Estimating Customer and Time

In this paper we establish a joint central limit theorem for customer and time averages by applying a martingale central limit theorem in a Markov framework. The limiting values of the two averages appear in the translation terms. This central limit theorem helps to construct confidence intervals for estimators and perform statistical tests. It thus helps determine which finite average is a more asymptotically efficient estimator of its limit. As a basis for testing for PASTA (Poisson arrivals see time averages), we determine the variance constant associated with the central limit theorem for the difference between the two averages when PASTA holds. OR/MS Subject Classification: Queues, limit theorems: central limit theorems for customer and time averages. Queues, statistical inference: estimating customer and time averages. Statistics, estimation: averages over time and at embedded points. This paper is concerned with estimating limiting time averages, customer (embedded) averages and their differences. When the two limiting averages are known to agree, i.e., when arrivals see time averages (ASTA); see Melamed and Whitt (1990a) and references cited there, we want to know which finite average is a more asymptotically efficient estimator (i.e., produces smaller confidence intervals with large samples). When the two limiting averages need not agree, we want to estimate their difference and be able to test for ASTA. To illustrate what happens when ASTA is known to hold, we give two examples. Example 1: The Workload in the M/M/1 Queue. Let {U(t) : t ≥ 0 } be the continuous-time workload (or virtual waiting time) process in an M/M/1 queue with service rate 1 and arrival rate ρ < 1. Let {N(t) : t ≥ 0 } be the Poisson arrival process with associated arrival times {T n : n ≥ 1 }. Then {U(T n − ) : n ≥ 1 } is the sequence of waiting times (before beginning service). It is well known that the time average V(t) = t 1 _ _ ∫ 0 t U(s) ds , t > 0 , (1) and the customer average W(t) = N(t) 1 _ ___ k = 1 Σ N(t) U(T k − ) , N(t) > 0 , (2) both converge with probability one (w.p.1) to ρ /( 1 − ρ) as t → ∞, so that we have ASTA. There is no need to estimate the limit in this case, but it is interesting to ask which estimator tends to produce smaller confidence intervals. We can easily decide, because it is known that central limit theorems (CLTs) hold, i.e., t 1/2 [V(t) − v] = = > N( 0 , σv ) (3) and