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October  2017, 13(4): 1945-1973. doi: 10.3934/jimo.2017026

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

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

* Corresponding author

Received  September 2015 Published  April 2017

Fund Project: 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.

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.

Citation: Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026
##### References:

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The reviewing process of the paper was handled by Wuyi Yue and Yutaka Takahashi as Guest Editors.

##### References:
Mean response for a customer of class $p$
Mean waiting time for a customer of class $p$
State transition diagram for a customer of class $p$ until service completion.
Second moment of the response time for a customer of class $p$
Third moment of the response time for a customer of class $p$
State transition diagram for a customer of class $p$ until service preemption or completion.
Second moment of the waiting time for a customer of class $p$
Third moment of the waiting time for a customer of class $p$
Covariance of the total waiting time and the total service time for a customer of class $p$
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
 $\lambda$ $E [W _4 ^\circ]$ $E [W _4]$ $E [( W _4 ^\circ ) ^2]$ $E [( W _4 ) ^2]$ $E [( W _4 ^\circ ) ^3]$ $E [( W _4 ) ^3]$ 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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