Transient Analysis of D ( t ) / M ( t ) / 1 Queuing System with Applications to Computing Airport Delays

This thesis is motivated by the desire to estimate air traffic delays at airports under a range of assumptions about the predictability of (a) inter-arrival times of demands (arrivals and departures) and (b) service times of aircraft movements (landings and takeoffs). It consists of two main parts. In the first, a transient analysis of a D(t)/M(t)/1 queuing system is presented. The reason for focusing on such a system is that it may be useful in evaluating some of the benefits of a future Air Traffic Management (ATM) system, such as the Next Generation Air Transportation System (NGATS or NextGen) currently being developed in the United States. One of the main features of these future ATM systems will be high predictability and regularity of the inter-arrival times of airport demands, i.e., a nearly deterministic demand process. This will be achieved through significant reductions in aircraft trajectory uncertainty, with the expectation that airport delays will also decrease substantially as a result. We develop a novel, computationallyefficient numerical approach for solving D(t)/M(t)/1 queuing systems with general, dynamic demand and service rates. We also discuss the complexity of the approach and some characteristics of the resulting solutions. In the second part of the thesis, we use a set of models of dynamic queuing systems, in addition to our D(t)/M(t)/1 model to explore the range of values that airport delays may assume under different sets of assumptions about the level of uncertainty associated with demand inter-arrival times and with service times. We thus compute airport delays under different queuing systems in a dynamic setting (where demand and service rates are time-varying) to capture the entire range of uncertainties expected during the deployment of various future ATM system technologies. The specific additional models we consider are: a deterministic D(t)/D(t)/1 model in which it is assumed that airport demands for landings and takeoffs occur at exactly as scheduled; and a M(t)/Ek(t)/1 model which, because of the "richness" of the family of Erlang distributions, Ek, can be used to approximate most M(t)/G(t)/1 models that may arise in airport applications. It can be seen that these models, when used together, provide bounds on estimated airport delays, with the D(t)/D(t)/1 model most likely to offer a lower bound and the M(t)/M(t)/1 model (i.e., the special case of M(t)/Ek(t)/1 with k = 1), an upper bound. We show through a set of examples based on a few of the busiest airports in the United States that: the closeness of the delay estimates provided by the different models depend on the level of congestion at an airport and the relative shapes of the dynamic profiles of capacity and demand at the airport; the difference (on a "percentage" basis) between the estimates provided by the deterministic model and the stochastic ones is largest for uncongested airports and decreases as the level of congestion increases; D(t)/M(t)/1 and M(t)/D(t)/1 produce estimates of the same order of magnitude, and reflect delays in the presence of "moderate" uncertainty at an airport; and delays under a D(t)/M(t)/1 queuing system are always higher than under a M(t)/D(t)/1 system. Thesis Supervisor: Amedeo Odoni Title: Professor of Aeronautics and Astronautics, Professor of Civil and Environmental Engineering