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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

Hideaki Takagi

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  October 2018 Early access  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.

Keywords: Multiserver queue, Poisson arrival, last-come first-served, preemptive-resume, preemptive-loss, impatient customers, service completion, abandonment, birth-and-death process, first passage time, absorbing state.
Mathematics Subject Classification: Primary: 60K25, 90B22; Secondary: 60K20, 68M20.
Citation: Hideaki Takagi. Times until service completion and abandonment in an M/M/$ m$ preemptive-resume LCFS queue with impatient customers. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1701-1726. doi: 10.3934/jimo.2018028
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]

G. P. Klimow, Bedienungsprozesse, Birkhäuser, Basel, 1979.

[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]

G. P. Klimow, Bedienungsprozesse, Birkhäuser, Basel, 1979.

[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.

Figure 1.  State transitions for the customer behavior until service completion or abandonment
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Figure 2.  Mean number of customers in service and the probability of service completion
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Figure 3.  Mean number of waiting customers and the probability of abandonment
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Figure 4.  Mean number of customers in the system and the mean time until departure
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Figure 5.  Second and third moments of the time until departure
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Figure 6.  Conditional mean times until service completion and abandonment
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Figure 7.  Conditional second moments of the times until service completion and abandonment
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Figure 8.  Probabilities of service completion and abandonment and moments of the time until departure in an M/M/$ m $ preemptive-loss LCFS system
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Figure 9.  Means and second moments of the times until service completion and abandonment in an M/M/$ m $ preemptive-loss LCFS system
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Figure 10.  System and customer performance measures in an M/M/$ m $ preemptive LCFS queue with patient customers only
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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
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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
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 $.
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(a) M/M/$ 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 $.
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