Tij = Tj Ti = in Software

Generation Quick Response Code in Software Tij = Tj Ti =

(12-9)

It should be carefully noted that the only difference between Eqs (12-8) and (12-9) is in the first term of the equation, where Vi and Vj are interchanged

Example Problem 12-3 This problem will solve Example Problem 12-2 using the error-free analytical equations developed above It is necessary to determine the arrival capacity of the runway in an error-free context where aircraft separations are maintained in the airspace along the common approach path between the entry gate and the arrival threshold There are four possible interarrival cases, a leading A and a trailing A, a leading B and a trailing B, a leading A and a trailing B, and a leading B and a trailing A These cases are governed by Eqs (12-7) and (12-8) Equation (12-7) gives the minimum time between arrivals at the runway threshold when the leading aircraft is approaching the runway at an approach speed less than or equal to the approach speed of the trailing aircraft

If we have a leading A and a trailing A, or a leading B and a trailing B, both aircraft are traveling at the same speed Therefore, Eq (12-7) applies and we have for a type A following a type A Tij = and for a type B following a type B Tij = 3(3600) = 120 s 90 4(3600) = 120 s 120

When the leading aircraft is type B and the trailing aircraft is type A, Eq (12-7) also applies and we have Tij = 3(3600) = 90 s 120

When the leading aircraft is approaching the runway at an approach speed greater than the approach speed of the trailing aircraft, the minimum time between arrivals at the runway threshold is given by Eq (12-8) This is the case when a type B follows a type A aircraft Therefore, we have Tij = 1 5(3600) 1 (3600) = 220 s + 7 120 90 120

The ordered queue consists of the pairs of arrivals B-A, A-A, A-B, and B-A Therefore, we have the following interarrival matrix and probability matrix based upon the actual queue of arriving aircraft given: [Tij ] : A Trailing A 120 B 220 Leading B 90 120

From Eq (12-5), since in the error-free case [Tij] is equal to [Mij], we have that the expected value of the interarrival time is E( Tij) = 025(120) + 050(90) + 025(220) + 000(120) = 130 s From Eq (12-6) for the arrival capacity of the runway we then have Ca = 3600 = 28 operations per hour 130

This agrees exactly with the results of this problem done by the time-space diagram method in Example Problem 12-2

The above models represent the situation of a perfect system with no errors To take care of position errors, a buffer time is added to the minimum separation time to ensure that the minimum interarrival separations are maintained The size of the buffer depends upon the probability of violation of the minimum separation rules which is acceptable Figure 12-10 shows the position of the trailing aircraft as it approaches the runway threshold In the top portion of this illustration, the trailing aircraft is sequenced so as its mean position is exactly determined by the minimum separation between the leading and trailing aircraft However, if the aircraft position is a random variable there is an equal probability that it can be either ahead or behind schedule Naturally if it is ahead of schedule the minimum separation criterion will be violated If the position error is normally distributed, then the shaded area of the bell-shaped curve would correspond to a probability of violation of the minimum separation rule of 50 percent Therefore, in order to lower this probability of violation, the aircraft may be scheduled to arrive at this position later by building in a buffer to the minimum separation criterion as shown in the bottom portion of the illustration In this case, only when the aircraft is so far ahead