Waiting Times when Service Times are Stable Laws: Tamed and Wild

  • Donald P. Gaver
  • Patricia A. Jacobs
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 19)


In various applications of service system or queueing theory, there may arise a need to consider service times, S, of great variability, i.e., that seem to possess nearly Pareto tails:
$$ P\{ S > x\} \equiv 1 - {{F}_{S}}(x) = O({{x}^{{ - a}}}) $$
as x →∞, where α is small enough so that no moments, E[S k ], k ≥ 1, are finite. In this chapter, we examine certain aspects of such problems for M/G/1 systems, focusing on service times that are describable by positive stable laws. In view of Theorem 1 of Feller ([6], p. 448), it is impossible to ignore the class of stable law models to represent the behavior of (15.1); there is the additional fact that stable laws approximate the distributions of sums of many long-tailed independent random variables, e.g., the sum of a number of activities that constitute service. But there is the problem that without finite first and second moments, at a minimum, classical queue-theoretic results do not directly apply.


Service Time Arrival Rate Waiting Time Traffic Intensity Busy Period 
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© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Donald P. Gaver
  • Patricia A. Jacobs

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