Management Science and Financial Engineering

The Korean Operations Research and Management Science Society (한국경영과학회)
권호 표지

Stationary Waiting Times in m-node Tandem Queues with Communication Blocking

  • Seo, Dong-Won (College of Management and International Relations, Kyung Hee University) ;
  • Lee, Ho-Chang (College of Management and International Relations, Kyung Hee University) ;
  • Ko, Sung-Seok (Department of Industrial Engineering, Konkuk University)
  • Published : 2008.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this study, we consider stationary waiting times in a Poisson driven single-server m-node queues in series. We assume that service times at nodes are independent, and are either deterministic or non-overlapped. Each node excluding the first node has a finite waiting line and every node is operated under a FIFO service discipline and a communication blocking policy (blocking before service). By applying (max, +)-algebra to a corresponding stochastic event graph, a special case of timed Petri nets, we derive the explicit expressions for stationary waiting times at all areas, which are functions of finite buffer capacities. These expressions allow us to compute the performance measures of interest such as mean, higher moments, or tail probability of waiting time. Moreover, as applications of these results, we introduce optimization problems which determine either the biggest arrival rate or the smallest buffer capacities satisfying probabilistic constraints on waiting times. These results can be also applied to bounds of waiting times in more general systems. Numerical examples are also provided.

Keywords

References

  1. Ayhan, H. and D-W. Seo, "Laplace Transform and Moments of Waiting Times in Poisson Driven (Max, +)-Linear Systems," Queueing Systems 37, 4 (2001), 405-438 https://doi.org/10.1023/A:1010845618420
  2. Ayhan, H. and D-W. Seo, "Tail Probability of Transient and Stationary Waiting Times in (Max, +)-Linear Systems," IEEE Transactions on Automatic Control 47, 1 (2002), 151-157 https://doi.org/10.1109/9.981736
  3. Baccelli, F., G. Cohen, G. J. Olsder, and J-P. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, John Wiley and Sons 1992
  4. Baccelli, F., S. Hasenfuss, and V. Schmidt, "Expansions for Steady State Characteristics in (Max, +) Linear Systems," Stochastic Models 14 (1998), 1-24
  5. Baccelli, F. and V. Schmidt, "Taylor Series Expansions for Poisson Driven (Max, +) Linear Systems," Annals of Applied Probability 6, 1 (1996), 138-185 https://doi.org/10.1214/aoap/1034968069
  6. Seo, D.-W., "Application of (Max, +)-algebra to the Waiting Times in Deterministic 2-node Tandem Queues with Blocking," J. KORMS 30, 1 (2005), 149-159
  7. Seo, D.-W., "Application of (Max, +)-algebra to the Waiting Times in Deterministic 3-node Tandem Queues with Blocking," J. KORMS 30, 2 (2005), 73-80
  8. Seo, D.-W. and B.-K. Song, "Application of (Max, +)-algebra to Optimal Buffer Size in Deterministic Queues in Series with Blocking," LNAI 3642 (2005), 671-677
  9. Shaked M. and J. G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, 1994
  10. Wan, Y.-W. and R. W. Wolff, "Bounds for Different Arrangements of Tandem Queues with Nonoverlapping Service Times," Management Science 39, 9 (1993), 1173-1178 https://doi.org/10.1287/mnsc.39.9.1173
  11. Whitt, W., "The Best Order for Queues in Series," Management Science 31, 4 (1985), 475-487 https://doi.org/10.1287/mnsc.31.4.475