Modelling Systems with a Finite-Capacity Queue: A Theoretical Investigation into the Design and Operation of Real Systems

This study investigates M/M/n:(m/FIFO) systems with a limited queue capacity (incorporating both “waiting and rejection”). This category of systems can be considered to be mixed-service systems. They operate as queuing systems for customers admitted to the system awaiting service, as well as systems that implement rejection or loss for customers who are denied when the system is full (when all servers and the buffer capacity are occupied). The correlation between the system size and a set of performance measures is analysed for the given arrival and service rates. The system size is determined based on a threshold rate of rejected customers. The correlation between the buffer size and the utilisation factor has direct relevance in the design of real systems (e.g., when the dynamics of the arrival rate can be estimated, it provides a solution for phasing the building of physical waiting places for a specific service capacity). In addition, the analysis of customer rejection probability and average waiting time as a function of the effective utilisation factor could yield practical insights for designing and operating real systems. The second part of this study presents a model for optimising the size of a multi-server system with a finite queue capacity. Initially, the number of servers is determined, assuming that the existing situation does not allow for an increase in the buffer capacity. Then, the case in which both server and buffer capacities become decision variables is presented. The operating losses (which are more straightforward to measure than the related costs) are used as an optimisation criterion.