisson distribution with mixing measure ν, meaning that the law of
M (t ) may be represented as
Q ( A) =
[0,∞)
Qγ ( A) dν (γ ) ,
(15)
where Qγ is the law of a random variable with a Poisson distribution with parameter
γ ∈ [0, ∞). Indeed, suppose that Q is defined by (15). Then standard
measuretheoretic arguments (cf. [16, Pr. III.2.1]) show that
R
exp (iθ x ) dQ (x ) =
exp (iθ x ) dQγ (x ) dν (γ )
[0,∞) R
=
[0,∞)
so Q is actually the law of M (t ).
We may use these observations about modulated infiniteserver queues and mixed
Poisson distributions to prove the following intuitive lemma.
Lemma 8.2 Let A and B be measurable subsets of R. If P (φt ( J ) ∈ B) > 0, then
inf Qγ ( A) ≤ P (M (t ) ∈ A  φt ( J ) ∈ B) ≤ sup Qγ ( A) ,
γ ∈B γ ∈B
where Qγ is the law of a random variable with a Poisson distribution with parameter
γ ∈ [0, ∞).
Proof Denote ∗ = {φt ( J ) ∈ B}. If P ( ∗) > 0, we define a new probability space
( ∗, F ∗, P∗) by taking F ∗ = {F ∩ ∗  F ∈ F } and P∗ (F ) = P (F ) /P ( ∗) for
F ∈ F ∗.
For each random function X defined on ( , F , P) we denote its restriction to
∗ by X ∗, which is a random function on ( ∗, F ∗, P∗). Using the independence
assumptions (stated at the beginning of this section), it is easy to verify that J ∗ is an
independent càdlàg stochastic process, Y ∗ is an independent standard Poisson process
and Z ∗ (1, j ) , Z ∗ (2, j ) , . . . is a sequence of independent, identically distributed
random variables for each j ∈ E . Moreover, the map (ω, j ) → Z ∗ (k, j ) (ω) is
Z∗ ⊗B (E ) /B ([0, ∞]) measurable and Z ∗ (k, j ) has cumulative distribution function
Fj .
Following the procedure described in this section, we construct the modulated
infiniteserver queue ( J ∗, Z ∗, λ, μ) on ( ∗, F ∗, P∗) and we denote the number of
jobs in the system at time t ≥ 0 by K (t ). Lemma 8.1 implies that K (t ) has a Poisson
distribution with random parameter φt ( J ∗).
It is easy to verify that
P∗ (K (t ) ∈ A)
∞
k=1
= P∗
1{τk∗≤t} − 1{τk∗≤t}1{Z∗(k,J ∗(τk∗))≤ [τk∗,t) μ(J ∗(r)) dr} ∈ A
= P∗
= P {M (t ) ∈ A} ∩ ∗ /P
= P (M (t ) ∈ A  φt ( J ) ∈ B) .
∗
1{τk ≤t} − 1{τk ≤t}1{Z(k,J (τk ))≤ [τk ,t) μ(J (r)) dr} ∈ A
Since K (t ) has a Poisson distribution with random parameter φt ( J ∗), it follows
that P (M (t ) ∈ A  φt ( J ) ∈ B) = [0,∞) Qγ ( A) dν∗ (γ ), where ν∗ is the law of
φt ( J ∗). But we have P∗ (φt ( J ∗) ∈ R \ B) = 0, so P (M (t ) ∈ A  φt ( J ) ∈ B) =
B Qγ ( A) dν∗ (γ ). Now observe that
inf Qγ ( A) =
γ ∈B
inf Qβ ( A) dν∗ (γ )
B β∈B
≤
≤
B
Qγ ( A) dν∗ (γ )
sup Qβ ( A) dν∗ (γ )
B β∈B
= γin∈fB Qγ ( A) ,
which completes the proof.
