# American Institute of Mathematical Sciences

October  2018, 14(4): 1701-1726. doi: 10.3934/jimo.2018028

## Times until service completion and abandonment in an M/M/$m$ preemptive-resume LCFS queue with impatient customers

 Professor Emeritus, University of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan

The reviewing process of this paper was handled by Yutaka Takahashi and Wuyi Yue

Received  January 2017 Revised  June 2017 Published  February 2018

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 the academic year 2016

We consider an M/M/$m$ preemptive-resume last-come first-served (PR-LCFS) queue without exogenous priority classes of impatient customers. We focus on analyzing the time interval from the arrival to either service completion or abandonment for an arbitrary customer. We formulate the problem as a one-dimensional birth-and-death process with two absorbing states, and consider the first passage times in this process. We give explicit expressions for the probabilities of service completion and abandonment. Furthermore, we present sets of recursive computational formulas for calculating the mean and second moment of the times until service completion and abandonment. The two special cases of a preemptive-loss system and an ordinary M/M/$m$ queue with patient customers only, both incorporating the preemptive LCFS discipline, are treated separately. We show some numerical examples in order to demonstrate the computation of theoretical formulas.

Citation: Hideaki Takagi. Times until service completion and abandonment in an M/M/$m$ preemptive-resume LCFS queue with impatient customers. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1701-1726. doi: 10.3934/jimo.2018028
##### References:
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##### References:
 [1] R. B. Cooper, Introduction to Queueing Theory, 2$^{nd}$ edition, Elsevier North Holland, New York, 1981. Google Scholar [2] N. Gautam, Analysis of Queues: Methods and Applications, CRC Press, Boca Raton, Florida, 2012.Google Scholar [3] B. V. Gnedenko and I. N. Kovalenko, Introduction to Queueing Theory, 2$^{nd}$ edition, revised and supplemented. Translated by Samuel Kotz, Springer-Verlag, New York, 1994.Google Scholar [4] F. Iravani and B. Balcio${\tilde {\rm g}}$lu, On priority queues with impatient customers, Queueing Systems, 58 (2008), 239-260. doi: 10.1007/s11134-008-9069-6. Google Scholar [5] D. L. Jagerman, Difference Equations with Applications to Queues, Marcel Dekker, New York, 2000. Google Scholar [6] O. Jouini, Analysis of a last come first served queueing system with customer abandonment, Computers & Operations Research, 39 (2012), 3040-3045. doi: 10.1016/j.cor.2012.03.009. Google Scholar [7] O. Jouini and A. Roubos, On multiple priority multi-server queues with impatience, Journal of the Operational Research Society, 65 (2014), 616-632. Google Scholar [8] G. P. Klimow, Bedienungsprozesse, Birkhäuser, Basel, 1979. Google Scholar [9] V. G. Kulkarni, Modeling and Analysis of Stochastic Systems, Chapman & Hall, Boca Raton, Florida, 1995. Google Scholar [10] A. Mandelbaum and S. Zeltyn, Service engineering in action: The Palm/Erlang-A queue, with applications to call centers, in Advances in Services Innovations (eds. D. Spath and K. -P. Fähnrich), Springer, (2007), 17-45Google Scholar [11] A. Myskja and O. Espvik (editors), Tore Olaus Engset, 1865-1943, The Man Behind the Formula, Tapir