Air traffic management: High-low traffic intensity analysis

European Journal of Operational Research 80 (1995) 45-58 North-Holland

45

Theory and Methodology

Air traffic management: High-low traffic intensity analysis Enio E. Velazco Department of Management, Worcester Polytechnic Institute, Worcester, MA 01609, USA Received December 1990; revised January 1993

Abstract: The Federal Aviation Administration (FAA) is responsible for administering the Air Traffic Control (ATC) system. The primary purposes of the ATC system are the preventing of collisions, and expediting and maintaining of an orderly flow of air traffic while recognizing the varying requirements of any National Airspace System (NAS) user. Nationwide management of air traffic flow can be aided by using simulation and mathematical models to estimate and track measures of performance of interest. In this paper, a queueing based methodology is proposed to analyze airports' operations during high and low traffic intensity conditions. Keywords: Queueing; Air traffic; Traffic analysis

I. Introduction

The Air Traffic Control (ATC) system, which is administered by the Federal Aviation administration (FAA), is responsible for the preventing of collisions, and the expediting and maintaining of an orderly flow of air traffic. The ATC system must, in addition, recognize the varying requirements of any National Airspace System (NAS) user. Users such as private, commercial and military aircraft, airports, etc. have all different requirements. For example, private and military aircraft tend to request more flexible schedules and routes than commercial aircraft. The NAS can be viewed as a network of airports interconnected by hundreds of miles of airways. The continental United States' sky has been divided into twenty airspaces and each airspace into several sectors. Each airspace is controlled by an air-route traffic control center (ARTCC). A particular ARTCC has responsibility within its airspace for en-route traffic, transition traffic (climbing to and descending from cruising altitude), and back-up traffic. The back-up traffic constitutes all traffic that must be held outside the area of jurisdiction of an airport's terminal area approach controllers, as they are currently handling the maximum agreed traffic. Aircraft are not permitted to penetrate the airspace of another ARTCC or sector unless prior coordination has taken place. All coordinations must take place ahead of the aircraft's movement. The rationale is to ensure that no aircraft is transferred from one controlling authority to another until the airspace is clear to receive it, according to the appropriate standards of separation. If controllers cannot Correspondence to: Dr. E.E. Velazco, Department of Management, Worcester Polytechnic Institute, Worcester, MA 01609, USA.

0377-2217/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 3 ) E 0 0 4 5 - Y

46

E.E. Velazco / Air traffic management

• tl o

q~,-/'/lO o •

i¢

o/

UNITEO STATES (MalnlQnd)

Figure 1. Primaryairports comply with the separation standards, they will delay the incoming aircraft by requesting a reduction in the cruising speed or a change of flight path. Since an incoming aircraft that has been delayed cannot, generally, be expected to depart on time, the delay phenomenon is propagated throughout the NAS network. A multiplicative delay effect occurs when the incoming delayed aircraft feed into other flights. Operations Research techniques, Queueing Theory in particular, can be used to analyze the NAS system. The NAS can be modeled as an open queueing network. An open queueing network is composed of a finite number of stations or queues, each ready to perform service. Customers (aircraft) seeking service (landing or take-off) enter the network at some station (airport) according to some arrival process and go from station to station according to some routing discipline receiving service at each station they visit, and eventually leave the system. An aircraft is assumed to leave the system after it has landed and cleared the runway at its final destination. Figures 1 and 2 depict the primary airports in the continental United States and a network representation of a part of the system respectively. The complexity of this system led the FAA, using private contractors, to develop many computer models to analyze capacity and delay problems at specific airports; e.g., ADSIM and RDSIM. More sophisticated simulation models were also developed. For instance, SIMMOD is designed to study the effects of changes in local or regional air traffic conditions; i.e., at the ARTCC authority level. However, a mean to address system wide air traffic management was missing. In the fall of 1987, the FAA, through

