Analysis of BMAP/MSP/1 queue with MAP generated negative customers and disasters. (English) Zbl 07706315
Summary: This paper studies an infinite-buffer single server queueing model with negative customers and disasters. The ordinary (positive) customers arrive in the system according to the batch Markovian arrival process and join the queue for service. The negative customer and disaster are characterized by two independent Markovian arrival processes. A negative customer removes the positive customer undergoing service, if any, and a disaster makes the system empty by simultaneously removing all the positive customers present in the system. We obtain the steady-state vector generating function (VGF) of the distribution of the number of positive customers in the system at arbitrary epoch. Further, the inversion of the VGF is carried out using the roots method to obtain the distribution. Moreover, the probability vectors at post-departure and pre-arrival epochs are also obtained. The results obtained throughout the analysis are computationally tractable, as illustrated by few numerical examples. Finally, we discuss the impact of the correlation of the arrival process of negative customers and disasters on the performance of the system through some graphical representations.
MSC:
| 62-XX | Statistics |
Keywords:
batch Markovian arrival process; disaster; G-queue; Markovian service process; negative customers; vector generating functionReferences:
| [1] | Abbas, K.; Aïssani, D., Strong stability of the embedded Markov chain in an \(####\) queue with negative customers, Applied Mathematical Modelling, 34, 10, 2806-12 (2010) · Zbl 1201.90055 |
| [2] | Atencia, I.; Moreno, P., The discrete-time \(####\) queue with negative customers and disasters, Computers & Operations Research, 31, 9, 1537-48 (2004) · Zbl 1107.90330 |
| [3] | Barbhuiya, F. P.; Kumar, N.; Gupta, U. C., Batch renewal arrival process subject to geometric catastrophes, Methodology and Computing in Applied Probability, 21, 1, 69-83 (2019) · Zbl 1411.90088 · doi:10.1007/s11009-018-9643-2 |
| [4] | Boxma, O. J.; Perry, D.; Stadje, W., Clearing models for \(####\) queues, Queueing Systems, 38, 3, 287-306 (2001) · Zbl 1024.90020 |
| [5] | Chae, K. C.; Park, H. M.; Yang, W. S., A \(####\) queue with negative and positive customers, Applied Mathematical Modelling, 34, 6, 1662-71 (2010) · Zbl 1193.60107 |
| [6] | Chakravarthy, S. R., The batch Markovian arrival process: A review and future work, Advances in Probability Theory and Stochastic Processes, 1, 21-49 (2001) |
| [7] | Chakravarthy, S. R., A disaster queue with Markovian arrivals and impatient customers, Applied Mathematics and Computation, 214, 1, 48-59 (2009) · Zbl 1170.60330 · doi:10.1016/j.amc.2009.03.081 |
| [8] | Chakravarthy, S. R., Wiley encyclopedia of operations research and management science, Markovian arrival processes (2010) |
| [9] | Chen, A.; Renshaw, E., The \(####\) queue with mass exodus and mass arrivals when empty, Journal of Applied Probability, 34, 1, 192-207 (1997) · Zbl 0876.60079 |
| [10] | Dharmaraja, S.; Di Crescenzo, A.; Giorno, V.; Nobile, A. G., A continuous-time Ehrenfest model with catastrophes and its jump-diffusion approximation, Journal of Statistical Physics, 161, 2, 326-45 (2015) · Zbl 1329.60302 · doi:10.1007/s10955-015-1336-4 |
| [11] | Dharmaraja, S.; Kumar, R., Transient solution of a Markovian queuing model with heterogeneous servers and catastrophes, Opsearch, 52, 4, 810-26 (2015) · Zbl 1365.90099 · doi:10.1007/s12597-015-0209-6 |
| [12] | Dudin, A.; Karolik, A. V., \(####\) queue with Markovian input of disasters and non-instantaneous recovery, Performance Evaluation, 45, 1, 19-32 (2001) · Zbl 1013.68035 |
| [13] | Dudin, A.; Semenova, O., A stable algorithm for stationary distribution calculation for a \(####\) queueing system with markovian arrival input of disasters, Journal of Applied Probability, 41, 2, 547-56 (2004) · Zbl 1057.60084 |
