Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T10:22:44.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic expansions of moments of the waiting time in closed and open processor-sharing systems with multiple job classes

Published online by Cambridge University Press:  01 July 2016

Debasis Mitra  and
J. A. Morrison
Debasis Mitra*
Affiliation:
Bell Laboratories
J. A. Morrison*
Affiliation:
Bell Laboratories
*
∗Postal address: Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, U.S.A.
∗Postal address: Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of queueing networks. We give effective methods for computing, for each of possibly several job-classes, the second moment of the equilibrium waiting time for closed systems as well as for open systems. Both open and closed systems have a CPU operating under the processor-sharing (‘time-slicing') discipline in which service-time requirements may depend on job-class. The closed system also includes a bank of terminals grouped according to job-classes, with the class structure allowing distinctions in the user's behavior in the terminal. In the contrasting open system, the job streams submitted to the CPU are Poisson with rate parameters dependent on job-classes.

Our results are exact for the open system and, for the closed system, in the form of an asymptotic series in inverse powers of a parameter N. In fact, the result for open networks is simply the first term in the asymptotic series. For larger closed systems, the parameter N is larger and thus fewer terms of the series need be computed to achieve a desired degree of accuracy. The complexity of the calculations for the asymptotic expansions is polynomial in number of classes and, importantly, independent of the class populations. Only the results on the single-class systems, closed and open, were previously known.

Keywords

QUEUEING NETWORKSQUEUEING THEORYWAITING-TIME MOMENTS
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 4 , December 1983 , pp. 813 - 839
Copyright
Copyright © Applied Probability Trust 1983 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Boxma, O. J., Kelly, F. P. and Konheim, A. G. (1983) The product form for sojourn time distributions in cyclic exponential queues. J. Assoc. Comput. Mach. CrossRefGoogle Scholar
[2] Coffman, E., Muntz, R. R. and Trotter, H. (1970) Waiting time distributions for processor-sharing systems. J. Assoc. Comput. Mach. 17, 123–130.Google Scholar
[3] Cohen, J. W. (1979) The multiple phase service network with generalized processor sharing. Acta Informatica 12, 245–284.CrossRefGoogle Scholar
[4] Daduna, H. (1982) Passage times for overtake-free paths in Gordon-Newell networks. Adv. Appl. Prob. 14, 672–686.CrossRefGoogle Scholar
[5] Fayolle, G., Mitrani, I. and Iasnogorodski, R. (1980) Sharing a processor among many job classes. J. Assoc. Comput. Mach. 27, 519–532.Google Scholar
[6] Feller, W. (1968) An Introduction to Probability Theory and its Applications, 1. Wiley, New York.Google Scholar
[7] Gaver, D. P., Latouche, G. and Jacobs, P. A. (1982) Stochastic models for response times in processor-sharing computer models.Google Scholar
[8] Gordon, W. J. and Newell, G. F. (1967) Closed queueing system with exponential servers. Operat. Res. 15, 254–265.Google Scholar
[9] Harrison, P. G. (1980) Cyclic time distribution in tree-like queueing networks. Report, Department of Computing, Imperial College, London.Google Scholar
[10] Horduk, A. and Van Dijk, N. (1981) Stationary probabilities for networds of queues. Report, Institute of Applied Mathematics and Computer Science, University of Leiden.Google Scholar
[11] Kelly, F. P. and Pollett, P. K. (1983) Sojourn times in queueing networks. Adv. Appl. Prob. 15, 638–656.Google Scholar
[12] Kleinrock, L. (1976) Queueing Systems, Vol. II: Computer Applications. Wiley, New York.Google Scholar
[13] Lavenberg, S. S. and Reiser, M. (1980) Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers. J. Appl. Prob. 17, 1048–1061.CrossRefGoogle Scholar
[14] McKenna, J. and Mitra, D. (1982) Integral representations and asymptotic expansions for closed markovian queueing networks: normal usage. Bell System Tech. J. 61, 661–683.Google Scholar
[15] Marden, M. (1966) Geometry of Polynomials. American Mathematical Society, Providence, R.I. Google Scholar
[16] Melamed, B. (1982) Sojourn times in queueing networks. Math. Operat. Res. 7, 223–244.CrossRefGoogle Scholar
[17] Mitra, D. (1981) Waiting time distributions from closed queueing network models of shared-processor systems. In Proc. 8th Internat. Symp. Computer Performance, Modeling, Measurement and Evaluation, ‘Performance 81’, Amsterdam, 4-6 November 1981, North-Holland, Amsterdam, 113–131.Google Scholar
[18] Mitra, D. and Morrison, J. A. (1983) Asymptotic expansions of moments of the waiting time in a shared processor of an interactive system. SIAM J. Computing 12.Google Scholar
[19] Muntz, R. (1972) Waiting time distributions for round robin queueing systems. In Proc. Symp. Computer-Communications Network and Teletraffic, Polytechnic Institute of Brooklyn, April 1972.Google Scholar
[20] Ramakrishnan, K. G. and Mitra, D. (1982) An overview of PANACEA, a software package for analyzing markovian networks. Bell System Tech. J. 61, 2849–2872.CrossRefGoogle Scholar
[21] Sakota, M., Noguchi, S. and Oizumi, J. (1969) Analysis of a processor-shared queueing model for time sharing systems. In Proc. 2nd Hawaii Internat. Conf. Systems Sciences, 625–627.Google Scholar
[22] Salza, S. and Lavenberg, S. S. (1981) Approximating response time distributions in closed queueing network models of computer performance. In Proc. 8th Internat. Symp. Computer Performance, Modeling, Measurement and Evaluation, ‘Performance 81’, Amsterdam, 4-6 November 1981, North-Holland, Amsterdam, 133–146.Google Scholar
[23] Sauer, C. H. and Chandy, K. M. (1981) Computer Systems Performance Modeling. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
[24] Sevcik, K. C. and Mitrani, I. (1981) The distribution of queueing network states at input and output instants. J. Assoc. Comput. Mach. 28, 358–371.CrossRefGoogle Scholar