__..~

I-q-r~ II -~ ~ tif~)-.-

Figure 2. A queueingnetwork

47

E.E. Velazco / Air traffic management

Simulation Model Situation ~ , ~ ~ ~ ~ 4 ~ Under Study

Deci sion IsMade

l QueueingModel 1 Figure 3. Current decisionmakingmethodology its contract with the MITRE Corporation, has undertaken the development of a long-term analysis capability called NASPAC (National Airspace System Performance Analysis Capability) to study the system wide performance of the NAS. NASPAC is currently based on two models: a discrete event simulation that models the movement of individual aircraft through a nationwide network of airports, navigation fixes, routes, and sectors; and an analytical queueing model. The simulation model can estimate the effects of proposed actions on system performance, not only at the airport where the action is taken, but also at other airports that interact with this airport, and on the national system as a whole. The simulation model could be used to analyze system's performance at the ARTCC authority level, however, its results are not as accurate as, for example, SIMMOD's. The queueing network model is a very simple model that resembles a Jackson type network. Despite its simplicity, the queueing model offers a few important advantages over the simulation model: • It generally requires less time to prepare the inputs necessary to evaluate a new scenario. • The execution time of the model is significantly shorter. Because of these factors, the turnaround time needed to analyze a specific question is much shorter for the queueing model than for the simulation model. The preceding observation was realized earlier in Frolow et al. (1988). Frolow and his colleagues maintain that the decision maker would have both simulation and queueing models outputs before making a decision about air traffic flow in the NAS. Figure 3 depicts the current decision-making methodology. Readers should be awared that due to mathematical complexity, exact results are only known for Jackson and BCMP networks; cf. Jackson (1963) and Baskett et al. (1975). If the network is large (several hundreds of queues), calculation of the normalization constant can pose problems and can even be impossible. Consequently, much effort has been directed toward developing approximation methods; cf. Gelenbe et al. (1980), Stewart (1978), Chandy et al. (1975) and Marie (1978). There are some queueing network software packages in the market; Snowdon and Ammons (1988) offer an excellent survey. These packages are nonsimulated-based computational tools that have proven worthy in the analysis and design of computer, telecommunications and manufacturing systems. Suffice to say, these packages still have many restrictive assumptions that render them ineffective in the analysis of the NAS.

2. A new methodology

A slight modification in the current methodology is proposed in this work, and it mainly involves the queueing-model block of Figure 3. Rather than having a Jackson-like queueing network model to analyze and synthesize the NAS, a more realistic model is proposed. The new model calls for an iterative aggregation/disaggregation analytical analysis of the NAS. Figure 4 shows the proposed methodology.

E.E. Velazco / Air traffic management

48

i Simulation Model Situation Under Study

~

,

~

]

~

[

Decision IsMade

I [ SystemProblemJ / [ \.... [--1 Figure 4. Proposeddecisionmakingmethodology The aggregation/disaggregation (a/d) analysis methodology can be expressed mathematically as follows. Consider a problem, P, under study on a complex system A, and assume the solution, x *, can be obtained via a solution-procedure operator f. Then, fA(P) = x *

(1)

where fA indicates that the solution-procedure operator has been taken with respect to the complex system A. The a / d methodology proposes to study a smaller, less complex system B instead of A that retains much of the information relative to the solution of P and yields a solution y, i.e. fa(P) = y .

(2)

The smaller system B is defined as

B

(3)

where "~1 is an intermediate-step solution in an iterative solution process of P on A. This iterative process converges to x*, ~ also denotes the 'current' state of A during the iterative solution process. Equations (2) and (3) make up the aggregation step. Once y has been obtained, we can update the state of A to ~7r Hopefully,

(4) Then, the iterative solution process is start anew from "~1" This step is called the disaggregation step. Starting the iterative process at £1 yields a new intermediate-step solution £2 that is used subsequently in (3) and (2). This a / d process is stopped when [ x * - ~ , [-~ 0. The iterative a / d method serves to purposes: (i) It is a theoretical tool to analyze complex systems in terms of separate subsystems. (ii) It is a computational method to speed up the iterative process on A. In the NAS context, B represents a particular airport's airspace, and A the entire NAS or even a large sector of it. Iterative a / d methods are also found in Social Sciences (Economics, Markov decision processes), in Queueing Theory (Markov chains) and in Applied Mathematics (projection method, defect correction and algebraic multigrid methods). Some of these applications are found in Marie (1982) and Lavenberg (1983).

E.E. Velazco / Air traffic management

49

3. The airport's capacity At any particular airport we distinguish two flows of aircraft, namely: the in-bound and out-bound flights. After the successful transfer of an aircraft between the controller at the corresponding ARTCC and the controller at the terminal area, the en-route aircraft contacts the control tower while approaching to obtain a landing clearance, see Figure 5. If the clearance is denied, as generally happens at major airports, the aircraft is asked to fly up to a certain flight level and remain there, following a constant pattern called the holding pattern, awaiting further clearance. When several aircraft are awaiting, they form what is called a holding stack; each aircraft flies at different flight levels following approximately the same enlarged elliptic pattern at 1000 feet intervals, cf. Field (1985). Meanwhile on the ground, outbound aircraft are being subject to a similar phenomenon at the airport's taxiways. At due time, the airport's air traffic controller will give a landing or departing clearance that will vary the aircraft order in the holding stack or waiting line at the taxiway respectively. Independently, both queues are FIFO. The preceding paragraphs give a general idea of what takes place at the terminal area level. The merging of flight paths and the splitting of takeoffs into one, or a few approach or departure paths respectively contributes to aircraft delay as well. As expected, statistics reveal that most delays occur at the airport's control zone, cf. Newell (1979). During the management and control stage of the NAS, traffic engineers need to know the minimum and maximum levels of air traffic airports can or are encountering. This information is of great importance when planning, say, next day's flight schedule, or redistributing traffic flow. Design engineers can also make use of this information when attempting to establish a reasonable airport capacity for a major expansion or new construction. Since the air traffic is not constant, working with an average value instead of the maximum and minimum could result in underestimation of the NAS's performance. Thus, we choose to analyze the capacity of an airport at the two ends of the operational spectrum, at peak and off-peak periods. Since the capacity of an airport is highly influenced by its runway capacity, we choose to model the runway utilization. For simplicity, the following assumptions are made, cf. Koopman (1972) and Bookbinder (1986). During peak periods on a single runway, controllers will alternate between landings and takeoffs to increase throughput. In contrast during off-peak periods, they will assign to landings a non-preemptive priority over takeoffs. These assumptions allow us to see a clear cut difference between the heavy and light traffic behavior of the runway operations; another reason to conduct a separate

/

Figure 5. An airport's airspace

E.E. Velazco / Air traffic management

50

analysis. In Sections 4 and 5, we concentrate on the application of the solution-procedure operator f as in (3) and (2). Section 6 gives a summary and the conclusions.

4. Off-peak period analysis Hereafter, the following assumptions are made. First, arrivals are assumed to take place according to a Poisson process. Service times have a general distribution. The former assumption is widely supported in the literature; e.g., Newell (1979), Koopman (1972), Newell (1982) and Bookbinder (1986). The latter one represents an improvement over the work reported in Koopman (1972). For modeling purposes, any aircraft requesting service will be referred to as a customer while the runway is the server. Priority 1 customers are arriving aircraft and priority 2 customers are departing aircraft. Our main goal here is to estimate the average waiting time of an aircraft wishing to either land in or takeoff from a particular airport. The service discipline is FIFO, the system's capacity is infinite. Because we are analyzing the light traffic operation, the probability of our capacity assumption not being true is negligible. Let Pi be the probability that an arrival has priority i (i = 1 if landing, i = 2 if takeoff). We also have two Poisson distributions describing the arrival process with parameters A1 and A2 respectively. The random variables describing the duration of the service times have pdf H 1 and /-/2, with ~1 and ~b2 as their Laplace-Stieltjes transforms. The first moments of H 1 and H 2 are a 1 and a 2 respectively. If we were to disregard priorities in this model, notice that for a Poisson process of density A,

H(x) = Prob[a service time < x ] =plHl(x) +pzHz(x)

(5)

and a = E[service time]

=plal

+P2a2

where Pl

= A1/A,

P2 = A2/A and A =

A 1 + A 2.

