- #1
Mehmood_Yasir
- 68
- 2
Homework Statement
Pedestrian are arriving to a signal for crossing road with an arrival rate of ##\lambda## arrivals per minute. Whenever the first Pedestrian arrives at signal, he exactly waits for time ##T##, thus we say the first Pedestrian arrives at time ##0##. When time reaches ##T##, light flashes and all other Pedestrians who have arrived within ##T## also cross the road together with the first Pedestrian. Same process repeats.
What is the PDF of wait time of Pedestrians i.e, ##1^{st}##, ##2^{nd}##, ##3^{rd},...,##, pedestrians?
2. Homework Equations
Poisson arrival time density function is ##f_{X_k} (t)=\frac{{(\lambda t)}^{k} e^{-\lambda t}}{k!}=Erlang(k,\lambda; t)## distributed
The Attempt at a Solution
As said, the first pedestrian arrive at time 0 and exactly wait for ##T## time. After time ##T##, all pedestrian arrived at the crossing will cross the street together with the first one.
Now, only see the wait time of each pedestrian to find the density function of wait time.
As the first one wait exactly for time ##T## minutes, thus we can say its wait time density function is just an impulse function shifted at ##T## i.e., ##f_{W1} (t)=\delta (t-T)##.
Since the probability density function of arrival time of the second pedestrian following Poisson process is ##Erlang(1,\lambda)=e^{-\lambda t}## distributed, he waits for ##T-E[X_1|X_1<T]## where ##E[X_1|X_1<T]## is the mean value of the conditional arrival time of the second pedestrian given it is less than ##T##. What is the pdf of wait time of the second pedestrian? Can I say that the arrival time of the second pedestrian follows ##Erlang(1,\lambda; t)=e^{-\lambda t}## distribution, then the wait time pdf should be ##Erlang(1,\lambda; t)=e^{-\lambda (t-(T-E[X_1|X_1<T]))}##. Similarly, what about the third, fourth,... Pedestrians wait time density function?