Article Contents

2023, Volume 19, Issue 6: 4615-4640. Doi: 10.3934/jimo.2022143

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Performance analysis of an infinite-buffer batch-size-dependent bulk service queue with server breakdown and multiple vacation

Sourav Pradhan^, and
Prasenjit Karan

Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur-440010, India

^*Corresponding author: Sourav Pradhan
^*Corresponding author: Sourav Pradhan

Received: December 2021

Revised: May 2022

Early access: August 2022

Published: June 2023

Abstract / Introduction Full Text(HTML) Figure(6) / Table(6) Related Papers Cited by

Abstract

In several manufacturing systems, server's breakdown is a common phenomenon for which renovation of the service station is required and has a definite effect on the system. The server's vacation is a significant tool to deal with the systems where a common server is shared by several users. The principal goal of this paper is to analyze a batch-service queueing model with server's breakdown, multiple vacations and batch-size-dependent service time. To be more precise, the center of attention is on the queue length together with server content distribution for which we obtain the bivariate probability generating function using supplementary variable technique. Server content distribution is very worthy for the manufacturing systems. The probabilities have been extracted and used to acquire the arbitrary epoch probabilities. In order to upgrade the qualitative aspects of the concerned model, diverse numerical results and graphical observations are illustrated.

Keywords:

Mathematics Subject Classification: Primary: 60K25, 68M20.

Citation:

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Figure 1. Total system cost versus service threshold value

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Figure 2. Total system cost versus arrival rate

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Figure 3. Average queue/system length versus breakdown probability

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Figure 4. Average waiting time in the queue/system versus breakdown probability

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Figure 5. Average queue/system length versus renovation rate

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Figure 6. Average waiting time in the queue/system versus renovation rate

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Table 1. Probability distributions at service completion epoch ($ p_{n, r}^+, q_n^+, \gamma_n^+ $) of an $ M/G^{(a, b)}_n/1 $ queue with service time $ \sim D $, vacation time $ \sim M $, renovation time $ \sim HE_2 $, $ \sigma = 0.6 $, $ \lambda = 0.08 $, $ a = 5 $, $ b = 12 $, $ \mu_i = \frac{1.5}{i+1} $, $ \rho = 0.627333 $

$ n $	$ p_{n, 5}^+ $	$ p_{n, 6}^+ $	$ p_{n, 7}^+ $	$ p_{n, 8}^+ $	$ p_{n, 9}^+ $	$ p_{n, 10}^+ $	$ p_{n, 11}^+ $	$ p_{n, 12}^+ $	$ p_n^+ $	$ q_n^+ $	$ \gamma_n^+ $
0	0.007077	0.003196	0.001409	0.000611	0.000263	0.000113	0.000048	0.000020	0.012741	0.002910	0.002665
1	0.018873	0.010228	0.005260	0.002610	0.001265	0.000603	0.000284	0.000148	0.039275	0.012533	0.009949
2	0.025164	0.016366	0.009819	0.005570	0.003037	0.001609	0.000835	0.000534	0.062936	0.029336	0.019634
3	0.022368	0.017457	0.012219	0.007921	0.004859	0.002861	0.001632	0.001299	0.070620	0.050256	0.027538
4	0.014912	0.013965	0.011405	0.008450	0.005831	0.003815	0.002394	0.002403	0.063178	0.071406	0.031126
5	0.007953	0.008938	0.008515	0.007210	0.005598	0.004069	0.002810	0.003618	0.048714	0.051931	0.030441
10	0.000035	0.000099	0.000204	0.000337	0.000471	0.000580	0.000645	0.004223	0.006597	0.010566	0.010236
15	0.000000	0.000000	0.000000	0.000001	0.000003	0.000007	0.000012	0.001178	0.001203	0.002149	0.002259
25	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000051	0.000051	0.000088	0.00010
35	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000002	0.000002	0.000003	0.000004
$ \geq $40	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000

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Table 2. Probability distributions at arbitrary epoch ($ p_{n, r}, q_n, \gamma_n $) of an $ M/G^{(a, b)}_n/1 $ queue

$ n $	$ p_{n, 5} $	$ p_{n, 6} $	$ p_{n, 7} $	$ p_{n, 8} $	$ p_{n, 9} $	$ p_{n, 10} $	$ p_{n, 11} $	$ p_{n, 12} $	$ p_n^{queue} $	$ q_n $	$ \gamma_n $
0	0.031480	0.024983	0.019102	0.014285	0.010552	0.007748	0.005673	0.004151	0.122210	0.002578	0.001653
1	0.025211	0.021586	0.017355	0.013418	0.010132	0.007547	0.005579	0.007140	0.125249	0.011100	0.006175
2	0.016853	0.016150	0.014094	0.011568	0.009123	0.007013	0.005302	0.009185	0.127471	0.025983	0.012196
3	0.009424	0.010352	0.010035	0.008936	0.007509	0.006062	0.004759	0.010378	0.129095	0.044512	0.017123
4	0.004471	0.005714	0.006247	0.006130	0.005572	0.004795	0.003964	0.010767	0.130283	0.063244	0.019375
5	0.001829	0.002745	0.003419	0.003735	0.003713	0.003443	0.003030	0.010433	0.097320	0.045995	0.018972
10	0.000003	0.000012	0.000032	0.000066	0.000110	0.000162	0.000212	0.004268	0.020655	0.009358	0.006427
15	0.000000	0.000000	0.000000	0.000000	0.000000	0.000001	0.000002	0.000974	0.004310	0.001904	0.001427
25	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000041	0.000185	0.000078	0.000064
35	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000001	0.000007	0.000003	0.000003
$ \geq $40	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
$ L $=8.501607, $ L_q $=4.115442, $ L_{s} $=8.224059
$ P_{busy} $=0.533333, $ W $=10.627009, $ W_q $=5.144302

