Efficient Computation of Probabilities of Events Described by Order Statistics and Applications to Queue Inference

Abstract

This paper derives recursive algorithms for efficiently computing event probabilities related to order statistics and applies the results in a queue inferencing setting. Consider a set of N i.i.d. random variables in [0, 1]. When the experimental values of the random variables are arranged in ascending order from smallest to largest, one has the order statistics of the set of random variables. Both a forward and a backward recursive O(N3) algorithm are developed for computing the probability that the order statistics vector lies in a given N-rectangle. The new algorithms have applicability in inferring the statistical behavior of Poisson arrival queues, given only the start and stop times of service of all N customers served in a period of continuous congestion. The queue inference results extend the theory of the “Queue Inference Engine” (QIE), originally developed by Larson in 1990 (Larson, R. C. 1990. The queue inference engine: Deducing queue statistics from transactional data. Management Sci. 36(5) 586–601, and 1991. The queue inference engine: Addendum. Management Sci. 37(6) 1062.). The methodology is extended to a third O(N3) algorithm, employing both forward and backward recursion, that computes the conditional average probability (averaged over all N customers in a congestion period) that the in-queue wait is less than t minutes, given the departure time data and assuming first come, first served service. To our knowledge, this result is the first O(N3) exact algorithm for computing points on the in-queue waiting time distribution function, conditioned on the start and stop time data. The paper concludes with an extension to the computation of certain correlations of in-queue waiting times. Illustrative computational results are included throughout. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.