More Web Proxy on the site http://driver.im/

article

Single-Server Queues with Markov-Modulated Arrivals and Service Speed

Author:

Tetsuya TakineAuthors Info & Claims

Queueing Systems: Theory and Applications, Volume 49, Issue 1

Pages 7 - 22

https://doi.org/10.1007/s11134-004-5553-9

Published: 01 January 2005 Publication History

Abstract

This paper considers single-server queues with several customer classes. Arrivals of customers are governed by the underlying continuous-time Markov chain with finite states. The distribution of the amount of work brought into the system on arrival is assumed to be general, which may differ with different classes. Further, the service speed depends on the state of the underlying Markov chain. We first show that given such a queue, we can construct the corresponding new queue with constant service speed by means of a change of time scale, and the time-average quantities of interest in the original queue are given in terms of those in the new queue. Next we characterize the joint distribution of the length of a busy period and the number of customers served during the busy period in the original queue. Finally, assuming the FIFO service discipline, we derive the Laplace--Stieltjes transform of the actual waiting time distribution in the original queue.

References

[1]

{1} O.J. Boxma and I.A. Kurkova, The M/G/1 queue with two service speeds, Adv. in Appl. Probab. 33 (2001) 520-540.

[2]

{2} O.J. Boxma and T. Takine, The M/G/1 FIFO queue with several customer classes, Queueing Systems 45 (2003) 185-189.

[3]

{3} M. Eisen and M. Tainiter, Stochastic variations in queueing processes, Oper. Res. 11 (1963) 922-927.

Digital Library

[4]

{4} A. Graham, Kronecker Products and Matrix Calculus with Applications (Ellis Horwood, Chichester, 1981).

[5]

{5} S. Halfin, Steady-state distribution for the buffer content of an M/G/1 queue with varying service rate, SIAM J. Appl. Math. 23 (1972) 356-363.

[6]

{6} Q.-M. He, Queues with marked customers, Adv. in Appl. Probab. 28 (1996) 567-587.

[7]

{7} A.A. Kherani and A. Kumar, Stochastic models for throughput analysis of randomly arriving elastic flows in the Internet, in: IEEE INFOCOM 2002, IEEE, Piscataway, NJ (2002) pp. 1014-1023.

[8]

{8} J.G. Kim and M.M. Krunz, Bandwidth allocation in wireless networks with guaranteed packet-loss performance, IEEE/ACM Trans. Networking 8 (2000) 337-349.

Digital Library

[9]

{9} G. Latouche and T. Takine, Markov renewal fluid queues, J. Appl. Probab. (to appear).

[10]

{10} M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications (Marcel Dekker, New York, 1989).

[11]

{11} R. Nunez-Queija, A queueing model with varying service rate for ABR, in: Computer Performance Evaluation: Modelling Techniques and Tools, Lecture Notes in Computer Science, Vol. 1469 (Springer, New York, 1998) pp. 93-104.

[12]

{12} P. Purdue, The M/M/1 queue in a Markovian environment, Oper. Res. 22 (1974) 562-569.

Digital Library

[13]

{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, Stochastic Models 10 (1994) 183-204.

[14]

{14} T. Takine, A nonpreemptive priority MAP/G/1 queue with two classes of customers, J. Oper. Res. Soc. J. 39 (1996) 266-290.

[15]

{15} T. Takine, The nonpreemptive priority MAP/G/1 queue, Oper. Res. 47 (1999) 917-927.

Digital Library

[16]

{16} T. Takine, Queue length distribution in a FIFO single-server queue with multiple arrival streams having different service time distributions, Queueing Systems 39 (2001) 349-375.

[17]

{17} T. Takine, Matrix product-form solution for an LCFS-PR single-server queue with multiple arrival streams governed by a Markov chain, Queueing Systems 42 (2002) 131-151.

[18]

{18} U. Yechiali, A queueing-type birth-and-death process defined on a continuous-time Markov chain, Oper. Res. 21 (1973) 604-609.

Digital Library

[19]

{19} U. Yechiali and P. Naor, A queueing problems with heterogeneous arrival and service, Oper. Res. 19 (1971) 722-734.

