class6atc - Too Close for Comfort Geometrical Probability...

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Too Close for Comfort : Geometrical Probability in the Sky 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 λ E (per minute). Likewise, northbound planes would arrive under an independent Poisson process with parameter λ 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
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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*).
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