Too Close for Comfort tude of 35,000 feet at junction J(Figure 1). In the absence of air-traffic control, the times at which eastbound planes would arrive at the junction would reflect a Poisson process with parameter E(per minute). Likewise, northbound planes would arrive under an independent Poisson process with parameter AN. All planes move at a speed of 600 miles per hour along their routes Figurel: Eastbound and Northbound Air Routes Cross at J The Federal Aviation Administration thinks it dangerous if two planes cruising at the same altitude get within 5 miles of one another(in which case they are said to conflict). The idea is that, if a conflict arises, the planes are traveling so fast that they could collide if one of them deviates from its course. with the faa standard in mind we calculate the probabilities of three interesting events E: the chance that an eastbound plane that has just reached J is in conflict at that moment with a northbound plar N the chance that a northbound plane that has just reached j is at that moment in conflict with an eastbound plane
Too Close for Comfort Suppose that two aerial routes--one Eastbound and one Northbound--cross at an altitude of 35,000 feet at junction J (Figure 1). In the absence of air-traffic control, the times at which eastbound planes would arrive at the junction would reflect a Poisson process with parameter E (per minute). Likewise, northbound planes would arrive under an independent Poisson process with parameter N. All planes move at a speed of 600 miles per hour along their routes. J Figure1:Eastbound and Northbound Air Routes Cross at J The Federal Aviation Administration thinks it dangerous if two planes cruising at the same altitude get within 5 miles of one another (in which case they are said to conflict). The idea is that, if a conflict arises, the planes are traveling so fast that they could collide if one of them deviates from its course. With the FAA standard in mind, we calculate the probabilities of three interesting events: E: the chance that an eastbound plane that has just reached J is in conflict at that moment with a northbound plane N: the chance that a northbound plane that has just reached J is at that moment in conflict with an eastbound plane
EE: the chance that a given eastbound plane that passes through J is at any time conflict with a northbound plane that passes through J P(EI To find P(E), we note that the conflict occurs if, at the time the eastbound plane reaches J, there is a northbound plane within five miles of J. If E* is the complement of E, then E* requires that there be no northbound plane within five miles of the junction. It is easier to find P(es)than P(e), so we will do so and then invoke the rule p(e)=1-p(es) We arent told anything about planes that are not at the junction, so how can we determine whether an aircraft is within five miles of J? We can exploit the clue that planes travel at 600 miles per hour(which works out to ten miles per minute, or one mile every six seconds ). Suppose that a plane is north of j and within five miles of it. Then the plane must have passed through J within the last thirty seconds. Similarly, if a northbound plane is still south of J but less than five miles away, it will reach J within the next thirty seconds Thus, if an eastbound plane reaches J at time t, there will be a conflict at t if a northbound plane passes through J between t-0.5(in minutes)and t +0.5. And there will be no conflict if no northbound plane reaches t over the interval(t-0.5, t+0.5) e can therefore write P(E*)= P(no northbound arrivals at J over(t-0.5, t+0. 5))=exp(-AN) and thus that P(E)=1-exp(-AN) P(N:
EE: the chance that a given eastbound plane that passes through J is at any time in conflict with a northbound plane that passes through J. P(E) To find P(E), we note that the conflict occurs if, at the time the eastbound plane reaches J, there is a northbound plane within five miles of J. If E* is the complement of E, then E* requires that there be no northbound plane within five miles of the junction. It is easier to find P(E*) than P(E), so we will do so and then invoke the rule P(E) = 1 - P(E*). We aren’t told anything about planes that are not at the junction, so how can we determine whether an aircraft is within five miles of J? We can exploit the clue that planes travel at 600 miles per hour (which works out to ten miles per minute, or one mile every six seconds). Suppose that a plane is north of J and within five miles of it. Then, the plane must have passed through J within the last thirty seconds. Similarly, if a northbound plane is still south of J but less than five miles away, it will reach J within the next thirty seconds. Thus, if an eastbound plane reaches J at time t, there will be a conflict at t if a northbound plane passes through J between t-0.5 (in minutes) and t + 0.5. And there will be no conflict if no northbound plane reaches t over the interval (t-0.5,t+0.5). We can therefore write: P(E*) = P(no northbound arrivals at J over (t-0.5,t+0.5)) = exp(-N) and thus that P(E) = 1 - exp(-N) P(N):
