Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T05:02:28.741Z Has data issue: false hasContentIssue false
  • English
  • Français

On the ordering of tandem queues with exponential servers

Published online by Cambridge University Press:  14 July 2016

Tapani Lehtonen
Tapani Lehtonen*
Affiliation:
Helsinki School of Economics
*
∗Postal address: Helsinki School of Economics, Runebergink. 22–24, 00100 Helsinki 10, Finland.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider tandem queues which have a general arrival process. The queueing system consists of s (s ≧ 2) single-server service stations and the servers have exponential service-time distributions. Firstly we give a new proof for the fact that the departure process does not depend on the particular allocation of the servers to the stations. Secondly, considering the service rates, we prove that the departure process becomes stochastically faster as the homogeneity of the servers increases in the sense of a given condition. It turns out that, given the sum of the service rates, the departure process is stochastically fastest in the case where the servers are homogeneous.

Keywords

EXPONENTIAL DISTRIBUTIONDEPARTURE PROCESSSTOCHASTIC ORDER(WEAK) MAJORIZATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 23 , Issue 1 , March 1986 , pp. 115 - 129
Copyright
Copyright © Applied Probability Trust 1986 

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

Arjas, E. and Lehtonen, T. (1978) Approximating many server queues by means of single server queues. Math. Operat. Res. 3, 205–223.CrossRefGoogle Scholar
Bhat, U. N. and Shalaby, M. (1979) Approximation techniques in the solution of queueing problems. Naval Res. Logist. Quart. 26, 311–326.CrossRefGoogle Scholar
Burke, P. J. (1956) The output of a queueing system. Operat. Res. 4, 699–704.CrossRefGoogle Scholar
Jackson, R. R. P. (1954) Queueing systems with phase-type service. Operat. Res. Quart. 5, 109–120.CrossRefGoogle Scholar
Jacobs, O. R. Jr and Schach, S. (1972) Stochastic order relationships between GI/G/k systems. Ann. Math. Statist. 43, 1623–1633.CrossRefGoogle Scholar
Jakuschev, J. F. (1973) On two node queueing systems in series with renewal arrival process (in Russian). Tech. Cybernet. 3, 47–51.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Krämer, W. (1973) Total waiting time distribution function and the fate of a customer in a system with two queues in series. In Proc. 7th Internat. Teletraffic Congress .Google Scholar
Lehtonen, T. (1983) Stochastic comparisons for many server queues with non-homogeneous exponential servers. Opsearch 20, 1–15.Google Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
Naumov, V. A. (1976) On the independent behavior of subsystems of complicated systems (in Russian). In Papers of the 3rd Seminar Meeting on Queueing Theory , Puschtschino, Publishing House of the Moscow State University, 169–177.Google Scholar
Niu, S. C. (1981) On the comparison of waiting times in tandem queues. J. Appl. Prob. 18, 707–714.CrossRefGoogle Scholar
Pinedo, ?. (1982) On the optimal order of stations in tandem queues. Proc. Conf. Applied Probability, and Computer Science: The Interface , ed. Disney, R. L. and Ott, T. J., Birkhaüser, Boston.Google Scholar
Pinedo, M. and Wolff, R. W. (1982) A comparison of tandem queues with dependent and independent service times. Operat. Res. 30, 464–479.CrossRefGoogle Scholar
Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.CrossRefGoogle Scholar
Stoyan, D. (1977) Qualitative Eigenschaften und Abschätzungen stochastischer Modelle. Academie-Verlag, Berlin.CrossRefGoogle Scholar
Stoyan, D. and Stoyan, H. (1969) Monotonieeigenschaften der Kundenwartezeiten im Modell GI/G/1. Z. angew. Math. Mech. 49, 729–734.CrossRefGoogle Scholar
Tembe, S. V. and Wolff, R. W. (1974) The optimal order of service in tandem queues. Operat. Res. 22, 824–832.CrossRefGoogle Scholar
Weber, R. R. (1979) The interchangeability of tandem ./M/1 queues in series. J. Appl. Prob. 16, 690–695.CrossRefGoogle Scholar