1 1.231J/16.781J/ESD.224J Fall 2004 ASSIGNMENT # 1 (Out: September 16, 2004; due: October 5, 2004) Problem 1 (40 points) The following information is given about air traffic at a particular runway: (a) Aircraft are classified into 3 types: heavy (H), large (L), and small (S). (b) Relevant aircraft characteristics are as follows: Aircraft type Approach speed (knots) Mix (%) Runway occupancy time on landing (seconds) H 150 30 60 L 120 50 50 S 90 20 40 (c) The length of the final approach to the runway is 6 n. miles. (d) The minimum separation requirements (in nautical miles) between consecutive landing aircraft on final approach are given by the matrix below (rows indicate the leading aircraft and columns the trailing aircraft): H L S H 4 5 6* L 2.5 2.5 4 S 2.5 2.5 2.5 [*= This particular separation applies only when the leading aircraft is at the runway threshold; in other words, when the leading heavy airplane reaches the runway, the trailing small airplane must be at least 6 n. miles behind it; note that all the other separation requirements in the matrix apply throughout the final approach.] (e) Two landing aircraft cannot occupy the runway simultaneously. (f) The deviation from assigned separations between successive airborne aircraft on final approach is described by the probability density function (pdf) of Figure 1. Time 0 on this Figure corresponds to the assigned arrival time of the second of the two aircraft at some given point. For example, if a landing aircraft B is supposed to be 100 seconds behind another aircraft A at the time when A reaches the runway threshold, then, according to Figure 1, B will actually be between 80 and 120 seconds behind A and the distance will be distributed statistically as a random variable with a triangular pdf. [We2shall assume, as an approximation, that the deviations for successive pairs of aircraft are statistically independent; in other words, if for the pair A-B (“A followed by B”) the aircraft B is, e.g., 12 seconds "early" (i.e. at -12 seconds in Figure 1) this does not have implications for where aircraft C is, relative to B, in the next pair of landing aircraft B-C.] (g) It is desired by Air Traffic Control (ATC) to leave sufficient additional separation ("buffer") between successive landing aircraft so as to have a probability of violating the minimum airborne separation requirements (see 4 above) equal to 0.10. Note that the same size buffer will be used for all aircraft pairs. Fig. 1 f (t) [prob'y density function]T-200+20t(seconds) Part 1 (10 points): Find the runway’s capacity (arrivals per hour) when it is used only for arrivals. [Note that this is a simple application of the model we presented in the class, with the complication that you now have to compute the length of the buffer b on the basis of the information provided in (f) and (g). By contrast, in the lecture notes you were simply told that b=10 seconds.] [Note: In all the remaining parts of this problem assume that there is no additional separation (“buffer”) between arriving aircraft; in other words, now b = 0. Stated differently, we assume that air traffic controllers can always achieve the minimum required separation between landing aircraft, as dictated by the constraints in (c) and (d).] Assume now that during peak departure periods, ATC sends some Type S aircraft to this runway for take-off. Specifically, they try to insert as many type S departures as possible, between consecutive arrivals while observing the following rules: (h) The departures will not in any way affect the arrival rate; in other words, separations between successive arrivals, as shown under (d), will not be increased in any way in order to accommodate departures. (i) A departure inserted between two arrivals can begin its takeoff roll only after the leading arrival has exited the runway and must lift off the runway before the trailing arrival touches down on the runway. Note that the time needed for an arriving aircraft to exit the runway is given by the runway occupancy time on arrival shown under (c). Assume also that the time during which an arriving aircraft occupies the runway,3provides sufficient time for a departing aircraft of Type S to enter the runway and get ready to begin its take-off roll. (In other words, the takeoff roll can begin immediately after the preceding arriving aircraft exits the runway.) (j) The runway occupancy time on takeoff of type S aircraft is 50 seconds. This is the time from beginning of takeoff roll to lifting off the runway. (k) If two or more departures are to be inserted between a pair of arriving aircraft, the minimum separation time between consecutive departures of type S aircraft is 50 seconds. Part 2 (10 points): Compute the number of departures of type S aircraft that can be performed per hour on the runway under the conditions and rules just described. This airport is sometimes forced to use only a single runway during IFR weather periods. Thus, the runway must accommodate both landings and takeoffs during these periods. Assume that the traffic mix (H, L, S) for departing aircraft is identical with the traffic mix for arriving aircraft. The following rules/assumptions apply: (l) The local air traffic controllers use an operations-sequencing strategy of alternating landings and takeoffs on the runway. This means that during periods of continuous demand a landing is always followed by a takeoff, which is then followed by a landing, etc. Thus, when the minimum required time gap between two landing aircraft, i and j, is not sufficient to insert a takeoff, the time gap will be increased by ATC appropriately, so that a takeoff can be inserted. (m) The minimum separation requirement (in seconds) between two departing aircraft is 90 seconds; in other words, a takeoff cannot be initiated within less than 90 seconds of the initiation of the preceding takeoff. [Note: Under our operations sequencing strategy, exactly one arrival always takes place between two departures.] (n) Takeoffs wait next to the threshold of the runway. As soon as a landing aircraft crosses the runway threshold, the next departing aircraft enters the runway and prepares for the takeoff run. It takes 30 seconds for a departing aircraft to enter the runway and set up for take-off. (Note that, in the meanwhile, the arriving aircraft that just landed is