A $ {BMAP/BMSP/1} $ queue with Markov dependent arrival and Markov dependent service batches

Achyutha Krishnamoorthy; Anu Nuthan Joshua

doi:10.3934/jimo.2020101

Article Contents

2021, Volume 17, Issue 5: 2925-2941. Doi: 10.3934/jimo.2020101

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A $ {BMAP/BMSP/1} $ queue with Markov dependent arrival and Markov dependent service batches

Achyutha Krishnamoorthy^1, and
Anu Nuthan Joshua^2,

1.
Centre for Research in Mathematics, C.M.S. College, Kottayam-686001, India
2.
Department of Mathematics, Union Christian College, Aluva-683102, India

Received: November 2019

Early access: May 2020

Published: September 2021

The first author is supported by UGC, Govt. of India, Emeritus Fellow(EMERITUS 2017-18 GEN 10822(SA-II)) and DST, Indo-Russian project: INT/RUS/RSF/P-15

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Abstract

Batch arrival and batch service queueing systems are of importance in the context of telecommunication networks. None of the work reported so far consider the dependence of consecutive arrival and service batches. Batch Markovian Arrival Process($ BMAP $) and Batch Markovian Service Process ($ BMSP $) take care of the dependence between successive inter-arrival and service times, respectively. However in real life situations dependence between consecutive arrival and service batch sizes also play an important role. This is to regulate the workload of the server in the context of service and to restrict the arrival batch size when the flow is from the same source. In this paper we study a queueing system with Markov dependent arrival and service batch sizes. The arrival and service batch sizes are assumed to be finite. Further, successive inter-arrival and service time durations are also assumed to be correlated. Specifically, we consider a $ BMAP/BMSP/1 $ queue with Markov dependent arrival and Markov dependent service batch sizes. The stability of the system is investigated. The steady state probability vectors of the system state and some important performance measures are computed. The Laplace-Stieltjes transform of waiting time and idle time of the server are obtained. Some numerical examples are provided.

Keywords:

Queueing system,
batch Markovian arrival process,
batch Markovian service process,
Markov dependent arrival/service batch sizes.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

Table 1. Transition rate submatrices within level 0

$ \rm{ From} $	$ \rm{To} $	$ \rm{ Rate} $
$ (0, p, n_1, n_2 = 0(k)) $	$ (0, p, n_1, n_2 = 0(k)) $	$ \textbf I_s \otimes D_0 $
$ (0, p, n_1, n_2) $	$ (0, p, n_1, n_2) $	$ S_0 \oplus D_0 $
$ (0, p, n_1, n_2) $	$ (0, p + m_1, m_1, n_2) $	$ \textbf I_s \otimes p_{n_1 m_1}D_{c} $
$ (0, p, n_1, n_2 = 0(k)) $	$ (0, p + m_1 - k, m_1, k) $	$ \textbf I_s \otimes p_{n_1 m_1} D_{c} $
$ (0, p, n_1, n_2 = 0(k)) $	$ (0, p+m_1, m_1, 0(k)) $	$ \textbf I_s \otimes p_{n_1 m_1} D_{c} $
$ (0, p, n_1, n_2 ) $	$ (0, p, n_1, m_2 = 0(k)) $	$ q_{n_2 k} S_d \otimes \textbf I_r $
$ (0, p, n_1, n_2) $	$ (0, p-m_2, n_1, m_2) $	$ q_{n_2 m_2} S_d \otimes \textbf I_r $

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Table 2. Transition rate submatrices except those within level 0

