Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-01T04:17:48.491Z Has data issue: false hasContentIssue false
  • English
  • Français

MR/GI/1 queues by positively correlated arrival stream

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

R. Szekli ,
R. L. Disney  and
S. Hur
R. Szekli*
Affiliation:
Wroclaw University
R. L. Disney*
Affiliation:
Texas A & M University
S. Hur*
Affiliation:
Texas A & M University
*
∗ Postal address: Mathematical Institute, Wrocław University, 50-384 Wrocław, pl. Grunwaldzki 2/4, Poland.
∗∗ Postal address: Department of Industrial Engineering, Texas A & M University, College Station, TX 77843-3131, USA.
∗∗ Postal address: Department of Industrial Engineering, Texas A & M University, College Station, TX 77843-3131, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times are shown.

Keywords

MARKOV RENEWAL QUEUESCORRELATIONSASSOCIATIONSTOCHASTIC ORDERING

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 497 - 514
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

Barlow, R. E. and Proschan, F. (1981) Statistical Theory of Life Testing: Probability Models. Silver Spring, Maryland.Google Scholar
ÇInlar, E. (1967) Queues with semi-Markovian arrivals. J. Appl. Prob. 4, 365–379.Google Scholar
ÇInlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Daley, D. J. and Rolski, T. (1992) Finiteness of waiting-time moments in general stationary single-server queues. Ann. Appl. Prob. 2, 987–1008.CrossRefGoogle Scholar
Disney, R. L. and Kiessler, P. C. (1987) Traffic Processes in Queueing Networks, A Markov Renewal approach. The Johns Hopkins University Press. Baltimore.Google Scholar
Hadley, G. (1961) Linear Algebra. Addison-Wesley, Reading, Mass.Google Scholar
Jagerman, D. and Melamed, B. (1992a) The transition and autocorrelation structure of TES processes. Part I: general theory. Commun. Statist.-Stoch. Models 8, 193–219.CrossRefGoogle Scholar
Jagerman, D. and Melamed, B. (1992b) The transition and autocorrelation structure of TES processes. Part II: special cases. Commun. Statist.-Stoch. Models 8, 499–527.CrossRefGoogle Scholar
Karlin, S. and Rinott, Y. (1980) Classes of orderings of measures and related correlation inequalities. I, Multivariate totally positive distributions. J. Multivariate Anal. 10, 467–498.CrossRefGoogle Scholar
Leon, R. and Lynch, J. (1983) Preservation of life distribution classes under reliability operations. Math. Operat. Res. 8, 159–169.Google Scholar
Livny, M., Melamed, B. and Tsiolis, A. K. (1992) The impact of autocorrelation on queueing systems. Management Sci.Google Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
Meester, L. and Shanthikumar, J. G. (1992) Regularity of stochastic processes: a theory based on directional convexity. Prob. Eng. Inf. Sci.Google Scholar
Neuts, M. F. (1981) Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press. Baltimore.Google Scholar
Patuwo, B. E., Disney, R. L. and Mcnickle, D. C. (1991) The effect of correlated arrivals on queues.Google Scholar
Rolski, T. (1983) Comparison theorems for queues with dependent interarrival times. In Modelling and Performance Evaluation, Proceedings of the International Seminar, Paris, 42–67.Google Scholar
Rolski, T. (1986) Upper bounds for single server queues with doubly stochastic Poisson arrivals. Math. Operat. Res. 11, 442–450.Google Scholar
Rolski, T. (1989) Queues with nonstationary arrivals. QUESTA 5, 113–130.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, Berlin.Google Scholar
Szekli, R., Disney, R. L. and Hur, S. (1992) On performance comparison of MR/GI/1 queues. Technical Report, INEN/OR/WP/14/6-92, Texas A&M University, College Station, TX.Google Scholar
Tchen, A. (1980) Inequalities for distributions with given marginals. Ann. Prob. 8, 811–827.CrossRefGoogle Scholar