Digital Library

Cited By

Huang JZhang HZhang J(2016)A Unified Approach to Diffusion Analysis of Queues with General Patience-Time DistributionsMathematics of Operations Research10.1287/moor.2015.077241:3(1135-1160)Online publication date: 1-Aug-2016
https://dl.acm.org/doi/10.1287/moor.2015.0772
Takine T(2016)Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/$$\infty $$ź and BMAP/M/c+M queuesQueueing Systems: Theory and Applications10.1007/s11134-016-9482-184:1-2(49-77)Online publication date: 1-Oct-2016
https://dl.acm.org/doi/10.1007/s11134-016-9482-1
Thorsdottir HVerloop I(2016)Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplinesQueueing Systems: Theory and Applications10.1007/s11134-016-9477-y83:1-2(29-55)Online publication date: 1-Jun-2016
https://dl.acm.org/doi/10.1007/s11134-016-9477-y
Show More Cited By

Recommendations

Analysis of $$\hbox {M}^{\mathrm {x}}/\hbox {G}/1$$Mx/G/1 queues with impatient customers

This paper considers a batch arrival $$\hbox {M}^{\mathrm {x}}/\hbox {G}/1$$Mx/G/1 queue with impatient customers. We consider two different model variants. In the first variant, customers in the same batch are assumed to have the same patience time, ...
Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline ...
Tail asymptotics of the waiting time and the busy period for the $${{\varvec{M/G/1/K}}}$$ queues with subexponential service times

We study the asymptotic behavior of the tail probabilities of the waiting time and the busy period for the $$M/G/1/K$$ queues with subexponential service times under three different service disciplines: FCFS, LCFS, and ROS. Under the FCFS discipline, the result on ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Queueing Systems: Theory and Applications

Queueing Systems: Theory and Applications Volume 49, Issue 1

January 2005

79 pages

ISSN:0257-0130

Issue’s Table of Contents

Copyright © Copyright © 2005 Springer Science + Business Media, Inc.

Publisher

J. C. Baltzer AG, Science Publishers

United States

Publication History

Published: 01 January 2005

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

10
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 18 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Huang JZhang HZhang J(2016)A Unified Approach to Diffusion Analysis of Queues with General Patience-Time DistributionsMathematics of Operations Research10.1287/moor.2015.077241:3(1135-1160)Online publication date: 1-Aug-2016
https://dl.acm.org/doi/10.1287/moor.2015.0772
Takine T(2016)Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/$$\infty $$ź and BMAP/M/c+M queuesQueueing Systems: Theory and Applications10.1007/s11134-016-9482-184:1-2(49-77)Online publication date: 1-Oct-2016
https://dl.acm.org/doi/10.1007/s11134-016-9482-1
Thorsdottir HVerloop I(2016)Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplinesQueueing Systems: Theory and Applications10.1007/s11134-016-9477-y83:1-2(29-55)Online publication date: 1-Jun-2016
https://dl.acm.org/doi/10.1007/s11134-016-9477-y
Dorsman JVlasiou MZwart B(2015)Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speedsQueueing Systems: Theory and Applications10.1007/s11134-014-9422-x79:3-4(293-319)Online publication date: 1-Apr-2015
https://dl.acm.org/doi/10.1007/s11134-014-9422-x
Baumann HSandmann W(2012)Steady state analysis of level dependent quasi-birth-and-death processes with catastrophesComputers and Operations Research10.1016/j.cor.2011.05.00339:2(413-423)Online publication date: 1-Feb-2012
https://dl.acm.org/doi/10.1016/j.cor.2011.05.003
Kim CDudin AKlimenok VKhramova V(2009)Erlang loss queueing system with batch arrivals operating in a random environmentComputers and Operations Research10.1016/j.cor.2007.10.02236:3(674-697)Online publication date: 1-Mar-2009
https://dl.acm.org/doi/10.1016/j.cor.2007.10.022
Fralix BAdan I(2009)An infinite-server queue influenced by a semi-Markovian environmentQueueing Systems: Theory and Applications10.1007/s11134-008-9100-y61:1(65-84)Online publication date: 1-Jan-2009
https://dl.acm.org/doi/10.1007/s11134-008-9100-y
Perel EYechiali U(2008)Queues where customers of one queue act as servers of the other queueQueueing Systems: Theory and Applications10.1007/s11134-008-9097-260:3-4(271-288)Online publication date: 1-Dec-2008
https://dl.acm.org/doi/10.1007/s11134-008-9097-2
D'Auria B(2008)M/M/∞ queues in semi-Markovian random environmentQueueing Systems: Theory and Applications10.1007/s11134-008-9068-758:3(221-237)Online publication date: 1-Mar-2008
https://dl.acm.org/doi/10.1007/s11134-008-9068-7
Masuyama HTakine T(2007)Multiclass Markovian fluid queuesQueueing Systems: Theory and Applications10.1007/s11134-007-9019-856:3-4(143-155)Online publication date: 1-Aug-2007
https://dl.acm.org/doi/10.1007/s11134-007-9019-8

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents