Modeling of Flight Delay State-Space Model - Advanced Research on Computer Education, Simulation and Modeling - page 27

tasks consecutively each day. Assume d denotes a departure event and a denotes an
arrival event. Then the discrete event sequence of an aircraft performs in a day can be
written as d 1 a 1 d 2 a 2 …d n a n , where the state of the next event only depends on the state
of the current event, not on the state of the past event. The discrete events sequence is
a Markov chain. Therefore, the relationship among states can be represented in a
state-space model, which can be expressed as:
x i+1 =x i +u i +w i (1)

y i =x i +v i (2)
Where x i denotes the state variable, u i denotes the system input, y i denotes the meas-
urements, w i and v i denote the process and measurement noise, respectively. The
system model (1) describes the evolution of the state variables over the sequence,
whereas the measurement model (2) represents how measurements relate to the state
variables. If an aircraft accomplish n flight tasks, then we have i=1,…,2n. When i is
an odd number, x i denotes a departure delay state or an arrival delay state, vise versa.
Flight delay in this paper represents the difference between the actual flight time
and the scheduled flight time. Random factors such as weather, baggage check-ins,
and mechanical failures may result in a delayed flight. On the other hand, an early
flight task completion is achievable through planning methods and strategies. Flight
delays caused by these uncertainties can be added to the model as u i . Additionally, air
turnaround time and ground turnaround time correspond to two uncorrelated proc-
esses. Values of u i for different models should be estimated in two delay states. How-
ever, the relationship between the uncertainties and flight delays are not to be repre-
sented by any mathematical models, which leaves the calculation of u i a key problem
in establishment of the state-space model.

2.2 Modeling of the System Input

In general, x i is the departure delay from an upstream airport, u i is represented as the
delay in air. When u i <0, it is actually denoted as flight time compensation. Earlier
statistics show that the longer itinerary duration a flight is to take, the more compen-
sation the flight can obtain. Therefore, a more effective way to represent u i is given as
following:

u i =sf i *r i (3)

Where sf i denotes the scheduled flight time between airports, r i denotes the delay of
per scheduled flight time, or delay rate. The density distribution of the delay rate is
shown in Fig.1.
The delay rates vary significantly in distribution, decreasing sharply as a function of
the distance from the center, which suggests us to use a finite mixture model [5] to de-
scribe the delay rate distribution. The density distribution g of delay rate is modeled as a
function with m mixed components. The mixture density of the ith point is written as:

m

∑

g(r Θ)= αψ(r θ )

(4)

i j j i j

j=1