Abstract
We study the asymptotics of the stationary sojourn time Z of a “typical customer” in a tandem of single-server queues. It is shown that in a certain “intermediate” region of light-tailed service time distributions, Z may take a large value mostly due to a large value of a single service time of one of the customers. Arguments used in the paper also allow us to obtain an elementary proof of the logarithmic asymptotics for the tail distribution of the stationary sojourn time in the whole class of light-tailed distributions.
Similar content being viewed by others
References
Loynes, R.M., The Stability of a System of Queues in Series, Proc. Cambridge Philos. Soc., 1964, vol. 60, no. 3, pp. 569–574.
Baccelli, F. and Foss, S., Ergodicity of Jackson-type Queueing Networks, Queueing Systems Theory Appl., 1994, vol. 17, no. 1–2, pp. 5–72.
Bertsimas, D., Paschalidis, I.Ch., and Tsitsiklis, J.N., On the Large Deviations Behavior of Acyclic Networks of G/G/1 queues, Ann. Appl. Probab., 1998, vol. 8, no. 4, pp. 1027–1069.
Ganesh, A.J., Large Deviations of the Sojourn Time for Queues in Series, Ann. Oper. Res., 1998, vol. 79, no. 1, pp. 3–26.
Zachary, S. and Foss, S.G., On the Exact Distributional Asymptotics for the Supremum of a Random Walk with Increments in a Class of Light-Tailed Distributions, Sibirsk. Mat. Zh., 2006, vol. 47, no. 6, pp. 1265–1274 [Siberian Math. J. (Engl. Transl.), 2006, vol. 47, no. 6, pp. 1034–1041].
Baccelli, F. and Foss, S., Moments and Tails in Monotone-Separable Stochastic Networks, Ann. Appl. Probab., 2004, vol. 14, no. 2, pp. 612–650.
Baccelli, F. and Foss, S., On the Saturation Rule for the Stability of Queues, J. Appl. Probab., 1995, vol. 32, no. 2, pp. 494–507.
Pakes, A.G., On the Tails of Waiting-Time Distributions, J. Appl. Probab., 1975, vol. 12, no. 3, pp. 555–564.
Bertoin, J. and Doney, R.A., Some Asymptotic Results for Transient Random Walks, Adv. in Appl. Probab., 1996, vol. 28, no. 1, pp. 207–226.
Ganesh, A., O’Connell, N., and Wischik, D. Big Queues, Lect. Notes Math., vol. 1838, Berlin: Springer, 2004.
Borovkov, A.A., Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Moscow: Nauka, 1972. Translated under the title Stochastic Processes in Queueing Theory, New York: Springer, 1976.
Borovkov, A.A. and Mogul’skii, A.A., The Second Rate Function and the Asymptotic Problems of Renewal and Hitting the Boundary for Multidimensional Random Walks, Sibirsk. Mat. Zh., 1996, vol. 37, no. 4, pp. 745–782 [Siberian Math. J. (Engl. Transl.), 1996, vol. 37, no. 4, pp. 647–682].
Lelarge, M., Tail Asymptotics for Monotone-Separable Networks, J. Appl. Probab., 2007, vol. 44, no. 2, pp. 306–320.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.G. Foss, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 4, pp. 93–108.
Supported in part by the EPSRC, Grant EP/E033717/1.
Rights and permissions
About this article
Cite this article
Foss, S.G. On the exact asymptotics for the stationary sojourn time distribution in a tandem of queues with light-tailed service times. Probl Inf Transm 43, 353–366 (2007). https://doi.org/10.1134/S0032946007040084
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1134/S0032946007040084