Academic Press, Trondheim, Norway, 2002Google Scholar [12] C. Palm, Etude des délais d'attente, Ericsson Technics, 5 (1937), 39-56, cited in [14]. Google Scholar [13] C. Palm, Research on telephone traffic carried by full availability groups, Tele, 1 (1957), 1-107 (English translation of results first published in 1946 in Swedish in the same journal, then entitled Tekniska Meddelanden från Kungliga Telegrafstyrelsen.), cited in [10] and [14].Google Scholar [14] J. Riordan, Stochastic Service Systems, John Wiley & Sons, New York, 1962. Google Scholar [15] S. Subba Rao, Queuing with balking and reneging in M/G/1 systems, Metrika, (1967/68), 173-188. doi: 10.1007/BF02613493. Google Scholar [16] H. Takagi, Waiting time in the M/M/$m / ( m + c )$ queue with impatient customers, International Journal of Pure and Applied Mathematics, 90 (2014), 519-559. doi: 10.12732/ijpam.v90i4.13. Google Scholar [17] H. Takagi, Waiting time in the M/M/$m$ FCFS nonpreemptive priority queue with impatient customers, International Journal of Pure and Applied Mathematics, 97 (2014), 311-344. doi: 10.12732/ijpam.v97i3.5. Google Scholar [18] H. Takagi, Waiting time in the M/M/$m$ LCFS nonpreemptive priority queue with impatient customers, Annals of Operations Research, 247 (2016), 257-289. doi: 10.1007/s10479-015-1876-7. Google Scholar [19] H. Takagi, Times to service completion and abandonment in the M/M/$m$ preemptive LCFS queue with impatient customers QTNA'16, 2016, Wellington, New Zealand, ACM ISBN 978-1-4503-4842-3/16/12. doi: 10.1145/3016032.3016036. Google Scholar [20] H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, 3$^{rd}$ edition, Academic Press, San Diego, California, 1998. Google Scholar [21] W. Whitt, Engineering solution of a basic call-center model, Management Science, 51 (2005), 221-235. doi: 10.1287/mnsc.1040.0302. Google Scholar [22] R. W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice Hall, Englewood Cliffs, New Jersey, 1989. Google Scholar
State transitions for the customer behavior until service completion or abandonment
Mean number of customers in service and the probability of service completion
Mean number of waiting customers and the probability of abandonment
Mean number of customers in the system and the mean time until departure
Second and third moments of the time until departure
Conditional mean times until service completion and abandonment
Conditional second moments of the times until service completion and abandonment
Probabilities of service completion and abandonment and moments of the time until departure in an M/M/$m$ preemptive-loss LCFS system
Means and second moments of the times until service completion and abandonment in an M/M/$m$ preemptive-loss LCFS system
System and customer performance measures in an M/M/$m$ preemptive LCFS queue with patient customers only
Numerical example for the probabilities and moments of the times until service completion and abandonment
