Abstract
In this paper we consider a single server queueing model with under general bulk service rule with infinite upper bound on the batch size which we call group clearance. The arrivals occur according to a batch Markovian point process and the services are generally distributed. The customers arriving after the service initiation cannot enter the ongoing service. The service time is independent on the batch size. First, we employ the classical embedded Markov renewal process approach to study the model. Secondly, under the assumption that the services are of phase type, we study the model as a continuous-time Markov chain whose generator has a very special structure. Using matrix-analytic methods we study the model in steady-state and discuss some special cases of the model as well as representative numerical examples covering a wide range of service time distributions such as constant, uniform, Weibull, and phase type.
Article PDF
Similar content being viewed by others
References
Abolnikov L, Dukhovny A (2003) Optimization in HIV screening problems. Int J Stoch Anal 16(4):361–374
Arumuganathan R, Jeyakumar S (2005) Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times. Appl Math Model 29(10):972–986
Ayyappan G, Karpagam S (2018) An M[X]/G(a,b)/1 queueing system with breakdown and repair, stand-by server, multiple vacation and control policy on request for re-service. Mathematics 6(6):101. http://www.mdpi.com/2227-7390/6/6/101
Ayyappan G, Nirmala M (2018) An M[X]/G(a,b)/1 queue with breakdown and delay time to two phase repair under multiple vacation. Applications and Applied Mathematics: An International Journal 13 (2):639–663
Baba Y (1996) A bulk service GI/M/1 queue with service rates depending on service batch size. J Oper Res Soc Jpn 39(1):25–35
Baetens J, Steyaert B, Claeys D, Bruneel H (2016) System occupancy of a two-class batch-service queue with class-dependent variable server capacity. In: International conference on analytical and stochastic modeling techniques and applications, Springer, pp 32–44. https://doi.org/10.1007/978-3-319-43904-4_3
Baetens J, Claeys D, Steyaert B, Bruneel H (2017) System performance of a variable-capacity batch-service queue with geometric service times and customer-based correlation. In: 31st european conference on modelling and simulation, ECMS 2017, vol 31, pp 649–655. http://www.scs-europe.net/dlib/2017/2017-0649.htm
Baetens J, Steyaert B, Claeys D, Bruneel H (2018) Delay analysis of a two-class batch-service queue with class-dependent variable server capacity. Math Methods Oper Res 88(1):37–57. https://doi.org/10.1007/s00186-017-0627-8
Baetens J, Steyaert B, Claeys D, Bruneel H (2019) Analysis of a batch-service queue with variable service capacity, correlated customer types and generally distributed class-dependent service times. Perform Eval 135:102012. https://linkinghub.elsevier.com/retrieve/pii/S0166531618302542
Bailey NTJ (1954) On queueing processes with bulk service. J R Stat Soc Series B Stat Methodol 16(1):80–87
Banerjee A, Gupta UC (2012) Reducing congestion in bulk-service finite-buffer queueing system using batch-size-dependent service. Perform Eval 69 (1):53–70. https://linkinghub.elsevier.com/retrieve/pii/S0166531611001350
Banerjee A, Gupta UC, Goswami V (2014) Analysis of finite-buffer discrete-time batch-service queue with batch-size-dependent service. Comput Ind Eng 75:121–128. https://linkinghub.elsevier.com/retrieve/pii/S0360835214001892
Banerjee A, Gupta UC, Chakravarthy SR (2015) Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service. Comput Oper Res 60:138–149. https://doi.org/10.1016/j.cor.2015.02.012. http://linkinghub.elsevier.com/retrieve/pii/S030505481500043X
Banik AD (2015) Single server queues with a batch Markovian arrival process and bulk renewal or non-renewal service. J Syst Sci Syst Eng 24(3):337–363. https://doi.org/10.1007/s11518-015-5268-y
