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Matrix product-form solutions for Markov chains with a tree structure

Part of: Special processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Raymond W. Yeung  and
Bhaskar Sengupta
Raymond W. Yeung*
Affiliation:
The Chinese University of Hong Kong
Bhaskar Sengupta*
Affiliation:
C & C Research Laboratories, NEC USA
*
* Postal address: Department of Information Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. e-mail:whyeung@ie.cuhk.hk
** Postal address: C & C Research Labs., NEC USA, 4 Independence Way, Princeton, NJ 08540, USA. e-mail:bhaskar@ccrl.nj.nec.com
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Abstract

We have two aims in this paper. First, we generalize the well-known theory of matrix-geometric methods of Neuts to more complicated Markov chains. Second, we use the theory to solve a last-come-first-served queue with a generalized preemptive resume (LCFS-GPR) discipline. The structure of the Markov chain considered in this paper is one in which one of the variables can take values in a countable set, which is arranged in the form of a tree. The other variable takes values from a finite set. Each node of the tree can branch out into d other nodes. The steady-state solution of this Markov chain has a matrix product-form, which can be expressed as a function of d matrices Rl,· ··, Rd. We then use this theory to solve a multiclass LCFS-GPR queue, in which the service times have PH-distributions and arrivals are according to the Markov modulated Poisson process. In this discipline, when a customer's service is preempted in phase j (due to a new arrival), the resumption of service at a later time could take place in a phase which depends on j. We also obtain a closed form solution for the stationary distribution of an LCFS-GPR queue when the arrivals are Poisson. This result generalizes the known result on a LCFS preemptive resume queue, which can be obtained from Kelly's symmetric queue.

Keywords

MATRIX-GEOMETRIC METHODSMULTICLASS LCFS-GPR QUEUE

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 26 , Issue 4 , December 1994 , pp. 965 - 987
Copyright
Copyright © Applied Probability Trust 1994 

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