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A Duality Relation Between the Workload and Attained Waiting Time in FCFS G/G/s Queues

Part of: Special processes

Published online by Cambridge University Press:  30 January 2018

Yi-Ching Yao
Yi-Ching Yao*
Affiliation:
Academia Sinica and National Chengchi University
*
∗ Postal address: Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan. Email address: yao@stat.sinica.edu.tw
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Abstract

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Sengupta (1989) showed that, for the first-come–first-served (FCFS) G/G/1 queue, the workload and attained waiting time of a customer in service have the same stationary distribution. Sakasegawa and Wolff (1990) derived a sample path version of this result, showing that the empirical distribution of the workload values over a busy period of a given sample path is identical to that of the attained waiting time values over the same period. For a given sample path of an FCFS G/G/s queue, we construct a dual sample path of a dual queue which is FCFS G/G/s in reverse time. It is shown that the workload process on the original sample path is identical to the total attained waiting time process on the dual sample path. As an application of this duality relation, we show that, for a time-stationary FCFS M/M/s/k queue, the workload process is equal in distribution to the time-reversed total attained waiting time process.

Keywords

dualityworkload processsample pathreverse time

MSC classification

Primary: 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 300 - 307
Copyright
Copyright © Applied Probability Trust 2013 

References

Cohen, J. W. (1991). On the attained waiting time. Adv. Appl. Prob. 23, 660–661.Google Scholar
Sakasegawa, H. and Wolff, R. W. (1990). The equality of the virtual delay and attained waiting time distribution. Adv. Appl. Prob. 22, 257–259.Google Scholar
Sengupta, B. (1989). An invariance relationship for the G/G/1 queue. Adv. Appl. Prob. 21, 956–957.Google Scholar
Toyoizumi, H. (1997). Sengupta's invariant relationship and its application to waiting time inference. J. Appl. Prob. 34, 795–799.Google Scholar
Yamazaki, G. and Miyazawa, M. (1991). The equality of the workload and total attained waiting time in average. J. Appl. Prob. 28, 238–244.Google Scholar