Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-19T08:09:46.697Z Has data issue: false hasContentIssue false
  • English
  • Français

Concavity of the throughput of tandem queueing systems with finite buffer storage space

Published online by Cambridge University Press:  01 July 2016

Ludolf E. Meester  and
J. George Shanthikumar
Ludolf E. Meester*
Affiliation:
University of California, Berkeley
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
*
∗Postal address: Department of Statistics, University of California, Berkeley, CA 94720, USA.
∗∗Postal address: Walter A. Haas School of Business, University of California, Berkeley, CA 94720, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

Keywords

BLOCKINGSAMPLE PATH CONSTRUCTIONSTOCHASTIC CONCAVITYBUFFER ALLOCATION
Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 22 , Issue 3 , September 1990 , pp. 764 - 767
Copyright
Copyright © Applied Probability Trust 1990 

References

Anantharam, V. and Tsoucas, P. (1990) Stochastic concavity of throughput in series of queues with finite buffers Adv. Appl. Prob. 22, 761–763.Google Scholar
Hillier, F. S., Boling, R. W. and So, K. C. (1986) Toward characterizing the optimal allocation of storage space in production line systems with variable operation times. Technical report, Department of Operations Research, Stanford University, Stanford, CA.Google Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Perros, H. G. (1986) A survey of queueing networks with blocking, Technical report, Department of Computer Science, North Carolina State University, Raleigh, NC 27695.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1988) Stochastic convexity and its applications. Adv. Appl. Prob. 20, 427–446.Google Scholar
Shanthikumar, J. G. and Yao, D. D. (1991) Strong stochastic convexity: Closure properties and applications. J. Appl. Prob. 28, to appear.Google Scholar
Yamazaki, G. and Sakasegawa, H. (1975) Properties of duality in tandem queueing systems. Ann. Inst. Statist. Math. 27, 201–212.Google Scholar