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 [8]. The methodology is extended to a third O(N 3 ) algorithm, employing both forward and backward recursion, that computes the conditional probability that a random customer of the N served waited in queue less than r minutes, given the observed customer departure times 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.