The reasoning is the same as for P(E), so we can write PO xp(- It might seem surprising that P(E)and P(n)differ, given that each conflict we are considering involves one eastbound and one northbound plane. If, however, 2N> NE (for example), then more northbound planes reach J per hour than do eastbound planes Thus, if equal numbers of northbound and eastbound planes face conflicts, the percentage of conflicts is lower for northbound planes passing through J than eastbound ones. And P(E)and P(N)reflect these percentages P(EE: The reader might be wondering: what is the difference between P(EE)and P(e)? The definitions of the events differ a bit: P(E)requires that a conflict be in progress when an eastbound plane reaches J; P(EE)requires some east/north conflict, but allows for the possibility that the conflict is already over(or has not yet begun) when the eastbound plane passes through J. Still, does this distinction really matter? Well, yes. Suppose that, when an eastbound plane arrives at J, there is a northbound plane six miles north of J. The two planes are not then in conflict. But consider the situation twelve seconds earlier, when the northbound plane was four miles north of J and the eastbound plane two miles west of it. The Pythagorean theorem reminds us that the two planes were v20=4.5 miles apart at that time(i. e. that they were in conflict, even though they no longer are) hrough a combination of geometry and calculus, one can reach a potent conclusion EE occurs if, at the moment the eastbound plane reaches J, there is a northbound aircraft within 10/2=7. 1 miles of the junction either way. (If the plane is between 5 and 7.1 miles away, then the conflict has already ended or not yet started. Because planes travel
The reasoning is the same as for P(E), so we can write: P(N) = 1 - exp(-N) It might seem surprising that P(E) and P(N) differ, given that each conflict we are considering involves one eastbound and one northbound plane. If, however, N > E (for example), then more northbound planes reach J per hour than do eastbound planes. Thus, if equal numbers of northbound and eastbound planes face conflicts, the percentage of conflicts is lower for northbound planes passing through J than eastbound ones. And P(E) and P(N) reflect these percentages. P(EE): The reader might be wondering: what is the difference between P(EE) and P(E)? The definitions of the events differ a bit: P(E) requires that a conflict be in progress when an eastbound plane reaches J; P(EE) requires some east/north conflict, but allows for the possibility that the conflict is already over (or has not yet begun) when the eastbound plane passes through J. Still, does this distinction really matter? Well, yes. Suppose that, when an eastbound plane arrives at J, there is a northbound plane six miles north of J. The two planes are not then in conflict. But consider the situation twelve seconds earlier, when the northbound plane was four miles north of J and the eastbound plane two miles west of it. The Pythagorean theorem reminds us that the two planes were 20 = 4.5 miles apart at that time (i.e. that they were in conflict, even though they no longer are). Through a combination of geometry and calculus, one can reach a potent conclusion: EE occurs if, at the moment the eastbound plane reaches J, there is a northbound aircraft within 10/2 7.1 miles of the junction either way. (If the plane is between 5 and 7.1 miles away, then the conflict has already ended or not yet started.) Because planes travel
ten miles per minute, 7. 1 miles corresponds to. 71 minutes of flight. Therefore, the eastbound aircraft reaching J at time t will face a conflict if a northbound plane passes hrough J over the interval(t-.71, t+ 71). In consequence, P(EE)=1-P(EE*)=1-exp(-1.42入E) This exercise assumed the absence of air-traffic control. and random arrivals at j under Poisson processes. In reality, aircraft arrival times at junctions would never be left to chance alone. These calculations suggest the frequency at which potentially hazardous situations would arise based on activity levels in the air-traffic system(and thus, the magnitude of the task facing of the task facing the controllers). The agnificence with which the controllers perform this task is suggested by a statistic:over he 1990s, 5 billion passengers travelled in commercial jet aircraft in the United States The number killed in midair collisions was zero
ten miles per minute, 7.1 miles corresponds to .71 minutes of flight. Therefore, the eastbound aircraft reaching J at time t will face a conflict if a northbound plane passes through J over the interval (t-.71,t+.71). In consequence, P(EE) = 1 - P(EE*) = 1 - exp(-1.42E) This exercise assumed the absence of air-traffic control, and random arrivals at J under Poisson processes. In reality, aircraft arrival times at junctions would never be left to chance alone. These calculations suggest the frequency at which potentially hazardous situations would arise based on activity levels in the air-traffic system (and thus, the magnitude of the task facing of the task facing the controllers). The magnificence with which the controllers perform this task is suggested by a statistic: over the 1990’s, 5 billion passengers travelled in commercial jet aircraft in the United States. The number killed in midair collisions was zero