 Parameter setting: $m = 5, \mu = 1, \theta = 2$, and $\lambda = 10$ ($\rho = 2$ and $\theta = 2$). $k$ $P _k \{ {\rm Sr} \}$ $P _k \{ {\rm Ab} \}$ $E [ T _k ]$ $E [ T _k , {\rm Sr} ]$ $E [ T _k , {\rm Ab} ]$ $E [ T _k ^2 ]$ $E [ T _k ^2 , {\rm Sr} ]$ $E [ T _k ^2 , {\rm Ab} ]$ $E [ T _k ^3 ]$ 0 0.48730 0.51270 0.74365 0.19074 0.55291 0.93439 0.14509 0.78930 1.61923 1 0.43604 0.56396 0.71802 0.16108 0.55693 0.87910 0.12145 0.75765 1.50083 2 0.37451 0.62549 0.68726 0.13062 0.55663 0.81788 0.09902 0.71886 1.37535 3 0.29966 0.70034 0.64983 0.10014 0.54969 0.74997 0.07831 0.67166 1.24242 4 0.20717 0.79283 0.60358 0.07105 0.53254 0.67463 0.05990 0.61473 1.10179 5 0.09089 0.90911 0.54544 0.04579 0.49965 0.59124 0.04432 0.54692 0.95333 6 0.05093 0.94907 0.52546 0.03324 0.49223 0.55870 0.03623 0.52247 0.89240 7 0.03314 0.96686 0.51657 0.02601 0.49056 0.54257 0.03117 0.51141 0.86061 8 0.02376 0.97624 0.51188 0.02138 0.49050 0.53326 0.02764 0.50562 0.84136 9 0.01819 0.98181 0.50910 0.01820 0.49090 0.52729 0.02502 0.50227 0.82847 10 0.01459 0.98541 0.50730 0.01588 0.49142 0.52317 0.02297 0.50020 0.81921 11 0.01211 0.98789 0.50605 0.01411 0.49194 0.52017 0.02132 0.49885 0.81223 12 0.01031 0.98969 0.50516 0.01273 0.49243 0.51788 0.01995 0.49794 0.80675 13 0.00896 0.99104 0.50448 0.01161 0.49287 0.51609 0.01879 0.49730 0.80232 14 0.00790 0.99210 0.50395 0.01069 0.49326 0.51464 0.01780 0.49684 0.79865 15 0.00707 0.99293 0.50353 0.00991 0.49362 0.51345 0.01693 0.49652 0.79556 16 0.00638 0.99362 0.50319 0.00925 0.49394 0.51245 0.01617 0.49628 0.79292 17 0.00582 0.99418 0.50291 0.00868 0.49423 0.51159 0.01549 0.49611 0.79062 18 0.00534 0.99466 0.50267 0.00819 0.49449 0.51086 0.01488 0.49598 0.78860 19 0.00494 0.99506 0.50247 0.00775 0.49472 0.51022 0.01433 0.49589 0.78682 20 0.00459 0.99541 0.50229 0.00736 0.49493 0.50965 0.01383 0.49583 0.78522
 Parameter setting: $m = 5, \mu = 1, \theta = 2$, and $\lambda = 10$ ($\rho = 2$ and $\theta = 2$). $k$ $P _k \{ {\rm Sr} \}$ $P _k \{ {\rm Ab} \}$ $E [ T _k ]$ $E [ T _k , {\rm Sr} ]$ $E [ T _k , {\rm Ab} ]$ $E [ T _k ^2 ]$ $E [ T _k ^2 , {\rm Sr} ]$ $E [ T _k ^2 , {\rm Ab} ]$ $E [ T _k ^3 ]$ 0 0.48730 0.51270 0.74365 0.19074 0.55291 0.93439 0.14509 0.78930 1.61923 1 0.43604 0.56396 0.71802 0.16108 0.55693 0.87910 0.12145 0.75765 1.50083 2 0.37451 0.62549 0.68726 0.13062 0.55663 0.81788 0.09902 0.71886 1.37535 3 0.29966 0.70034 0.64983 0.10014 0.54969 0.74997 0.07831 0.67166 1.24242 4 0.20717 0.79283 0.60358 0.07105 0.53254 0.67463 0.05990 0.61473 1.10179 5 0.09089 0.90911 0.54544 0.04579 0.49965 0.59124 0.04432 0.54692 0.95333 6 0.05093 0.94907 0.52546 0.03324 0.49223 0.55870 0.03623 0.52247 0.89240 7 0.03314 0.96686 0.51657 0.02601 0.49056 0.54257 0.03117 0.51141 0.86061 8 0.02376 0.97624 0.51188 0.02138 0.49050 0.53326 0.02764 0.50562 0.84136 9 0.01819 0.98181 0.50910 0.01820 0.49090 0.52729 0.02502 0.50227 0.82847 10 0.01459 0.98541 0.50730 0.01588 0.49142 0.52317 0.02297 0.50020 0.81921 11 0.01211 0.98789 0.50605 0.01411 0.49194 0.52017 0.02132 0.49885 0.81223 12 0.01031 0.98969 0.50516 0.01273 0.49243 0.51788 0.01995 0.49794 0.80675 13 0.00896 0.99104 0.50448 0.01161 0.49287 0.51609 0.01879 0.49730 0.80232 14 0.00790 0.99210 0.50395 0.01069 0.49326 0.51464 0.01780 0.49684 0.79865 15 0.00707 0.99293 0.50353 0.00991 0.49362 0.51345 0.01693 0.49652 0.79556 16 0.00638 0.99362 0.50319 0.00925 0.49394 0.51245 0.01617 0.49628 0.79292 17 0.00582 0.99418 0.50291 0.00868 0.49423 0.51159 0.01549 0.49611 0.79062 18 0.00534 0.99466 0.50267 0.00819 0.49449 0.51086 0.01488 0.49598 0.78860 19 0.00494 0.99506 0.50247 0.00775 0.49472 0.51022 0.01433 0.49589 0.78682 20 0.00459 0.99541 0.50229 0.00736 0.49493 0.50965 0.01383 0.49583 0.78522
Numerical example for the probabilities and moments of the times until service completion and abandonment in special cases
