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.

Development of approaches for early detection of cancer requires a comprehensive understanding of the cellular functions that lead to cancer, as well as implementing strategies for population-wide early detection. Cell functions are supported by proteins that are produced by active or expressed genes. Identifying cancer biomarkers, i.e., the genes that are expressed and the corresponding proteins present only in a cancer state of the cell, can lead to its use for early detection of cancer and for developing drugs. There are approximately 30,000 genes in the human genome producing over 500,000 proteins, thereby posing significant analytical challenges in linking specific genes to proteins and subsequently to cancer. Along with developing diagnostic strategies, effective population-wide implementation of these strategies is dependent on the behavior and interaction between entities that comprise the cancer care system, like patients, physicians, and insurance policies. Hence, obtaining effective early cancer detection requires developing models for a systemic study of cancer care. In this research, we develop models to address some of the analytical challenges in three distinct areas of early cancer detection, namely proteomics, genomics, and disease progression. The specific research topics (and models) are: 1) identification and quantification of proteins for obtaining biomarkers for early cancer detection (mixed integer-nonlinear programming (MINLP) and wavelet-based model), 2) denoising of gene values for use in identification of biomarkers (wavelet-based multiresolution denoising algorithm), and 3) estimation of disease progression time of colorectal cancer for developing early cancer intervention strategies (computational probability model and an agent-based simulation).