Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T23:15:15.290Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected]
** Postal address: C & C Research Labs., NEC USA, 4 Independence Way, Princeton, NJ 08540, USA. e-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[2] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, New York.Google Scholar
[3] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[4] Loynes, R. M. (1962) The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.CrossRefGoogle Scholar
[5] Neuts, M. F. (1981) Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD.Google Scholar
[6] Sengupta, B. (1989) Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue. Adv. Appl. Prob. 21, 159–180.CrossRefGoogle Scholar
[7] Takine, T., Sengupta, B. and Yeung, R. W. (1994) A generalization of the matrix M/G/1 paradigm for Markov chains with a tree structure. Submitted for publication.Google Scholar
[8] Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.Google Scholar
[9] Tweedie, R. L. (1982) Operator-geometric stationary distributions for Markov chains, with applications to queueing models. Adv. Appl. Prob. 14, 368–391.CrossRefGoogle Scholar
[10] Walrand, J. (1988) An Introduction to Queueing Networks. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[11] Wolff, R. A. (1982) Poisson arrivals see time averages. Operat. Res. 30, 223–231.Google Scholar