Runway queuing model

We propose an analytical queuing model for estimating the queuing delays during the departure service process of an aircraft. The proposed approach can be used to model the whole runway system, or each individual runway, depending on the data availability, and the desired level of modeling.

We define as the service rate, µ(t), the number of departing aircraft that can take off from the runway(s) modeled per 15-minute interval. Four fundamental assumptions are made in our attempt to model the service process in the runway system:

1. The demand is given by the known runway schedule, calculated by Module 1 of the model.

Thus, aircraft arrive at the queuing system according to the known runway schedule.

2. The service rate is assumed to follow a time-dependent (dynamic) Erlang distribution.

3. There is finite queuing space for the departing aircraft to wait in.

4. Aircraft in the departure queue are served on a First-Come-First-Served (FCFS) basis.

This framework is attractive because it models both the mean and variance of the departure process throughput, which have been shown to be important when calculating queuing delays in the context of airport systems [53, 54]. In queuing theory, this type of system would be denoted as an D(t)/Ek(t)/1 with finite queuing space C. In our case, as it will be explained later, the finite queuing space C is introduced for computational tractability and not for modeling the finite

available space for aircraft to queue around the departure runway(s).

In terms of the queuing literature for airport systems modeling, this model fits into the “Micro Model” literature. Each period of time of a day that the runway configuration considered is in use is divided into time windows of equal duration, ∆, each of 15 minutes, and indexed with i = 1, 2, . . . , T . For each time period i, a throughput distribution, of type Erlang with rate k_iµ_i and shape ki, is provided based on the operating conditions in the airport. Each flight of the runway schedule is indexed with index l. During each time window i, we have (from Module 1) the set of arrival times at the departure queue of aircraft in this time window, S(i). Their arrival times are provided by Module 1 and are assumed known a priori, but are not necessarily uniformly spaced in each time window.

We also note here that Module 2 models one departure runway in the case of EWR (Runway 4L/22R). Thus, the modeled queuing system maps to the physical queuing process at Runway 4L/22R. If there are multiple runways and the runway assignments are known, we can model each of them with a queuing system. An application of such a model for the departure process at Detroit Metropolitan Wayne County Airport (DTW) can be found in other work [76]. By contrast, in Appendix G, we use a single queuing system to model the queuing process at the main runway (27L) and the secondary runway (35) at PHL. Similarly in Appendix H, we use a single queuing system for modeling the departure runways 18C/36C and 18L/36R at CLT. The approximation of modeling two departure runways with a single queuing system is driven by the lack of runway assignments in the ASPM data. However, the predictive power of the model remains strong, as can be verified in Appendix G and in Appendix H.

We model this system with a discrete-time Markov Chain. A service completion of an Erlang process with shape k and rate kµ is represented with k stages of exponentially distributed random variables with rate kµ. We call each such stage a stage-of-work. Each stage of the Markov chain q denotes that there are q stages-of-work to be completed at the runway, i.e., there are min(1, q) aircraft in service and max(d(q − k)/ke, 0) aircraft in the departure queue.

We summarize the notation used in this section:

• l: Index of each aircraft.

• ∆: Duration of each time window.

• i: Index of each time window.

• ki: Shape parameter of the Erlang service time distribution during time window i.

• k_iµ_i: Rate parameter of the Erlang service time distribution during time window i.

• C: Queuing space of the queuing system, measured in units of aircraft.

• Ql : stages-of-work to be completed for emptying the system immediately after the l^th air-craft’s arrival. This means that there are Q_l stages-of-work to be completed (including its service time) until the l^th aircraft exits the system.

• pQl(j) : Probability that Ql takes the value j.

• cl : Inter-arrival time at the system between the (l − 1)^th and l^th aircraft.

• Cl=Pl

j=1cj: time of arrival of the l^th aircraft at the system.

• Sl: Total time that aircraft l spends in the queuing system, including time-in-service.

• Dl: Queuing delay that aircraft l experiences.

• Z0: Number of aircraft in queue at the beginning of the first time window.

• fν(x): P.m.f. of random variable X, drawn from a Poisson distribution with parameter ν.

• Fν(x): C.d.f. of random variable X, drawn from a Poisson distribution with parameter ν.

Static service process distribution

Here, we assume that that service process is static and is described by a single Erlang distribution with parameters (k, kµ) at all time windows. We observe the system at the epochs of arrival, Cl, of each aircraft l. In Figure 4-5, we show an example of the state of the Markov chain that an aircraft encounters, assuming that there are exactly 3 stages-of-work to be completed before the arrival of l^th aircraft in the system, and an Erlang distribution with shape 2 for the departure throughput. Upon the arrival of l^th aircraft, 2 stages-of-work are added and the system (instantaneously) transitions to state 5.

