Abstract
We consider a MAP/PH/1 queue with two priority classes and nonpreemptive discipline, focusing on the asymptotic behavior of the tail probability of queue length of low-priority customers. A sufficient condition under which this tail probability decays asymptotically geometrically is derived. Numerical methods are presented to verify this sufficient condition and to compute the decay rate of the tail probability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abate, J., Whitt, W.: Asymptotics for M/G/1 low-priority waiting-time tail probabilities. Queueing Syst. 25, 173–233 (1997)
Berger, A.W., Whitt, W.: Effective bandwidths with priorities. IEEE/ACM Trans. Netw. 6, 447–460 (1998)
Berger, A.W., Whitt, W.: Extending the effective-bandwidth concept to networks with priority classes. IEEE Commun. Mag. 2–7 (1998)
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Science. Academic Press, San Diego (1979)
Bertsimas, D., Paschalidis, I.C., Tsitsiklis, J.N.: Asymptotic buffer overflow probabilities in multiclass multiplexers: An optimal control approach. IEEE Trans. Autom. Control 43, 315–335 (1998)
Delas, S., Mazumdar, R.R., Rosenberg, C.P.: Tail asymptotics for HOL priority queues handling a large number of independent stationary sources. Queueing Syst. 40, 183–204 (2002)
Elwalid, A.I., Mitra, D.: Analysis, approximations and admission control of a multi-service multiplexing system with priorities. In: Proc. IEEE INFOCOM’95, Boston, MA, 2–6 April, pp. 463–472 (1995)
Falkenberg, E.: On the asymptotic behavior of the stationary distribution of Markov chains of M/G/1-type. Stoch. Models 10, 75–97 (1994)
Foley, R.D., McDonald, D.R.: Bridges and networks: exact asymptotics. Ann. Appl. Probab. 15, 542–586 (2005)
Glynn, P.W., Whitt, W.: Logarithmic asymptotics for steady-state tail probability in a single-server queues. Appl. Probab. Adv. 31, 131–156 (1999)
Graham, A.: Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Chichester (1986)
Haque, L., Liu, L., Zhao, Y.Q.: Sufficient conditions for a geometric tail in a QBD process with countably many levels and phases. Stoch. Models 77–99 (2005)
Hashida, O., Takahashi, Y.: A discrete-time priority queue with switched batch Bernoulli process inputs and constant service times. In: Jensen, A., Iversen, V.B. (eds.) Teletraffic and Datatraffic in a Period of Change, pp. 521–526. North Holland, Amsterdam (1991)
Khamisy, A., Sidi, M.: Discrete-time priority queueing systems with two-state Markov modulated arrival process. In: INFOCOM 91, pp. 1456–1463 (1991)
Kingman, J.C.F.: A convexity property of positive matrices. Q. J. Math. 12, 283–284 (1961)
Latouche, G., Ramaswami, V.: Introduction to matrix analytic methods in stochastic modeling. In: ASA-SIAM. (1999)
Li, H., Miyazawa, M., Zhao, Y.Q.: Geometric decay in a QBD process with countable background states with applications to a join-the-shortest-queue model. Stoch. Models 413–438 (2007)
Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Stoch. Models 7, 1–46 (1991)
Lucantoni, D.M., Meier-Hellstern, K.S., Neuts, M.F.: A single-server queue with server vacations and a class of non-renewal arrival process. Adv. Appl. Probab. 22, 676–705 (1990)
Miyazawa, M., Zhao, Y.Q.: The stationary tail asymptotics in the GI/G/1 type queue with countably many background states. Adv. Appl. Probab. 36, 1231–1251 (2004)
Neuts, M.F.: A versatile Markovian point process. J. Appl. Probab. 16, 764–779 (1979)
Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore (1981)
Neuts, M.F.: Structured stochastic matrices of the M/G/1 type and their applications. Marcel Dekker, New York (1989)
Sidi, M., Segall, A.: Structured priority queueing systems with applications to packet-radio networks. Perform. Eval. 3, 265–275 (1983)
Subramanian, V., Srikant, R.: Tail probabilities of low-priority waiting times and queue lengths in MAP/GI/1 queue. Queueing Syst. 34, 215–236 (2000)
Takahashi, Y., Fujimoto, K., Makimoto, N.: Geometric decay of the steady-state probabilities in a Quasi-Birth-And Death process with a countable number of phases. Stoch. Models 17, 1–24 (2001)
Takine, T., Sengupta, B., Hasegawa, T.: An analysis of a discrete-time queue for broadband ISDN with priorities among traffic classes. IEEE Trans. Commun. 42, 1837–1845 (1994)
Takine, T.: A nonpreemptive priority MAP/G/1 queue with two classes of customers. J. Oper. Res. Soc. Jpn. 39, 266–290 (1996)
Takine, T.: The nonpreemptive priority MAP/G/1 queue. Oper. Res. 47, 917–927 (1999)
Tweedie, R.L.: Operator-geometric stationary distribution for Markov chains, with application to queueing models. Adv. Appl. Probab. 14, 392–433 (1982)
Xue, J., Alfa, A.S.: Tail probability of low-priority queue length in a discrete-time priority BMAP/PH/1 queue. Stoch. Models 21, 799–820 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xue, J., Alfa, A.S. Geometric tail of queue length of low-priority customers in a nonpreemptive priority MAP/PH/1 queue. Queueing Syst 69, 45–76 (2011). https://doi.org/10.1007/s11134-011-9221-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-011-9221-6