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Processor-sharing and random-service queues with semi-Markovian arrivals

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

De-An Wu  and
Hideaki Takagi
De-An Wu*
Affiliation:
University of Tsukuba
Hideaki Takagi*
Affiliation:
University of Tsukuba
*
∗Postal address: Doctoral Program in Policy and Planning Sciences, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba-shi, Ibaraki 305-8573, Japan.
∗∗Postal address: Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennoudai, Tsukuba-shi, Ibaraki 305-8573, Japan. Email address: [email protected]
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Abstract

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We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-l customer who, upon his arrival, meets k customers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-l customer who, upon his arrival, meets k+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

Keywords

Queuesemi-Markov arrival processprocessor sharingrandom servicesojourn timewaiting time

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K15: Markov renewal processes, semi-Markov processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 478 - 490
Copyright
© Applied Probability Trust 2005 

References

Andersson, M., Cao, J., Kihl, M. and Nyberg, C. (2003). Performance modeling of an Apache web server with bursty arrival traffic. In Proc. Internat. Conf. Internet Comput. (Las Vegas, June 2003), Vol. 2, pp. 508–514.Google Scholar
Çinlar, E. (1967). Queues with semi-Markovian arrivals. J Appl. Prob. 4, 365–379.CrossRefGoogle Scholar
Coffman, E. G. Jr., Muntz, R. R. and Trotter, H. (1970). Waiting time distributions for processor-sharing systems. J. Assoc. Comput. Mach. 17, 123–130.CrossRefGoogle Scholar
Cohen, J. W. (1982). The Single Server Queue (North-Holland Ser. Appl. Math. Mech. 8), 2nd edn. North-Holland, Amsterdam.Google Scholar
Cohen, J. W. (1984). On processor sharing and random service. J. Appl. Prob. 21, 937.CrossRefGoogle Scholar
Ding, W. (1991). A unified correlated input process model for telecommunication networks. In Teletraffic and Datatraffic in a Period of Change, eds Jensen, A. and Iversen, V. B., Elsevier, Amsterdam, pp. 539–544.Google Scholar
Gantmacher, F. R. (2000). The Theory of Matrices, Vol. 1. AMS, Providence, RI.Google Scholar
Gantmacher, F. R. (2000). The Theory of Matrices, Vol. 2. AMS, Providence, RI.Google Scholar
Heffes, H. (1980). A class of data traffic processes–covariance function characterization and related queuing results. Bell System Tech. J. 59, 897–929.CrossRefGoogle Scholar
Jagerman, D. and Sengupta, B. (1991). The GI/M/1 processor-sharing queue and its heavy traffic analysis. Commun. Statist. Stoch. Models 7, 379–395.CrossRefGoogle Scholar
Kherani, A. and Kumar, A. (2002). Stochastic models for throughput analysis of randomly arriving elastic flows in the Internet. In Proc. IEEE INFOCOM (New York, July 2002), Vol. 2, pp. 1014–1023.Google Scholar
Loynes, R. M. (1962). Stationary waiting-time distribution for single server queue. Ann Math. Statist. 33, 1323–1339.CrossRefGoogle Scholar
Neuts, M. F. (1966). The single server queue with Poisson input and semi-Markov service times. {J. Appl. Prob.} 3, 202–230.CrossRefGoogle Scholar
Ott, T. J. (1984). The sojourn time distribution in the M/G/1 queue with processor sharing J. Appl. Prob.} 21, 360–378.CrossRefGoogle Scholar
Ramaswami, V. (1984). The sojourn time in the GI/M/1 queue with processor sharing. J. Appl. Prob. 21, 437–442.CrossRefGoogle Scholar
Sengupta, B. (1991). An approximation for the sojourn-time distribution for the GI/G/1 processor sharing discipline queue. Commun. Statist. Stoch. Models 8, 379–395.Google Scholar
Takács, L. (1962). Introduction to the Theory of Queues. Oxford University Press.Google Scholar
Yagyu, S. and Takagi, H. (2002). A queueing model with input of MPEG fame sequence and interfering traffic. J. Operat. Res. Soc. Japan 45, 317–338.Google Scholar
Yashkov, S. F. (1983). A derivation of response time distribution for an M/G/1 processor-sharing queue. Problems Control Inf. Theory 12, 133–148.Google Scholar
Yashkov, S. F. (1987). Processor-sharing queues: some progress in analysis. Queueing Systems 2, 1–17.CrossRefGoogle Scholar