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Principal component analysis with drop rank covariance matrix
A $ {BMAP/BMSP/1} $ queue with Markov dependent arrival and Markov dependent service batches
1. | Centre for Research in Mathematics, C.M.S. College, Kottayam-686001, India |
2. | Department of Mathematics, Union Christian College, Aluva-683102, India |
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.
References:
[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. Google Scholar |
[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. Google Scholar |
[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. |
show all references
References:
[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. Google Scholar |
[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. Google Scholar |
[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. |
NCA (2a) | NCA (2b) | ZCA | PCA (3a) | PCA (3b) | |
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NCS(2a) | |||||
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PCS |
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ZCS | |||||
PCS |
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) | |
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NCS(2a) NCA(2b) | |||||||||
NCS(2a) ZCA | |||||||||
NCS(2a) PCA(3a) | |||||||||
NCS(2a) PCA(3b) |
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) | |
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NCS(2a) PCA(3b) |
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) | |
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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) | |
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NCS(2a) PCA(3b) |
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) | |
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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) | |
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ZCS ZCA | |||||||||
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B2(1, 2) | ||||||||
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ZCS ZCA | ||||||||
ZCS PCA(3a) | ||||||||
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NCS(2a) PCA(3b) |
B2(1, 2) | ||||||||
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ZCS ZCA | ||||||||
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PCS ZCA | ||||||||
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NCS(2a) PCA(3b) |
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