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

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

## Abstract

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:
$$P\{ S > x\} \equiv 1 - {{F}_{S}}(x) = O({{x}^{{ - a}}})$$
(15.1)
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.

## Keywords

Service Time Arrival Rate Waiting Time Traffic Intensity Busy Period
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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