| [14] | Dudin, A.; Nishimura, S., A \(####\) queueing system with Markovian arrival input of disasters, Journal of Applied Probability, 36, 3, 868-81 (1999) · Zbl 0949.60100 |
| [15] | Economou, A.; Fakinos, D., A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes, European Journal of Operational Research, 149, 3, 625-40 (2003) · Zbl 1030.60069 · doi:10.1016/S0377-2217(02)00465-4 |
| [16] | Gail, H. R.; Hantler, S. L.; Taylor, B. A., Spectral analysis of \(####\) type Markov chains, Advances in Applied Probability, 28, 1, 114-65 (1996) · Zbl 0845.60092 |
| [17] | Gelenbe, E., Random neural networks with negative and positive signals and product form solution, Neural Computation, 1, 4, 502-10 (1989) · doi:10.1162/neco.1989.1.4.502 |
| [18] | Gupta, U. C.; Singh, G.; Chaudhry, M. L., An alternative method for computing system-length distributions of \(####\) queues using roots, Performance Evaluation, 95, 60-79 (2016) |
| [19] | Gupta, U. C.; Kumar, N.; Barbhuiya, F. P.; Joshua, V.; Varadhan, S.; Vishnevsky, V., Applied Probability and Stochastic Processes, Infosys Science Foundation Series, A queueing system with batch renewal input and negative arrivals, 143-57 (2020), Singapore: Springer, Singapore · Zbl 1473.60133 |
| [20] | Harrison, P. G.; Pitel, E., Sojourn times in single-server queues by negative customers, Journal of Applied Probability, 30, 4, 943-63 (1993) · Zbl 0787.60116 · doi:10.2307/3214524 |
| [21] | Harrison, P. G.; Pitel, E., \(####\) queues with negative arrival: An iteration to solve a fredholm integral equation of the first kind, MASCOTS’95. Proceedings of the Third International Workshop on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems, 423-26 (1995) |
| [22] | Harrison, P. G.; Pitel, E., The \(####\) queue with negative customers, Advances in Applied Probability, 28, 2, 540-66 (1996) · Zbl 0861.60088 |
| [23] | He, Q. M.; Neuts, M. F., Markov chains with marked transitions, Stochastic Processes and Their Applications, 74, 1, 37-52 (1998) · Zbl 0932.60078 · doi:10.1016/S0304-4149(97)00109-9 |
| [24] | Jain, G.; Sigman, K., A pollaczek-khintchine formula for \(####\) queues with disasters, Journal of Applied Probability, 33, 4, 1191-200 (1996) · Zbl 0867.60082 |
| [25] | Krishnamoorthy, A.; Pramod, P. K.; Chakravarthy, S. R., Queues with interruptions: A survey, Top, 22, 1, 290-320 (2014) · Zbl 1305.60095 · doi:10.1007/s11750-012-0256-6 |
| [26] | Kumar, B. K.; Arivudainambi, D., Transient solution of an \(####\) queue with catastrophes, Computers & Mathematics with Applications, 40, 10-11, 1233-40 (2000) · Zbl 0962.60096 |
| [27] | Kumar, B. K.; Madheswari, S. P., Transient behaviour of the \(####\) queue with catastrophes, Statistica, 62, 1, 129-36 (2002) · Zbl 1188.60048 |
| [28] | Kumar, N.; Gupta, U. C., Analysis of batch Bernoulli process subject to discrete-time renewal generated binomial catastrophes, Annals of Operations Research, 287, 1, 257-83 (2020) · Zbl 1444.60082 · doi:10.1007/s10479-019-03410-z |
| [29] | Kumar, N.; Gupta, U. C., A renewal generated geometric catastrophe model with discrete-time Markovian arrival process, Methodology and Computing in Applied Probability, 22, 3, 1293-324 (2020) · Zbl 1460.60099 · doi:10.1007/s11009-019-09768-8 |
| [30] | Kumar, N.; Gupta, U. C., Analysis of a population model with batch Markovian arrivals influenced by Markov arrival geometric catastrophes, Communications in Statistics - Theory and Methods, 50, 13, 3137-58 (2021) · Zbl 07530972 · doi:10.1080/03610926.2019.1682166 |
| [31] | Kumar, N.; Barbhuiya, F. P.; Gupta, U. C., Analysis of a geometric catastrophe model with discrete-time batch renewal arrival process, RAIRO - Operations Research, 54, 5, 1249-68 (2020) · Zbl 1448.60177 · doi:10.1051/ro/2019074 |
