Single-Server Queueing System with Markov-Modulated Arrivals and Service Times

Key words: Markov-modulated queues, waiting time, heavy traffic.Markov-modulated queueing systems are those in which the input process or service mechanism is influenced by an underlying Markov chain. Several models for such systems have been investigated. In this paper we present heavy traffic analysis of single queueing system with Poisson arrival process whose arrival rate is a function of the state of Markov chain and service times depend on the state of the same Markov chain at the epoch of arrivals

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

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