# Poisson process & renewal processes

• May 14th 2009, 06:00 PM
Last_Singularity
Poisson process & renewal processes
Question: "Traffic on Snyder Hill Road is a Poisson process with rate 1 car per minute. That is, the times between cars are independent exponentials \$\displaystyle t_1, t_2, . . .\$ with mean 1. Let \$\displaystyle T_k = t_i + ... + t_k\$ be the time the k-th car passes. A turtle needs two minutes to cross the road. Let \$\displaystyle N = min\{i : t_i > 2\}\$

(a) Find the expected value of \$\displaystyle T_{N-1}\$, which is the time until turtle starts to cross, and the expected value of \$\displaystyle T_N\$, the time the next car passes

(b) Consider a renewal reward process with \$\displaystyle r_i = 1\$ if \$\displaystyle t_i > 2\$ and \$\displaystyle r_i = 0\$ otherwise. At what rate are rewards earned?"

Thanks a bunch - I really appreciate it!
• May 15th 2009, 07:02 AM
cl85
Just to get you started, what is the distribution of N? Then use total expectation: \$\displaystyle E[T] = E[E[T|N]]\$