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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. R. Gantmacher, "The Theory of Matrices, Vol. 2,'' Translated by K. A. Hirsch Chelsea Publishing Co., New York, 1959.  Google Scholar

[5]

Q.-M. He, Queues with marked customers, Adv. Appl. Prob., 28 (1996), 567-587. doi: 10.2307/1428072.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

E. Çinlar, "Introduction to Stochastic Processes," Prentice-Hall, Englewood Cliffs, N.J., 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. R. Gantmacher, "The Theory of Matrices, Vol. 2,'' Translated by K. A. Hirsch Chelsea Publishing Co., New York, 1959.  Google Scholar

[5]

Q.-M. He, Queues with marked customers, Adv. Appl. Prob., 28 (1996), 567-587. doi: 10.2307/1428072.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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