Estimating the Parameters of Mixed Shifted Negative Binomial Distributions via an EM Algorithm

Discrete phase-type (DPH) distributions have one property that is not shared by continuous phase-type (CPH) distributions, i.e., representing a deterministic value as a DPH random variable. This property distinguishes the application of DPH in stochastic modeling of real-life problems such as stochastic scheduling where service time random variables should be compared with a deadline that is usually a constant value. In this paper, we consider a restricted class of DPH distributions, called Mixed Shifted Negative Binomial (MSNB) and show its flexibility in producing a wide range of variances as well as its adequacy in fitting fat-tailed distributions. These properties render MSNB applicable to represent data on certain types of service time. Therefore, we adapt an expectation-maximization (EM) algorithm to estimate the parameters of MSNB distributions that accurately fit trace data. To present the applicability of the proposed algorithm, we use it to fit real operating room times as well as a set of benchmark traces generated from continuous distributions as case studies. Finally, we illustrate the efficiency of the proposed algorithm by comparing its results to that of two existing algorithms in the literature. We conclude that our proposed algorithm outperforms other DPH algorithms in fitting trace data and distributions.