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I would appreciate a hint on this problem:

A pedestrian wishes to cross a single lane of fast-moving traffic. Suppose the number of vehicles that have passed by time $t$ is a Poisson process of rate $\lambda$ and suppose it takes time $a$ to walk across the lane. Assuming the pedestrian can foresee correctly the times at which vehicles will pass by, how long on average does it take to cross over safely?

Thoughts: I know that I have to find E($T$) where $T$ is the time taken to cross over. I also know that if $J_1$ is the time the first vehicle arrives then I need to find a connection between E(T) and $\{J_1>a\}$ and $\{J_1<a\}$. I am stuck here and don't know how to proceed. Any help will be appreciated.

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    $\begingroup$ so you need to find the first 'break' in traffic of length greater $a$? Use the distribution of the distance between two consecutive points. $\endgroup$
    – swmo
    Oct 23, 2014 at 8:27
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    $\begingroup$ Exploit the homogeneity of the process. With each vehicle one of two things can happen: either a time gap of at least $a$ occurs, in which case the length of time to cross equals the current wait plus $a$, or else the gap--call it $X$--is smaller than $a$. In the latter case it's the same as if the whole process has restarted. Thus, the expected waiting time is increased in this case by the expectation of $X$ conditional on $X\lt a$ (which you need to compute in terms of $\lambda$). Thus you will obtain a recursive formula for $\mathbb{E}(T)$ (which is easily solved). $\endgroup$
    – whuber
    Oct 23, 2014 at 17:44

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As @whuber said, we have (calling $T$ the wating time and $J_1$ the first cas that arrives) $$E(T)= E(E(T|J_1)) = 0 P(J_1>a) + \int_0^a (u + E(T)) f_{J_1}(u) du.$$ Solving, one should get $$E(T)= \frac{e^{\lambda a}(1-e^{-\lambda a})}{\lambda}- a .$$

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