We turn now to calculating the probabilities p_Ql(j). p^l_ij is defined as the probability of the following event: There are i stages-of-work to be completed immediately after the arrival of the l^th aircraft, given that there were j stages-of-work to be completed immediately after the arrival of (l − 1)^th aircraft:

p^l_ij = Pr(Q_l= i|Q_l−1 = j) (4.11)

3

Figure 4-5: Markov chain transition at the time of arrival of the l^th aircraft.

If the system is at state Q_l−1 when (l − 1)^thaircraft arrives, it evolves during time c_l similarly to a Poisson process:

Equation (4.12) can be explained as follows: Define as an event, the completion of a stage-of-work in the Markov chain of the system. The time between each pair of consecutive events has an exponential distribution with parameter (kµ). Each of the inter-arrival times of events is identically distributed and independent of other inter-arrival times. Thus, the probability of the completion of i < j stages-of-work (out of j) in time cl is given by the Poisson distribution with parameter (kµc_l). We note that up to j stages-of-work can be completed, and thus we can have up to j events during time cl. The condition i ≤ j does not impact the probability distribution of i < j events.

Each event is independent of its previous and its future ones. The condition i ≤ j impacts the probability distribution of having exactly i = j events (emptying the queuing system), which is given by the probability of having j, j + 1, j + 2, . . . Poisson arrivals (or one minus the probability of having 1, 2, . . . , j − 1 events).

For the third case of Equation (4.12) we have: If the system is in a state higher than kC − k after the arrival of the (l − 1)^th aircraft, we enforce that there will be space for the l^th aircraft when it arrives. Thus, for k(C − 1) < i ≤ kC, the system transitions to state j with the sum

of the probability of 0, 1, . . . , i − j + k events. In other words, we enforce the completion of as many stages-of-work as necessary to make space for the arrival of l^th aircraft. We note that this is different from standard queuing systems modeling convention, where the extra customer is rejected from the system if there is no queuing space. Our convention gives an opportunity for biasing the performance of the system, since selecting a very low value of parameter C can

“boost” the throughput of the system, by forcing stage-of-work completions at each arrival epoch.

Thus, selecting the queuing space parameter C is governed by tradeoffs between computational tractability and accuracy. Numerical experiments suggest that setting C = 100 is a good choice for modeling the runway queuing system. In other words, for C = 100, the system behaves as if it had infinite queuing space.

During time cl, the system cannot get to a state higher than i since there are no arrivals of aircraft in the system. We also note, that the calculated probabilities of having 0, 1, . . . , j events during time cl map to the probabilities p^l_(j+k)j, p^l_(j+k−1)j, . . . , p^l_kj because the arrival of l^th aircraft at time C_l instantaneously adds k stages-of-work.

Using the Poisson distribution notation introduced, Equation (4.12) can be written in the following form, where ν = kµc_l:

Using Equation (4.12) we calculate the probabilities p_Ql:

p_Ql(i) =

The starting condition for the calculation is the queue that the first aircraft arrival encounters

(Z₀). If the queue is empty at the beginning of the time period modeled, Z₀ = 0, and the first aircraft will bring the system to state k upon its arrival. A non-empty queue starting condition is interesting for runway configuration changes, where aircraft that pushed back before the runway configuration change are lined up at the new departure runway when its use commences.

pQ0(i) =







1 if i = k · Z0

0 otherwise

(4.16)

By inspection, we note that Matrix (4.13) is a stochastic matrix, as its columns add up to 1 for all values of k, and ν. Thus for a valid p_Ql−1 vector, the multiplication of Equation (4.15) returns a valid probability vector pQ_l.

To summarize, Equation (4.16) gives the state of the queue at the first aircraft’s arrival at time C1. The state of the queue at the times of arrival of aircraft 2, 3, . . . is calculated with Equation (4.15).

Dynamic service process distribution

For dynamic service time distributions, Equation (4.15) is still valid by using the appropriate service time distribution parameters (ki, kiµi) for the time window i, in which inter-arrival time cl falls:

ν_i = k_iµ_ic_l (4.17)

p_Ql = P (ν_i)p_Ql−1 (4.18)

A subtlety here is that the state space of the queuing system can change. For example, if we have two throughput distributions, with shapes k₁ and k₂, where k₁ 6= k₂, the space of the chain of the queuing system corresponding to the first distribution will be {0, 1, . . . , k1· C} and the second {0, 1, . . . , k2· C}. Transitioning from the throughput distribution of the former to that of the latter can be done only by approximately mapping the probabilities of states {0, 1, . . . , k1· C} to those of {0, 1, . . . , k2· C}.