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

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

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  • 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.

    Mathematics Subject Classification: Primary: 60K25, 90B22; Secondary: 60K20, 68M20.

    Citation:

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  • Figure 1.  State transitions for the customer behavior until service completion or abandonment

    Figure 2.  Mean number of customers in service and the probability of service completion

    Figure 3.  Mean number of waiting customers and the probability of abandonment

    Figure 4.  Mean number of customers in the system and the mean time until departure

    Figure 5.  Second and third moments of the time until departure

    Figure 6.  Conditional mean times until service completion and abandonment

    Figure 7.  Conditional second moments of the times until service completion and abandonment

    Figure 8.  Probabilities of service completion and abandonment and moments of the time until departure in an M/M/$ m $ preemptive-loss LCFS system

    Figure 9.  Means and second moments of the times until service completion and abandonment in an M/M/$ m $ preemptive-loss LCFS system

    Figure 10.  System and customer performance measures in an M/M/$ m $ preemptive LCFS queue with patient customers only

    Table 1.  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
     | Show Table
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    Table 2.  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 16.536111 0.00441 5.07289 36.2624 351.529
    1 0.13994 1.15743 3.38325 18.951712 0.00264 5.57289 42.5852 432.445
    2 0.20991 1.22303 3.83497 22.690813 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 $.
     | Show Table
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