Fitting birth-and-death queueing models to data

Given measurements of the number of customers in a queueing system over a finite time interval, it is natural to try to fit a stationary birth-and-death process model, because it is remarkably tractable, even when the birth and death rates depend on the state in an arbitrary way. Natural estimators of the birth (death) rate in each state are the observed number of transitions up (down) from that state divided by the total time spent in that state. It is tempting to validate the model by comparing the steady-state distribution of the model based on those estimated rates to the empirical steady-state distribution recording the proportion of time spent in each state. However, it is inappropriate to draw strong conclusions from a close fit to the same data, because these two distributions are necessarily intimately related, even if the model assumptions are not nearly satisfied. We elaborate by (i) establishing stochastic comparisons between these two fitted distributions using likelihood-ratio stochastic ordering and (ii) quantifying their difference.