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 zn([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], [not partial differential]zn([theta],[omega])/[not partial differential][theta] exists for every n=1,2,..., and as N-->[infinity],[summation operator][infinity]n=1([not partial differential]zn([theta],[omega])/[not partial differential][theta])/N-->[not partial differential]Z([theta])/[not partial differential][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.