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Multithreshold control of the $$BMAP/G/1$$ queuing system with map flow of Markovian disasters. (English. Russian original) Zbl 1195.90032
Autom. Remote Control 68, No. 1, 95-108 (2007); translation from Avtom. Telemekh. 2007, No. 1, 105-120 (2007).
Summary: Consideration is given to the $$BMAP/G/1$$ queuing system with a controlled mode of operation and flow of disasters interrupting servicing and emptying the system. The multithreshold strategies are used to control the mode of operation. For a fixed control strategy, determined are the stationary distribution of the system state probabilities at the instants of servicing completion and the performance characteristics such as the mean times between the customer departures, mean fraction of using a mode, mean number of customers lost in unit time because of disasters, and the probability that an arbitrary customer will be safely serviced. A numerical example illustrating the results is presented.

##### MSC:
 90B22 Queues and service in operations research 60K25 Queueing theory (aspects of probability theory)
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##### References:
 [1] Rykov, V.V., On the Monotonicity Conditions for the Optimal Policies to Control the Queuing Systems, Avtom. Telemekh., 1999, no. 9, pp. 92–106. [2] Rykov, V.V., Monotone Control of Queuing Systems with Heterogeneous Servers, Queuing Syst., 2001, vol. 37, no. 4, pp. 391–403. · Zbl 1017.90026 · doi:10.1023/A:1010893501581 [3] Tijms, H.C., On the Optimality of a Switch-over Policy for Controlling the Queue Size in a M/G/1 Queue with Variable Service Rate, Lecture Notes Comput. Sci., 1976, vol. 40, pp. 736–742. · doi:10.1007/3-540-07622-0_506 [4] Artalejo, J.R. and Economou, A., Optimal Control and Performance Analysis of an M x/M/1 Queue with Batches of Negative Customers, RAIRO Oper. Res., 2004, vol. 38, pp. 121–151. · Zbl 1092.90013 · doi:10.1051/ro:2004016 [5] Nobel’, R., Regenerative Approach for Analysis of the M X/G/1 Queue with Two Kinds of Servicing, Avtomat. Vychisl. Tekhn., 1998, no. 1, pp. 3–14. [6] Nobel, R.D. and Tijms, H.C., Optimal Control for a M x/G/1 Queue with Two Service Modes, Eur. J. Oper. Res., 1999, vol. 113, no. 3, pp. 610–619. · Zbl 0947.90028 · doi:10.1016/S0377-2217(98)00085-X [7] Dudin, A.N. and Nishimura, S., Optimal Control for a BMAP/G/1 Queue with Two Service Modes, Math. Probl. Eng., 1999, vol. 5, pp. 255–273. · Zbl 0954.60084 · doi:10.1155/S1024123X99001088 [8] Dudin, A.N., Optimal Hysteresis Control of a Nonreliable BMAP/SM/1 System with Two Modes of Operation, Avtom. Telemekh., 2002, no. 10, pp. 58–72. [9] Dudin, A., Optimal Multithreshold Control for a BMAP/G/1 Queue with N Service Modes, Queuing Syst., 1998, vol. 30, pp. 273–287. · Zbl 0919.90063 · doi:10.1023/A:1019121222439 [10] Kim, C.S., Klimenok, V., Birukov, A., and Dudin, A., Optimal Multi-threshold Control by the BMAP/SM/1 Retrial System, Ann. Oper. Res., 2006, vol. 141, no. 1. · Zbl 1100.60050 [11] Lucantoni, D.M., New Results on the Single Server Queue with a Batch Markovian Arrival Process, Commun. Stat.: Stochastic Models, 1991, vol. 7, pp. 1–46. · Zbl 0733.60115 · doi:10.1080/15326349108807174 [12] Gelenbe, E., Random Neural Networks with Negative and Positive Signals and Product Form Solution, Neural Comput., 1989, vol. 1, pp. 502–510. · doi:10.1162/neco.1989.1.4.502 [13] Bocharov, P.P. and Vishnevskii, V.M., G-networks: Development of the Theory of Multiplicative Networks, Avtom. Telemekh., 2003, no. 5, pp. 46–74. [14] Artalejo, J., G-networks: A Versatile Approach for Work Removal in Queuing Networks, Eur. J. Oper. Res., 2000, vol. 126, pp. 233–249. · Zbl 0971.90007 · doi:10.1016/S0377-2217(99)00476-2 [15] Bocharov, P.P., Gavrilov, E.V., and Pechinkin, A.V., Exponential Queuing Network with Dependent Servicing, Negative Customers, and Changes in the Customer Type, Avtom. Telemekh., 2004, no. 7, pp. 35–59. · Zbl 1074.90003 [16] Shin, Y.W. and Choi, B.D., A Queue with Positive and Negative Arrivals Governed by a Markov Chain, Prob. Eng. Inform. Sci., 2003, vol. 17, pp. 487–501. · Zbl 1047.60095 · doi:10.1017/S0269964803174049 [17] Dudin, A.N. and Nishimura, S., A BMAP/SM/1 Queuing System with Markovian Arrival of Disasters, J. Appl. Prob., 1999, vol. 36, no. 3, pp. 868–881. · Zbl 0949.60100 [18] Dudin, A.N. and Karolik, A.V., BMAP/SM/1 Queue with Markovian Input of Disasters and Noninstantaneous Recovery, Performance Evaluat., 2001, vol. 45, pp. 19–32. · Zbl 1013.68035 · doi:10.1016/S0166-5316(00)00063-8 [19] Semenova, O.V., Optimal Control for a BMAP/SM/1 Queue with MAP-input of Disasters and Two Operation Modes, RAIRO Oper. Res., 2004, vol. 38, no. 2, pp. 153–171. · Zbl 1092.90018 · doi:10.1051/ro:2004017 [20] Klimenok, V.I. and Dudin, A.N., Ergodicity Condition for a Class of Multidimensional Markov Chains, in Conf. Prob. Theory, Math. Statistics, Applications, Minsk: Belarus. Gos. Univ., 2004, pp. 47–53. [21] Dudin, A.N. and Klimenok, V.I., Sistemy massovogo obsluzhivaniya s korrelirovannymi potokami (Queuing Systems with Correlated Flows), Minsk: Belarus. Gos. Univ., 2000. [22] Gail, H.R., Hantler, S.L., and Taylor, B.A., Spectral Analysis of M/G/1 and G/M/1 Type Markov Chains, Advances Appl. Prob., 1996, vol. 28, pp. 114–165. · Zbl 0845.60092 · doi:10.1017/S0001867800027300 [23] Skorokhod, A.V., Elementy teorii veroyatnostei i sluchainykh protsessov (Elements of the Probability Theory and Random Processes), Kiev: Vishcha Shkola, 1980. [24] Lucantoni, D.M. and Neuts, M.F., Some Steady-state Distributions for the BMAP/SM/1 Queue, Commun. Stat.: Stochastic Models, 1994, vol. 10, pp. 575–598. · Zbl 0804.60086 · doi:10.1080/15326349408807311
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