In the following example we will describe the main example of a modulated
infiniteserver queue in this paper, which is essentially the modulated M/M/∞ queue. It is
constructed such that job k has a service requirement with an exponential distribution
with parameter κ ( j ) if the background process is in state j upon its arrival, with κ
some continuous function. This example includes Model I and Model II as mentioned
in Sect. 1.
Example 8.3 Let J be a background process with state space E , as described above.
Let λ, μ and κ be continuous maps from E to [0, ∞) and let Z 1, Z 2, . . . be a sequence
of independent standard exponential random variables. Define
Z k (ω) /κ ( j ) if κ ( j ) > 0;
∞ if κ ( j ) = 0.
Clearly, F ∞Y, F∞
J and Z are independent in this example. Using that
(ω, j ) ∈
× E
Z k (ω) /κ ( j ) ∈ (a, ∞)
ω ∈
Z k (ω) ∈ (qa, ∞) × { j ∈ E  κ ( j ) ∈ (0, q)}
for a ∈ (0, ∞), it is readily verified that the map (ω, j ) →
B (E ) /B ([0, ∞]) measurable.
Z (k, j ) (ω) is Z ⊗
In this case, the number of jobs M (t ) in the infiniteserver queue ( J, λ, μ, Z ) has
a Poisson distribution with random parameter φt ( J ), where the parameter map φt is
given by
λ ( f (s)) e−κ( f (s)) st μ( f (r)) dr ds.
This follows immediately from Lemma 8.1 and the construction of Z .
The next example describes a nonexponential queue: the service requirement
distributions will be Pareto distributions. The background process will determine the shape
parameter of the Pareto distributions.
Example 8.4 Let J be a background process with state space E = R. Let λ and μ
be continuous maps from E to [0, ∞). Let κ be a continuous map from E to (0, ∞)
and let Z 1, Z 2, . . . be a sequence of independent random variables having a Pareto
distribution with scale parameter 1 and shape parameter 1, so that P Z 1 > x =
1/ (1 ∨ x ) for x ∈ R. Define
Z k (ω)
.
In a similar way as in the previous example one checks that F∞ J and Z are
indeY , F∞
pendent and that the map (ω, j ) → Z (k, j ) (ω) is Z ⊗ B (E ) /B ([0, ∞]) measurable.
Given these service requirements, the number of jobs M (t ) in the infiniteserver
queue ( J, λ, μ, Z ) has a Poisson distribution with random parameter φt ( J ), where
the parameter map φt is given by
0
s
φt ( f ) =
−κ( f (s))
As before, this follows from Lemma 8.1 and the construction of Z .
Appendix 2: Continuity and convergence in Skorokhod space
In the previous section we showed that M (t ) has a Poisson distribution with a
random parameter φt ( J ), where φt is the parameter map associated with the modulated
infiniteserver queue ( J, Z , λ, μ). For specific choices of the service requirements
Z (k, j ), the map φt enjoys several continuity and convergence properties. We explore
some of these properties in this section, mainly for the setup of Example 8.3.
Let E be a metric space with metric ρ. Let D ([0, ∞) ; E ) denote the space of càdlàg
functions f : [0, ∞) → E , i.e., lims↓t f (s) = f (t ) and lims↑t f (s) exists in E for
every t ≥ 0, where lims↑0 f (s) := f (0) by convention.
Define a metric d◦ on D ([0, ∞) ; E ) via
d◦ ( f, g) = λi∈nf γ (λ) ∨
0
e−u d ( f, g, λ, u) du .
denotes the space of increasing homeomorphisms of [0, ∞),
γ (λ) = sup log (λ (t ) − λ (s)) − log (t − s)
t>s≥0
and
d ( f, g, λ, u) =
sup [1 ∧ ρ ( f (t ∧ u) , g (λ (t ) ∧ u))] .
t∈[0,∞)
The metric d◦ induces the Skorokhod J1 topology. For more details, see [
13
] or [21].
Definition 9.1 A function fc ∈ D ([0, ∞) ; E ) is called a piecewise constant function
or a step function if there exist n ∈ N, finitely many time points 0 = t0 < t1 <
. . . < tn < ∞ and α0, . . . , αn ∈ E such that fc (t ) = αi for t ∈ ti , ti+1 and
i = 0, . . . , n − 1 and fc (t ) = αn for t ∈ [tn, ∞).