Banik AD, Chaudhry ML, Gupta UC (2008) On the finite buffer queue with renewal input and batch Markovian service process: GI/BMSP/1/N. Methodol Comput Appl Probab 10(4):559–575
Banik AD, Gupta UC, Chaudhry ML (2009) Finite-buffer bulk service queue under Markovian service process: GI/MSP(a,b)/1/N. Stoch Anal Appl 27(3):500–522
Bank B, Samanta SK (2020) Analytical and computational studies of the BMAP/G(a,Y )/1 queue. Communications in Statistics - Theory and Methods, pp 1–29
Bar-Lev SK, Parlar M, Perry D, Stadje W, der Duyn Schouten FAV (2007) Applications of bulk queues to group testing models with incomplete identification. Eur J Oper Res 183(1):226–237
Bladt M, Nielsen BF (2017) Matrix-exponential distributions in applied probability, volume 81 of Probability Theory and Stochastic Modelling. Springer, Boston
Chakravarthy SR (1992) A finite capacity GI/PH/1 queue with group services. Nav Res Logist 39(3):345–357
Chakravarthy SR (1993) Analysis of a finite MAP/G/1 queue with group services. Queueing Syst 13 (4):385–407. https://doi.org/10.1007/BF01149262
Chakravarthy SR (2001) The batch Markovian arrival process: A review and future work. In: Krishnamoorthy A et al.(eds) Advances in probability theory and stochastic processes. Notable Publications Inc., pp 21–39
Chakravarthy SR (2011) Markovian Arrival Processes, American Cancer Society. https://doi.org/10.1002/9780470400531.eorms0499
Chakravarthy SR (2015) Matrix-analytic queueing models, 2nd Edition, Birkhäuser, chap 8, pp 177–199
Chakravarthy SR, Dudin AN (2002a) A batch Markovian queue with a variable number of servers and group services. In: Latouche G, Taylor P (eds) Matrix-analytic methods - theory and applications. World Scientific Publishing Co., pp 63–88
Chakravarthy SR, Dudin AN (2002b) A multi-server retrial queue with BMAP arrivals and group services. Queueing Syst 42(1):5–31. https://doi.org/10.1023/A:1019989127190
Chakravarthy SR, Maity A, Gupta UC (2017) An ‘(s, s)’ inventory in a queueing system with batch service facility. Ann Oper Res 258(2):263–283
Charfi E, Gueguen C, Chaari L, Cousin B, Kamoun L (2017) Dynamic frame aggregation scheduler for multimedia applications in IEEE 802.11n networks. Trans Emerg Telecommun Technol 28(2): e2942
Chaudhry M, Templeton J (1983) A first course in bulk queues. Wiley, New York
Chaudhry ML, Gupta UC (1999) Modelling and analysis of M/G(a,b)/1/N queue–a simple alternative approach. Queueing Syst 31(1-2):95–100
Chaudhry ML, Banik AD, Pacheco A, Ghosh S (2016) A simple analysis of system characteristics in the batch service queue with infinite-buffer and Markovian service process using the roots method: \({{GI}}/c-{{MSP}}^{(a,b)}/1/\infty \). RAIRO - Operations Research 50(3):519–551
Cheng Y, Yang Y, Du DZ (2019) A class of asymptotically optimal group screening strategies with limited item participation. Discret Appl Math 270:83–95. https://linkinghub.elsevier.com/retrieve/pii/S0166218X19302938
Ilya C, Natalia N, Evgeny I (2017) Task scheduling in desktop grids: Open problems. Open Eng 7 (1):343. https://doi.org/10.1515/eng-2017-0038. https://www.degruyter.com/view/j/eng.2017.7.issue-1/eng-2017-0038/eng-2017-0038.xml
Claeys D, Laevens K, Walraevens J, Bruneel H (2010a) Complete characterisation of the customer delay in a queueing system with batch arrivals and batch service. Math Methods Oper Res 72(1):1–23. https://doi.org/10.1007/s00186-009-0297-2
Claeys D, Walraevens J, Laevens K, Bruneel H (2010b) A queueing model for general group screening policies and dynamic item arrivals. Eur J Oper Res 207(2):827–835. https://linkinghub.elsevier.com/retrieve/pii/S037722171000398X
Claeys D, Steyaert B, Walraevens J, Laevens K, Bruneel H (2013) Analysis of a versatile batch-service queueing model with correlation in the arrival process. Perform Eval 70(4):300–316. https://linkinghub.elsevier.com/retrieve/pii/S0166531612001344
