Abstract
A multi-server queueing system with a finite buffer, where a Batch Markovian Arrival Process (BMAP) arrives, is studied. The servicing time of a customer has a phase-type (PH) distribution. Customers are admitted to the system in accordance with the disciplines of partial admission, complete admission and complete rejection. Except standard (positive) customers, a MAP flow of negative customers arrives to the system. In a random way, a negative customer removes from the system one of the positive customers that are at the server. A stationary distribution of system state probabilities, the Laplace-Stieltjes transformation of the stationary distribution of waiting time, major characteristics of system performance are found. Numerical examples are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gelenbe, E., Random Neural Networks with Negative and Positive Signals and Product Form Solution, Neural Comput., 1989, vol. 1, pp. 502–510.
Artalejo, J., A Versatile Approach for Work Removal in Queueing Networks, Eur. J. Oper. Res., 2000, vol. 126, pp. 233–249.
Bocharov, P.P. and Vishnevski, V.M., G-networks Development of the Theory of Multiplicative Networks, Avtom. Telemekh., 2003, no. 5, pp. 46–74.
Vishnevski, V.M., Teoreticheskie osnovy proektirovaniya komp’yuternykh setei (Theoretical Fundamentals of Computer Network Design), Moscow: RITS “Tekhnosfera,” 2003.
Bocharov, P.P., Gavrilov, E.I., D’Apice, C., and Pechinkin, A.V., On Decomposition of Queueing Networks with Dependent Servicing and Negative Customers, Avtom. Telemekh., 2004, no. 1, pp. 97–116.
Bocharov, P.P., Gavrilov, E.V., and Pechinkin, A.V., Exponential Queueing Network with Dependent Servicing, Negative Customers, and Modification of the Customer Type, Avtom. Telemekh., 2004, no. 7, pp. 35–59.
Bocharov, P., Viskova, E., and Nadaev, E., Markovain Queueing System of Finite Capacity with Negative Customers in Discrete Time, Queues: Flows, Systems, Networks, 2005, no. 18, pp. 14–19.
Dudin, A.N., Kim, C.S., and Semenova, O.V., An Optimal Multithrreshold Control for the Input Flow of the GI/PH/1 with a BMAP Flaw of Negative Customers, Avtom. Telemekh., 2004, no. 9, pp. 71–84.
Semenova, O.V., An Optimal Threshold Control for a BMAP/SM/1 System with MAP Disaster Flow, Avtom. Telemekh., 2003, no. 9, pp. 89–102.
Li, Q. and Zhao, Y., A MAP/G/1 Queue with Negative Customers, Queueing Syst., 2004, vol. 47, pp. 5–43.
Shin, Y., Multi-server Retrial Queue with Negative Customers and Disasters, Proc. Fifth Int. Workshop on Retrial Queues, Choi, B., Ed., Seoul: Korea Univ., 2005, pp. 53–60.
Shin, Y.W. and Choi, B.D., A Queue with Positive and Negative Arrivals Governed by a Markov Chain, Prob. Engin. Inform. Sci., 2003, no. 17, pp. 487–501.
Anisimov, V. and Artalejo, J., Analysis of Markov Multiserver Retrial Queues with Negative Arrivals, Queueing Syst., 2001, vol. 39, pp. 157–182.
Chaplygin, V.V., Stationary Characteristics of a Queueing System G/MSP/n/r with the Flow of Negative Customers, Inform. Pross., 1999, vol. 5, no. 1, pp. 1–19.
Asmussen, S., Applied Probability and Queues, New York: Springer-Verlag, 2003.
Lucantoni, D., New Results on the Single Server Queue with a Batch Markovian Arrival Process, Commun. Statist.-Stochastic Models., 1991, vol. 7, pp. 1–46.
Bocharov, P.P. and Pechinkin, A.V., Teoriya massovogo obsluzhivaniya (Queueing Theory), Moscow: Ross. Univ. Druzhby Narodov, 1995.
Neuts, M., Structured Stochastic Matrices of M/G/1 Type and Their Applications, New York: Marcel Dekker, 1989.
Graham, A., Kronecker Products and Matrix Calculus with Applications, Cichester: Ellis Horwood, 1981.
Khinchin, A.Ya., Raboty po matematicheskoi teorii massovogo obsluzhivaniya (On Queueing Mathematical Theory), Moscow: Fizmatgiz, 1963.
Dudin, A. and Klimenok, V., Multi-dimensional Quasitoeplitz Markov Chain, J. Appl. Math. Stochast. Anal., 1999, vol. 12, pp. 393–415.
Kemeni, J., Snell, J., and Knapp, A., Denumerable Markov Chains, New York: Van Nostrand, 1966.
Klimenok, V., Kim, C., Orlovsky, D., and Dudin, A., Lack of Invariant Property of the Erlang Loss Model in Case of MAP Input, Queueing Syst., 2005, vol. 49, pp. 187–213.
Author information
Authors and Affiliations
Additional information
Original Russian Text © C.S. Kim, V.I. Klimenok, D.S. Orlovskii, 2006, published in Avtomatika i Telemekhanika, 2006, No. 12, pp. 106–122.
The authors thank the Korean Research Foundation (KRF) for sponsoring this research, Republic of Korea, project no. KRF-2005-212-C00001.
Rights and permissions
About this article
Cite this article
Kim, C.S., Klimenok, V.I. & Orlovskii, D.S. Multi-server queueing system with a Batch Markovian Arrival Process and negative customers. Autom Remote Control 67, 1958–1973 (2006). https://doi.org/10.1134/S0005117906120083
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0005117906120083