$ \rm{ From} $	$ \rm{To} $	$ \rm{ Rate} $
$ (0, p, n_1 , n_2) $	$ (1, p + m_1- q, m_1, n_2) $	$ \textbf I_s \otimes p_{n_1 m_1}D_{c} $
$ (1, p, n_1, n_2) $	$ (0, q + p- m_2, n_1, m_2) $	$ q_{n_2 m_2} S_d \otimes \textbf I_r $
$ (l, p, n_1, n_2) $	$ (l+1, p + m_1 - q, m_1, n_2) $	$ \textbf I_s \otimes p_{n_1 m_1} D_{c} $
$ (l, p, n_1, n_2) $	$ (l, p, n_1, n_2) $	$ S_0 \oplus D_0 $
$ (l, p, n_1, n_2) $	$ (l, p + m_1, m_1, n_2) $	$ \textbf I_s \otimes p_{n_1 m_1} D_{c} $
$ (l, p, n_1, n_2) $	$ (l, p- m_2, n_1, m_2) $	$ q_{n_2 m_2} S_d \otimes \textbf I_r $
$ (l, p, n_1, n_2) $	$ (l-1, q + p- m_2, n_1, m_2) $	$ q_{n_2 m_2} S_d \otimes \textbf I_r $

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Table 3. Expected queue length under various arrival and service processes

	NCA (2a)	NCA (2b)	ZCA	PCA (3a)	PCA (3b)
NCS(2b)	$ {4.5898} $	$ {5.7659} $	$ {3.1824} $	$ {42.5934} $	$ {179.1428} $
NCS(2a)	$ {3.3006} $	$ {4.3856} $	$ {2.1585} $	$ {41.1375} $	$ {177.7154} $
ZCS	$ {1.1070} $	$ {1.1905} $	$ {0.9351} $	$ {8.1323} $	$ {40.2739} $
PCS	$ {40.5835} $	$ {42.1713} $	$ {32.6996} $	$ {82.5983} $	$ {214.3443} $

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Table 4. Expected idle time in $ (0, 0, 1, 0(1), 1, 1), (0, 0, 1, 0(2), 1, 1), (0, 0, 1, 0(3), 1, 1) $ respectively and arrival phase changes to 1, 2, 3 respectively at the end of idle time under various arrival and service processes

	L(1, 1)	M(1, 1)	N(1, 1)	L(1, 2)	M(1, 2)	N(1, 2)	L(1, 3)	M(1, 3)	N(1, 3)
ZCS NCA(2a)	$ {0.0200} $	$ {1.5725} $	$ {0.7685} $	$ {0} $	$ {0} $	$ {0} $	$ {1.9756} $	$ {0.4426} $	$ {2.5034} $
ZCS NCA(2b)	$ {0.2199} $	$ {1.2456} $	$ {1.1507} $	$ {0.2274} $	$ {0.1551} $	$ {0.2979} $	$ {1.4797} $	$ {0.8040} $	$ {1.7725} $
ZCS ZCA	$ {1.0333} $	$ {1.8600} $	$ {2.7280} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $
ZCS PCA(3a)	$ {1.8098} $	$ {2.2052} $	$ {2.3602} $	$ {1.8682} $	$ {2.3471} $	$ {2.5799} $	$ {2.5189} $	$ {4.2899} $	$ {6.0942} $
ZCS PCA(3b)	$ {1.9756} $	$ {1.5247} $	$ {5.1267} $	$ {0} $	$ {0} $	$ {0} $	$ {0.0200} $	$ {0.0514} $	$ {0.0965} $
PCS NCA(2a)	$ {0.0200} $	$ {1.5725} $	$ {0.7685} $	$ {0} $	$ {0} $	$ {0} $	$ {1.9756} $	$ {0.4426} $	$ {2.5034} $
PCS NCA(2b)	$ {0.2199} $	$ {1.2456} $	$ {1.1507} $	$ {0.2274} $	$ {0.1551} $	$ {0.2979} $	$ {1.4797} $	$ {0.8040} $	$ {1.7725} $
PCS ZCA	$ {1.0333} $	$ {1.8600} $	$ {2.7280} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $
PCS PCA(3a)	$ {1.8098} $	$ {2.2052} $	$ {2.3602} $	$ {1.8682} $	$ {2.3471} $	$ {2.5799} $	$ {2.5189} $	$ {4.2899} $	$ {6.0942} $
PCS PCA(3b)	$ {1.9756} $	$ {1.5247} $	$ {5.1267} $	$ {0} $	$ {0} $	$ {0} $	$ {0.0200} $	$ {0.0514} $	$ {0.0965} $
NCS(2a) NCA(2a)	$ {0.0200} $	$ {1.5725} $	$ {0.7685} $	$ {0} $	$ {0} $	$ {0} $	$ {1.9756} $	$ {0.4426} $	$ {2.5034} $
NCS(2a) NCA(2b)	$ {0.2199} $	$ {1.2456} $	$ {1.1507} $	$ {0.2274} $	$ {0.1551} $	$ {0.2979} $	$ {1.4797} $	$ {0.8040} $	$ {1.7725} $
NCS(2a) ZCA	$ {1.0333} $	$ {1.8600} $	$ {2.7280} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $
NCS(2a) PCA(3a)	$ {1.8098} $	$ {2.2052} $	$ {2.3602} $	$ {1.8682} $	$ {2.3471} $	$ {2.5799} $	$ {2.5189} $	$ {4.2899} $	$ {6.0942} $
NCS(2a) PCA(3b)	$ {1.9756} $	$ {1.5247} $	$ {5.1267} $	$ {0} $	$ {0} $	$ {0} $	$ {0.0200} $	$ {0.0514} $	$ {0.0965} $

