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Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

Published online by Cambridge University Press:  14 July 2016

J. George Shanthikumar  and
Ushio Sumita
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
Ushio Sumita*
Affiliation:
University of Rochester
*
∗Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
∗∗Postal address: William E. Simon Graduate School of Business Administration, University of Rochester, Rochester, NY 14627, USA.
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Abstract

We consider a discrete-time stochastic process {Wn, n≧0} governed by i.i.d random variables {ξ n} whose distribution has support on (–∞,∞) and replacement random variables {Rn} whose distributions have support on [0,∞). Given Wn, Wn+ 1 takes the value Wn + ζ n+ 1 if it is non-negative. Otherwise Wn+ 1 takes the value Rn +1 where the distribution of Rn+ 1 depends only on the value of Wn + ζn +1. This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn = 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

Keywords

QUEUES WITH SERVER VACATIONS OR PRIORITYERGODIC DECOMPOSITION THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 3 , September 1989 , pp. 552 - 565
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

Partially supported by NSF Grant ECS-8601210.

Partially suported by NSF Grant ECS-8600992 and by the IBM Program of Support for Education in Management of Information Systems.

References

Cinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Cooper, R. B. (1970) Queues served in cyclic order: waiting times. Bell Syst. Tech. J. 49, 399.Google Scholar
Doshi, B. (1985) A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times. J. Appl. Prob. 22, 419–428.Google Scholar
Doshi, B. (1986) Queueing systems with vacations — a survey. Queueing Systems 1, 29–66.Google Scholar
Gaver, D. P. (1962) A waiting line with interrupted service, including priorities. J. R. Statist. Soc. B24, 73–90.Google Scholar
Gelenbe, E. and Iasnogorodski, R. (1980) A queue with server of walking type (autonomous service). Ann. Inst. H. Poincaré 16, 63–73.Google Scholar
Heyman, D. P. (1968) Optimal operating policies for M/G/1 queueing systems. Operat. Res. 16, 362–382.CrossRefGoogle Scholar
Heyman, D. P. (1977) The T-policy for the MIG/1 queue. Management Sci. 23, 775–778.CrossRefGoogle Scholar
Jaiswal, N. K. (1968) Priority Queues. Academic Press, New York.Google Scholar
Keilson, J. (1965a) Green's Function Methods in Probability Theory. Griffin, London.Google Scholar
Keilson, J. (1965b) The role of Green's functions in congestion theory. In Proc. Symp. Congestion Theory, University of North Carolina Press, Chapel Hill.Google Scholar
Keilson, J. (1969) A queue model for interrupted communication. Opsearch 6, 59–67.Google Scholar
Keilson, J. and Servi, L. (1986) Oscillating random walk models for G/G/1 vacation systems with Bernoulli schedules. J. Appl. Prob. 23, 790–802.Google Scholar
Keilson, J. and Sumita, U. (1983) Evaluation of the total time in system in a preempt/resume priority queue via a modified Lindley process. Adv. Appl. Prob. 15, 840–856.Google Scholar
Kingman, J. F. C. (1964) The stochastic theory of regenerative events. Z. Wahrscheinlichkeitsth. 2, 180–224.Google Scholar
Lemoine, A. (1975) Limit theorems for generalized single server queues: the exceptional system. SIAM J. Appl. Math. 28, 596–606.Google Scholar
Levy, Y. and Yechiali, U. (1975) Utilization of idle time in an M/G/1 queueing system. Management Sci. 22, 202–211.Google Scholar
Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.Google Scholar
Minh, D. (1980) Analysis of the exceptional queueing system by use of regenerative processes and analytical methods. Math. Operat. Res. 5, 147–159.CrossRefGoogle Scholar
Neuts, M. F. (1986) Generalizations of the Pollaczek–Khinchin integral equation in the theory of queueing systems. Adv. Appl. Prob. 18, 952–990.Google Scholar
Pakes, A. G. (1972) A GI/M/1 queue with a modified service mechanism. Ann. Inst. Statist. Math. 24, 589–597.Google Scholar
Pakes, A. G. (1973) On the busy period of the modified GI/G/1 queue. J. Appl. Prob. 10, 192–197.Google Scholar
Shanthikumar, J. G. (1980) Some analyses on the control of queues using level crossings of regenerative processes. J. Appl. Prob. 17, 814–821.Google Scholar
Shanthikumar, J. G. (1982) Level crossing analysis of some variants of GI/M/1 queues. Opsearch 19, 148–159.Google Scholar
Sumita, U. and Liu Sheng, O. (1988) Analysis of query processing in distributed database systems with fully replicated files: A hierarchical approach. Performance Evaluation 8, 223–238.Google Scholar
Teghem, J. Jr. (1986) Control of the service processes in queueing system. Eur. J. Operat. Res. 23, 141–158.CrossRefGoogle Scholar
Van Der Duyn Schouten, F. A. (1978) An M/G/1 queueing model with vacation times. Z. Operat. Res. 22, 95–105.Google Scholar
Welch, P. D. (1964) On a generalized M/G/1 queueing process in which the first customer of each busy period receives exceptional service. Operat. Res. 12, 736–752.Google Scholar
Yadin, M. and Naor, P. (1963) Queueing systems with a removable service station. Operat. Res. Quart. 14, 393–405.CrossRefGoogle Scholar