Some consequences of estimating parameters for the M/M/1 queue

An important interface between stochastic models and actual systems comes in estimating values for model parameters using “real world” data. This interface between models and systems is studied for one of the most elementary stochastic systems, the M/M/1 queue. Estimating arrival rates and service rates results in a notable discrepancy between the state distribution for the model (estimated parameters) and the state distribution for the actual system (known parameters). Also, the expected number of customers in the model is infinite regardless of the (unknown) value of the actual traffic intensity. The truth of this assertion is obvious if one allows estimated traffic intensities to equal or exceed one. However, it is shown that the mean for the model is infinite even if the estimated traffic intensity is restricted to be strictly less than one.