Abstract
In this study, by applying (max, +)-algebra to a stochastic event graph, a special case of timed Petri nets, we consider characteristics of waiting times in Poisson driven single-server 2 queues in series with a finite buffer and having constant service times at each queue. We show that the sojourn time does not depend on the finite buffer capacity and also derive the explicit expressions of waiting times at all areas of the system as a function of the finite buffer capacity, which allow one to compute and compare waiting times under two blocking policies. Moreover, an optimization problem which determines the smallest buffer capacity satisfying a predetermined probabilistic constraint on waiting times is considered as an application of these results.
This work was supported by grant No.(R01-2004-000-10948-0) from the Basic Research Program of the Korea Science & Engineering Foundation.
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References
Ayhan, H., Seo, D.W.: Laplace Transform and Moments of Waiting Times in Poisson Driven (Max,+)-Linear Systems. Queueing Systems 37(4), 405–438 (2001)
Ayhan, H., Seo, D.W.: Tail Probability of Transient and Stationary Waiting Times in (Max,+)-Linear Systems. IEEE Transactions on Automatic Control 47(1), 151–157 (2002)
Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.-P.: Synchronization and Linearity: An Algebra for Discrete Event Systems. John Wiley & Sons, Chichester (1992)
Baccelli, F., Hasenfuss, S., Schmidt, V.: Transient and Stationary Waiting Times in (Max, +) Linear Systems with Poisson Input. Queueing Systems 26, 301–342 (1997)
Baccelli, F., Hasenfuss, S., Schmidt, V.: Expansions for Steady State Characteristics in (Max,+) Linear Systems. Stochastic Models 14, 1–24 (1998)
Baccelli, F., Schmidt, V.: Taylor Series Expansions for Poisson Driven (Max,+) Linear Systems. Annals of Applied Probability 6(1), 138–185 (1996)
Grassmann, W.K., Drekic, S.: An Analytical Solution for a Tandem Queue with Blocking. Queueing Systems 36, 221–235 (2000)
Labetoulle, J., Pujolle, G.: A Study of Queueing Networks with Deterministic Service and Application to Computer Networks. Acta Informatica 7, 183–195 (1976)
Nakade, K.: New Bounds for Expected Cycle Times in Tandem Queues with Blocking. European Journal of Operations Research 125, 84–92 (2000)
Wan, Y.-W., Wolff, R.W.: Bounds for Different Arrangements of Tandem Queues with Nonoverlapping Service Times. Management Science 39(9), 1173–1178 (1993)
Whitt, W.: The Best Order for Queues in Series. Management Science 31(4), 475–487 (1985)
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Seo, DW., Song, BK. (2005). Application of (Max, +)-Algebra to the Optimal Buffer Size in Poisson Driven Deterministic Queues in Series with Blocking. In: Ślęzak, D., Yao, J., Peters, J.F., Ziarko, W., Hu, X. (eds) Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing. RSFDGrC 2005. Lecture Notes in Computer Science(), vol 3642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11548706_71
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DOI: https://doi.org/10.1007/11548706_71
Publisher Name: Springer, Berlin, Heidelberg
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