1 1.231J/16.781J/ESD.224J Fall 2007 ASSIGNMENT # 1 (Out: September 13; due: September 27) Problem 1: A Capacity Envelope (32 points) In this problem you will compute the capacity envelope (Section 10.6 of the textbook) of a single runway under a specified set of somewhat simplified conditions. The following information is given about air traffic at that runway: (a) Aircraft can be classified into 3 types: heavy (H), large/medium (L), and small (S). (b) Some relevant aircraft characteristics are as shown in Table 1 below: Aircraft type Approach speed (knots) Mix (%) Runway occupancy time on landing (seconds) H 150 30 70 L 135 40 60 S 105 30 50 Table 1 (c) The length of the final approach to the runway is 6 n. miles. (d) The minimum separation requirements (in nautical miles) between successive landing aircraft on final approach are given by Table 2 below (rows indicate the leading aircraft and columns the following aircraft): H L S H 4 5 6* L 2.5 2.5 4* S 2.5 2.5 2.5 [*= These separations apply only when the leading aircraft is at the runway threshold; all the other separations apply throughout the final approach] Table 2 (e) A “buffer time” of 15 seconds (see Section 10.5 of de Neufville and Odoni textbook) is added to all the minimum acceptable separation times between successive landings to account for uncertainty in spacing aircraft.2(f) The minimum separation requirements (in seconds) between successive departing aircraft are given by Table 3 below (rows indicate the leading aircraft and columns the trailing aircraft): H L S H 90 120 120 L 90 90 90 S 60 60 60 Table 3 As in Section 10.6, we shall approximate the capacity envelope as a polygon defined by four points 1 – 4. Part 1: Find Point 1 (“all arrivals” point): assuming this runway is used for arrivals only, what is its (maximum throughput) capacity. Part 2: Find Point 4 (“all departures” point): assuming this runway is used for departures only, what is its (maximum throughput) capacity. [No buffer times are added for departures.] Part 3: Find Point 2 (“free departures” point) under a set of assumptions described below. [Note that, under the simplification introduced in (g) below, the “free departures” consist of S aircraft only.] Please also note that the number of S aircraft given a “free departure” cannot exceed 30% of the “all arrivals” capacity (why?). (g) During peak departure periods, ATC sends some Type S aircraft (only) 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) – (k). (h) The departures will not in any way affect the arrival rate; in other words, separations between successive arrivals, as shown in Table 2, will not be increased in any way in order to accommodate departures. (Assume also that the 15-second buffer time – see (e) above – continues to be added to the separation between each pair of consecutive landing aircraft. (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 (b). Assume also that the time during which an arriving aircraft occupies the runway, provides 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.3 (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 60 seconds, as shown in Table 3. Part 4: Find Point 3 (“alternating arrivals and departures” point) under a set of assumptions described below. Note that, in this case, the runway, by design, handles an equal number of landings and of takeoffs in each hour. (l) The local air traffic controllers use an operations-sequencing strategy of alternating landings and takeoffs on the runway, i.e., 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. (By ‘time gap’ we mean here the minimum separation time plus the 15-second buffer.) (m) 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 45 seconds for a departing aircraft of any type to enter the runway and set up for take-off. (Note that, in the meanwhile, the arriving aircraft that just landed is moving down the runway toward a runway exit.) (n) A takeoff run cannot begin until the preceding landing aircraft has cleared the runway. (o) Once a takeoff run begins, the runway occupancy time for all departing aircraft (time from the beginning of the takeoff run to clearing the runway) is equal to 60 seconds. [Making runway occupancy times on takeoff equal for all departing aircraft is a reasonable approximation that simplifies calculations.] (p) The separation between two consecutive takeoffs must be at least 90 seconds in all cases (i.e., disregard for now the separation matrix shown in Table 3). (q) A landing aircraft is not allowed to cross the runway threshold unless the runway is clear of departing aircraft, i.e., the preceding departure has lifted off. (Note that this is the only "departure-followed-by-arrival" separation requirement.) [A helpful example regarding Part 4: Consider a pair S-H of consecutive arrivals (“arrival of S followed by arrival of H”): The separation between the arrivals in this case will be 75 seconds [=(2.5/150)x3600 +15]. But to insert a departure, we need 110 seconds (50 for the arriving S aircraft to leave the runway plus 60 for the takeoff run of the departing aircraft. Thus, the required separation between the S and