The convergence property of sample derivatives in closed Jackson queuing networks

A stochastic system such as a queuing network can be specified by system parameters and a random vector which represents the random effect involved in the system. For each realization of the random vector, the system performance measure as a function of system parameters is called a sample performance function. The derivative of the sample performance function of the system throughput in a finite period with respect to a mean service time in a queuing network can be obtained using perturbation analysis based on only one trajectory of the network. This paper studies the sample performance functions of closed Jackson queuing networks. It proves that the elasticity of the sample performance function of the throughput in a finite period with respect to the mean service time converges in mean to that of the mean throughput in steady-state as the number of customers served (or, equivalently, the length of the period) goes to infinity.