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Table 5. Expected idle time in $ (0, 0, 1, 0(1), 1, 2), (0, 0, 1, 0(2), 1, 2), (0, 0, 1, 0(3), 1, 2) $ respectively and arrival phase changes to 1, 2, 3 respectively at the end of idle time under various arrival and service processes

	L(2, 1)	M(2, 1)	N(2, 1)	L(2, 2)	M(2, 2)	N(2, 2)	L(2, 3)	M(2, 3)	N(2, 3)
ZCS NCA(2a)	$ {0.0100} $	$ {0.7881} $	$ {0.3977} $	$ {0} $	$ {0} $	$ {0} $	$ {0.9878} $	$ {0.2292} $	$ {1.8764} $
ZCS NCA(2b)	$ {0.0567} $	$ {0.5166} $	$ {0.5367} $	$ {0.0351} $	$ {0.0530} $	$ {0.2016} $	$ {0.5964} $	$ {0.3021} $	$ {1.2421} $
ZCS ZCA	$ {0.5333} $	$ {1.3600} $	$ {2.2280} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $
ZCS PCA(3a)	$ {1.9276} $	$ {2.5820} $	$ {2.8420} $	$ {1.9727} $	$ {2.6957} $	$ {3.0442} $	$ {1.4113} $	$ {3.6018} $	$ {5.8677} $
ZCS PCA(3b)	$ {0.9878} $	$ {2.5447} $	$ {4.1548} $	$ {0} $	$ {0} $	$ {0} $	$ {0.0100} $	$ {0.0336} $	$ {0.0707} $
PCS NCA(2a)	$ {0.0100} $	$ {0.7881} $	$ {0.3977} $	$ {0} $	$ {0} $	$ {0} $	$ {0.9878} $	$ {0.2292} $	$ {1.8764} $
PCS NCA(2b)	$ {0.0567} $	$ {0.5166} $	$ {0.5367} $	$ {0.0351} $	$ {0.0530} $	$ {0.2016} $	$ {0.5964} $	$ {0.3021} $	$ {1.2421} $
PCS ZCA	$ {0.5333} $	$ {1.3600} $	$ {2.2280} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $
PCS PCA(3a)	$ {1.9276} $	$ {2.5820} $	$ {2.8420} $	$ {1.9727} $	$ {2.6957} $	$ {3.0442} $	$ {1.4113} $	$ {3.6018} $	$ {5.8677} $
PCS PCA(3b)	$ {0.9878} $	$ {2.5447} $	$ {4.1548} $	$ {0} $	$ {0} $	$ {0} $	$ {0.0100} $	$ {0.0336} $	$ {0.0707} $
NCS(2a) NCA(2a)	$ {0.0100} $	$ {0.7881} $	$ {0.3977} $	$ {0} $	$ {0} $	$ {0} $	$ {0.9878} $	$ {0.2292} $	$ {1.8764} $
NCS(2a) NCA(2b)	$ {0.0567} $	$ {0.5166} $	$ {0.5367} $	$ {0.0351} $	$ {0.0530} $	$ {0.2016} $	$ {0.5964} $	$ {0.3021} $	$ {1.2421} $
NCS(2a) ZCA	$ {0.5333} $	$ {1.3600} $	$ {2.2280} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $
NCS(2a) PCA(3a)	$ {1.9276} $	$ {2.5820} $	$ {2.8420} $	$ {1.9727} $	$ {2.6957} $	$ {3.0442} $	$ {1.4113} $	$ {3.6018} $	$ {5.8677} $
NCS(2a) PCA(3b)	$ {0.9878} $	$ {2.5447} $	$ {4.1548} $	$ {0} $	$ {0} $	$ {0} $	$ {0.0100} $	$ {0.0336} $	$ {0.0707} $

