Poisson process & renewal processes

**Last_Singularity** · May 14th 2009, 06:00 PM

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 $t_1, t_2, . . .$ with mean 1. Let $T_k = t_i + ... + t_k$ be the time the k-th car passes. A turtle needs two minutes to cross the road. Let $N = min\{i : t_i > 2\}$

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

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

Thanks a bunch - I really appreciate it!

**cl85** · May 15th 2009, 07:02 AM

Just to get you started, what is the distribution of N? Then use total expectation: $E[T] = E[E[T|N]]$

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