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Table 3. Probability distributions at service completion epoch of an $ M/G^{(a, b)}_n/1 $ queue with service time$ \sim $mixed, vacation time $ \sim HE_2 $, renovation time$ \sim M $, $ \lambda = 1.25 $, $ a = 4 $, $ b = 9 $, $ \mu_4 = 0.375000 $, $ \mu_5 = 0.300000 $, $ \mu_6 = 0.079365 $, $ \mu_7 = 0.068027 $, $ \mu_8 = 0.166667 $, $ \mu_9 = 0.150000 $, $ \rho = 0.127778 $

$ n $	$ p_{n, 4}^+ $	$ p_{n, 5}^+ $	$ p_{n, 6}^+ $	$ p_{n, 7}^+ $	$ p_{n, 8}^+ $	$ p_{n, 9}^+ $	$ p_n^+ $	$ q_n^+ $	$ \gamma_n^+ $
0	0.034180	0.007894	0.001682	0.000434	0.000176	0.000078	0.044446	0.116290	0.008889
1	0.019531	0.005263	0.001096	0.000297	0.000158	0.000072	0.026420	0.193773	0.008247
2	0.008370	0.002631	0.000714	0.000204	0.000071	0.000053	0.012047	0.233049	0.005158
3	0.003188	0.001169	0.000466	0.000140	0.000021	0.000037	0.005024	0.251047	0.002724
4	0.001138	0.000487	0.000304	0.000096	0.0000004	0.000025	0.002054	0.064234	0.001319
5	0.000390	0.000194	0.000199	0.000066	0.000000	0.000016	0.000865	0.016541	0.000613
10	0.000001	0.000001	0.000024	0.000010	0.000000	0.000002	0.000038	0.000020	0.000019
15	0.000000	0.000000	0.000003	0.000001	0.000000	0.000000	0.000004	0.000000	0.000002
$ \geq $20	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000

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Table 4. Probability distributions at arbitrary epoch of an $ M/G^{(a, b)}_n/1 $ queue

$ n $	$ p_{n, 4} $	$ p_{n, 5} $	$ p_{n, 6} $	$ p_{n, 7} $	$ p_{n, 8} $	$ p_{n, 9} $	$ p_n^{queue} $	$ q_n $	$ \gamma_n $
0	0.081646	0.024554	0.007868	0.002370	0.000640	0.000194	0.227867	0.099533	0.011059
1	0.033047	0.011458	0.005140	0.001629	0.000245	0.000181	0.228035	0.166072	0.010260
2	0.012219	0.004910	0.003361	0.001120	0.000067	0.000134	0.228122	0.199890	0.006417
3	0.004284	0.002000	0.002200	0.000771	0.000014	0.000092	0.228173	0.215419	0.003389
4	0.001450	0.000788	0.001442	0.000531	0.000002	0.000062	0.061510	0.055591	0.001641
5	0.000479	0.000303	0.000946	0.000366	0.000000	0.000041	0.017332	0.014432	0.000763
10	0.000001	0.000002	0.000116	0.000057	0.000000	0.000005	0.000226	0.000018	0.000024
15	0.000000	0.000000	0.000014	0.000009	0.000000	0.000000	0.000027	0.000000	0.000002
$ \geq $20	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
$ L $=2.722981, $ L_q $=1.763711, $ L_{s} $=4.574273
$ P_{busy} $=0.209709, $ W $=18.153210, $ W_q $=11.758079

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Table 5. Performance measures for different probability of server's breakdown

$\sigma $	$ L_{q}$	$ L $	$ L_{s}$	$ W_{q} $	$W$
0.0	12.430488	18.920051	14.601525	6.215244	9.460026
0.05	12.541887	19.037950	14.616153	6.270943	9.518975
0.10	12.658263	19.160957	14.631074	6.329131	9.580479
0.15	12.779974	19.289436	14.646302	6.389987	9.644718
0.20	12.907417	19.423787	14.661850	6.453708	9.711894
0.25	13.041025	19.564453	14.677733	6.520513	9.782227
0.30	13.181282	19.711922	14.693964	6.590641	9.855961
0.35	13.328720	19.866734	14.710560	6.664360	9.933367
0.40	13.483936	20.029492	14.727539	6.741968	10.014746
0.45	13.647590	20.200867	14.744919	6.823795	10.100434
0.50	13.820427	20.381610	14.762720	6.910213	10.190805
0.55	14.003278	20.572562	14.780964	7.001639	10.286281
0.60	14.197083	20.774674	14.799673	7.098542	10.387337
0.65	14.402906	20.989018	14.818873	7.201453	10.494509
0.70	14.621952	21.216812	14.838590	7.310976	10.608406

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Table 6. Performance measures for different arrival rates

$ \lambda $	$ L_{q} $	$ L $	$ L_{s} $	$ W_{q} $	$ W $	$ P_{busy} $	$ P_{idle} $
1.75	11.310239	16.925046	14.438075	6.462994	9.671455	0.388889	0.611111
2.00	13.181282	19.711922	14.693964	6.590641	9.855961	0.444444	0.555556
2.25	15.621999	23.083005	14.922485	6.943111	10.259113	0.500000	0.500000
2.50	18.966311	27.371646	15.129603	7.586524	10.948658	0.555556	0.444444
2.75	23.855139	33.217037	15.319478	8.674596	12.078923	0.0.611111	0.388889
3.00	31.703060	42.033143	15.495129	10.567687	14.011048	0.666667	0.333333
3.25	46.372536	57.681670	15.658821	14.268472	17.748206	0.722222	0.277778
3.50	83.526050	95.824502	15.812296	23.864586	27.378429	0.777778	0.222222

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