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Table 6. Expected idle time in $ (0, 0, 1, 0(1), 1, 3), (0, 0, 1, 0(2), 1, 3), (0, 0, 1, 0(3), 1, 3) $ respectively and arrival phase changes to 1, 2, 3 respectively at the end of idle time under various arrival and service processes

	L(3, 1)	M(3, 1)	N(3, 1)	L(3, 2)	M(3, 2)	N(3, 2)	L(3, 3)	M(3, 3)	N(3, 3)
ZCS NCA(2a)	$ {0.0044} $	$ {0.0168} $	$ {1.2523} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {1.5682} $	$ {0.7559} $
ZCS NCA(2b)	$ {0.0483} $	$ {0.1888} $	$ {1.0304} $	$ {0.0032} $	$ {0.1781} $	$ {0.1684} $	$ {0.0088} $	$ {1.1688} $	$ {0.9376} $
ZCS ZCA	$ {0.2000} $	$ {1.0267} $	$ {1.8947} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $
ZCS PCA(3a)	$ {0.0176} $	$ {0.1166} $	$ {0.2426} $	$ {0.0214} $	$ {0.1252} $	$ {0.2598} $	$ {0.0891} $	$ {0.2358} $	$ {0.4843} $
ZCS PCA(3b)	$ {0} $	$ {0.0159} $	$ {0.0449} $	$ {0} $	$ {0} $	$ {0} $	$ {0.0044} $	$ {0.0080} $	$ {0.0120} $
PCS NCA(2a)	$ {0.0044} $	$ {0.0168} $	$ {1.2523} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {1.5682} $	$ {0.7559} $
PCS NCA(2b)	$ {0.0483} $	$ {0.1888} $	$ {1.0304} $	$ {0.0032} $	$ {0.1781} $	$ {0.1684} $	$ {0.0088} $	$ {1.1688} $	$ {0.9376} $
PCS ZCA	$ {0.2000} $	$ {1.0267} $	$ {1.8947} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $
PCS PCA(3a)	$ {0.0176} $	$ {0.1166} $	$ {0.2426} $	$ {0.0214} $	$ {0.1252} $	$ {0.2598} $	$ {0.0891} $	$ {0.2358} $	$ {0.4843} $
PCS PCA(3b)	$ {0} $	$ {0.0159} $	$ {0.0449} $	$ {0} $	$ {0} $	$ {0} $	$ {0.0044} $	$ {0.0080} $	$ {0.0120} $
NCS(2a) NCA(2a)	$ {0.0044} $	$ {0.0168} $	$ {1.2523} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {1.5682} $	$ {0.7559} $
NCS(2a) NCA(2b)	$ {0.0483} $	$ {0.1888} $	$ {1.0304} $	$ {0.0032} $	$ {0.1781} $	$ {0.1684} $	$ {0.0088} $	$ {1.1688} $	$ {0.9376} $
NCS(2a) ZCA	$ {0.2000} $	$ {1.0267} $	$ {1.8947} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $	$ {0} $
NCS(2a) PCA(3a)	$ {0.0176} $	$ {0.1166} $	$ {0.2426} $	$ {0.0214} $	$ {0.1252} $	$ {0.2598} $	$ {0.0891} $	$ {0.2358} $	$ {0.4843} $
NCS(2a) PCA(3b)	$ {0} $	$ {0.0159} $	$ {0.0449} $	$ {0} $	$ {0} $	$ {0} $	$ {0.0044} $	$ {0.0080} $	$ {0.0120} $

