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 $\lambda_i$ by a factor $N$ and the rates $\nu_{ij}$ 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 $N$ tends to $\infty$. In particular, it is shown that the logarithmic asymptotics correspond to those of a Poisson distribution with an appropriate mean

Central Limit Theorems for Markov-modulated infinite-server queues

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 an independently evolving Markovian background process. Scaling the arrival rates $\lambda_i$ by a factor $N$ and the rates $q_{ij}$ of the background process by a factor $N^\alpha$, with $\alpha \in \mathbb R^+$, we establish a central limit theorem as $N$ tends to $\infty$. We find different scaling regimes, which depend on the specific value of $\alpha$. Remarkably, for $\alpha<1$, we find a central limit theorem in which the centered process has to be normalized by $N^{{1-}\alpha/2}$ rather than $\sqrt{N}$; in the expression for the variance deviation matrices appear

Markov-modulated infinite-server queues: approximations by time-scaling

This paper studies an infinite-server queue in a semi-Markov environment: the queue's input rate is modulated by a semi-Markovian background process, and the service times are assumed to be exponentially distributed. The primary objective of this paper is to propose approximations for the queue-length distribution, based on time-scaling arguments. The analysis starts with an explicit analysis of the cases in which the transition times of the modulating semi-Markov process are either all deterministic or all exponential. We use these results to obtain approximations under time-scalings; there we consider subsequently a quasi-stationary regime (in which time is slowed down) and fluid-scaling regime (in which time is sped up) are examined. Notably, in the latter regime, the limiting distribution of the number of customers present is Poisson, irrespective of the distribution of the transition times. The accuracy of the resulting approximations is illustrated by several numerical experiments, that moreover give an indication of the speed of convergence in the both regimes, for various distributions of the transition times. The last section derives conditions under which the distribution of the number of customers present is Poisson (in the exact sense, i.e., not in a limiting regime)

Time-scaling limits for Markov-modulated infinite-server queues

In this paper we study semi-Markov modulated M/M/$\infty$ 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 $N$, and then $N$ is sent to $\infty$; 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

Refined large deviations asymptotics for Markov-modulated infinite-server systems

Many networking-related settings can be modeled by Markov-modulated infinite-server systems. In such models, the customers’ arrival rates and service rates are modulated by a Markovian background process; additionally, there are infinitely many servers (and consequently the resulting model is often used as a proxy for the corresponding many-server model). The Markov-modulated infinite-server model hardly allows any explicit analysis, apart from results in terms of systems of (ordinary or partial) differential equations for the underlying probability generating functions, and recursions to obtain all moments. As a consequence, recent research efforts have pursued an asymptotic analysis in various limiting regimes, notably the central-limit regime (describing fluctuations around the average behavior) and the large-deviations regime (focusing on rare events). Many of these results use the property that the number of customers in the system obeys a Poisson distribution with a random parameter. The objective of this paper is to develop techniques to accurately approximate tail probabilities in the large-deviations regime. We consider the scaling in which the arrival rates are inflated by a factor N, and we are interested in the probability that the number of customers exceeds a given level Na. Where earlier contributions focused on so-called logarithmic asymptotics of this exceedance probability (which are inherently imprecise), the present paper improves upon those results in that exact asymptotics are established. These are found in two steps: first the distribution of the random parameter of the Poisson distribution is characterized, and then this knowledge is used to identify the exact asymptotics. The paper is concluded by a set of numerical experiments, in which the accuracy of the asymptotic results is assessed