• Previous Article
    Catastrophe equity put options under stochastic volatility and catastrophe-dependent jumps
  • JIMO Home
  • This Issue
  • Next Article
    Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy
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

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.
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, (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. 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. doi: 10.1239/jap/1082999085.

[4]

F. R. Gantmacher, "The Theory of Matrices, Vol. 2,'', Translated by K. A. Hirsch Chelsea Publishing Co., (1959).

[5]

Q.-M. He, Queues with marked customers,, Adv. Appl. Prob., 28 (1996), 567. 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. 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. 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. 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. 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. 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. 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. 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. 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, (1994).

show all references

References:
[1]

E. Çinlar, "Introduction to Stochastic Processes,", Prentice-Hall, (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. 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. doi: 10.1239/jap/1082999085.

[4]

F. R. Gantmacher, "The Theory of Matrices, Vol. 2,'', Translated by K. A. Hirsch Chelsea Publishing Co., (1959).

[5]

Q.-M. He, Queues with marked customers,, Adv. Appl. Prob., 28 (1996), 567. 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. 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. 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. 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. 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. 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. 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. 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. 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, (1994).

[1]

Naoto Miyoshi. On the stationary LCFS-PR single-server queue: A characterization via stochastic intensity. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 713-725. doi: 10.3934/naco.2011.1.713

[2]

Yongjiang Guo, Yuantao Song. The (functional) law of the iterated logarithm of the sojourn time for a multiclass queue. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-28. doi: 10.3934/jimo.2018192

[3]

Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026

[4]

Yutaka Sakuma, Atsushi Inoie, Ken’ichi Kawanishi, Masakiyo Miyazawa. Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time. Journal of Industrial & Management Optimization, 2011, 7 (3) : 593-606. doi: 10.3934/jimo.2011.7.593

[5]

Dequan Yue, Jun Yu, Wuyi Yue. A Markovian queue with two heterogeneous servers and multiple vacations. Journal of Industrial & Management Optimization, 2009, 5 (3) : 453-465. doi: 10.3934/jimo.2009.5.453

[6]

Shaojun Lan, Yinghui Tang, Miaomiao Yu. System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435

[7]

Sin-Man Choi, Ximin Huang, Wai-Ki Ching. Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment. Journal of Industrial & Management Optimization, 2012, 8 (2) : 299-323. doi: 10.3934/jimo.2012.8.299

[8]

Tuan Phung-Duc. Single server retrial queues with setup time. Journal of Industrial & Management Optimization, 2017, (3) : 1329-1345. doi: 10.3934/jimo.2016075

[9]

Bara Kim, Jeongsim Kim. Explicit solution for the stationary distribution of a discrete-time finite buffer queue. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1121-1133. doi: 10.3934/jimo.2016.12.1121

[10]

Zsolt Saffer, Wuyi Yue. A dual tandem queueing system with GI service time at the first queue. Journal of Industrial & Management Optimization, 2014, 10 (1) : 167-192. doi: 10.3934/jimo.2014.10.167

[11]

Yung Chung Wang, Jenn Shing Wang, Fu Hsiang Tsai. Analysis of discrete-time space priority queue with fuzzy threshold. Journal of Industrial & Management Optimization, 2009, 5 (3) : 467-479. doi: 10.3934/jimo.2009.5.467

[12]

Tuan Phung-Duc, Ken'ichi Kawanishi. Multiserver retrial queue with setup time and its application to data centers. Journal of Industrial & Management Optimization, 2019, 15 (1) : 15-35. doi: 10.3934/jimo.2018030

[13]

Gábor Horváth, Zsolt Saffer, Miklós Telek. Queue length analysis of a Markov-modulated vacation queue with dependent arrival and service processes and exhaustive service policy. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1365-1381. doi: 10.3934/jimo.2016077

[14]

Thomas Demoor, Joris Walraevens, Dieter Fiems, Stijn De Vuyst, Herwig Bruneel. Influence of real-time queue capacity on system contents in DiffServ's expedited forwarding per-hop-behavior. Journal of Industrial & Management Optimization, 2010, 6 (3) : 587-602. doi: 10.3934/jimo.2010.6.587

[15]

Michiel De Muynck, Herwig Bruneel, Sabine Wittevrongel. Analysis of a discrete-time queue with general service demands and phase-type service capacities. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1901-1926. doi: 10.3934/jimo.2017024

[16]

Bart Feyaerts, Stijn De Vuyst, Herwig Bruneel, Sabine Wittevrongel. The impact of the $NT$-policy on the behaviour of a discrete-time queue with general service times. Journal of Industrial & Management Optimization, 2014, 10 (1) : 131-149. doi: 10.3934/jimo.2014.10.131

[17]

Gopinath Panda, Veena Goswami. Effect of information on the strategic behavior of customers in a discrete-time bulk service queue. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-20. doi: 10.3934/jimo.2019007

[18]

Ahmed M. K. Tarabia. Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs. Journal of Industrial & Management Optimization, 2011, 7 (4) : 811-823. doi: 10.3934/jimo.2011.7.811

[19]

Dequan Yue, Wuyi Yue, Guoxi Zhao. Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states. Journal of Industrial & Management Optimization, 2016, 12 (2) : 653-666. doi: 10.3934/jimo.2016.12.653

[20]

Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]