922 research outputs found
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
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
Analysis of Markov-modulated infinite-server queues in the central-limit regime
This paper focuses on an infinite-server queue modulated by an independently
evolving finite-state Markovian background process, with transition rate matrix
. Both arrival rates and service rates are depending
on the state of the background process. The main contribution concerns the
derivation of central limit theorems for the number of customers in the system
at time , in the asymptotic regime in which the arrival rates
are scaled by a factor , and the transition rates by a
factor , with . The specific value of
has a crucial impact on the result: (i) for the system
essentially behaves as an M/M/ queue, and in the central limit theorem
the centered process has to be normalized by ; (ii) for ,
the centered process has to be normalized by , with the
deviation matrix appearing in the expression for the variance
A functional central limit theorem for a Markov-modulated infinite-server queue
The production of molecules in a chemical reaction network is modelled as a
Poisson process with a Markov-modulated arrival rate and an exponential decay
rate. We analyze the distributional properties of , the number of molecules,
under specific time-scaling; the background process is sped up by ,
the arrival rates are scaled by , for large. A functional central limit
theorem is derived for , which after centering and scaling, converges to an
Ornstein-Uhlenbeck process. A dichotomy depending on is observed. For
the parameters of the limiting process contain the deviation
matrix associated with the background process.Comment: 4 figure
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
Dynamic Service Rate Control for a Single Server Queue with Markov Modulated Arrivals
We consider the problem of service rate control of a single server queueing
system with a finite-state Markov-modulated Poisson arrival process. We show
that the optimal service rate is non-decreasing in the number of customers in
the system; higher congestion rates warrant higher service rates. On the
contrary, however, we show that the optimal service rate is not necessarily
monotone in the current arrival rate. If the modulating process satisfies a
stochastic monotonicity property the monotonicity is recovered. We examine
several heuristics and show where heuristics are reasonable substitutes for the
optimal control. None of the heuristics perform well in all the regimes.
Secondly, we discuss when the Markov-modulated Poisson process with service
rate control can act as a heuristic itself to approximate the control of a
system with a periodic non-homogeneous Poisson arrival process. Not only is the
current model of interest in the control of Internet or mobile networks with
bursty traffic, but it is also useful in providing a tractable alternative for
the control of service centers with non-stationary arrival rates.Comment: 32 Pages, 7 Figure
The single server semi-markov queue
A general model for the single server semi-Markov queue is studied. Its solution is reduced to a matrix factorization problem. Given this factorization, results are obtained for the distributions of actual and virtual waiting times, queue lengths both at arrival epochs and in continuous time, the number of customers during a busy period, its length and the length of a busy cycle. Two examples are discussed for which explicit factorizations have been obtained
- …