Poisson processes Two oneway subway lines the A train line a

Poisson processes: Two one-way subway lines, the A train line and the B train line, intersect at a transfer station. A trains and B trains arrive at the station according to independently operating. The rate of Train A is 3 Trains/hr, the rate of Train B is 6 Trains/hr. We assume that passenger boarding and unboarding occurs almost instantaneously, not unlike true rush-hour conditions in many cities throughout the world! a. What is the probability that the station handles exactly 9 trains during any given hour? b. If an observer counts the number of trains that the station handles each hour, starting at 8:00 A.M. on Tuesday, what is the expected number of hours until he or she will first count exactly 9 trains during an hour that commences on the hour? (e.g., 9: 00 A.M., 10: 00 A.M., 2: 00 P.M.)

Solution

Poission distribution you that so I am not explaining much in this part.

But I will help you to give a method for how this problem can you solve.

mean (a)= 3 t/hr

mean(b) = 6 t/hr

Now exactly 9 train per hour,

P(t=9) = P(a=9)*P(b=0)+P(a=9)*P(b=0)+P(a=8)*P(b=1)+P(a=7)*P(b=2)+P(a=6)*P(b=3)+P(a=5)*P(b=4)+P(a=4)*P(b=5)+P(a=3)*P(b=6)+P(a=2)*P(b=7)+P(a=1)*P(b=8)+P(a=0)*P(b=9)= 0.131

All the probabilities are found by excel with poission distribution formula.

avg time per train for a = 20 min/train

avg time per train for b = 10 min/train

avg time combine t = 20*10/(20+10) = 20/3 min/train = 1/9 hr/train