Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue

Hideaki Takagi

doi:10.3934/jimo.2017026

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2017, Volume 13, Issue 4: 1945-1973. Doi: 10.3934/jimo.2017026

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Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue

Hideaki Takagi^,

Professor Emeritus, University of Tsukuba, Faculty of Engineering, Information and Systems, 1-1-1 Tennoudai, Tsukuba-shi, Ibaraki 305-8573, Japan

* Corresponding author
* Corresponding author

Received: September 2015

Published: November 2017

The author is supported by the Grant-in-Aid for Scientific Research (C) No. 26330354 from the Japan Society for the Promotion of Science (JSPS) in 2015.

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Abstract

We present a unified and refined analysis of the response time and waiting time in the M/M/$ m $ FCFS preemptive-resume priority queueing system in the steady state by scrutinizing and extending the previous studies such as Brosh (1969), Segal (1970), Buzen and Bondi (1983), Tatashev (1984), and Zeltyn et al. (2009). In particular, we analyze the durations of interleaving waiting times and service times during the response time of a tagged customer of each priority class that is preempted by the arrivals of higher-priority class customers. Our new contribution includes the explicit formulas for the second and third moments of the response time and the third moment of the waiting time. We also clarify the dependence between the waiting time and the total service time. Numerical examples are shown in order to demonstrate the computation of theoretical formulas.

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Mathematics Subject Classification: Primary: 60K25; Secondary: 68M28, 90B22.

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Figure 1. Mean response for a customer of class $ p $

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Figure 2. Mean waiting time for a customer of class $ p $

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Figure 3. State transition diagram for a customer of class $ p $ until service completion.

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Figure 4. Second moment of the response time for a customer of class $ p $

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Figure 5. Third moment of the response time for a customer of class $ p $

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Figure 6. State transition diagram for a customer of class $ p $ until service preemption or completion.

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Figure 7. Second moment of the waiting time for a customer of class $ p $

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Figure 8. Third moment of the waiting time for a customer of class $ p $

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Figure 9. Covariance of the total waiting time and the total service time for a customer of class $ p $

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Table 1. Mean and the second and third moments of the initial waiting time $ W _p ^\circ $ and the waiting time $ W _p $ for a customer of class $ p = 4 $

$ \lambda $	$ E [W _4 ^\circ] $	$ E [W _4] $	$ E [( W _4 ^\circ ) ^2] $	$ E [( W _4 ) ^2] $	$ E [( W _4 ^\circ ) ^3] $	$ E [( W _4 ) ^3] $
1	0.00113	0.00306	0.00076	0.00208	0.00084	0.00233
2	0.02843	0.06234	0.03404	0.07668	0.06965	0.15964
3	0.21468	0.37324	0.51808	0.92966	2.12874	3.88666
4	1.38528	1.86222	9.00433	12.3304	95.5844	131.932
4.5	4.69227	5.47392	69.7454	81.9532	1623.17	1911.54
4.8	16.1018	17.1546	634.213	676.740	37923.0	40476.7

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References

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