D’Arienzo MP, Dudin AN, Dudin SA, Manzo R (2019) Analysis of a retrial queue with group service of impatient customers. J Ambient Intell Humaniz Comput 11(6):2591–2599. https://doi.org/10.1007/s12652-019-01318-x
Dudin A, Chakravarthy SR (2002a) Optimal hysteretic control for the BMAP/G/1 system with single and group service modes. Ann Oper Res 112(1):153–169. https://doi.org/10.1023/A:1020985106453
Dudin AN, Chakravarthy SR (2002b) A single server retrial queuing model with batch arrivals and group services. In: Artalejo JR, Krishnamoorthy A (eds) Advances in stochastic modelling. Notable Publications Inc., New Jersey
Dudin AN, Chakravarthy SR (2003) Multi-threshold control of the BMAP/SM/1/K queue with group services. J Appl Math Stoch Anal 16 (4):327–347. https://www.hindawi.com/archive/2003/276047/abs/
Germs R, van Foreest N (2013) Analysis of finite-buffer state-dependent bulk queues. OR Spectrum 35(3):563–583
Gold H, Tran-Gia P (1993) Performance analysis of a batch service queue arising out of manufacturing system modelling. Queueing Syst 14(3-4):413–426
Grippa P, Schilcher U, Bettstetter C (2019) On access control in cabin-based transport systems. IEEE Trans Intell Transp Syst 20(6):2149–2156
Gupta GK, Banerjee A, Gupta UC (2020) On finite-buffer batch-size-dependent bulk service queue with queue-length dependent vacation. Qual Technol Quant Manag 17(5):501–527
Gupta G, Banerjee A (2018) On M/G(a, b)/1/N queue with batch size-and queue length-dependent service. In: International conference on mathematics and computing, Springer Nature, pp 249–262 . https://doi.org/10.1007/978-981-13-2095-8_20
Gupta G, Banerjee A (2019) Steady state analysis of system size-based balking in M/Mb/1 queue. Int J Math Oper Res 14:319
He QM (2014) Fundamentals of matrix-analytic methods, Springer, New York
Hébuterne G, Rosenberg C (1999) Arrival and departure state distributions in the general bulk-service queue. Nav Res Logist (NRL) 46(1):107–118
Ivashko E, Rumyantsev A, Chernov I, Ponomarev V, Shabaev A (2018) Survey on deduplication techniques in flash-based storage. In: Proceedings of the 22nd conference of open innovations association FRUCT, vol 426, pp 25–33
Jensen A (1953) Markoff chains as an aid in the study of Markoff processes. Scand Actuar J 1953(sup1):87–91. https://doi.org/10.1080/03461238.1953.10419459
Jeyakumar S, Senthilnathan B (2017) Modelling and analysis of a bulk service queueing model with multiple working vacations and server breakdown. RAIRO - Operations Research 51(2):485–508
Kendall DG (1953) Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann of Math Stat 24(3):338–354
Lohse S, Pfuhl T, Berkó-Göttel B, Rissland J, Geißler T, Gärtner B, Becker SL, Schneitler S, Smola S (2020) Pooling of samples for testing for SARS-CoV-2 in asymptomatic people. The Lancet Infectious Diseases https://doi.org/10.1016/S1473-3099(20)30362-5
Maity A, Gupta UC (2015) Analysis and optimal control of a queue with infinite buffer under batch-size dependent versatile bulk-service rule. OPSEARCH 52(3):472–489. https://doi.org/10.1007/s12597-015-0197-6
Mazalov VV, Nikitina NN, Ivashko EE (2014) Hierarchical two-level game model for tasks scheduling in a desktop grid. In: 2014 6th international congress on ultra modern telecommunications and control systems and workshops (ICUMT), pp 541–545
Neuts M, Li JM (1996) An algorithm for the P(n,t) matrices of a continuous BMAP. In: Chakravarthy SR, Alfa AS (eds) Matrix-analytic Methods in Stochastic Models. Marcel Dekker, pp 7–19
Neuts MF (1967) A general class of bulk queues with Poisson input. Ann of Math Stat 38(3):759–770
Neuts MF (1981) Matrix-Geometric solutions in stochastic models. Johns Hopkins University Press, Baltimore
Neuts MF, Chandramouli Y (1987) Statistical group testing with queueing involved. Queueing Syst 2(1):19–39
