## Stochastic Queueing Models for Air Transportation Systems with Scheduled Arrivals

- Author(s): Nikoleris, Anastasios Nikolaos
- Advisor(s): Hansen, Mark
- et al.

## Abstract

In this thesis we examine a queueing system with a single server under 4D trajectory (4DT) aircraft operations. In the definition of 4DT, we consider time as the fourth dimension. Thus, under 4DT's aircraft are assigned scheduled times of arrival at a fix, which they meet with some stochastic error. We start our analysis by assuming a normal distribution for the error, and that aircraft enter service according to a First-Scheduled-First-Served (FSFS) queue discipline. We develop a recursive queueing model that employs the Clark approximation method to analytically estimate the mean and variance of aircraft delays. The accuracy of the approximation method is assessed through simulation experiments, which indicate good accuracy of the Clark method in estimating total system delays. Using a wide range of representative demand and capacity scenarios at seven major US airports, we compare the model estimates for average queueing delay per flight with those from a deterministic queueing model. We find that the estimates of expected queueing delay from the stochastic and the deterministic model are strongly correlated and very similar, except for cases of low airport utilization, characterized by average deterministic queueing delay over all arrivals smaller than one minute.

Next, we study a simplified situation in which a sequence of aircraft with the same 4DT execution accuracy are assigned scheduled times of arrival at a fix with constant excess time separation between them. Under these assumptions, the expected delay to a flight from imperfect trajectory adherence - which we term stochastic delay - depends on the excess time separation, or buffer, expressed as a ratio to the trajectory imprecision, as well as the place of the flight in the sequence. As the buffer goes up, the stochastic delay goes down, but at the cost of increased deterministic delay from reduced capacity. If stochastic delay costs more than deterministic delay, then the optimum buffer is greater than zero, but quite small under plausible cost ratios and trajectory precision levels.

We also explore the effect of queue disciplines that give priority to aircraft equipped with avionics that enable them to execute 4D trajectories with high precision. We find that by switching from a FSFS to a Best-Equipped-Best-Served (BEBS) policy, total delay in the system can remain at the same level, while achieving significant delay savings for equipped aircraft.

The basic queueing model is extended to analyze a system for aircraft landings at a single runway under 4D trajectory-based operations. The server of this queueing system is the runway threshold, at which aircraft are assigned scheduled times of arrival. In accord with evidence from a variety of sources, we assume a Gumbel distribution for aircraft's stochastic lateness and for runway occupancy times (RT). We propose an approximate solution to analytically estimate the mean and variance of queueing delays, the accuracy of which is demonstrated through simulation experiments.

Next, we study again a simplified situation, in which a sequence of aircraft with the same capability of adherence to 4D trajectories, minimum headway, and runway occupancy time distribution are metered at a constant rate. We investigate the relationship between buffer time between scheduled arrivals and expected loss in system efficiency, defined as the weighted sum of delays due to imperfect trajectory adherence and varying RT - which we term stochastic delay - and due to reduced throughput. We find that if stochastic delay costs more than deterministic delay, then the minimum expected loss in efficiency is attained for buffer values greater than zero. Under conditions that we consider realistic, such a buffer would reduce planned throughput about 15% compared to the maximum possible under deterministic conditions. It is also shown that when RT is the binding factor that determines throughput, stochastic delays are substantially higher, compared to a situation where minimum headway is the principal constraint. For the former cases, it is found that the marginal benefit from improving trajectory precision diminishes when the standard deviation of adherence error is less than 0.5 times that of RT. Therefore, when runway throughput is controlled primarily by aircraft's time to exit the runway, improving 4D trajectory precision yields delay savings, but at a decreasing rate.

In order to investigate the potential increase in runway throughput from implementation of 4D trajectories, we study the case of paired arrivals at the San Francisco International Airport. A mathematical model is developed that estimates the scheduling headway between two consecutive pairs of landing aircraft. This headway minimizes the time interval between consecutive arrival pairs, while allowing sufficient time for a pair of departing aircraft to take-off in the meantime. Results derived from a simplified case study indicate potential increases in landing throughput by two aircraft per hour for every second of decrease in the standard deviation of adherence error. Additionally, we compute the upper bound in landings brought by almost perfect adherence to 4DT's, which appears as high as 81 aircraft/hour, an increase of 47% from current level of 55 aircraft/hour.