×

Exact tail asymptotics for a discrete-time preemptive priority queue. (English) Zbl 1312.60114

Summary: In this paper, we consider a discrete-time preemptive priority queue with different service completion probabilities for two classes of customers, one with high-priority and the other with low-priority. This model corresponds to the classical preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server. Due to the possibility of customers’ arriving and departing at the same time in a discrete-time queue, the model considered in this paper is more complicated than the continuous-time model. In this model, we focus on the characterization of the exact tail asymptotics for the joint stationary distribution of the queue length of the two types of customers, for the two boundary distributions and for the two marginal distributions, respectively. By using generating functions and the kernel method, we get the exact tail asymptotic properties along the direction of the low-priority queue, as well as along the direction of the high-priority queue.

MSC:

60K25 Queueing theory (aspects of probability theory)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
90B22 Queues and service in operations research
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alfa A S. Matrix-geometric solution of discrete time MAP/PH/1 priority queue. Nav. Res. Logist., 45: 23-50 (1998) · Zbl 0902.90061 · doi:10.1002/(SICI)1520-6750(199802)45:1<23::AID-NAV2>3.0.CO;2-N
[2] Alfa A S, Liu B, He Q M. Discrete-time analysis of MAP/PH/1 multiclass general preemptive priority queue. Nav. Res. Logist., 50: 662-682 (2003) · Zbl 1044.90012 · doi:10.1002/nav.10083
[3] Bender E. Asymptotic methods in enumeration. SIAM Review, 16: 485-513 (1974) · Zbl 0294.05002 · doi:10.1137/1016082
[4] Drekic S, Woolford D G. A preemptive priority queue with balking. Eur. J. Oper. Res., 164: 387-401 (2005) · Zbl 1068.90030 · doi:10.1016/j.ejor.2004.01.017
[5] Fayolle G, Iasnogorodski R, Malyshev V. Random Walks in the Quarter-Plane. Springer-Verlag, New York, 1999 · Zbl 0932.60002 · doi:10.1007/978-3-642-60001-2
[6] Flajolet F, Sedgewick R. Analytic Combinatorics. Cambridge University Press, Cambridge, 2009 · Zbl 1165.05001 · doi:10.1017/CBO9780511801655
[7] Gail H R, Hantler S L, Taylor B A. Analysis of a non-preemptive priority multiserver queue. Appl. Probab., 20: 852-879 (1988) · Zbl 0671.60095 · doi:10.2307/1427364
[8] Gail H R, Hantler S L, Taylor B A. On preemptive Markovian queue with multiple servers and two priority classes. Math. Oper. Res., 17: 365-391 (1992) · Zbl 0762.90028 · doi:10.1287/moor.17.2.365
[9] Hunter J J. Mathematical Techniques of Applied Probability, Volume 2, Discrete Time Models: Techniques and Applications. Academic Press, New York, 1983 · Zbl 0539.60065
[10] Isotupa K P S, Stanford D A. An infinite-phase quasi-birth-and-death model for the non-preemptive priority M/PH/1 queue. Stoch. Models, 18: 378-410 (2002) · Zbl 1009.60080
[11] Kao E P S, Narayanan K S. Computing steady-state probabilities of a non-preemptive priority multiserver queue. ORSA J. Comput., 2: 211-218 (1990) · Zbl 0760.60085 · doi:10.1287/ijoc.2.3.211
[12] Li H, Zhao Y Q. Exact tail asymptotics in a priority queue-characterizations of the preemptive model. Queueing Syst., 63: 355-381 (2009) · Zbl 1209.90116 · doi:10.1007/s11134-009-9142-9
[13] Li H, Zhao Y Q. Tail asymptotics for a generalized two-demand queueing model-a kernel method. Queueing Syst., 69: 77-100 (2011) · Zbl 1235.60132 · doi:10.1007/s11134-011-9227-0
[14] Li H, Tavakoli J, Zhao Y Q. Analysis of exact tail asymptotics for singular random walks in the quarter plane. Queueing Syst., 74: 151-179 (2013) · Zbl 1273.60054 · doi:10.1007/s11134-012-9321-y
[15] Miller D R. Computation of steady-state probabilites for M/M/1 priority queues. Oper. Res., 29: 945-958 (1981) · Zbl 0468.60089 · doi:10.1287/opre.29.5.945
[16] Takine T. A nonpreemptive priority MAP/G/1 queue with two classes of customers. J. Oper. Res. Soc. Jpn., 39: 266-290 (1996) · Zbl 0870.90063
[17] 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) · Zbl 1069.60085 · doi:10.1081/STM-200056025
[18] Xu C. Tail Asymptotics for a Discrete-Time Priority Preemptive Queueing System. Master Thesis, Carleton University, 2010
[19] Zhao J A, Li B, Cao X R, Ahmad I. A matrix-analytic solution for the DBMAP/PH/1 priority queue, Queueing Syst., 53: 127-145 (2006) · Zbl 1104.60059 · doi:10.1007/s11134-006-8306-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.