The set of step functions in D ([0, ∞) ; E ) is denoted by S ([0, ∞) ; E ).
Proposition 9.2 Let f ∈ D ([0, ∞) ; E ). For all T > 0 and
function fc ∈ S ([0, ∞) ; E ) such that
> 0 there exists a step
sup ρ ( f (t ) , fc (t )) < .
t∈[0,T ]
Proof This is derived in the same way as [21, Th. 12.2.2].
Corollary 9.3 The set S ([0, ∞) ; E ) is dense in D ([0, ∞) ; E ).
Consequently, every continuous function on D ([0, ∞) ; E ) is completely determined
by its behavior on the set of step functions.
Now we will investigate properties of the parameter map under the assumptions of
Example 8.3. Let λ : E → [0, ∞), κ : E → [0, ∞) and μ : E → [0, ∞) be continuous.
For t ≥ 0, we would like to show that the function φt : D ([0, ∞) ; E ) → [0, ∞)
defined by equation (16) is continuous. Note that φt has the form
φt ( f ) =
0
λ ( f (s)) e−κ( f (s)) st μ( f (r)) dr ds
in this case and that it is the parameter map obtained in Example 8.3.
First, we observe that the map cλ : D ([0, ∞) ; E ) → D ([0, ∞) ; R) defined via
cλ ( f ) (t ) = λ ( f (t )) is continuous, because λ is continuous. Similarly, the functions
cκ and cμ are continuous.
Next, let f, g ∈ D ([0, ∞) ; R). Then pointwise multiplication of f and g is defined
via ( f g) (t ) = f (t ) g (t ). This is a measurable map which is continuous at ( f, g) if
f or g is continuous (cf. [20, Th. 4.2]).
Finally, let f ∈ D ([0, ∞) ; R). Then the map ψ : D ([0, ∞) ; R) → D ([0, ∞) ; R)
t
defined via ψ (t ) = 0 f (s) ds is continuous. This follows almost immediately from
the definition of ψ and the characterization in [13, Pr. 3.5.3].
0
e−κ( fn(s)) st μ( fn(r)) dr ds →
e−κ( f (s)) st μ( f (r)) dr ds
0
as fn → f in D ([0, ∞) ; E ). But this follows from repeated applications of the first
three observations.
Hence, the map φt must be continuous. Observe that continuity of λ, κ and μ is
crucial to obtain this result. We summarize these findings in the following lemma.
Lemma 9.4 Let λ : E → [0, ∞), κ : E → [0, ∞) and μ : E → [0, ∞) be
continuous. Then the function φt : D ([0, ∞) ; E ) → [0, ∞) as defined in equation (16) is
continuous. Consequently, the parameter map obtained in Example 8.3 is continuous.
Another property of the map φt as defined in Eq. (16) is described in Lemma 9.6.
We will use the following easy lemma in the proof of Lemma 9.6.
Lemma 9.5 Let x , y ∈ (−∞, 0]. If 0 ≤ α ≤ β <
1 − e−βx−y.
∞, then (ex )α − (ey )α
Lemma 9.6 Let λ : E → [0, ∞), κ : E → [0, ∞) and μ : E → [0, ∞) be continuous
and let φt : D ([0, ∞) ; E ) → [0, ∞) be defined by Eq. (16).
Let f, g ∈ D ([0, ∞) ; E ) and assume that there exists a finite set E ∗ ⊂ E
such that f (s) ∈ E ∗ and g (s) ∈ E ∗ for all s ∈ [0, t ] and that the set A =
{s ∈ [0, t ]  f (s) = g (s)} has Lebesgue measure . Then
φt ( f ) − φt (g) ≤ λ+ (1 + t ) 1 − e− κ+μ+ +
,
(18)
where λ+ = maxx∈E∗ λ (x ), μ+ = maxx∈E∗ μ (x ) and κ+ = maxx∈E∗ κ (x ).
