409 research outputs found
Large deviations of an infinite-server system with a linearly scaled background process
This paper studies an infinite-server queue in a Markov environment, that is, an infinite-server queue with arrival rates and service times depending on the state of a Markovian background process. We focus on the probability that the number of jobs in the system attains an unusually high value. Scaling the arrival rates ¿i¿i by a factor NN and the transition rates ¿ij¿ij of the background process as well, a large-deviations based approach is used to examine such tail probabilities (where NN tends to 88). The paper also presents qualitative properties of the system’s behavior conditional on the rare event under consideration happening. Keywords: Queues; Infinite-server systems; Markov modulation; Large deviation
Rare event analysis of Markov-modulated infinite-server queues: a Poisson limit
This article studies an infinite-server queue in a Markov environment, that is, an infinite-server queue with arrival rates and service times depending on the state of a Markovian background process. Scaling the arrival rates (i) by a factor N and the rates (ij) of the background process by N1+E (for some E>0), the focus is on the tail probabilities of the number of customers in the system, in the asymptotic regime that N tends to . In particular, it is shown that the logarithmic asymptotics correspond to those of a Poisson distribution with an appropriate mean
Scaling limits for infinite-server systems in a random environment
This paper studies the effect of an overdispersed arrival process on the
performance of an infinite-server system. In our setup, a random environment is
modeled by drawing an arrival rate from a given distribution every
time units, yielding an i.i.d. sequence of arrival rates
. Applying a martingale central limit theorem, we
obtain a functional central limit theorem for the scaled queue length process.
We proceed to large deviations and derive the logarithmic asymptotics of the
queue length's tail probabilities. As it turns out, in a rapidly changing
environment (i.e., is small relative to ) the overdispersion
of the arrival process hardly affects system behavior, whereas in a slowly
changing random environment it is fundamentally different; this general finding
applies to both the central limit and the large deviations regime. We extend
our results to the setting where each arrival creates a job in multiple
infinite-server queues
Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation
We consider a linear stochastic fluid network under Markov modulation, with a
focus on the probability that the joint storage level attains a value in a rare
set at a given point in time. The main objective is to develop efficient
importance sampling algorithms with provable performance guarantees. For linear
stochastic fluid networks without modulation, we prove that the number of runs
needed (so as to obtain an estimate with a given precision) increases
polynomially (whereas the probability under consideration decays essentially
exponentially); for networks operating in the slow modulation regime, our
algorithm is asymptotically efficient. Our techniques are in the tradition of
the rare-event simulation procedures that were developed for the sample-mean of
i.i.d. one-dimensional light-tailed random variables, and intensively use the
idea of exponential twisting. In passing, we also point out how to set up a
recursion to evaluate the (transient and stationary) moments of the joint
storage level in Markov-modulated linear stochastic fluid networks
Rare event analysis of Markov-modulated infinite-service queues: A Poisson limit
This paper studies an infinite-server queue in a Markov environment, that is, an infinite-server
queue with arrival rates and service times depending on the state of a Markovian background
process. Scaling the arrival rates by a factor and the rates of the background process by N^{1+\vareps}
(for some \vareps > 0), the focus is on the tail probabilities of the number of customers in the system, in
the asymptotic regime that tends to . In particular, it is shown that the logarithmic asymptotics
correspond to those of a Poisson distribution with an appropriate mean
Tail asymptotics of a Markov-modulated infinite-server queue
This paper analyzes large deviation probabilities related to the number of customers in
a Markov modulated infinite-server queue, with state-dependent arrival and service rates.
Two specific scalings are studied: in the first, just the arrival rates are linearly scaled by (for large ),
whereas in the second
in addition the Markovian background process is sped up by a factor , for some .
In both regimes, (transient and stationary) tail probabilities decay essentially exponentially,
where the associated decay rate corresponds to that of the probability
that the sample mean of i.i.d.\ Poisson random variables
attains an atypical value
Time-scaling limits for Markov-modulated infinite-server queues
In this paper we study semi-Markov modulated M/M/ queues, which are to be understood as infinite-server systems in which the Poisson input rate is modulated by a Markovian background process (where the times spent in each of its states are assumed deterministic), and the service times are exponential. Two specific scalings are considered, both in terms of transient and steady-state behavior. In the former the transition times of the background process are divided by , and then is sent to ; a Poisson limit is obtained. In the latter both the transition times and the Poissonian input rates are scaled, but the background process is sped up more than the arrival process; here a central-limit type regime applies. The accuracy and convergence rate of the limiting results are demonstrated with numerical experiments
Functional Large Deviations for Cox Processes and Queues, with a Biological Application
We consider an infinite-server queue into which customers arrive according to
a Cox process and have independent service times with a general distribution.
We prove a functional large deviations principle for the equilibrium queue
length process. The model is motivated by a linear feed-forward gene regulatory
network, in which the rate of protein synthesis is modulated by the number of
RNA molecules present in a cell. The system can be modelled as a tandem of
infinite-server queues, in which the number of customers present in a queue
modulates the arrival rate into the next queue in the tandem. We establish
large deviation principles for this queueing system in the asymptotic regime in
which the arrival process is sped up, while the service process is not scaled.Comment: 36 pages, 2 figures, to appear in Annals of Applied Probabilit
Markov-modulated infinite-server queues driven by a common background process
International audienceThis paper studies a system with multiple infinite-server queues which are modulated by a common background process. If this background process, being modeled as a finite-state continuous-time Markov chain, is in state j, then the arrival rate into the i-th queue is λi,j, whereas the service times of customers present in this queue are exponentially distributed with mean µ −1 i,j ; at each of the individual queues all customers present are served in parallel (thus reflecting their infinite-server nature). Three types of results are presented: in the first place (i) we derive differential equations for the probability generating functions corresponding to the distributions of the transient and stationary numbers of customers (jointly in all queues), then (ii) we set up recursions for the (joint) moments, and finally (iii) we establish a central limit theorem in the asymptotic regime in which the arrival rates as well as the transition rates of the background process are simultaneously growing large
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