The FIFO single-server queue with disasters andmultiple Markovian arrival streams

Yoshiaki Inoue; Tetsuya Takine

doi:10.3934/jimo.2014.10.57

Article Contents

2014, Volume 10, Issue 1: 57-87. Doi: 10.3934/jimo.2014.10.57

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The FIFO single-server queue with disasters and multiple Markovian arrival streams

Yoshiaki Inoue^1, and
Tetsuya Takine^1,

1.
Department of Information and Communications Technology, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita 565-0871

Received: September 2012

Revised: June 2013

Published: January 2014

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Abstract

We consider a FIFO single-server queue with disasters and multiple Markovian arrival streams. When disasters occur, all customers are removed instantaneously and the system becomes empty. Both the customer arrival and disaster occurrence processes are assumed to be Markovian arrival processes (MAPs), and they are governed by a common underlying Markov chain with finite states. There are $K$ classes of customers, and the amounts of service requirements brought by arriving customers follow general distributions, which depend on the customer class and the states of the underlying Markov chain immediately before and after arrivals. For this queue, we first analyze the first passage time to the idle state and the busy cycle. We then obtain two different representations of the Laplace-Stieltjes transform of the stationary distribution of work in system, and discuss the relation between those. Furthermore, using the result on the workload distribution, we analyze the waiting time and sojourn time distributions, and derive the joint queue length distribution.

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Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22.

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