Abstract
Managers often desire to assign resources to minimize balking in service systems. Discrete event simulations are often used to study alternative assignments. When the distribution of the number of balking events in a business day is approximated by a Poisson distribution, the objective becomes that of selecting a population corresponding to the smallest mean number of unfortunate events. A procedure for selecting a Poisson population with the smallest mean is described in which the selection is carried out based on a random sample of size n from each population. Examples of a bank lobby and manufacturing process are used to illustrate this procedure.
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Appendices
Appendix A: SIMAN listing examples
Appendix B: Proof of Equation (4)
For a fixed value of k, and the most general configuration, λ[1]⩽λ[2]⩽⋯⩽λ[k], the P(CS) for the proposed selection rule is given by (2). Now, fixing 
and summing over all possible values of 
we can show that (2) becomes,
For a fixed λ[1], the P(CS) is a strictly increasing function of differences λ[i]−λ[1] and a strictly decreasing function of ratios λ[1]/λ[i] (i=2, 3, …, k). Refer to Corollaries 3.1 and 3.2 of Mulekar and Sobel (1999) for the proof. We can lower the P(CS) defined for the general configuration by moving each 
closer to 
(i=3, 4, …, k). Thus we can use configuration Ω*={λ[1]+δ1*=λ[2]=⋯=λ[k] and λ[1]/δ2*=λ[2]=⋯=λ[k]} to search for the LFC, that is, the configuration for which the P(CS) is the lowest. Since samples are drawn independently of each other, from Equation (B.1), the P(CS) under the configuration Ω* becomes
Note that the P(CS) under the configuration Ω*, given by (B.2) is at most equal to the PCS under the general configuration given by (B.1). Suppose m denotes the number of populations tied with 
for the smallest value of the estimated event rate. In the equation (B.2), successive terms account for the possibility of m=0, 1, 2, …, k−1 tied populations. The conditional probability of correct selection, given m populations tied with the best one is 1/(1+m), and 
gives the number of different ways in which m populations can be tied for the best place. Now using the following three results,
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the selection procedure proposes taking random samples of equal sizes (n) from all the populations,
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the sum of independent observations is a sufficient statistic for estimating the event rate, and
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the distribution of sum of independent Poisson random variates is also a Poisson random variable
we can rewrite Equation (B.2) as follows:
Following the proof of Theorem 3.2 by Mulekar and Sobel (1999), to find the LFC for a fixed λ[1], we can restrict our attention to points on two lines, namely, λ[2]=δ1*+λ[1] and λ[2]=λ[1]/δ1*. The minimum of the P(CS) is achieved where these two lines meet, that is, at the point given by (3). Now to obtain the infimum of the probability of correct selection, insert the LFC given by Equation (3) in Equation (B.3). Hence we get the desired result.
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Mulekar, M., Matejcik, F. On selecting a process with the smallest number of unfortunate events. J Oper Res Soc 57, 416–422 (2006). https://doi.org/10.1057/palgrave.jors.2602001
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DOI: https://doi.org/10.1057/palgrave.jors.2602001