Proof Clearly, we have
φt ( f ) − φt (g) ≤
A
λ ( f (s)) e−κ( f (s)) st μ( f (r)) dr
−λ (g (s)) e−κ(g(s)) st μ(g(r)) dr ds
+
[0,t]\A
−λ (g (s)) e−κ(g(s)) st μ(g(r)) dr ds.
λ ( f (s)) e−κ( f (s)) st μ( f (r)) dr
Denote the first integral on the righthand side by I1 and the second integral on the
righthand side by I2. It is easy to see that I1 is bounded above by λ+. We may find
an upper bound for I2 as follows. For s ∈ [0, t ] \ A we have
It follows that
≤ λ+ 1 − e−κ+ μ+ .
sup
s∈[0,t] s
= λ ( f (s)) e−κ( f (s)) st μ( f (r)) dr − λ ( f (s)) e−κ( f (s)) st μ(g(r)) dr
≤ λ+
e− st μ( f (r)) dr κ( f (s)) − e− st μ(g(r)) dr κ( f (s))
To obtain the last inequality, we apply Lemma 9.5 and use that
μ ( f (r )) dr −
μ (g (r )) dr ≤
λ ( f (s)) e−κ( f (s)) st μ( f (r)) dr − λ (g (s)) e−κ(g(s)) st μ(g(r)) dr ds
Combining the upper bounds for I1 and I2 proves the lemma.
To conclude this section, we provide a lemma asserting the continuity of the map
φt as defined by Eq. (17). The continuity is established using the same arguments as
above.
Lemma 9.7 Let λ : E → [0, ∞) and μ : E → [0, ∞) be continuous. Then the map
φt : D ([0, ∞) ; E ) → [0, ∞) defined by
0
− f (s)
is continuous. Consequently, the parameter map obtained in Example 8.4 is
continuous.
Appendix 3: Properties of Poisson random variables
For γ ≥ 0, let P0 (γ ) , P1 (γ ) , P2 (γ ) , . . . denote a sequence of i.i.d. random variables
that have a Poisson distribution with parameter γ . In this section, we will fix an arbitrary
x ∈ R, δ > 0, λ ≥ 0 and > 0 and define λ− = max {0, λ − } and λ+ = λ + .
Recall that B+ (λ, ) = B (λ, ) ∩ R+.
We would like to prove a large deviations lower bound for
lim inf inf
n→∞ γ ∈B+(λ, ) n
1
log P
1 n
n
i=1
s
s
t
and
Proof Let 0 ≤ γ− ≤ γ+ < ∞. For y ∈ R it holds that
inf
γ ∈B+(λ, )
P
1 n
n
i=1
inf
γ ∈(B(λ, )∩B[x,δ])∪ λ−,λ+
P
Pi (γ ) ∈ B (x , δ) .
1 n
n
i=1
P ( P0 (γ+) = y) ≥ P ( P0 (γ−) = y)
if y ≥ γ+ ≥ γ−
P ( P0 (γ+) = y) ≤ P ( P0 (γ−) = y)
if γ+ ≥ γ− ≥ y.
Of course, the difficulty here is the presence of the infimum over a range of parameters.
We will show in Proposition 10.1 that this infimum may be taken over certain restricted
subsets of B+ (λ, ). For each of these subsets we will provide a large deviations lower
bound, from which we will derive a lower bound when the infimum is taken over
B+ (λ, ). This is the content of Proposition 10.3.
Proposition 10.1 For all x ∈ R, δ > 0, λ ≥ 0 and
> 0 it holds that
Because we are working with i.i.d. Poisson random variables, we may write
P
1 n
n
Pi (γ ) ∈ B (x , δ)
= P ( P0 (nγ ) ∈ (n (x − δ) , n (x + δ))) .
Now the statement of the proposition is an easy consequence of the Eqs. (20), (21)
and (22) combined.
