Problem 5: (20 Points) In a bus station M57 and M72 can stop and you can take ei

Question

Problem 5: (20 Points) In a bus station M57 and M72 can stop and you can take either of them to go to your both is Poisson. destination. The arrival headway for M5 is 4 minutes and for M72 is 6 minutes and arrival distribution of a) If you arrive to the station, what is the expected wait time for you to get on any of these bus? b) If the arrival rate of the passenger for bus 57 is 2 passenger per minute, what is average length of c) If each bus can only board 2 passenger at the stop location, and if the arrival rate of the passenger queue of the people waiting to board on M57? for bus 57 is 5 passenger per minute, what is average length of queue of the people waiting to board on M57 after 30 minutes? (assume that at the beginning of the inspection time nobody was in the line)

Explanation / Answer

a) Let us assume that you reach station om time

Now , M57 time is 4 minutes

M72 time is 6 minutes

Assuming you reach on time

Hence , Now wait time for next coming bus is M72

Since , Arrival distribution is Poisson

You will wait for M72 which will come in 2 minutes

Hence wait time is 2 minutes

b)

Assuming No passenger leaving in M72

Since 2 passenger per minute hence

For 3 minutes 6 passengers

At 4th minute M57 arrives

Hence 6 passengers board on M57 and 2 passengers arrives at station for wait.

Hence , this continues resulting in

6 passengers queue

C)

This is simple

Let us take M57 arrives after every 4 minutes

Per minute 5 passengers arrive

And per 4 minute 2 passengers boards

This means 2/4 passengers per minute leave

Hence , for 30 minutes

30( Invoming - Outgoing)

=30(5-0.5)

=135

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