On the $ BMAP_1, BMAP_2/PH/g, c $ retrial queueing system

Yi Peng; Jinbiao Wu

doi:10.3934/jimo.2020124

Article Contents

2021, Volume 17, Issue 6: 3373-3391. Doi: 10.3934/jimo.2020124

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On the $ BMAP_1, BMAP_2/PH/g, c $ retrial queueing system

Yi Peng^1, and
Jinbiao Wu^2, ,

1.
School of Mathematical Science, Changsha Normal University, Changsha 410100, Hunan, China
2.
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

^* Corresponding author: Jinbiao Wu
^* Corresponding author: Jinbiao Wu

Received: December 2019

Revised: April 2020

Early access: June 2020

Published: November 2021

The first author is supported by Provincial Natural Science Foundation of Hunan under Grant 2019JJ50677 and the Program of Hehua Excellent Young Talents of Changsha Normal University. The second author is supported by Provincial Natural Science Foundation of Hunan under Grant 2020JJ4760

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Abstract

In this paper, we consider the BMAP/PH/c retrial queue with two types of customers where the rate of individual repeated attempts from the orbit is modulated according to a Markov Modulated Poisson Process. Using the theory of multi-dimensional asymptotically quasi-Toeplitz Markov chain, we obtain the algorithm for calculating the stationary distribution of the system. Main performance measures are presented. Furthermore, we investigate some optimization problems. The algorithm for determining the optimal number of guard servers and total servers is elaborated. Finally, this queueing system is applied to the cellular wireless network. Numerical results to illustrate the optimization problems and the impact of retrial on performance measures are provided. We find that the performance measures are mainly affected by the two types of customers' arrivals and service patterns, but the retrial rate plays a less crucial role.

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Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22.

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Figure 1. $ L_{orb} $ as function of the parameter $ g $ with $ (c, \lambda_o, \lambda_h) = (10, 2, 2) $

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Figure 2. Dependence of the blocking probability for originating calls on the value $ g $ with $ (c, \lambda_o, \lambda_h) = (10, 2, 2) $

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Figure 3. Dependence of the blocking probability for handoff calls on the value $ g $ with $ (c, \lambda_o, \lambda_h) = (10, 2, 2) $

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Table 1. The stationary join distribution of the system with $(c, g, \lambda_o, \lambda_h, \lambda_r) = (8, 6, 2, 2, 2)$

$i$ $\backslash$ $b$	0	1	2	3	4	5	6	7	8	$sum$
0	0.0467	0.1413	0.2136	0.2148	0.1604	0.0923	0.0376	0.0016	0.0001	0.9084
1	0.0001	0.0006	0.0020	0.0047	0.0091	0.0149	0.0208	0.0013	0.0001	0.0536
2	0.0000	0.0000	0.0002	0.0008	0.0023	0.0054	0.0109	0.0008	0.0000	0.0206
3	0.0000	0.0000	0.0000	0.0002	0.0007	0.0022	0.0055	0.0005	0.0000	0.0092
4	0.0000	0.0000	0.0000	0.0001	0.0002	0.0009	0.0028	0.0003	0.0000	0.0043
5	0.0000	0.0000	0.0000	0.0000	0.0001	0.0004	0.0014	0.0001	0.0000	0.0020
	1.0e-003 $\times$
6	0.0000	0.0001	0.0007	0.0055	0.0334	0.1661	0.6986	0.0667	0.0051	0.9761
	1.0e-003 $\times$
7	0.0000	0.0000	0.0002	0.0019	0.0130	0.0731	0.3474	0.0337	0.0027	0.4721
	1.0e-003$\times$
8	0.0000	0.0000	0.0001	0.0007	0.0052	0.0325	0.1724	0.0169	0.0014	0.2291
	1.0e-004$\times$
9	0.0000	0.0000	0.0002	0.0025	0.0211	0.1460	0.8533	0.0846	0.0069	1.1145
	1.0e-004$\times$
10	0.0000	0.0000	0.0001	0.0009	0.0087	0.0660	0.4217	0.0421	0.0035	0.5429
$sum$	0.0468	0.1419	0.2158	0.2206	0.1728	0.1162	0.0800	0.0047	0.0002	0.999

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Table 2. The the optimal value $ g^* $ for the optimization problem (I) with $ (c, \lambda_o, p_0) = (20, 10, 0.0001) $

$ \lambda_r $ $ \backslash $ $ \lambda_h $	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	20
1	18	18	18	17	17	17	16	16	16	16	15	15	15	14	14	13
10	18	18	18	17	17	17	16	16	16	16	15	15	15	14	14	13
20	18	18	18	17	17	17	16	16	16	16	15	15	15	14	14	13

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Table 3. The optimal value $ c^* $ for the optimization problem (II) with $ (\lambda_r, p_1, p_2) = (10, 0.001, 0.0001) $

$ \lambda_h $ $ \backslash $ $ \lambda_o $	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	20
1	8	11	14	16	18	20	22	24	26	28	29	31	33	35	37	45
5	10	13	15	17	19	21	23	25	27	29	30	32	34	36	38	46
10	13	15	17	19	21	23	25	27	29	31	32	34	36	37	39	47

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