Define n,,(i) to be the waiting time of the n-th customer if it has priority i, and ~:n(i) to be the number of customers of priority i in the system immediately after the n-th departure (i --- 1, 2). We are interested in finding the stationary distribution of {nn(i)}. One can show that if Aa < 1, there is a unique stationary distribution, whereas if Aa > 1, there is no stationary distribution. Moreover if Aia i < 1, the limiting distribution lim~__,®P{n,,(i) < x} exists, for both i = 1 and 2, is independent of the initial distribution and agrees with the stationary distribution. If Aia i >_ 1, then lim

n ---, w

P{nn(i ) _
(6)

irrespective of the initial distribution, cf. Takfics (1962). Finally, the stationary distribution of {nn(i)} will be denoted by Wi*(x), its Laplace-Stieltjes transform

by oo

~2" : f e -sx dW/*(x), R e ( s ) > 0,

(7)

"0

and its r-moment by oo

Wi:g(r)= fO xr d W / * ( x ) .

(8)

E.E. Velazco /Air traffic management

51

4.1. Priority-1 customers In this subsection, we will obtain the formulas for the generating function of the number of aircraft waiting to land and the Laplace-Stieltjes transform for their waiting time during steady state. Proposition 1. / f 21a 1 < 1

lim P { ~ . ( 1 ) r/--~ oo

+ ~(2)

and 2a < 1, a stationary process exists, and = O} = 1 - 2 a = P0.

(9)

The above proposition is true because the zero state can be dealt with irrespective of the priorities. Thus, the value of P is readily obtained from an M / G / 1 model, cf. Takfics (1962) and Kleinrock (1975). Now, for a stationary process let v,(z)

(lO)

=

be the appropriate probability generating function. Theorem 1. f f

21a 1 < 1 and 2 a < 1 then,

u , ( z ) : ~1(21 -- 21Z) / t

UI(O)

UI(Z) Z

+P0~~'1 ) + I//2(21 - 2 1 Z ) { U I ( O ) - P o - ~ } .

(11)

Proof. See the A p p e n d s . Proposition 2.

UI(Z) has the form shown below: ~bl(AI-AIZ)[~Z-I+aAI(1-Z)]+Z~2(AI-A1Z)[I--~]

Ul( Z ) =

hi

Z - ¢q(,h - 2 1 Z )

(12)

In equation (11), we know that UI(1)= 1 and P0 = 1 - 2a but we do not know what /]1(0) is. Now, multiply (11) by Z and isolate UI(Z). If we let then Z = 1 and use L'Hopital's rule along with ¢q(0) = ~2(0)= 1, ~ ( 0 ) = - a 1, ~b~(0)= - a z, we obtain /.]1(0) : 1 - 21a.

(13)

Equation (12) is readily obtained from (13). To calculate the waiting time of priority-1 customers (landing aircraft), we define 1 if the n-th departing customer has priority 1, Xn(1) = 0 otherwise,

G~(Z) = E[Ze~"] xn(1) = 1] . e[x(1 ) = 1]. Proposition 3.

G I ( Z ) = ~/1(A1 - AlZ) [(UI(Z) - UI(0 ) ) / / z --t-PoA,/A].

(14)

Equation (14) follows (11). Notice that

U,(Z) :E[Z'~"[Xn(1 ) = 1]P[Xn(1 ) : 11 + E[Z'~'>IXn(1):01 "P[Xn(1 ) = 0 I.

(15)

Let Z = 1 in (10), and G,(1) = 1 - u,(0)

+PoXl/X =All/2.

(16)

E.E. Velazco / Air traffic management

52

Then, G I ( Z ) / G I ( 1 ) is the generating function of the number of customers present in the system immediately after the departure of a customer of priority 1. Since the number of customers with priority 1 present in the system immediately after the departure of a customer of priority 1 is equal to the number of customers of priority 1 arriving during the waiting time and the service time of the departing customer, then the following is true:

G I ( Z ) / G , ( 1 ) = f2?(A, - A,Z)~b,(A, -- A~Z). Proposition 4. f f S = A l

/2?(8) =

- •1 Z in

G,[1 - S / A l l

(17)

(13),

for Re(S) <0,

(18)

Evidently {r/n*(1)} has a unique limiting and stationary distribution if and only if {~:n(1)} has one. From (14) we can show the validity of Theorem 2.

Theorem 2. I f Ala I < 1 and A2a 2 < 1, then O? =

P0 + A2(1 - ~ 2 ( S ) ) / S (1 - O , ( s ) ) / S

for R e ( S ) > 0.

