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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

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    * Corresponding author
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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  • 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.

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

    Citation:

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

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

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

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

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

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

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

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

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

    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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  • [1] I. Brosh, Preemptive priority assignment in multichannel systems, Operations Research, 17 (1969), 526-535.  doi: 10.1287/opre.17.3.526.
    [2] J. P. Buzen and A. B. Bondi, The response times of priority classes under preemptive resume in M/M/m queues, Operations Research, 31 (1983), 456-465.  doi: 10.1287/opre.31.3.456.
    [3] R. B. Cooper, Introduction to Queueing Theory, 2nd edition, Elsevier North Holland, New York, 1981.
    [4] M. Fujiki, (Japanese) Fundamental theory and application on communication traffic. 5 queueing theory (part 2), Transactions of the Institute of Electronics and Communication Engineers of Japan, 55 (1972), 1194-1200. 
    [5] D. P. Gaver Jr., A waiting line with interrupted service, including priorities, Journal of the Royal Statistical Society, Series B (Methodological), 24 (1962), 73-90. 
    [6] V. G. Kulkarni, Modeling and Analysis of Stochastic Systems, Chapman & Hall, Boca Raton, Florida, 1995.
    [7] M. Segal, A multiserver system with preemptive priorities, Operations Research, 18 (1970), 316-323.  doi: 10.1287/opre.18.2.316.
    [8] H. Takagi, Detailed analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue, in Queueing Theory and Network Applications (eds. T. V. Do, Y. Takahashi, W. Yue and V.-Ha Nguen), Springer, (2016), 3–17.
    [9] H. Takagi, Analysis of the response and waiting times in the M/M/m LCFS preemptive-resume priority queue, International Journal of Pure and Applied Mathematics, 109 (2016), 325-370.  doi: 10.12732/ijpam.v.109i2.12.
    [10] A. G. Tatashev, Calculation of the distribution of the waiting time in a multiple-channel queueing system with fixed priorities, Engineering Cybernetics, 22 (1984), 59-62, (Originally published in Tekhnicheskaya Kibernetika, 1983, 163--166). 
    [11] H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, 3rd edition, Academic Press, San Diego, California, 1998.
    [12] H. White and L. S. Christie, Queuing with preemptive priorities or with breakdown, Operations Research, 6 (1958), 79-95.  doi: 10.1287/opre.6.1.79.
    [13] S. ZeltynZ. Feldman and S. Wasserkrug, Waiting and sojourn times in a multi-server queue with mixed priorities, Queueing Systems, 61 (2009), 305-328.  doi: 10.1007/s11134-009-9110-4.
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