Niranjan SP, Chandrasekaran VM, Indhira K (2018) Performance characteristics of a batch service queueing system with functioning server failure and multiple vacations. J Phys Conf Ser 1000:012112. https://doi.org/10.1088/1742-6596/1000/1/012112
O’Mullane W, Li N, Nieto-Santisteban M, Szalay A, Thakar A, Gray J (2005) Batch is back: CasJobs, serving multi-TB data on the Web. In: IEEE international conference on Web services (ICWS’05), IEEE, pp 33–40
Panda G, Goswami V (2020) Effect of information on the strategic behavior of customers in a discrete-time bulk service queue. J Ind Manag Optim 16 (3):1369–1388. https://doi.org/10.3934/jimo.2019007
Panda G, Banik AD, Guha D (2018) Stationary analysis and optimal control under multiple working vacation policy in a GI/M(a, b)/1 queue. J Syst Sci Complexity 31 (4):1003–1023. https://doi.org/10.1007/s11424-017-6172-y
Powell WB, Humblet P (1986) The bulk service queue with a general control strategy: Theoretical analysis and a new computational procedure. Oper Res 34(2):267–275
Pradhan S, Gupta UC (2017) Modeling and analysis of an infinite-buffer batch-arrival queue with batch-size-dependent service: \(M^{X}/G_{n}^{(a,b )}/ 1\). Perform Eval 108:16–31. https://linkinghub.elsevier.com/retrieve/pii/S0166531616303078
Pradhan S, Gupta UC (2019) Analysis of an infinite-buffer batch-size-dependent service queue with Markovian arrival process. Ann Oper Res 277(2):161–196. https://doi.org/10.1007/s10479-017-2476-5
Pradhan S, Gupta UC, Samanta S (2016) Queue-length distribution of a batch service queue with random capacity and batch size dependent service: \(M / {G}_{r}^{Y}/1\). OPSEARCH 53(2):329–343. https://doi.org/10.1007/s12597-015-0231-8
Sasikala S, Indhira K (2016) Bulk service queueing models – a survey. Int J Pure Appl Math 106(6):43–56. https://acadpubl.eu/jsi/2016-106-6-7-8/2016-106-6/6/6.pdf
Saxena A, Claeys D, Bruneel H, Zhang B, Walraevens J (2018) Modeling data backups as a batch-service queue with vacations and exhaustive policy. Comput Commun 128:46–59. https://linkinghub.elsevier.com/retrieve/pii/S0140366418302901
Sikdar K, Gupta UC (2005) Analytic and numerical aspects of batch service queues with single vacation. Comput Operat Res 32(4):943–966
Vadivu AS, Arumuganathan R (2015) Cost analysis of MAP/G(a, b)/1/N queue with multiple vacations and closedown times. Qual Technol Quant Manage 12(4):605–626, publisher: Taylor & Francis. https://doi.org/10.1080/16843703.2015.11673438
Xie M, Xia L, Xu J (2020) On M/G[b]/1/K queue with multiple state-dependent vacations: A real problem from media-based cache in hard disk drives. Perform Eval 139:102085. https://linkinghub.elsevier.com/retrieve/pii/S0166531620300055
Yu M, Alfa AS (2015) Algorithm for computing the queue length distribution at various time epochs in DMAP/G(1, a, b)/1/N queue with batch-size-dependent service time. Eur J Oper Res 244(1):227–239. https://linkinghub.elsevier.com/retrieve/pii/S0377221715000764
Yu M, Tang Y (2018) Analysis of the sojourn time distribution for M/GL/1 queue with bulk-service of exactly size L. Methodol Comput Appl Probab 20 (4):1503–1514. https://doi.org/10.1007/s11009-018-9635-2
Zee DJVD, Harten AV, Schuur P (2001) On-line scheduling of multi-server batch operations. IIE Trans 33(7):569–586
Zeng Y, Xia CH (2017) Optimal bulking threshold of batch service queues. J Appl Prob 54(2):409–423. https://www.cambridge.org/core/article/optimal-bulking-threshold-of-batch-service-queues/C1DD49670E8DFF9B2E55F82BF049BC29
Acknowledgements
AR thanks Dr. Natalia Nikitina for helpful discussions. The study of AR was partially supported by RFBR, projects 18-07-00147, 18-07-00156, 18-37-00094, 19-07-00303, 19-57-45022.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chakravarthy, S.R., Shruti & Rumyantsev, A. Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance. Methodol Comput Appl Probab 23, 1551–1579 (2021). https://doi.org/10.1007/s11009-020-09828-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-020-09828-4