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Optimal impulse control of a mean-reverting inventory with quadratic costs
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 |
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.
References:
[1] |
R. B. Cooper,
Introduction to Queueing Theory, 2$ ^{nd}$ edition, Elsevier North Holland, New York, 1981. |
[2] |
N. Gautam,
Analysis of Queues: Methods and Applications, CRC Press, Boca Raton, Florida, 2012. |
[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. |
[4] |
F. Iravani and B. Balcio$ {\tilde {\rm g}}$lu,
On priority queues with impatient customers, Queueing Systems, 58 (2008), 239-260.
doi: 10.1007/s11134-008-9069-6. |
[5] |
D. L. Jagerman,
Difference Equations with Applications to Queues, Marcel Dekker, New York, 2000. |
[6] |
O. Jouini,
Analysis of a last come first served queueing system with customer abandonment, Computers & Operations Research, 39 (2012), 3040-3045.
doi: 10.1016/j.cor.2012.03.009. |
[7] |
O. Jouini and A. Roubos,
On multiple priority multi-server queues with impatience, Journal of the Operational Research Society, 65 (2014), 616-632.
|
[8] | |
[9] |
V. G. Kulkarni,
Modeling and Analysis of Stochastic Systems, Chapman & Hall, Boca Raton, Florida, 1995. |
[10] |
A. Mandelbaum and S. Zeltyn, Service engineering in action: The Palm/Erlang-A queue, with applications to call centers, in Advances in Services Innovations (eds. D. Spath and K. -P. Fähnrich), Springer, (2007), 17-45 |
[11] |
A. Myskja and O. Espvik (editors),
Tore Olaus Engset, 1865-1943, The Man Behind the Formula, Tapir Academic Press, Trondheim, Norway, 2002 |
[12] |
C. Palm,
Etude des délais d'attente, Ericsson Technics, 5 (1937), 39-56, cited in [14].
|
[13] |
C. Palm, Research on telephone traffic carried by full availability groups,
Tele, 1 (1957), 1-107 (English translation of results first published in 1946 in Swedish in the same journal, then entitled Tekniska Meddelanden från Kungliga Telegrafstyrelsen.), cited in [10] and [14]. |
[14] |
J. Riordan,
Stochastic Service Systems, John Wiley & Sons, New York, 1962. |
[15] |
S. Subba Rao,
Queuing with balking and reneging in M/G/1 systems, Metrika, (1967/68), 173-188.
doi: 10.1007/BF02613493. |
[16] |
H. Takagi,
Waiting time in the M/M/$ m / ( m + c ) $ queue with impatient customers, International Journal of Pure and Applied Mathematics, 90 (2014), 519-559.
doi: 10.12732/ijpam.v90i4.13. |
[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. |
[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. |
[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. |
[20] |
H. M. Taylor and S. Karlin,
An Introduction to Stochastic Modeling, 3$ ^{rd}$ edition, Academic Press, San Diego, California, 1998. |
[21] |
W. Whitt,
Engineering solution of a basic call-center model, Management Science, 51 (2005), 221-235.
doi: 10.1287/mnsc.1040.0302. |
[22] |
R. W. Wolff,
Stochastic Modeling and the Theory of Queues, Prentice Hall, Englewood Cliffs, New Jersey, 1989. |
show all references
The reviewing process of this paper was handled by Yutaka Takahashi and Wuyi Yue
References:
[1] |
R. B. Cooper,
Introduction to Queueing Theory, 2$ ^{nd}$ edition, Elsevier North Holland, New York, 1981. |
[2] |
N. Gautam,
Analysis of Queues: Methods and Applications, CRC Press, Boca Raton, Florida, 2012. |
[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. |
[4] |
F. Iravani and B. Balcio$ {\tilde {\rm g}}$lu,
On priority queues with impatient customers, Queueing Systems, 58 (2008), 239-260.
doi: 10.1007/s11134-008-9069-6. |
[5] |
D. L. Jagerman,
Difference Equations with Applications to Queues, Marcel Dekker, New York, 2000. |
[6] |
O. Jouini,
Analysis of a last come first served queueing system with customer abandonment, Computers & Operations Research, 39 (2012), 3040-3045.
doi: 10.1016/j.cor.2012.03.009. |
[7] |
O. Jouini and A. Roubos,
On multiple priority multi-server queues with impatience, Journal of the Operational Research Society, 65 (2014), 616-632.
|
[8] | |
[9] |
V. G. Kulkarni,
Modeling and Analysis of Stochastic Systems, Chapman & Hall, Boca Raton, Florida, 1995. |
[10] |
A. Mandelbaum and S. Zeltyn, Service engineering in action: The Palm/Erlang-A queue, with applications to call centers, in Advances in Services Innovations (eds. D. Spath and K. -P. Fähnrich), Springer, (2007), 17-45 |
[11] |
A. Myskja and O. Espvik (editors),
Tore Olaus Engset, 1865-1943, The Man Behind the Formula, Tapir Academic Press, Trondheim, Norway, 2002 |
[12] |
C. Palm,
Etude des délais d'attente, Ericsson Technics, 5 (1937), 39-56, cited in [14].
|
[13] |
C. Palm, Research on telephone traffic carried by full availability groups,
Tele, 1 (1957), 1-107 (English translation of results first published in 1946 in Swedish in the same journal, then entitled Tekniska Meddelanden från Kungliga Telegrafstyrelsen.), cited in [10] and [14]. |
[14] |
J. Riordan,
Stochastic Service Systems, John Wiley & Sons, New York, 1962. |
[15] |
S. Subba Rao,
Queuing with balking and reneging in M/G/1 systems, Metrika, (1967/68), 173-188.
doi: 10.1007/BF02613493. |
[16] |
H. Takagi,
Waiting time in the M/M/$ m / ( m + c ) $ queue with impatient customers, International Journal of Pure and Applied Mathematics, 90 (2014), 519-559.
doi: 10.12732/ijpam.v90i4.13. |
[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. |
[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. |
[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. |
[20] |
H. M. Taylor and S. Karlin,
An Introduction to Stochastic Modeling, 3$ ^{rd}$ edition, Academic Press, San Diego, California, 1998. |
[21] |
W. Whitt,
Engineering solution of a basic call-center model, Management Science, 51 (2005), 221-235.
doi: 10.1287/mnsc.1040.0302. |
[22] |
R. W. Wolff,
Stochastic Modeling and the Theory of Queues, Prentice Hall, Englewood Cliffs, New Jersey, 1989. |










Parameter setting: |
|||||||||
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: |
|||||||||
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 |
(a) M/M/ |
|||||||||||
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/ |
|||||||||||
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 |
(a) M/M/ |
|||||||||||
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/ |
|||||||||||
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 |
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