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Table 7. $ A_1(j, j') $ and $ B_1(j, j') $

	$ A (1, 1) $	$ A (1, 2) $	$ A(2, 1) $	$ A(2, 2) $	$ B(1, 1) $	B(1, 2)	$ B(2, 1) $	$ B(2, 2) $
ZCS NCA(2a)	$ {0} $	$ {0.3000} $	$ {0} $	$ {0.1000} $	$ {0} $	$ {0.3400} $	$ {0} $	$ {0.1150} $
ZCS NCA(2b)	$ {0} $	$ {0.3000} $	$ {0} $	$ {0.1000} $	$ {0} $	$ {0.3400} $	$ {0} $	$ {0.1150} $
ZCS ZCA	$ {0} $	$ {0.3000} $	$ {0} $	$ {0.1000} $	$ {0} $	$ {0.3400} $	$ {0} $	$ {0.1150} $
ZCS PCA(3a)	$ {0} $	$ {0.3000} $	$ {0} $	$ {0.1000} $	$ {0} $	$ {0.3400} $	$ {0} $	$ {0.1150} $
ZCS PCA(3b)	$ {0} $	$ {0.3000} $	$ {0} $	$ {0.1000} $	$ {0} $	$ {0.3400} $	$ {0} $	$ {0.1150} $
PCS NCA(2a)	$ {0.0838} $	$ {0.0093} $	$ {0.0784} $	$ {2.0463} $	$ {0.0963} $	$ {0.0110} $	$ {0.0974} $	$ {2.3432} $
PCS NCA(2b)	$ {0.0838} $	$ {0.0093} $	$ {0.0784} $	$ {2.0463} $	$ {0.0963} $	$ {0.0110} $	$ {0.0974} $	$ {2.3432} $
PCS ZCA	$ {0.0838} $	$ {0.0093} $	$ {0.0784} $	$ {2.0463} $	$ {0.0963} $	$ {0.0110} $	$ {0.0974} $	$ {2.3432} $
PCS PCA(3a)	$ {0.0838} $	$ {0.0093} $	$ {0.0784} $	$ {2.0463} $	$ {0.0963} $	$ {0.0110} $	$ {0.0974} $	$ {2.3432} $
PCS PCA(3b)	$ {0.0838} $	$ {0.0093} $	$ {0.0784} $	$ {2.0463} $	$ {0.0963} $	$ {0.0110} $	$ {0.0974} $	$ {2.3432} $
NCS(2a) NCA(2a)	$ {0.3631} $	$ {0.1072} $	$ {0.1397} $	$ {0.0389} $	$ {0.4172} $	$ {0.1227} $	$ {0.1618} $	$ {0.0453} $
NCS(2a) NCA(2b)	$ {0.3631} $	$ {0.1072} $	$ {0.1397} $	$ {0.0389} $	$ {0.4172} $	$ {0.1227} $	$ {0.1618} $	$ {0.0453} $
NCS(2a) ZCA	$ {0.3631} $	$ {0.1072} $	$ {0.1397} $	$ {0.0389} $	$ {0.4172} $	$ {0.1227} $	$ {0.1618} $	$ {0.0453} $
NCS(2a) PCA(3a)	$ {0.3631} $	$ {0.1072} $	$ {0.1397} $	$ {0.0389} $	$ {0.4172} $	$ {0.1227} $	$ {0.1618} $	$ {0.0453} $
NCS(2a) PCA(3b)	$ {0.3631} $	$ {0.1072} $	$ {0.1397} $	$ {0.0389} $	$ {0.4172} $	$ {0.1227} $	$ {0.1618} $	$ {0.0453} $

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Table 8. $ A_2(j, j') $ and $ B_2(j, j') $

