On estimation problems for the G/G/∞ queue

Abstract

We consider estimation problems in G/G/∞ queue under incomplete information. Specifically, we are interested in scenarios where it is infeasible to track each individual job in the system and only aggregate statistics are known or observable. We first show that the minimum expected square estimator for the queue length process can be written as the sum of the minimum square estimators for an indicator function associated with each job, which is simply the survival function of the service time variable for each job. We also obtain tight lower and upper bounds on the time average of the square estimation error. Next we look at the inverse problem of estimating the service time distribution when the observed process is only the queue length process. We develop an on-line stochastic-optimization based estimation algorithm for the service time distribution and study its convergence under different parameter settings.