The method of moments for higher moments and the usefulness of formula manipulation systems
Though often not explicitly stated, the method of moments is one of the heavily used methods in queueing analysis. By tracing a typical, tagged customer as it proceeds through the system expected values for a wide range of systems, especially single server systems with Poisson arrivals, have been obtained in the last decades. This paper investigates the suitability of the method of moments to derive higher moments. Starting from general considerations, the problems in dealing with higher moments are insulated, and solutions are presented. These results give a deep insight into the nature of the method of moments.
It is difficult to handle the complex formulae for higher moments without the help of a formula manipulation system such as SCRATCHPAD II The tool was particularly useful to make first steps to find hypothetical formulae, then start proof (e.g. by induction) and to perform complex derivations, e.g. for comparison. The use of the tool gives deep insight into the potential use of computer algebra systems for queueing analysis.
Keywordsqueueing theory method of moments computer algebra
Unable to display preview. Download preview PDF.
- 1.A. Cobham: Priority Assignments in Waiting Line Problems, Operations Research, 2, 70–76, 1954.Google Scholar
- 1a.A. Cobham: Priority Assignment — A Correction, Operations Research, 3, 547, 1955.Google Scholar
- 2.R.W. Conway, W.L. Maxell, L.W. Miller: Theory of Scheduling, Addison-Wesley, Reading, Mass., 1967.Google Scholar
- 3.P. Dauphin: Queueing System Analysis Using Formula Manipulation, Master Thesis, University of Erlangen-Nürnberg, Institute for Mathematical Machines and Data Processing VII, Chair for Computer Architecture and Performance Evaluation (in German).Google Scholar
- 4.T.M. O'Donovan: Distribution of Attained and Residual Service General Queuing Systems, Operations Research, 22, 570–575, 1974.Google Scholar
- 5.W. Feller: An Introduction to Probability Theory and Its Applications Volume II, 2.Ed., John Wiley & Sons, New York et al., 1971.Google Scholar
- 6.D. Gross, C. Harris: Fundamentals of Queuing Theory, 2.Ed., John Wiley & Sons, New York et al., 1985.Google Scholar
- 7.U. Herzog: Performance Analysis of Computer Systems I: Analysis of Elementary Structures with Non-Marcovian Traffic (Part 2), Script Fernuniversität — Gesamtschule-Hagen, 1989 (in German).Google Scholar
- 8.L. Kleinrock: Queueing Systems Volume I: Theory, John Wiley & Sons, New York et al., 1975.Google Scholar
- 9.J.D.C. Little: A Proof of the Queueing Formula L=λW, Operations Research, 9, 383–387, 1961.Google Scholar
- 10.A. Mandelbaum, U. Yechiali: The Conditional Residual Service Time in the M/G/1 Queue, appeared in: A. Mandelbaum, U. Yechiali: Individual Optimization in the M/GI/1 Queue, Technical Report, Dept. of Statistics, University of Tel Aviv, Tel Aviv, 1979.Google Scholar
- 11.K. Marshall, R. Wolff: Customer Average and Time Average — Queue Lengths and Waiting Times, Journal of Applied Probability. 8, 535–542, 1971.Google Scholar
- 12.T.E. Phipps, jr.: Machine Repair as a Priority Waiting-Line Problem, Operations Research, 4, 76–85, 1956.Google Scholar
- 13.L. Takács: Delay Distributions for one Line with Poisson Input, General Holding Times and Various Orders of Service, Bell Systems Technical Journal, 42, 505–519, 1963.Google Scholar
- 14.D.M.G. Wishart: Queueing Systems in which the Discipline is’ Last-come First-served', Operations Research, 8, 591–599, 1960.Google Scholar
- 15.R.W. Wolff: Poisson Arrivals see Time Averages, Operations Research, 20, 223–231, 1982.Google Scholar