INTERCHANGEABILITY OF EXPECTATION AND DIFFERENTIATION OF WAITING-TIMES IN GI/G/1 QUEUES

A family of stable GI/G/1 queues, whose service time distributions depend on a real-valued parameter, theta, is considered. Let z(n) (theta, omega) denote a realization of the waiting time of the nth customer in the theta-dependent queue, for a sample sequence omega in the underlying probability space. Let Z(theta) denote the expected value of waiting time in the theta-dependent queue, that is, the queue with the theta-dependent service time distribution. Under appropriate conditions, the following will be shown: (1) Z is a continuously differentiable function of theta; (2) for almost every omega, partial derivative z(n)(theta, omega)/partial derivative theta exists for every n = 1, 2,..., and as N --> infinity, SIGMA(n=1)infinity (partial derivative z(n)(theta, omega)/partial derivative theta)/ N --> partial derivative Z(theta)/partial derivative theta. These properties are important in simulation-based optimization of functions of theta, involving the average customer's waiting time in GI/G/1 queues.