System Contents Versus System Delay for Discrete-Time Queueing Systems with Renewal Arrivals

摘要

This paper concerns discrete-time queueing systems operating with a first-come-first-served (FCFS) queueing discipline. Customers arrive in the system according to a renewal process; the inter-arrival times form a family of independent and identically distributed (i.i.d.) random variables. We establish a relationship between the probability generating function (pgf) of the system delay of an arbitrary customer and the pgf of the system contents during an arbitrary slot. Based on this result we derive relationships between the mean and variance, the mass function and the tail probabilities of the respective distributions. These relationships are valid irrespective of the characteristics of the servicing process, i.e., irrespective the number of servers, the distribution of the service times and possible correlation between the service times of consecutive customers.