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USING SINGULARITY ANALYSIS TO APPROXIMATE TRANSIENT CHARACTERISTICS IN QUEUEING SYSTEMS

Published online by Cambridge University Press:  16 February 2009

Joris Walraevens ,
Dieter Fiems  and
Marc Moeneclaey
Joris Walraevens
Affiliation:
Department of Telecommunications and Information Processing (IR07), Ghent University, B-9000 Gent, Belgium E-mail: [email protected], [email protected], [email protected]
Dieter Fiems
Affiliation:
Department of Telecommunications and Information Processing (IR07), Ghent University, B-9000 Gent, Belgium E-mail: [email protected], [email protected], [email protected]
Marc Moeneclaey
Affiliation:
Department of Telecommunications and Information Processing (IR07), Ghent University, B-9000 Gent, Belgium E-mail: [email protected], [email protected], [email protected]
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Abstract

In this article, we develop a simple method to approximate the transient behavior of queueing systems. In particular, it is shown how singularity analysis of a known generating function of a transient sequence of some performance measure leads to an approximation of this sequence. To illustrate our approach, several specific transient sequences are investigated in detail. By means of some numerical examples, we validate our approximations and demonstrate the usefulness of the technique.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 23 , Issue 2 , April 2009 , pp. 333 - 355
Copyright
Copyright © Cambridge University Press 2009

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References

1.Abate, J., Choudhury, G. & Whitt, W. (1995). Calculating the M/G/1 busy-period density and LIFO waiting-time distribution by direct numerical inversion. Operations Research Letters 18(3): 113–119.Google Scholar
2.Abate, J. & Whitt, W. (1992). Numerical inversion of probability generating functions. Operations Research Letters 12(4): 245–251.Google Scholar
3.Abate, J. & Whitt, W. (1992). Solving probability transform functional equations for numerical inversion. Operations Research Letters 12(5): 275–281.CrossRefGoogle Scholar
4.Adan, I., Van Leeuwaarden, J. & Winands, E. (2006). On the application of Rouché's theorem in queueing theory. Operations Research Letters 34(3): 355–360.CrossRefGoogle Scholar
5.Asrin, M. & Kamoun, F. (1998). A transient discrete-time analysis of the ATM multiplexer. Performance Evaluation 32(3): 153–183.Google Scholar
6.Bender, E. (1974). Asymptotic methods in enumeration. SIAM Review 16(4): 485–515.CrossRefGoogle Scholar
7.Blanc, J. (2002). On the numerical inversion of busy-period related transforms. Operations Research Letters 30(1): 33–42.Google Scholar
8.Bruneel, H. (1991). Exact derivation of transient behavior for buffers with random output interruptions. Computers Networks and ISDN Systems 22: 277–285.CrossRefGoogle Scholar
9.Bruneel, H., Steyaert, B., Desmet, E. & Petit, G. (1994). Analytic derivation of tail probabilities for queue lengths and waiting times in ATM multiserver queues. European Journal of Operational Research 76(3): 563–572.Google Scholar
10.Choudhury, G., Lucantoni, D. & Whitt, W. (1994). Multidimensional transform inversion with applications to the transient M/G/1 queue. Annals of Applied Probability 4(3): 719–740.CrossRefGoogle Scholar
11.Drmota, M. (1997). Systems of functional equations. Random Structures & Algorithms 10 (1–2): 103–124.Google Scholar
12.Flajolet, P. & Odlyzko, A. (1990). Singularity analysis of generating functions. SIAM Journal on Discrete Mathematics 3(2): 216–240.Google Scholar
13.Flajolet, P. & Sedgewick, R. (2009). Analytic Combinatorics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
14.Gluskin, E. (2003). Let us teach this generalization of the final-value theorem. European Journal of Physics 24(6): 591–597.CrossRefGoogle Scholar
15.Griffiths, J., Leonenko, G. & Williams, J. (2006). The transient solution to M/E−k/1 queue. Operations Research Letters 34(3): 349–354.Google Scholar
16.Kamoun, F. (2006). Performance analysis of a discrete-time queuing system with a correlated train arrival process. Performance Evaluation 63 (4–5): 315–340.Google Scholar
17.Parthasarathy, P. (1987). A transient solution to an M/M/1 queue: a simple approach. Advances in Applied Probability 19(4): 997–998.CrossRefGoogle Scholar
18.Takagi, H. (1991–1993). Queueing analysis: A foundation of performance evaluation. Amsterdam: North-Holland, Vols I–III.Google Scholar
19.Walraevens, J., Fiems, D. & Bruneel, H. (2008). Analysis of the transient delay in a discrete-time buffer with batch arrivals. In Proceedings of the 5th International Conference on Information Technology: New Generations (ITNG 2008)Las Vegas.Google Scholar
20.Walraevens, J., Fiems, D. & Bruneel, H. (2008). Time-dependent performance analysis of a discrete-time priority queue. Performance Evaluation 65(9): 641–652.Google Scholar
21.Walraevens, J., Steyaert, B. & Bruneel, H. (2003). Performance analysis of a single-server ATM queue with a priority scheduling. Computers and Operations Research 30(12): 1807–1829.Google Scholar