Some well-behaved estimators for the M=M=1 queue

Abstract It is known that, given the observed tra c intensityˆ ¡ 1, the expected value of the estimatorˆ =(1 −ˆ ) for the average number of customers =(1 − ) in a stationary M=M=1 queueing model is inÿnite (Schruben and Kulkarni, Oper. Res. Lett. 1 (1982) 75 -78). In this paper we generalize the above ÿndings to other system performance measures. Second, we show that, for the following four system performance measures: (a) mean waiting time in queue, (b) mean waiting time in system, (c) mean number of customers in queue and (d) mean number of customers in the system, estimators constructed by substituting parameter estimators for unknown parameters in the formula for the performance measure all have the undesirable properties that the expected value of the estimator does not exist and the estimator has inÿnite mean-squared error. Finally, we propose alternative estimators for these four system performance measures when ¡ 0, where 0 ¡ 1 is a known constant, and show that these alternative estimators are strongly consistent, asymptotically unbiased and have ÿnite variance and ÿnite mean squared error

Estimation of Measures in M/m/1 Queue

Maximum likelihood and uniform minimum variance unbiased estimators of steady-state probability distribution of system size, probability of at least ℓ customers in the system in steady state, and certain steady-state measures of effectiveness in the M/M/1 queue are obtained/derived based on observations on X, the number of customer arrivals during a service time. The estimators are compared using Asympotic Expected Deficiency (AED) criterion leading to recommendation of uniform minimum variance unbiased estimators over maximum likelihood estimators for some measures

ML and Umvu Estimation in the M/d/1 Queuing System

In the imbedded Markov chain (IMC) analysis of M/G/1 queuing system, X1, X2, …, Xn, … form a sequence of i.i.d random variables. where Xn denotes the number of customer arrivals during the service time of customer. In the M/D/1 queue, the distribution of common random variable X is the Poisson distribution with mean ρ, the traffic intensity. This fact is utilized for maximum likelihood (ML) and uniformly minimum variance unbiased (UMVU) estimation of traffic intensity, performance measures, transition probabilities of IMC, and correlation functions of departure process, based on a sample of fixed size n from P(ρ) distribution. Also, consistent asymptotic normality (CAN) property of ML estimators (MLEs) is established. The MLEs and UMVUEs are compared