(19)

Theorem 3. f f Ala 1 < 1 but Aa >__1 (i.e., A2a 2 >_ 1), (1 - h l a l ) O~'(S) =

1 - ~b2(S) ] a2S

l_A,(l_Ol(S))/S

(20)

Proof. The above conditions make the system behave as if it were an ordinary M IG l1 queueing system with extra work. The main characteristic of this system is that whenever the queue is emptied, the server turns around and starts doing some extra work. When he finishes, he would start servicing any new customers if there are any in the queue; otherwise he would turn around once again to do more extra work. In our case, because A2a 2 > 1, the takeoffs represent this extra work. The Laplace-Stieltjes transform of the waiting time of a priority 1 customer is equal to the Pollaczek-Khinchin formula for the waiting time in an M IG I1 times the transform of the distribution of the residual life. We have to take the residual life into account because landings have only a nonpreemptive priority over take-offs. 4.2. Priority-2 customers In this subsection, the takeoffs are analyzed. Formulas to estimate the number of aircraft waiting to takeoff and their associate delay during steady state conditions are provided. The following theorem is stated without proof, cf. Kleinrock (1975).

Theorem 4. f f A2a 2 _> 1 then {r/*(2)} has no stationary distribution. Let U2(Z) -- E[Z¢~.2~]. Then, the moments of the distribution function of the number of customers of priority 2 in the system immediately after the n-th departure can be calculated as follows.

Proposition 5. drU2(Z)dz r z=,

dre(Z)dz r z=,

drU'(Z)dz r z=,

(21)

E.E. Velazco / Air traffic management

53

where P(Z) =

(1 - Aa)O(A - AZ)(1 - Z ) ~b(A - A Z ) - Z

(22)

is the Pollaczek-Khinchin formula for the number of customers in an MIGI1 system, ~O(s)=.pltpl(S) + p2~b2(s), and UI(Z) is given by (8). Let Dl(x) = P[(a busy period of a customer of priority 1)
A,(s) = f e -sx d D l ( x ).

"0

Then, the following proposition is true for an M IG 11 system, cf. Takfics (1962).

Proposition 6. In an M IG I1 system, the Laplace-Stieltjes transform for the duration of the busy period must satisfy the following functional equation: '~l(S)

= I//l[ S "1- /~1 -- A I ' ~ I ( S ) ]

(23)

"

Define for an ordinary MIGI1 queueing system of only customers of priority 2, W2(X) to be the distribution of the waiting time, and O2(S) its Laplace-Stieltjes transform.

Theorem 5. If ha < 1, there is a stationary distribution for waiting time of priority-2 customers and its transform satisfies the following functional equation: ~.(]~ ( S )

: ,..Q2[ S + h I -- A I ' Y I ( S ) ]

(24)

.

Proof. The first part of the theorem follows from results of MIG[1, cf. Takfics (1962) and Kleinrock (1975). However, the second part is not trivial. We can observe that the waiting time for a customer of priority 2 is equal to the total service time of all customers before him irrespective of priority plus the length of all the busy periods generated by the number of customers of priority 1 that arrive during the total service time of all the customers before him. Let us assume that this number is j. Then the stationary waiting time distribution is W?(x)

=

• f~ e -xly

[j=0 0

J!

dWz(y ) ®D~I)(x)

(25)

where Di(X) represents the j-th iterated convolution of D1(X) and ® is the convolution operator. Since ¢¢

O~'(s) = f0 e-SX d W * ( x ) ,

(26)

(20) follows from (21).

5. Peak period analysis During peak periods, the terminal area's control zone is heavily crowded with landing and departing aircraft. At this point, the flow control center has probably already contacted all major airports having departing aircraft bound to this oversaturated terminal area, requesting that those flights be delayed. Although this measure positively affects the airport, it also causes a multiplicative-delay effect in the

