Applications of bulk queues to group testing models with incomplete identification

Abstract

A population of items is said to be ``group-testable'', (i) if the items can be classified as ``good'' and ``bad'', and (ii) if it is possible to carry out a simultaneous test on a batch of items with two possible outcomes: ``Success'' (indicating that all items in the batch are good) or ``failure'' (indicating a contaminated batch). In this paper, we assume that the items to be tested arrive at the group-testing centre according to a Poisson process and are served (i.e., group-tested) in batches by one server. The service time distribution is general but it depends on the batch size being tested. These assumptions give rise to the bulk queueing model M/G((m,M))/1, where m and M(> m) are the decision variables where each batch size can be between m and M. We develop the generating function for the steady-state probabilities of the embedded Markov chain. We then consider a more realistic finite state version of the problem where the testing centre has a finite capacity and present an expected profit objective function. We compute the optimal values of the decision variables (m, M) that maximize the expected profit. For a special case of the problem, we determine the optimal decision explicitly in terms of the Lambert function. (c) 2006 Elsevier B.V. All rights reserved.