Proposition 10.2 Let x ∈ R and δ > 0. If B+ (x , δ) = ∅, then
lim inf
n→∞ γ ∈B+[x,δ] n
1
log P
1 n
n
i=1
Pi (γ ) ∈ B (x , δ)
= 0.
γn− = n1 nγ , γn+ = n1 nγ and
Proof For a Borel set A ⊂ R, define pn ( Aγ ) = P n1 in=1 Pi (γ ) ∈ A . Now
suppose that B+ (x , δ) = ∅. Then the diameter of B+ (x , δ) is strictly positive and
bounded above by r = min {2δ, x + δ}.
Let Nr ∈ N be such that N1r < r2 . Then for all n ≥ Nr and γ ∈ B+ [x , δ] we define
γn∗ = min
γn−, γn+
∩ B (x , δ) .
Then max γ − γn− , γ − γn+ ≤ n1 < r2 and pn (B (x , δ)γ ) ≥ pn
each n ∈ N and each γ ∈ B+ [x , δ]. Using that n! ≤ nn+1/2e−n+1, we get
γ ∗ γ for
n
pn
γ ∗ γ
n
en(γn∗−γ )e−1 nγn∗ −1/2
nγ
nγ + 1
1
1 − n (x + δ) + 1
e−2 (n (x + δ))−1/2
(23)
The last equality is an application of Cramér’s Theorem for i.i.d. Poisson random
variables; the limit exists because B (x , δ) is a continuity set for the FenchelLegendre
transform corresponding to a Poisson distribution.
As shown in the inequalities (8) and (9), the Fenchel–Legendre transforms
corresponding to Poisson distributions are nicely ordered in some sense. This property
leads to the following propositions. Their proofs are elementary but tedious and are
therefore omitted.
for each n ∈ N and each γ ∈ B+ [x , δ]. This implies the statement.
Combined with Cramér’s Theorem in R, the two previous propositions enable us to
prove the following large deviations bound. Note that we prove an equality rather than
an inequality and that the limit exists.
Proposition 10.3 For all x ∈ R, δ > 0, λ ≥ 0 and
lim inf
n→∞ γ ∈B+(λ, ) n
1
log P
1 n
n
i=1
− a∈iBn(fx,δ)
Proof Define pn ( A  γ ) = P n1 in=1 Pi (γ ) ∈ A for Borel sets A ⊂ R and C =
(B (λ, ) ∩ B [x , δ]) ∪ λ−, λ+ . Thanks to Proposition 10.1 we may write
lim inf
n→∞ γ ∈B+(λ, ) n
1
1
log pn (B (x , δ)  γ ) = nl→im∞ γin∈fC n log pn (B (x , δ)  γ ) .
It follows from Proposition 10.2 that we may restrict the infimum to the set λ−, λ+ ,
so
1 1
nl→im∞ γin∈fC n log pn (B (x , δ)  γ ) = nl→im∞ γ ∈ mλ−i n,λ+ n log pn (B (x , δ)  γ )
=
min lim
γ ∈ λ−,λ+ n→∞ n
1
min
γ ∈ λ−,λ+
log pn (B (x , δ)  γ )
Proposition 10.4 Let F ⊂ R be closed and define f : [0, ∞) → [−∞, 0] via
f (γ ) = − inf
a∈F
If F ⊂ (−∞, 0), then f ≡ −∞. If F ∩ [0, ∞) = ∅, then f is realvalued and
continuous on (0, ∞). Additionally, limγ ↓0 f (γ ) = f (0), where f (0) = 0 if 0 ∈ F
and f (0) = ∞ if 0 ∈/ F . In any case, f −1 ([a, b]) is closed for all a, b ∈ (−∞, 0]
with a ≤ b.
Proposition 10.5 Let R ⊂ [0, ∞) be a nonempty, closed set. Let ψ : R → [0, ∞] be
a lower semicontinuous function. Then the function I : R → [0, ∞] defined via
I (a) = inf
γ ∈R
is a lower semicontinuous function.
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