Calibrating Infinite Server Queueing Models Driven By Cox Processes

This paper studies the problem of calibrating a $\text{Cox}/G/\infty$ infinite server queue to a dataset consisting of the number in the system and the age of the jobs currently in service, sampled at discrete time points. This calibration problem is complicated owing to the fact that the arrival intensity and the service time distribution must be jointly calibrated. Furthermore, maximizing the finite dimensional distribution (FDD) of the number-in-system process (which is the natural calibration objective) is intractable in this setting, since the computation of the FDDs involves an intractable integration over the path measure of the Cox input process. We derive an approximate inference procedure that maximizes a lower bound to the FDDs using stochastic gradient descent. This lower bound is tight when the calibrated parameters coincide with those of the ‘true’ model. We present extensive numerical experiments that demonstrate the efficacy and validity of the proposed method.