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.

[1]  A. Sterrett.  On the Detection of Defective Members of Large Populations , 1957 .

[2]  Jewgeni H. Dshalalow,et al.  A first passage problem and its applications to the analysis of a class of stochastic models , 1992 .

[3]  M. Neuts.  A General Class of Bulk Queues with Poisson Input , 1967 .

[4]  A. Macula.  Probabilistic nonadaptive group testing in the presence of errors and DNA library screening , 1999 .

[5]  Anthony J. Macula,et al.  Probabilistic Nonadaptive and Two-Stage Group Testing with Relatively Small Pools and DNA Library Screening , 1998, J. Comb. Optim..

[6]  Marcello Pagano,et al.  Screening for the Presence of a Disease by Pooling Sera Samples , 1994 .

[7]  D. Du,et al.  Combinatorial Group Testing and Its Applications , 1993 .

[8]  C. D. Litton,et al.  A First Course in Bulk Queues , 1983 .

[9]  Joseph L. Gastwirth,et al.  Group Testing for Sensitive Characteristics: Extension to Higher Prevalence Levels , 1994 .

[10]  M. L. Chaudhry,et al.  A first course in bulk queues , 1983 .

[11]  Marcel F. Neuts,et al.  DISTRIBUTION OF OCCUPATION TIME AND VIRTUAL WAITING TIME OF A GENERAL CLASS OF BULK QUEUES. , 1970 .

[12]  W H Swallow,et al.  Use of binomial group testing in tests of hypotheses for classification or quantitative covariables. , 2000, Biometrics.

[13]  Marcello Pagano,et al.  On the informativeness and accuracy of pooled testing in estimating prevalence of a rare disease: Application to HIV screening , 1995 .

[14]  Wolfgang Stadje,et al.  Multinomial Group Testing Models with Incomplete Identification , 2003 .

[15]  Ciro D'Apice,et al.  Queueing Theory , 2003, Operations Research.

[16]  John Reynolds,et al.  Mathematical Methods in Operations Research , 1970 .

[17]  Alexander Dukhovny,et al.  Markov chains with transition delta-matrix: ergodicityconditions, invariant probability measures and applications , 1991 .

[18]  Wolfgang Stadje,et al.  Group testing procedures with incomplete identification and unreliable testing results , 2006 .

[19]  Wolfgang Stadje,et al.  Optimal Group Testing with Processing Times and Incomplete Identification , 2004 .

[20]  Lawrence M. Wein,et al.  Pooled Testing for HIV Screening: Capturing the Dilution Effect , 2018, Oper. Res..

[21]  Wolfgang Stadje,et al.  HYPERGEOMETRIC GROUP TESTING WITH INCOMPLETE INFORMATION , 2003, Probability in the Engineering and Informational Sciences.

[22]  Alexander Dukhovny,et al.  Optimization in HIV screening problems , 2003 .

[23]  Robert M Corless,et al.  Some applications of the Lambert W  function to physics , 2000, Canadian Journal of Physics.