Evaluating the risk of unsatisfied random demand on a time interval

Abstract

This paper focuses on an important and very common problem and presents a theoretical framework for solving it: “determining the risk of unsatisfied request from users placing random demands on a time interval”. For the common case of a single source servicing a number of consumers, a closed-form solution has been derived for the risk of collision of random demands. Based on the closed-form solution, an efficient optimisation method has been developed for determining the optimal number of consumers that can be serviced by a single source, such that the probability of unsatisfied demand remains below a maximal tolerable level. A central part of the proposed theoretical framework is a general equation evaluating the risk of unsatisfied demand by the expected fraction of time of unsatisfied demand. The derived equation covers multiple sources servicing multiple consumers. Finally, the conducted parametric studies revealed an unexpected finding: the risk of collision of random demands on a time interval is practically insensitive to the standard deviations of the durations of demands. This surprising result provides the valuable opportunity to work with random demand times characterised by their means only, without supplying their probability distributions or variances.