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

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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  • 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.

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

    Citation:

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  • 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 $
     | Show Table
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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 $
     | Show Table
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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} $
     | Show Table
    DownLoad: CSV

    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} $
     | Show Table
    DownLoad: CSV

    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} $
     | Show Table
    DownLoad: CSV

    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} $
     | Show Table
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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} $
     | Show Table
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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} $
     | Show Table
    DownLoad: CSV
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