Approximate solutions for multi-server queuing systems with Erlangian service times and an application to air traffic management

Abstract

This thesis is concerned with approximations of certain M(t)/G(t)/n(t)/n(t) + q queueing systems. More specifically, we are interested in such systems under very general conditions such as time-varying demand and capacity, and high utilization, including occasional oversaturation. Conditions such as these cannot be addressed with existing methodologies. We focus on M(t)/G(t)/n(t)/n(t) + q systems that can be approximated fairly well by M(t)/E&(t)/n(t)/n(t) + q systems. The latter have a large number of system states, that increase with the system parameters k, n, q and the utilization ratio, and involve complicated state transition probabilities. We propose numerical methods to solve the corresponding Chapman-Kolmogorov equations, exactly and approximately We first describe the exact solution technique of M(t)/Ek(t)/n(t)/n(t) + q queueing systems. Then, we develop two heuristic solution techniques of M(t)/E&(t)/ndt)/n(t) + q queueing systems, and provide the corresponding complete state descriptions. We compare the exact and approximate results to validate our heuristics and to select the heuristic that best approximates the exact results in steady-state and under stationary conditions. We also propose two algorithms to vary the number of servers in the system, since many real-life problems involve such changes in response to variations in demand. Further results using our ELC heuristic show that our practical approach behaves well under nonstationary conditions, including varying capacity, and during the transient period to steady-state. We conclude that our heuristic approach is an excellent alternative for studying and analyzing M(t)/E&(t)/n(t)/n(t)+q models and, as a by-product, many M(t)/G(t)/n(t)/n(t) +q systems that arise in practice. Finally, we present an application of the M(t)/E&(t)/n(t)/n(t) + q queueing model in the context of Air Traffic Management. This model appears to be a reasonable approach to estimating delays and congestion in an en-route sector in the air traffic system and can be used as an important building block in developing an analytical model of the entire Air Traffic Management system.