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On the Interchangeability and Stochastic Ordering of Exponential Queues in Tandem with Blocking

Published online by Cambridge University Press:  27 July 2009

Xiuli Chao ,
Michael Pinedo  and
Karl Sigman
Xiuli Chao
Affiliation:
Center for Telecommunications Research and Department of Industrial Engineering and Operations Research Columbia University, New York, New York 10027
Michael Pinedo
Affiliation:
Center for Telecommunications Research and Department of Industrial Engineering and Operations Research Columbia University, New York, New York 10027
Karl Sigman
Affiliation:
Center for Telecommunications Research and Department of Industrial Engineering and Operations Research Columbia University, New York, New York 10027
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Extract

For a two-station tandem system with a general arrival process, exponential ervice times, and blocking, we show that the distribution of the departure process does not change when the two stations are interchanged. Blocking here means that for some fixed b≥1, any customer completing service at the first station when b customers are at the second station cannot enter the second queue, and the first station cannot start serving a new customer until a service completion occurs at the second station. This result remains true if arrivals are lost when there are a(a≥0) customers in the system. We also show that when the sum of the service rates is held constant, each departure epoch is stochastically minimized if the two rates are equal. For a and b infinite, our results reduce to those given by Weber [14] and Lehtonen [8]. Our proof is based on a coupling method first used by Lehtonen. The same results hold for a blocking mechanism in which a customer completing its service at the first station must restart service when b customers are present at the second.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 3 , Issue 2 , April 1989 , pp. 223 - 236
Copyright
Copyright © Cambridge University Press 1989

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References

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