Approximate queueing models for analyzing harbor terminal operations

The purpose of this paper is to present approximate queueing models to help assess the impacts of tug services on congested harbor terminals. The models are intended for harbors in which tug shortages are rare. A congested harbor terminal is modelled as a queueing system with m identical tugs (servers) and n identical berths (customers), and with general probability distributions of tug service time and berth cargo-handling time. The distribution of the number of berths in the system, a basic element to analyze the system performance, was established for two cases. For large m, the distribution was approximated by a binomial model and the respective accuracy tested. For small m, an approximate model for this distribution was developed. Particular emphasis was given to developing explicitly the probability of having one berth in queue and establishing the remaining probabilities of the distribution approximately. The model for small m was validated by means of simulation for various cases of harbor terminal operations exhibiting different ranges of the coefficient of variation of tug service time. The models were found to be reasonably accurate within a certain range covering situations in which tug shortages are in the order of 10% of the time or less.