On Bayesian Inference for Markovian Queueing Models

We are concerned with an M/M/- queueing model. Particularly, we focus on such models with join the shortest queue policy and N parallel queues. Assuming the steady-state case, a Bayesian paradigm is used in estimating the queueing system's parameters when full data or only queue lengths data are provided. We start with the case N=2 queues and found maximum likelihood estimators and Bayesian estimators for the system's parameters when full data about arrival times, departures times, and jobs' paths is provided. When only queue lengths data is provided, we estimate the traffic intensity of the JSQ-2 model using its asymptotic behaviour studied by Flatto [46]. Also, we provide Bayesian estimation for the traffic intensity using a new approximation to JSQ-2 queue length distribution. Indeed, a generalized bivariate geometric distribution (as proposed by [48]) is employed with an improper prior and a new proposed generalized bivariate beta distribution. When N is large enough, using the mean filed performance of the JSQ-N studied by Dawson et al. [47], we estimate the traffic intensity of JSQ-N model considering both maximum likelihood estimators and Bayesian estimators with different prior beliefs of the parameters. Numerical results are provided to show the accuracy of our estimators for JSQ-N for both cases N=2 and large enough N. Furthermore, we introduce a first study of privacy preserving analysis on a sensitive data released by a queueing model M/M/1. In fact, we propose, in the context of Differential Privacy [56], a Bayesian framework, inspired by [58], to generate private synthetic data, private service and arrival time rates from M/M/1 queueuing data. Keywords: Bayesian inference, Queueing models, Differential Privacy, Join the Shortest Queue (JSQ), Generalized bivariate beta distribution, Mean filed performance.