E.E. Velazco / Air traffic management

54

system. Because an aircraft that arrives late cannot be expected to depart on time, and given a delayed aircraft may represent several delayed flights. As of June 21, 1987, there was an average of 1267 delayed flights daily out of 17500 airline takeoffs and landings, or a little more than seven percent. However, bear in mind that FAA officials defined delays as flights that are more than 15 minutes late because they are held up air traffic controllers due to weather or too many planes in the sky. Thus, if your airplane is held up because of mechanical troubles, unreasonable scheduling, slow refueling, slow maintenance, late loading or unloading of baggage and passengers or because the flight crew fails to show up, it is not a delay in the eyes of the Federal Aviation Administration (FAA). In a heavy traffic situation, we recall controllers prefer to alternate between landings and takeoffs in a single runway to maximize throughput. So if the last operation in the runway involved a landing, then the third held aircraft in the stack probably would have to wait three times the average service time of a departing aircraft plus twice the average service time of a landing before receiving a landing clearance from the controller. In general, the k-th held aircraft in the stack would have an expected waiting time of EW(Lk) =ka + ( k - 1)b

(27)

where a is the average time for a takeoff, b is the average time for a landing and W~(k) stands for the waiting time of a landing aircraft. Let EWE(i) and EWT(i) be the expected waiting times for landing (L) and takeoffs (T) if the last operation was of type i; i = 1 if it was a landing, i = 2 if it was a takeoff. Furthermore, let M and N be the thresholds for the number of landing and departing aircraft at the terminal's control zone. The following propositions are true.

Proposition 7.

I f M = N = n, then

EWL(1 ) = ½(a(n + 1) + b ( n -

1)),

(28)

EWL(2 ) = EWT(1 ) = ½((a + b)(n - 1)),

(29)

EWT(2 ) = ½(a(n - 1) + b ( n + 1)).

(30)

Proposition 8.

I f M ~ N a n d M > N , then

[ N ( N + l) + 2 ( M - N ) N ] a EWE(1 ) =

+M(M-

1)b (31)

2M

[ N( N - 1) + 2( M - N ) N ] a + M( M - 1)b EWE(2 ) =

(32)

2M

EWT(1 ) = ½((a + b ) ( g -

1)),

EWT(2 ) = ½ ( ( N + 1)b + ( N -

(33) 1)a).

(34)

Proposition 9. If M--g N and M < N, then EWL(1) = ½((M+ 1)a + ( M - 1)b),

(35)

EWL(2) = ½ ( ( a + b ) ( M - 1 ) ) ,

(36)

EWL(1) EWL(2)

g(g-

1)a + [ M ( M -

1) + 2 ( N - M ) M ] b

2N

N(N-

1)a + [ M ( M + 1) + 2 ( N - M ) M ] b 2N

(37)

(38)

Propositions 7-9 are the direct result of the alternating sequence of landings and takeoffs. Recall that a is the average time for a takeoff and b the average time for a landing.

E.E. Velazco / Air traffic management

55

6. A numerical example Consider the data in Table 1. T h e departure processes are assumed to have a Poisson distribution with means a 1 = 1 for landings and a 2 = ~0 for take-offs. Maximum queue size is 8 for landings and take-offs; i.e., M = N = 8. Table 2 shows the periods of heavy traffic. Between midnight and 6:00 am the air traffic is very light; cf. NeweU (1981). For simplicity, we will characterize a peak and off-peak period, as the average of all the corresponding one-hour periods. Table 3 depicts this information. (Periods beginning at the eight, nine, ten, and fifteen through twenty-one hour were designated as peak periods.) Then, applying (24)-(26) where n = 8, yields EWL(1) = 10.2 minutes, EWL(2) = EW-r(1) = 8.9 minutes, EWT(2) = 10.1 minutes.

Table 1 Relevant data H o u r beginning

Intensity of arrivals

Intensity of take-offs

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

18 18 19 11 8 13 17 16 36 33 38 33 32 26 25 12 12

15 24 30 25 21 10 9 13 10 18 31 30 44 30 31 27 7

Table 2 Peak and off-peak periods Hour

Traffic intensity

Total traffic intensity

beginning

Landings (Pl)

Take-offs (Pz )

P = Pl + Pz

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

0.40 0.40 0.42 0.24 0.18 0.29 0.38 0.36 0.80 0.73 0.84 0.73 0.71 0.58 0.56 0.27 0.27

0.30 0.48 0.60 0.50 0.42 0.20 0.18 0.26 0.20 0.36 0.62 0.60 0.88 0.60 0.62 0.54 0.14

0.70 0.88 1.02 0.74 0.60 0.49 0.56 0.62 1.00 1.09 1.46 1.33 1.59 1.18 1.18 0.81 0.41

56

E.E. Velazco ~Air traffic management

Table 3 Typical peak and off-peakperiod characteristics A1 •~2 P1 02 p

Peak period

Off-peakperiod

27.1 27.3 0.60 0.55 1.15

13.7 14.6 0.30 0.30 0.60

During off-peak periods, we should expect that take-offs and landings receive clearance after 0.50 and 0.43 min. respectively. Subsequently, these results could be used as a baseline to analyze the air traffic flow in a sector or region of the NAS by studying the air traffic flow in neighboring airspaces (the a / d step.)

