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A complete solution for the optimal stochastic scheduling of a two-stage tandem queue with two flexible servers

  • Klaus Schiefermayr (a1) and Josef Weichbold (a2)

Abstract

We consider a two-stage tandem queue with two parallel servers and two queues. We assume that initially all jobs are present and that no further arrivals take place at any time. The two servers are identical and can serve both types of job. The processing times are exponentially distributed. After being served, a job of queue 1 joins queue 2, whereas a job of queue 2 leaves the system. Holding costs per job and per unit time are incurred if there are jobs holding in the system. Our goal is to find the optimal strategy that minimizes the expected total holding costs until the system is cleared. We give a complete solution for the optimal control of all possible parameters (costs and service times), especially for those parameter regions in which the optimal control depends on how many jobs are present in the two queues.

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Copyright

Corresponding author

Postal address: Fachbereich Mathematik, Fachhochschule Wels, Stelzhamerstr. 23, Wels, A-4600, Austria. Email address: k.schiefermayr@fh-wels.at
∗∗ Postal address: Institut für Stochastik, Universität Linz, Altenbergerstr. 69, Linz, A-4040, Austria. Email address: josef.weichbold@jku.at

References

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[1] Ahn, H., Duenyas, I. and Lewis, M. E. (2002). Optimal control of a two-stage tandem queuing system with flexible servers. Prob. Eng. Inf. Sci. 16, 453469.
[2] Ahn, H., Duenyas, I. and Zhang, R. Q. (1999). Optimal stochastic scheduling of a two-stage tandem queue with parallel servers. Adv. Appl. Prob. 31, 10951117.
[3] Ahn, H., Duenyas, I. and Zhang, R. Q. (2004). Optimal control of a flexible server. Adv. Appl. Prob. 36, 139170.
[4] Farrar, T. M. (1993). Optimal use of an extra server in a two station tandem queueing network. IEEE Trans. Automatic Control 38, 12961299.
[5] Hajek, B. (1984). Optimal control of two interacting service stations. IEEE Trans. Automatic Control 29, 491499.
[6] Pandelis, D. G. and Teneketzis, D. (1994). Optimal multiserver stochastic scheduling of two interconnected priority queues. Adv. Appl. Prob. 26, 258279.
[7] Rosberg, Z., Varaiya, P. P. and Walrand, J. C. (1982). Optimal control of service in tandem queues. IEEE Trans. Automatic Control 27, 600610.
[8] Schiefermayr, K. and Weichbold, J. (2005). The optimal control of a general tandem queue. To appear in Prob. Eng. Inf. Sci.

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