Steady-State Distribution for the Buffer Content of an $M /G/1$ Queue with Varying Service Rate

Halfin, S.

doi:10.1137/0123037

Cited by 24 publications

(9 citation statements)

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“…Queues with variable service speed arise naturally in practice, and therefore many classical works can be found, e.g., [3,5,12,18,19]. Recently, this kind of queues has been attracting considerable attention again, because of its applicability to telecommunication problems.…”

Section: Introductionmentioning

confidence: 98%

Single-Server Queues with Markov-Modulated Arrivals and Service Speed

Takine

2005

Queueing Syst

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This paper considers single-server queues with several customer classes. Arrivals of customers are governed by the underlying continuous-time Markov chain with finite states. The distribution of the amount of work brought into the system on arrival is assumed to be general, which may differ with different classes. Further, the service speed depends on the state of the underlying Markov chain. We first show that given such a queue, we can construct the corresponding new queue with constant service speed by means of a change of time scale, and the time-average quantities of interest in the original queue are given in terms of those in the new queue. Next we characterize the joint distribution of the length of a busy period and the number of customers served during the busy period in the original queue. Finally, assuming the FIFO service discipline, we derive the Laplace-Stieltjes transform of the actual waiting time distribution in the original queue.

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Section: Introductionmentioning

confidence: 98%

Single-Server Queues with Markov-Modulated Arrivals and Service Speed

Takine

2005

Queueing Syst

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“…In our case, the service speeds depend on an external environment that is governed by a Markov process. Analyses of single-server queueing models with Markov-modulated service speeds can be found in [17,27,29,30,37]. However, none of these papers concern themselves with the derivation of heavy-traffic asymptotics.…”

Section: Introductionmentioning

confidence: 99%

Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

2014

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We study a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traffic limits for the distributions of the joint workload, waiting-time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traffic limits. The marginals of these limiting distributions are shown to be exponential with rates that can be computed by matrix-analytic methods. Moreover, we show how to numerically compute the joint distributions, by viewing the limit processes as multi-dimensional semi-martingale reflected Brownian motions in the non-negative orthant.

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“…Gaver (1962) obtained the generating functions for the stationary waiting time and the number in the system in an M/G/1 queue. Avi-Itzhak and Naor (1963) derived the expected queue length for M/G/1 queue with server breakdown, also see, Halfin (1972), Fischer (1977, and. Green (1986, 1988).…”

Section: Queues With Random Service Interruptionsmentioning

confidence: 99%