The FIFO single-server queue with disasters and multiple Markovian arrival streams

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January 2014, 10(1): 57-87. doi: 10.3934/jimo.2014.10.57

The FIFO single-server queue with disasters and multiple Markovian arrival streams

Yoshiaki Inoue ^1, and Tetsuya Takine ^1,

1.	Department of Information and Communications Technology, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita 565-0871, Japan

Received September 2012 Revised June 2013 Published October 2013

Abstract
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We consider a FIFO single-server queue with disasters and multiple Markovian arrival streams. When disasters occur, all customers are removed instantaneously and the system becomes empty. Both the customer arrival and disaster occurrence processes are assumed to be Markovian arrival processes (MAPs), and they are governed by a common underlying Markov chain with finite states. There are $K$ classes of customers, and the amounts of service requirements brought by arriving customers follow general distributions, which depend on the customer class and the states of the underlying Markov chain immediately before and after arrivals. For this queue, we first analyze the first passage time to the idle state and the busy cycle. We then obtain two different representations of the Laplace-Stieltjes transform of the stationary distribution of work in system, and discuss the relation between those. Furthermore, using the result on the workload distribution, we analyze the waiting time and sojourn time distributions, and derive the joint queue length distribution.

Keywords: FIFO single-server queue, disasters, sojourn time, waiting time, workload, multiple Markovian arrivals, joint queue length..

Mathematics Subject Classification: Primary: 60K25; Secondary: 90B2.

Citation: Yoshiaki Inoue, Tetsuya Takine. The FIFO single-server queue with disasters and multiple Markovian arrival streams. Journal of Industrial and Management Optimization, 2014, 10 (1) : 57-87. doi: 10.3934/jimo.2014.10.57

References:

[1]	E. Çinlar, "Introduction to Stochastic Processes," Prentice-Hall, Englewood Cliffs, N.J., 1975.
[2]	A. Dudin and S. Nishimura, A BMAP/SM/1 queueing system with Markovian arrival input of disasters, J. Appl. Prob., 36 (1999), 868-881. doi: 10.1239/jap/1032374640.
[3]	A. Dudin and O. Semenova, A stable algorithm for stationary distribution calculation for a BMAP/SM/1 queueing system with Markovian arrival input of disasters, J. Appl. Prob., 41 (2004), 547-556. doi: 10.1239/jap/1082999085.
[4]	F. R. Gantmacher, "The Theory of Matrices, Vol. 2,'' Translated by K. A. Hirsch Chelsea Publishing Co., New York, 1959.
[5]	Q.-M. He, Queues with marked customers, Adv. Appl. Prob., 28 (1996), 567-587. doi: 10.2307/1428072.
[6]	D. P. Heyman and S. Stidham, Jr., The relation between customer and time averages in queues, Oper. Res., 28 (1980), 983-994. doi: 10.1287/opre.28.4.983.
[7]	G. Jain and K. Sigman, A Pollaczek-Khintchine formula for M/G/1 queues with disasters, J. Appl. Prob., 33 (1996), 1191-1200. doi: 10.2307/3214996.
[8]	H. Masuyama and T. Takine, Analysis and computation of the joint queue length distribution in a FIFO single-server queue with multiple batch Markovian arrival streams, Stoch. Models, 19 (2003), 349-381. doi: 10.1081/STM-120023565.
[9]	V. Ramaswami, A stable recursion for the steady state vector in Markov chains of M/G/1 type, Stoch. Models, 4 (1988), 183-188. doi: 10.1080/15326348808807077.
[10]	Y. W. Shin, BMAP/G/1 queue with correlated arrivals of customers and disasters, Oper. Res. Lett., 32 (2004), 364-373. doi: 10.1016/j.orl.2003.09.005.
[11]	T. Takine, Queue length distribution in a FIFO single-server queue with multiple arrival streams having different service time distributions, Queueing Syst., 39 (2001), 349-375. doi: 10.1023/A:1013961710829.
[12]	T. Takine, Matrix product-form solution for an LCFS-PR single-server queue with multiple arrival streams governed by a Markov chain, Queueing Syst., 42 (2002), 131-151. doi: 10.1023/A:1020152920794.
[13]	T. Takine and T. Hasegawa, The workload in the MAP/G/1 queue with state-dependent services: Its application to a queue with preemptive resume priority, Comm. Statist. Stochastic Models, 10 (1994), 183-204. doi: 10.1080/15326349408807292.
[14]	H. C. Tijms, "Stochastic Models, An Algorithmic Approach,'' Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1994. x+375 pp.

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References:

[1]	E. Çinlar, "Introduction to Stochastic Processes," Prentice-Hall, Englewood Cliffs, N.J., 1975.
[2]	A. Dudin and S. Nishimura, A BMAP/SM/1 queueing system with Markovian arrival input of disasters, J. Appl. Prob., 36 (1999), 868-881. doi: 10.1239/jap/1032374640.
[3]	A. Dudin and O. Semenova, A stable algorithm for stationary distribution calculation for a BMAP/SM/1 queueing system with Markovian arrival input of disasters, J. Appl. Prob., 41 (2004), 547-556. doi: 10.1239/jap/1082999085.
[4]	F. R. Gantmacher, "The Theory of Matrices, Vol. 2,'' Translated by K. A. Hirsch Chelsea Publishing Co., New York, 1959.
[5]	Q.-M. He, Queues with marked customers, Adv. Appl. Prob., 28 (1996), 567-587. doi: 10.2307/1428072.
[6]	D. P. Heyman and S. Stidham, Jr., The relation between customer and time averages in queues, Oper. Res., 28 (1980), 983-994. doi: 10.1287/opre.28.4.983.
[7]	G. Jain and K. Sigman, A Pollaczek-Khintchine formula for M/G/1 queues with disasters, J. Appl. Prob., 33 (1996), 1191-1200. doi: 10.2307/3214996.
[8]	H. Masuyama and T. Takine, Analysis and computation of the joint queue length distribution in a FIFO single-server queue with multiple batch Markovian arrival streams, Stoch. Models, 19 (2003), 349-381. doi: 10.1081/STM-120023565.
[9]	V. Ramaswami, A stable recursion for the steady state vector in Markov chains of M/G/1 type, Stoch. Models, 4 (1988), 183-188. doi: 10.1080/15326348808807077.
[10]	Y. W. Shin, BMAP/G/1 queue with correlated arrivals of customers and disasters, Oper. Res. Lett., 32 (2004), 364-373. doi: 10.1016/j.orl.2003.09.005.
[11]	T. Takine, Queue length distribution in a FIFO single-server queue with multiple arrival streams having different service time distributions, Queueing Syst., 39 (2001), 349-375. doi: 10.1023/A:1013961710829.
[12]	T. Takine, Matrix product-form solution for an LCFS-PR single-server queue with multiple arrival streams governed by a Markov chain, Queueing Syst., 42 (2002), 131-151. doi: 10.1023/A:1020152920794.
[13]	T. Takine and T. Hasegawa, The workload in the MAP/G/1 queue with state-dependent services: Its application to a queue with preemptive resume priority, Comm. Statist. Stochastic Models, 10 (1994), 183-204. doi: 10.1080/15326349408807292.
[14]	H. C. Tijms, "Stochastic Models, An Algorithmic Approach,'' Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1994. x+375 pp.

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