On comparison of steady-state infinitesimal perturbation analysis and likelihood ratio derivative estimates (Q6080669)

Consider a first-come-first-served (FCFS) M/M/1 queue with the average service time \(\theta\) and the arrival rate \(\lambda\). Let \(C_i\) denote the \(i\)-th customer arriving in the queue, \(S_i\) the service time of \(C_i\), \(A_i\) the inter-arrival time between \(C_i\) and \(C_{i+1}\), \(T_i\) the system time of \(C_i\). Denote the steady-state system time by \(T\). Let the traffic intensity \(\rho=\lambda\theta< 1\), so that the system is stable. The problem is to estimate \(dE[T]/d\theta\). Two of the most popular single-run stochastic derivative estimation techniques for \(dE[T]/d\theta\) are infinitesimal perturbation analysis (IPA) and likelihood ratio method (LR). The authors of this paper compare the variances of IPA and LR estimators for \(dE[T]/d\theta\). They derive explicit formulas for the IPA and LR derivative estimators for the steady-state system time of the M/M/1 queue and show that IPA indeed has smaller variance. They provide a theoretical result for the rule-of-thumb that IPA has lower variance than LR for steady-state performance measures.