 (a) M/M/$m$ preemptive-loss LCFS system: $m = 5, \mu = 1, \theta = \infty$, and $\lambda = 10$ ($\rho = 2$) $k$ $Q _k$ $P _k \{ {\rm Sr} \}$ $P _k \{ {\rm Ab} \}$ $E [ T _k ]$ $E [ T _k , {\rm Sr} ]$ $E [ T _k , {\rm Ab} ]$ $E [ T _k ^2 ]$ $E [ T _k ^2 , {\rm Sr} ]$ $E [ T _k ^2 , {\rm Ab} ]$ $E [ T _k ^3 ]$ 0 0.00068 0.43605 0.56395 0.43605 0.13781 0.29824 0.27561 0.07451 0.20111 0.22352 1 0.00677 0.37965 0.62035 0.37965 0.10798 0.27167 0.21597 0.05440 0.16157 0.16319 2 0.03384 0.31198 0.68802 0.31198 0.07783 0.23414 0.15567 0.03623 0.11944 0.10868 3 0.11279 0.22964 0.77036 0.22964 0.04839 0.18125 0.09678 0.02065 0.07613 0.06195 4 0.28198 0.12790 0.87210 0.12790 0.02143 0.10467 0.04286 0.00836 0.03450 0.02509 5 0.56395 0 1 0 0 0 0 0 0 0 (b) M/M/$m$ preemptive LCFS queue with patient customers only: $m = 5, \mu = 1, \theta = 0$, and $\lambda = 3$ ($\rho = 0.6$) $k$ $Q _k$ $E [ T _k ]$ $E [ T _k ^2 ]$ $E [ T _k ^3 ]$ $k$ $Q _k$ $E [ T _k ]$ $E [ T _k ^2 ]$ $E [ T _k ^3 ]$ 0 0.04665 1.11808 3.09648 16.5361 11 0.00441 5.07289 36.2624 351.529 1 0.13994 1.15743 3.38325 18.9517 12 0.00264 5.57289 42.5852 432.445 2 0.20991 1.22303 3.83497 22.6908 13 0.00159 6.07289 49.4081 524.721 3 0.20991 1.34111 4.59908 28.9123 14 0.00095 6.57289 56.7310 629.106 4 0.15743 1.57289 6.00215 40.1720 15 0.00057 7.07289 64.5539 746.350 5 0.09446 2.07289 8.82504 62.5736 16 0.00034 7.57289 72.8768 877.203 6 0.05668 2.57289 12.1479 91.0845 17 0.00021 8.07289 81.6997 1022.41 7 0.03401 3.07289 15.9708 126.455 18 0.00012 8.57289 91.0226 1182.73 8 0.02040 3.57289 20.2937 169.434 19 0.00007 9.07289 100.845 1358.91 9 0.01224 4.07289 25.1166 220.773 20 0.00004 9.57289 111.168 1151.69 10 0.00735 4.57289 30.4395 281.221 $E [ T _0 ^4 ] \approx 151.94$.
 (a) M/M/$m$ preemptive-loss LCFS system: $m = 5, \mu = 1, \theta = \infty$, and $\lambda = 10$ ($\rho = 2$) $k$ $Q _k$ $P _k \{ {\rm Sr} \}$ $P _k \{ {\rm Ab} \}$ $E [ T _k ]$ $E [ T _k , {\rm Sr} ]$ $E [ T _k , {\rm Ab} ]$ $E [ T _k ^2 ]$ $E [ T _k ^2 , {\rm Sr} ]$ $E [ T _k ^2 , {\rm Ab} ]$ $E [ T _k ^3 ]$ 0 0.00068 0.43605 0.56395 0.43605 0.13781 0.29824 0.27561 0.07451 0.20111 0.22352 1 0.00677 0.37965 0.62035 0.37965 0.10798 0.27167 0.21597 0.05440 0.16157 0.16319 2 0.03384 0.31198 0.68802 0.31198 0.07783 0.23414 0.15567 0.03623 0.11944 0.10868 3 0.11279 0.22964 0.77036 0.22964 0.04839 0.18125 0.09678 0.02065 0.07613 0.06195 4 0.28198 0.12790 0.87210 0.12790 0.02143 0.10467 0.04286 0.00836 0.03450 0.02509 5 0.56395 0 1 0 0 0 0 0 0 0 (b) M/M/$m$ preemptive LCFS queue with patient customers only: $m = 5, \mu = 1, \theta = 0$, and $\lambda = 3$ ($\rho = 0.6$) $k$ $Q _k$ $E [ T _k ]$ $E [ T _k ^2 ]$ $E [ T _k ^3 ]$ $k$ $Q _k$ $E [ T _k ]$ $E [ T _k ^2 ]$ $E [ T _k ^3 ]$ 0 0.04665 1.11808 3.09648 16.5361 11 0.00441 5.07289 36.2624 351.529 1 0.13994 1.15743 3.38325 18.9517 12 0.00264 5.57289 42.5852 432.445 2 0.20991 1.22303 3.83497 22.6908 13 0.00159 6.07289 49.4081 524.721 3 0.20991 1.34111 4.59908 28.9123 14 0.00095 6.57289 56.7310 629.106 4 0.15743 1.57289 6.00215 40.1720 15 0.00057 7.07289 64.5539 746.350 5 0.09446 2.07289 8.82504 62.5736 16 0.00034 7.57289 72.8768 877.203 6 0.05668 2.57289 12.1479 91.0845 17 0.00021 8.07289 81.6997 1022.41 7 0.03401 3.07289 15.9708 126.455 18 0.00012 8.57289 91.0226 1182.73 8 0.02040 3.57289 20.2937 169.434 19 0.00007 9.07289 100.845 1358.91 9 0.01224 4.07289 25.1166 220.773 20 0.00004 9.57289 111.168 1151.69 10 0.00735 4.57289 30.4395 281.221 $E [ T _0 ^4 ] \approx 151.94$.
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