Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T09:00:24.205Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotonicity results for MR/GI/1 queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Nicole Bäuerle
Nicole Bäuerle*
Affiliation:
University of Ulm
*
∗Postal address: Department of Mathematics VII, University of Ulm, D-89069 Ulm, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers queues with a Markov renewal arrival process and a particular transition matrix for the underlying Markov chain. We study the effect that the transition matrix has on the waiting time of the nth customer as well as on the stationary waiting time. The main theorem generalizes results of Szekli et al. (1994a) and partly confirms their conjecture. In this context we show the importance of a new stochastic ordering concept.

Keywords

MARKOV RENEWAL QUEUESSUPERMODULAR FUNCTIONSSTOCHASTIC ORDERING

MSC classification

Primary: 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 514 - 524
Copyright
Copyright © Applied Probability Trust 1997 

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

Bäuerle, N. (1994) Stochastic models with a Markovian environment. PhD dissertation. University of Ulm.Google Scholar
Chang, C., Chao, X., Pinedo, M. and Shanthikumar, J. (1991) Stochastic convexity for multidimensional processes and its application. Adv. Appl. Prob. 23, 210–228.Google Scholar
Daley, D. and Rolski, T. (1992) Finiteness of waiting-time moments in general stationary single-server queues. Ann. Appl. Prob. 2, 987–1008.Google Scholar
Kamae, T. and Krengel, U. (1978) Stochastic partial orderings. Ann. Prob. 6, 1044–1049.Google Scholar
Kamae, T., Krengel, U. and O'Brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
Loynes, R. (1962) The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 10, 497–520.Google Scholar
Marshall, A. and Olkin, I. (1979) Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
Meester, L. and Shanthikumar, J. (1993) Regularity of stochastic processes. Prob. Eng. Inf. Sci. 7, 343–360.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 inputs. Queueing Systems 5, 113–130.Google Scholar
Shaked, M. and Shanthikumar, J. (1990) Parametric stochastic convexity and concavity of stochastic processes. Ann. Inst. Statist. Math. 42, 509–531.Google Scholar
Shaked, M. and Shanthikumar, J. (1994) Stochastic Orders and Their Applications. Academic Press, New York.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.Google Scholar
Szekli, R., Disney, R. L. and Hur, S. (1994a) MR/GI/1 queues with positively correlated arrival stream. J. Appl. Prob. 31, 497–514.CrossRefGoogle Scholar
Szekli, R., Disney, R. L. and Hur, S. (1994b) On performance comparison of MR/GI/1 queues. Queueing Systems 17, 451–470.Google Scholar
Tchen, A. (1980) Inequalities for distributions with given marginals. Ann. Prob. 8, 814–827.Google Scholar