	$ A 2(1, 1) $	$ A2 (1, 2) $	$ A2(2, 1) $	$ A2(2, 2) $	$ B2(1, 1) $	B2(1, 2)	$ B2(2, 1) $	$ B2(2, 2) $
ZCS NCA(2a)	$ {0} $	$ {1.4212} $	$ {0} $	$ {0.4737} $	$ {0} $	$ {0.4599} $	$ {0} $	$ {0.1717} $
ZCS NCA(2b)	$ {0} $	$ {1.1870} $	$ {0} $	$ {0.3957} $	$ {0} $	$ {0.5642} $	$ {0} $	$ {0.2033} $
ZCS ZCA	$ {0} $	$ {1.9840} $	$ {0} $	$ {0.6613} $	$ {0} $	$ {1.5335} $	$ {0} $	$ {0.5363} $
ZCS PCA(3a)	$ {0} $	$ {2.5277} $	$ {0} $	$ {0.8426} $	$ {0} $	$ {2.3911} $	$ {0} $	$ {0.8288} $
ZCS PCA(3b)	$ {0} $	$ {3.7649} $	$ {0} $	$ {1.2550} $	$ {0} $	$ {2.9230} $	$ {0} $	$ {1.0221} $
PCS NCA(2a)	$ {0.3972} $	$ {0.0441} $	$ {0.3716} $	$ {9.6942} $	$ {0.1426} $	$ {0.0197} $	$ {0.2146} $	$ {3.4026} $
PCS NCA(2b)	$ {0.3317} $	$ {0.0368} $	$ {0.3103} $	$ {8.0964} $	$ {0.1693} $	$ {0.0220} $	$ {0.2257} $	$ {4.0690} $
PCS ZCA	$ {0.5544} $	$ {0.0615} $	$ {0.5187} $	$ {13.5328} $	$ {0.4477} $	$ {0.0550} $	$ {0.5298} $	$ {10.8229} $
PCS PCA(3a)	$ {0.7063} $	$ {0.0784} $	$ {0.6608} $	$ {17.2411} $	$ {0.6924} $	$ {0.0835} $	$ {0.7879} $	$ {16.7681} $
PCS PCA(3b)	$ {1.0521} $	$ {0.1168} $	$ {0.9843} $	$ {25.6805} $	$ {0.8533} $	$ {0.1048} $	$ {1.0087} $	$ {20.6271} $
NCS(2a) NCA(2a)	$ {1.7200} $	$ {0.5080} $	$ {0.6618} $	$ {0.1843} $	$ {0.6196} $	$ {0.1777} $	$ {0.2529} $	$ {0.0732} $
NCS(2a) NCA(2b)	$ {1.4365} $	$ {0.4243} $	$ {0.5528} $	$ {0.1539} $	$ {0.7351} $	$ {0.2128} $	$ {0.2949} $	$ {0.0844} $
NCS(2a) ZCA	$ {2.4011} $	$ {0.7091} $	$ {0.9239} $	$ {0.2573} $	$ {1.9419} $	$ {0.5664} $	$ {0.7671} $	$ {0.2174} $
NCS(2a) PCA(3a)	$ {3.0590} $	$ {0.9035} $	$ {1.1771} $	$ {0.3278} $	$ {3.0025} $	$ {0.8777} $	$ {1.1804} $	$ {0.3335} $
NCS(2a) PCA(3b)	$ {4.5564} $	$ {1.3457} $	$ {1.7533} $	$ {0.4882} $	$ {3.7009} $	$ {1.0794} $	$ {1.4617} $	$ {0.4143} $

