19 research outputs found

    Rare event analysis of Markov-modulated infinite-server queues: a Poisson limit

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    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

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    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 ΒΏiΒΏi by a factor NN and the transition rates ΒΏijΒΏij 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-service queues: A Poisson limit

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    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 λi\lambda_i by a factor NN and the rates νij\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 NN tends to ∞\infty. In particular, it is shown that the logarithmic asymptotics correspond to those of a Poisson distribution with an appropriate mean

    A functional central limit theorem for a Markov-modulated infinite-server queue

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    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 MM, the number of molecules, under specific time-scaling; the background process is sped up by NΞ±N^{\alpha}, the arrival rates are scaled by NN, for NN large. A functional central limit theorem is derived for MM, which after centering and scaling, converges to an Ornstein-Uhlenbeck process. A dichotomy depending on Ξ±\alpha is observed. For α≀1\alpha\leq1 the parameters of the limiting process contain the deviation matrix associated with the background process.Comment: 4 figure

    Explicit computations for some Markov modulated counting processes

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    In this paper we present elementary computations for some Markov modulated counting processes, also called counting processes with regime switching. Regime switching has become an increasingly popular concept in many branches of science. In finance, for instance, one could identify the background process with the `state of the economy', to which asset prices react, or as an identification of the varying default rate of an obligor. The key feature of the counting processes in this paper is that their intensity processes are functions of a finite state Markov chain. This kind of processes can be used to model default events of some companies. Many quantities of interest in this paper, like conditional characteristic functions, can all be derived from conditional probabilities, which can, in principle, be analytically computed. We will also study limit results for models with rapid switching, which occur when inflating the intensity matrix of the Markov chain by a factor tending to infinity. The paper is largely expository in nature, with a didactic flavor

    Pseudo steady-state period in non-stationary infinite-server queue with state dependent arrival intensity

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    An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period (PSSP). Distribution of duration of PSSP, its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation

    Tail asymptotics of a Markov-modulated infinite-server queue

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    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 NN (for large NN), whereas in the second in addition the Markovian background process is sped up by a factor N1+Ο΅N^{1+\epsilon}, for some Ο΅>0\epsilon>0. 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

    Analysis of Markov-modulated infinite-server queues in the central-limit regime

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    This paper focuses on an infinite-server queue modulated by an independently evolving finite-state Markovian background process, with transition rate matrix Q≑(qij)i,j=1dQ\equiv(q_{ij})_{i,j=1}^d. 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 tβ‰₯0t\ge 0, in the asymptotic regime in which the arrival rates Ξ»i\lambda_i are scaled by a factor NN, and the transition rates qijq_{ij} by a factor NΞ±N^\alpha, with α∈R+\alpha \in \mathbb R^+. The specific value of Ξ±\alpha has a crucial impact on the result: (i) for Ξ±>1\alpha>1 the system essentially behaves as an M/M/∞\infty queue, and in the central limit theorem the centered process has to be normalized by N\sqrt{N}; (ii) for Ξ±<1\alpha<1, the centered process has to be normalized by N1βˆ’Ξ±/2N^{{1-}\alpha/2}, with the deviation matrix appearing in the expression for the variance
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