Best Unbiased Estimation and CAN Property in the Stable M/M/1 Queue

Abstract

The Uniform Minimum Variance Unbiased (UMVU) estimators of ρ^ℓ, the probability of having ℓ or more customers, L, the expected system size, L _q , the expected number of customers in the queue, and , the expected number of customers in a non empty queue, are derived based on a random sample of fixed size n on system size at departure points from the geometric distribution on the support {0, 1, 2,…} with mean , which is the distribution of system size in M/M/1 queueing system in equilibrium. The derivations are based on application of Lehmann-Scheffe theorem. Also, CAN estimators of performance measures are derived. In addition the probability distribution of UMVU estimators are obtained.

Keywords:

• Consistent asymptotic normality
• M/M/1 queueing system
• Uniform minimum variance unbiased estimator

Acknowledgments

The first author is thankful to Council of Scientific and Industrial Research (CSIR), INDIA, for the initial research support. Also, the authors are thankful to a reviewer for his comments, which resulted in a better presentation in Sec. 4.