| [32] | Kumar, N.; Barbhuiya, F. P.; Gupta, U. C., Unified killing mechanism in a single server queue with renewal input, OPSEARCH, 57, 1, 246-59 (2020) · Zbl 07182119 · doi:10.1007/s12597-019-00408-w |
| [33] | Lucantoni, D. M., New results on the single server queue with a batch Markovian arrival process, Communications in Statistics. Stochastic Models, 7, 1, 1-46 (1991) · doi:10.1080/15326349108807174 |
| [34] | Lucantoni, D. M.; Meier-Hellstern, K. S.; Neuts, M. F., A single-server queue with server vacations and a class of non-renewal arrival processes, Advances in Applied Probability, 22, 3, 676-705 (1990) · doi:10.2307/1427464 |
| [35] | Neuts, M. F., A versatile Markovian point process, Journal of Applied Probability, 16, 4, 764-79 (1979) · Zbl 0422.60043 · doi:10.2307/3213143 |
| [36] | Park, H. M.; Yang, W. S.; Chae, K. C., Analysis of the \(####\) queue with disasters, Stochastic Analysis and Applications, 28, 1, 44-53 (2009) · Zbl 1181.60136 |
| [37] | Park, H. M.; Yang, W. S.; Chae, K. C., The \(####\) queue with negative customers and disasters, Stochastic Models, 25, 4, 673-88 (2009) · Zbl 1190.60088 |
| [38] | Ramaswami, V., The N/G/1 queue and its detailed analysis, Advances in Applied Probability, 12, 1, 222-61 (1980) · Zbl 0424.60093 · doi:10.2307/1426503 |
| [39] | Samanta, S. K.; Chaudhry, M. L.; Pacheco, A., Analysis of \(####\) queue, Methodology and Computing in Applied Probability, 18, 2, 419-40 (2016) · Zbl 1339.60135 |
| [40] | Semenova, O. V., An optimal threshold control for a \(####\) system with MAP disaster flow, Automation and Remote Control, 64, 9, 1442-54 (2003) · Zbl 1081.90021 |
| [41] | Semenova, O. V., Optimal control for a \(####\) queue with MAP-input of disasters and two operation modes, RAIRO-Operations Research, 38, 2, 153-71 (2004) · Zbl 1092.90018 |
| [42] | Semenova, O. V., Optimal hysteresis control for \(####\) queue with MAP-input of disasters, Quality Technology & Quantitative Management, 4, 3, 395-405 (2007) |
| [43] | Shortle, J. F., J. M., Thompson, D. Gross, and C. M. Harris. (2018) |
| [44] | Sudhesh, R.; Savitha, P.; Dharmaraja, S., Transient analysis of a two-heterogeneous servers queue with system disaster, server repair and customers’ impatience, Top, 25, 1, 179-205 (2017) · Zbl 1364.60124 · doi:10.1007/s11750-016-0428-x |
| [45] | Towsley, D.; Tripathi, S. K., A single server priority queue with server failures and queue flushing, Operations Research Letters, 10, 6, 353-62 (1991) · Zbl 0737.60086 · doi:10.1016/0167-6377(91)90008-D |
| [46] | Tweedie, R. L., Sufficient conditions for regularity, recurrence and ergodicity of Markov processes, Mathematical Proceedings of the Cambridge Philosophical Society, 78, 1, 125-36 (1975) · Zbl 0331.60047 · doi:10.1017/S0305004100051562 |
| [47] | Van Do, T., Bibliography on G-networks, negative customers and applications, Mathematical and Computer Modelling, 53, 1-2, 205-12 (2011) · doi:10.1016/j.mcm.2010.08.006 |
| [48] | Vishnevskii, V. M.; Dudin, A. N., Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks, Automation and Remote Control, 78, 8, 1361-403 (2017) · Zbl 1377.60089 · doi:10.1134/S000511791708001X |
| [49] | Wang, J.; Huang, Y.; Dai, Z., A discrete-time on-off source queueing system with negative customers, Computers & Industrial Engineering, 61, 4, 1226-32 (2011) · doi:10.1016/j.cie.2011.07.013 |
| [50] | Wu, H.; He, Q. M., Double-sided queues with marked Markovian arrival processes and abandonment, Stochastic Models, 37, 1, 23-58 (2021) · Zbl 1469.60295 · doi:10.1080/15326349.2020.1794898 |
| [51] | Yang, W. S.; Chae, K. C., A note on the \(####\) queue with Poisson negative arrivals, Journal of Applied Probability, 38, 4, 1081-85 (2001) · Zbl 0996.60102 |
| [52] | Zhou, W.-H., Performance analysis of discrete-time queue \(####\) with negative arrivals, Applied Mathematics and Computation, 170, 2, 1349-55 (2005) · Zbl 1080.60088 |
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