Hostname: page-component-89b8bd64d-4ws75 Total loading time: 0 Render date: 2026-05-09T02:52:08.386Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

Information

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

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Anantharam, V. (1987). Probabilistic proof of interchangeability of /M/l queues in series. Queueing Systems, Theory, and Application 12 (4).Google Scholar
Avi-Itzhak, B. & Yadin, M. (1965). A sequence of two servers with no intermediate queue. Management Science 11: 553–564.CrossRefGoogle Scholar
Chao, X. & Pinedo, M. (1988). On two servers in tandem with no intermediate buffer and batches arriving according to a Poisson process. Research Report, Department of Industrial Engineering and Operations Research, Columbia University, New York.Google Scholar
Dattatreya, E.S. (1978). Tandem queueing systems with blocking. Ph.D. Dissertation, Operations Research Center, University of California, Berkeley.Google Scholar
Greenberg, B.S. & Wolff, R.W. (1988). Optimal order of servers for tandem queues in light traffic. Management Science 34: 500–508.CrossRefGoogle Scholar
Hillier, F.S. & Boling, R.W. (1979). Optimal allocation of work in production line systems with variable operations times. Management Science 25: 721–729.CrossRefGoogle Scholar
Langaris, C. & Conolly, B. (1984). On the waiting time of a two-stage queueing system with blocking. Journal of Applied Probability 21: 628–638.CrossRefGoogle Scholar
Lehtonen, T. (1986). On the ordering of tandem queues with exponential servers. Journal of Applied Probability 23: 115–129.CrossRefGoogle Scholar
Melamed, B. (1986). A note on the reversibility and duality of some tandem blocking queueing systems. Management Science 32: 128–130.CrossRefGoogle Scholar
Muth, E.J. (1979). The reversibility property of production lines. Management Science 25: 152–158.CrossRefGoogle Scholar
Pinedo, M. (1982). On the optimal order of stations in tandem queues. In Disney, R.L. & Ott, T.J. (eds.), Applied probability-computer science: The interface, Boston, MA: Birkhauser, pp. 307–326.CrossRefGoogle Scholar
Tembe, S. & Wolff, R.W. (1974). Optimal order of servers in tandem queues. Operations Research 22: 824–832.CrossRefGoogle Scholar
Tsoucas, P. & Walrand, J. (1987). On the interchangeability and stochastic ordering of /M/1 queues in tandem. Advances in Applied Probability 19: 515–520.CrossRefGoogle Scholar
Weber, R.R. (1979). The interchangeability of /M/l queues in series. Journal of Applied Probability 16: 690–695.CrossRefGoogle Scholar
Wein, L.M. (1988). Ordering tandem queues in heavy traffic. Research Report, Sloan School of Management, Massachusetts Institute of Technology.Google Scholar