Question

Problem 2. Customers arrive at an ATM as a homogeneous Poisson process with rate 2. If the ATM is free, they use it for a fixed time T. If it is occupied they leave and don't come back. Let (t) be the counting process for how many people have started using the ATM. Explain why this is a renewal process and write down an expression for 2(t|H) and f(Si = T|H5_).

Answer 1

Solo N(4): Counting Process for how people have Thiven process in the problem is Te senewal procent because in this problem interonival times for the homogeneous Poisson Process are independently of identically distributed with an arbituary distribution. It doesn't follow a particular distribution. It may take any suitable arbiteary alist". Si = The arrival time of the ith event i iz1 14. Nezi if and only if Sust fance P(S. st) = P(NA) zi)

By using independent & increment assumptions, we have P(A. S p < t tdt) _ P {NH=1-1, a event in (t, t tdt)} + oldt) PEN(A)= -1} P { 1 event in - 6#*#$#) +old which gields, upon dividing by dit) & then letting it approach o, that delived expression for