Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-29T04:31:04.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Duality Results in the Theory of Queues

Published online by Cambridge University Press:  14 July 2016

Irwin Greenberg
Irwin Greenberg*
Affiliation:
Technische Hogeschool Delft, Netherlands, and New York University, Bronx, N. Y.
Get access Rights & Permissions [Opens in a new window]

Abstract

When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to the M/G/1 – G/M/l queues.

Type
Research Papers
Information
Journal of Applied Probability , Volume 6 , Issue 1 , April 1969 , pp. 99 - 121
Copyright
Copyright © Applied Probability Trust 

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

Bhat, U. N. (1965) Customer overflow in queues with finite waiting spaces. Aust. J. Statist. 7, 15–19.Google Scholar
Cohen, J. W. (1967) Single server queue with uniformly bounded virtual waiting time J. Appl. Prob. 5, 93–122.Google Scholar
Cohen, J. W. and Greenberg, I. (1968) Distribution of crossings of level K in a busy cycle of the M/G/l queue. Ann. Inst. Poincaré. B 4, 75–81.Google Scholar
Daley, D. J. (1964) Single server queueing systems with uniformly limited queueing time. J. Aust. Math. Soc. 4, 489–505.Google Scholar
Hasofer, A. M. (1966) The almost full dam with Poisson input. J. R. Statist. Soc. B 28, 329–335, 448–455.Google Scholar
Prabhu, N. U. (1964) Queues and Inventories. John Wiley, New York.Google Scholar
Shanbhag, D. N. (1966) On a duality principle in the theory of queues. Operat. Res. 14, 947–949.Google Scholar
Takács, L. (1965) Application of ballot theorems in the theory of queues. In Congestion Theory (Editors Smith, W. L. and Wilkinson, W. E.), U. of North Carolina Press, Chapel Hill. 337–393.Google Scholar