7. Summary and conclusions This paper is concerned with the air traffic management of NAS. Section 1 gives an introduction to the problem and a brief overview of NASPAC, probably the most serious effort to this problem. Section 2 discusses a new methodology involving an iterative aggregation/disaggregation analysis. The airport's capacity, an important component in defining system B in the a / d method is treated in Section 3. Sections 4 and 5 are concerned with the theoretical underpinnings of applying the solution-procedure operator. The iterative a / d method appears to be an appealing methodology to deal with the air traffic management of NAS. Advances in numerical optimization and super computing could allow for real-time applications, see Dembo et al. (1989). Additional work is needed to answer questions such as, in theory, proofs of convergence and estimates of the improvement over the convergence of the a / d step, and in practice, the definition of interactive partitions and strategies (how to aggregate and when?).

Acknowledgements I am grateful for the help of Bruce Jeckel from the Hopkins International Airport, Cleveland, and Homer Stamper from the ARTCC at Oberlin, Ohio. This work was partially funded by the Research & Development Council at WPI.

Appendix Proof of Theorem 1. In order to prove Theorem 1, we must consider four different cases: n 1 (x)

He(X)

H1(x)

~:.(1) > 0

~:.(1) = 0 ~:.(2) > 0

~:n(1) 0 ~:.(2) = 0

(a)

(b)

(c)

=

Pl

H2(x)

~n(1) 0 ~:.(2) = 0 =

(d)

P2

E.E. Velazco / Air traffic management

57

Figure (a) shows the case when the (n + 1)-st service time corresponds to a customer of priority 1 who was formerly at the head of the queue after the end of the n-th service period. Figure (b) depicts the case when at the end of the n-th service period there are no customers of priority 1 but there are some of 2. Thus, the duration of the next service period is given by H2(X). Figures (c) and (d) show the cases when the system empties after the n-th service period. Case (c) depicts the event of having a customer of priority 1 being serviced during the (n + 1)-st service period, which could only have occurred if the next arrival was of priority 1. Finally, figure (d) shows the case when the next arriving customer (to the empty system) is of priority 2. Since the process is stationary, studying the n-th or (n + 1)-st departure epoch should yield the same distribution. Therefore, U , ( Z ) = E[ Z ~ " ] . If the (n + 1)-st service consists of serving a customer with priority 1, two cases must be considered, (a) and (c). For case (a), the number of customers of priority 1 left in the system immediately after the (n + 1)-st departure is equal to the number of arrivals of priority-1 customers during the service of this (n + 1)-st customer, ~1(A1 - AIZ), and the number of customers of priority-1 customers left behind by the n-th departing customer assuming that ~n(1)> 0, i.e., we have to subtract the probability that G(1) = 0, ( G ( Z ) - G ( O ) ) / Z . Finally, because of independence, the generating function for the number of customers of type 1 immediately after the service time of a priority-1 customer in this case is ¢,,( A, - A , Z ) ( G ( Z ) - G ( O ) ) / Z .

(A.1)

For case (c), the number of priority-1 customers left behind by the (n + 1)-st departure is equal to the number of arrivals of priority- 1 customers during the service of this (n + 1)-st customer, ~b1(A1 - Al Z), times the probability that after the n-th departure the system was empty and the subsequent arrival has priority 1, PoA~/A. Finally then, the generating function for the number of customers of priority 1 immediately after the service time of a priority 1 customer in case (c) is qq( A, - A,Z) .Po A,/A.

(A.2)

Adding (A.1) and (A.2), we obtain the first term of the right-hand side in (12). A similar analysis of Figures (b) and (d) completes the proof.

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