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Insensitive Bounds for the Moments of the Sojourn Times in M/GI Systems Under State-Dependent Processor Sharing

Part of: Computer system organization Operations research and management science Stochastic processes Distribution theory - Probability Special processes

Published online by Cambridge University Press:  01 July 2016

Andreas Brandt  and
Manfred Brandt
Andreas Brandt*
Affiliation:
Humboldt-Universität zu Berlin
Manfred Brandt*
Affiliation:
Zuse Institute Berlin
*
∗ Postal address: Institut für Operations Research, Humboldt-Universität zu Berlin, Spandauer Strasse 1, D-10178 Berlin, Germany. Email address: [email protected]
∗∗ Postal address: Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustrasse 7, D-14195 Berlin, Germany. Email address: [email protected]
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Abstract

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We consider a system with Poisson arrivals and independent and identically distributed service times, where requests in the system are served according to the state-dependent (Cohen's generalized) processor-sharing discipline, where each request receives a service capacity that depends on the actual number of requests in the system. For this system, we derive expressions as well as tight insensitive upper bounds for the moments of the conditional sojourn time of a request with given required service time. The bounds generalize and extend corresponding results, recently given for the single-server processor-sharing system in Cheung et al. (2006) and for the state-dependent processor-sharing system with exponential service times by the authors (2008). Analogous results hold for the waiting times. Numerical examples for the M/M/m-PS and M/D/m-PS systems illustrate the given bounds.

Keywords

Poisson arrivalgeneral service timestate-dependent processor sharingCohen's generalized processor sharingconditional sojourn timeconditional waiting timemomentsinsensitive boundsmany serverM/GI/m-PSLaplace-Stieltjes transform

MSC classification

Primary: 60K25: Queueing theory 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service 60E15: Inequalities; stochastic orderings 60G10: Stationary processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 246 - 267
Copyright
Copyright © Applied Probability Trust 2010 

References

Ahlfors, L. V. (1979). Complex Analysis. McGraw-Hill, New York.Google Scholar
Bonald, T. and Proutière, A. (2002). Insensitivity in processor-sharing networks. Performance Evaluation 49, 193–209.Google Scholar
Borst, S., Boxma, O. and Jelenković, P. (2003). Reduced-load equivalence and induced burstiness in GPS queues with long-tailed traffic flows. Queueing Systems 43, 273–306.Google Scholar
Borst, S., Mandjes, M. and van Uitert, M. (2003). Generalized processor sharing queues with heterogeneous traffic classes. Adv. Appl. Prob. 35, 806–845.Google Scholar
Braband, J. (1994). Waiting time distributions for M/M/N processor sharing queues. Commun. Statist. Stoch. Models 10, 533–548.CrossRefGoogle Scholar
Brandt, A. and Brandt, M. (2008). Waiting times for M/M systems under state-dependent processor sharing. Queueing Systems 59, 297–319.Google Scholar
Brandt, A. and Brandt, M. (2009). Approximations for the second moments of sojourn times in M/GI systems under state-dependent processor sharing. ZIB Rep. 09-25, Konrad-Zuse-Zentrum für Informationstechnik Berlin. Available at http://opus.kobv.de/zib/volltexte/2009/1190/.Google Scholar
Brandt, M. and Brandt, A. (2010). On sojourn times in M/GI systems under state-dependent processor sharing. Queueing Systems 64, 167–201.Google Scholar
Cheung, S.-K., van den Berg, H. and Boucherie, R. (2006). Insensitive bounds for the moments of the sojourn time distribution in the M/G/1 processor-sharing queue. Queueing Systems 53, 7–18.Google 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.Google Scholar
Cohen, J. W. (1979). The multiple phase service network with generalized processor sharing. Acta Informatica 12, 245–284.Google Scholar
Guillemin, F., Robert, P. and Zwart, B. (2004). Tail asymptotics for processor-sharing queues. Adv. Appl. Prob. 36, 525–543.Google Scholar
Kitaev, M. Y. and Yashkov, S. F. (1978). Distribution of the conditional sojourn time of requests in shared-processor systems. Izv. Akad. Nauk SSSR Tekh. Kibernet. 4, 211–215.Google Scholar
Kleinrock, L. (1967). Time-shared systems: a theoretical treatment. J. Assoc. Comput. Mach. 14, 242–261.Google Scholar
Ott, T. J. (1984). The sojourn-time distribution in the M/G/1 queue with processor sharing. J. Appl. Prob. 21, 360–378.Google Scholar
Parekh, A. K. and Gallager, R. G. (1993). A generalized processor sharing approach to flow control in integrated services networks: the single-node case. IEEE/ACM Trans. Networking 1, 344–357.Google Scholar
Tolmachev, A. L. (1987). Insensitivity in queueing networks with generalized processor-sharing discipline. In Proc. 1st World Congr. Bernoulli Soc. (Tashkent, 1986), Vol. 1, VNU Science Press, Utrecht, pp. 283–286.Google Scholar
Van den Berg, J. L. and Boxma, O. J. (1991). The M/G/1 queue with processor sharing and its relation to a feedback queue. Queueing Systems 9, 365–401.Google Scholar
Yashkova, A. S. and Yashkov, S. F. (2003). Distribution of the virtual sojourn time in the M/G/1 processor sharing queue. Inf. Process. 3, 128–137.Google Scholar
Yashkov, S. F. (1983). A derivation of response time distribution for an M/G/1 processor-sharing queue. Prob. Control Inf. Theory 12, 133–148.Google Scholar
Yashkov, S. F. (1987). Processor-sharing queues: some progress in analysis. Queueing Systems 2, 1–17.Google Scholar
Yashkov, S. F. (1992). Mathematical problems in the theory of processor-sharing systems. J. Soviet Math. 58, 101–147.Google Scholar
Yashkov, S. F. and Yashkova, A. S. (2007). Processor sharing: a survey of the mathematical theory. Automation Remote Control 68, 1662–1731.Google Scholar
Zachary, S. (2007). A note on insensitivity in stochastic networks. J. Appl. Prob. 44, 238–248.Google Scholar
Zhang, Z. L., Towsley, D. and Kurose, J. (1994). Statistical analysis of generalized processor sharing scheduling discipline. ACM SIGCOMM Comput. Commun. Rev. 24, 68–77.Google Scholar