Basic Limit Theorems for Light Traffic Queues & Their Applications

Abstract

In this paper, we study some basic limit theorems characterizing the stationary behavior of light traffic queuing systems. Beginning with limit theorems for the simple M/M/1 queuing system, we demonstrate the methodology for applying these theorems for the benefit of service systems. The limit theorems studied here are dominant in the literature. Our contribution is primarily on the analysis leading to the application of these theorems in various problem situations for better operations. Relevant Examples are included to aid the application of the results studied in this work.

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Daman, O. and Sani, S. (2015) Basic Limit Theorems for Light Traffic Queues & Their Applications. American Journal of Operations Research, 5, 409-420. doi: 10.4236/ajor.2015.55034.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Medhi, J. (2003) Stochastic Models in Queuing Theory. Academic Press. An Imprint of Elsevier Science (USA). [2] Franken, P., Koonig, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Akademie-Verlag Publication, Germany. [3] Federgruen, A. and Tijms, H.C. (1980) Computation of the Stationary Distribution of the Queue Size in M/G/1 with Variable Service Rate. Journal of Applied Probability, 17, 515-522. http://dx.doi.org/10.2307/3213040 [4] Hoksad, P. (1978) Approximation for the M/G/m Queue. Journal of Operation Research, 26, 511-523. [5] Hoksad, P. (1979) On the Steady State Solution of the M/G/2 Queue. Advanced Applied Probability, 11, 240-255. http://dx.doi.org/10.2307/1426776 [6] Smith, J.M. (2002) M/G/C/K Blocking Probability Models and System Performance. Performance Evaluation, 52, 237-267. http://dx.doi.org/10.1016/S0166-5316(02)00190-6 [7] Tijms, H.C., Van Hoorn, M.H. and Federgruen, A. (1981) Approximation for the Steady State Probabilities in the M/G/C Queue. Advances in Applied Probability, 13, 186-206. http://dx.doi.org/10.2307/1426474 [8] Abate, A. and Whitt, W. (1994) Asymptotics for M/G/1 Low-Priority Waiting-Time Tail Probabilities. Queuing Systems, 25, 173-233. http://dx.doi.org/10.1023/A:1019104402024 [9] Leland, W.E., Taqqu, M.S., Willinger, W. and Wilson, D.V. (1994) On the Self-Similar Nature of Ethernet Traffic (Extended Version). IEEE/ACM Transactions on Networking, 2, 1-15. http://dx.doi.org/10.1109/90.282603 [10] Karagiannis, T., Molle, M., Faloutsos, M. and Broido, A. (2008) A Nonstationary View on Poisson Internet Traffic. Proceedings of IEEE INFOCOM, 84-89.