The variance of the number of customers in an infinite-server queueing system with an arbitrarily correlated arrival process

Motivated by a container storage problem at seaports, this paper examines the infinite-server queueing model with a (second order) stationary but arbitrarily correlated arrival process. It starts with a description of the application context, incuding an explanation of why non-standard inputs and service times arise in certain container storage problems. As further illustration, formulas characterizing the input and service processes for a particular situation are then presented and justified, together with the simple container accumulation variance formula that follows from the general results derived later. The paper then describes the general results. After reviewing the properties of a general arrival process, and based on minimal assumptions, it presents exact expressions for the variance of the number of customers in the system. (The expression for the mean is already well known.) Expressions are first derived for general independent service times and then for more general service processes that are of importance in port applications.