Service-time ages, residuals, and lengths in an M/GI/∞ service system

摘要

Important supplementary variables of a stationary M/GI/∞ service system are the service-time ages, residuals, and lengths of the customers present in the system at time 0. Our main result for a stationary system is that these times form a Poisson processes on R_+~3. Thus, by the order statistic property of Poisson processes, the three-dimensional vectors of the service-time ages, residuals, and lengths are independent and identically distributed. The joint distribution function of these three times is the same as the respective joint limiting distribution function of an inter-renewal time's age, residual, and length in a renewal process. The proof is based on a space-time Poisson representation of the M/GI/∞ system. A similar result is presented for a non-stationary system. Included is an ecological application concerning ages of trees in a forest.