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[1]	A. D. Banik, Analyzing state-dependent arrival in $GI/BMSP/1/\infty $ queues, Math. Comput. Modelling, 53 (2011), 1229-1246. doi: 10.1016/j.mcm.2010.12.007.
[2]	A. D. Banik, Queueing analysis and optimal control of $ BMAP/G^{(a, b)}/1/N $ and $ BMAP/MSP^{(a, b)}/1/N $ systems, Comput. Industrial Engineering, 57 (2009), 748-761. doi: 10.1016/j.cie.2009.02.002.
[3]	A. D. Banik, Stationary analysis of a $ BMAP/R/1 $ queue with $ R $-type multiple working vacations, Comm. Statist. Simulation Comput., 46 (2017), 1035-1061. doi: 10.1080/03610918.2014.990096.
[4]	A. D. Banik, M. L. Chaudhry and U. C. Gupta, On the finite buffer queue with renewal input and batch Markovian service process: $ GI/BMSP/1/N $, Methodol. Comput. Appl. Probab., 10 (2008), 559-575. doi: 10.1007/s11009-007-9064-0.
[5]	S. Ghosh and A. D. Banik, An algorithmic analysis of the $ BMAP/MSP/1 $ generalized processor-sharing queue, Comput. Oper. Res., 79 (2017), 1-11. doi: 10.1016/j.cor.2016.10.001.
[6]	S. Ghosh and A. D. Banik, Computing conditional sojourn time of a randomly chosen tagged customer in a $ BMAP/MSP/1 $ queue under random order service discipline, Ann. Oper. Res., 261 (2018), 185-206. doi: 10.1007/s10479-017-2534-z.
[7]	S. Ghosh and A. D. Banik, Efficient computational analysis of non-exhaustive service vacation queues: $ BMAP/R/1/N(\infty) $ under gated-limited discipline, Appl. Math. Model., 68 (2019), 540-562. doi: 10.1016/j.apm.2018.11.040.
[8]	C. Kim, V. I. Klimenok and A. N. Dudin, Analysis of unreliable $ BMAP/PH/N $ type queue with Markovian flow of breakdowns, Appl. Math. Comput., 314 (2017), 154-172. doi: 10.1016/j.amc.2017.06.035.
[9]	V. Klimenok, A. N. Dudin and K. Samouylov, Computation of moments of queue length in the $ BMAP/SM/1 $ queue, Oper. Res. Lett., 45 (2017), 467-470. doi: 10.1016/j.orl.2017.07.003.
[10]	V. Klimenok and O. Dudina, Retrial tandem queue with controllable strategy of repeated attempts, Quality Tech. Quantitative Manag., 14 (2017), 74-93. doi: 10.1080/16843703.2016.1189177.
[11]	V. Klimenok, O. Dudina, V. Vishnevsky and K. Samouylov, Retrial tandem queue with $ BMAP $ input and semi-Markovian service process, in International Conference on Distributed Computer and Communication Networks, Communications in Computer and Information Science, 700, Springer, Cham, 2017,159–173. doi: 10.1007/978-3-319-66836-9_14.
[12]	V. I. Klimenok, A. N. Dudin and K. E. Samouylov, Analysis of the $ BMAP/PH/N $ queueing systems with backup servers, Appl. Math. Model., 57 (2018), 64-84. doi: 10.1016/j.apm.2017.12.024.
[13]	D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Comm. Statist. Stochastic Models, 7 (1991), 1-46. doi: 10.1080/15326349108807174.
[14]	D. M. Lucantoni, K. S. Meier-Hellstern and M. F. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes, Adv. in Appl. Probab., 22 (1990), 676-705. doi: 10.2307/1427464.
[15]	M. F. Neuts, Versatile Markovian point process, J. Appl. Probab., 16 (1979), 764-779. doi: 10.2307/3213143.
[16]	M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach, Johns Hopkins Series in the Mathematical Sciences, 2, John Hopkins University Press, Baltimore, MD, 1981. doi: 10.2307/2287748.
[17]	G. Rama, R. Ramshankar, R. Sandhya, V. Sundar and R. Ramanarayanan, $ BMAP/M/C $ bulk service queue with randomly varying environment, IOSR J. Engineering, 5 (2015), 33-47.
[18]	V. Ramaswami, The $ N/G/1 $ queue and its detailed analysis, Adv. in Appl. Probab., 12 (1980), 222-261. doi: 10.2307/1426503.
[19]	S. K. Samanta, M. L. Chaudhry and A. Pacheco, Analysis of $ BMAP/MSP/1 $ queue, Methodol. Comput. Appl. Probab., 18 (2016), 419-440. doi: 10.1007/s11009-014-9429-0.
[20]	R. Sandhya, V. Sundar, G. Rama, R. Ramshankar and R. Ramanarayanan, $ BMAP/BMSP/1 $ queue with randomly varying environment, IOSR J. Engineering, 5 (2015), 01-12.
[21]	V. M. Vishnevskii and A. N. Dudin, Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks, Autom. Remote Control, 78 (2017), 1361-1403. doi: 10.1134/S000511791708001X.