Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-27T23:22:59.202Z Has data issue: false hasContentIssue false
  • English
  • Français

Markov-modulated single-server queueing systems

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

N. U. Prabhu  and
L. C Tang
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider single-server queueing systems that are modulated by a discrete-time Markov chain on a countable state space. The underlying stochastic process is a Markov random walk (MRW) whose increments can be expressed as differences between service times and interarrival times. We derive the joint distributions of the waiting and idle times in the presence of the modulating Markov chain. Our approach is based on properties of the ladder sets associated with this MRW and its time-reversed counterpart. The special case of a Markov-modulated M/M/1 queueing system is then analysed and results analogous to the classical case are obtained.

Keywords

MARKOV-MODULATED QUEUEING SYSTEMMARKOV RANDOM WALKSWAITING TIMEIDLE TIMETIMEREVERSIBILITY

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60K25: Queueing theory
Type
Part 3 Queueing Theory
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 169 - 184
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] Arjas, E. (1972) On the use of a fundamental identity in the theory of semi-Markov queues. Adv. Appl. Prob. 4, 271–284.Google Scholar
[2] Asmussen, S. (1991) Ladder heights and the Markov-modulated M/G/1 queue. Stoch. Proc. Appl. 37, 313–326.Google Scholar
[3] Feller, W. (1971) An Introduction to Probability Theory and Its Applications , Vol. II, 2nd edn. Wiley, New York.Google Scholar
[4] Heyde, C. C. (1967) A limit theorem for random walks with drift. J. Appl. Prob. 6, 129–134.Google Scholar
[5] Newbould, M. (1973) A classification of a random walk defined on a finite Markov chain. Z. Wahrscheinlichkeitsth. 26, 95–104.Google Scholar
[6] Prabhu, N. U. (1981) Stochastic Storage Processes. Springer-Verlag, New York.Google Scholar
[7] Prabhu, N. U. and Zhu, Y. (1989) Markov-modulated queueing systems. QUESTA 5, 215–245.Google Scholar
[8] Prabhu, N. U., Tang, L. C. and Zhu, Y. (1991) Some new results for the Markov random walk. J. Math. Phys. Sci. 25, 635–663.Google Scholar
[9] Presman, E. L. (1969) Factorization methods and boundary problems for sums of random variables given on Markov chains. Math. USSR Izvestija 3, 815–852.Google Scholar
[10] Takács, L. (1976) On fluctuation problems in the theory of queues. Adv. Appl. Prob. 8, 548583.Google Scholar
[11] Takács, L. (1978) On fluctuations of sums of random variables. In Studies in Probability and Ergodic Theory , ed. Rota, G.-C. Academic Press, New York.Google Scholar
[12] Tang, L. C. (1992) Limit theorems for Markov random walks. Technical report No. 1013, School of ORIE, Cornell University.Google Scholar