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

Hideaki Takagi

doi:10.3934/jimo.2018028

Article Contents

2018, Volume 14, Issue 4: 1701-1726. Doi: 10.3934/jimo.2018028

This issue Previous Article Optimal impulse control of a mean-reverting inventory with quadratic costs Next Article

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

Received: December 31, 2016

Revised: May 31, 2017

Early access: February 2018

Published: October 2018

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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Abstract

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.

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

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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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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	16.5361	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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