My question is about an example involving Poisson process inter arrival times.

First some background: A Poisson process refers to the number of arrivals at time $t$ denoted by $N_t$. The $n^{th}$ arrival time is denoted by $T_n$, $n$ a natural number.

Theorem: $$P(N_t=k) = \frac{e^{-\lambda t}(\lambda t)^k}{k!}, k=0,1,..., t \geq 0, \lambda \geq 0$$

Proposition: For any natural number $n$ and $t \geq 0$, $$P(T_n \leq t) = 1 - \sum_{k=0}^{n-1}\frac{e^{-\lambda t}(\lambda t)^k}{k!}$$ This is called the Erlang-n distribution.

Example, where I have my questions:

Taking a certain fixed point on a highway, let $U_1, U_2, ...$ be the successive inter arrival times of the vehicles at this point. [Does this mean that $U_n = T_{n+1} - T_n?$] Suppose $U_1, U_2, ...$ are independent and identically distributed random variables with the distribution $$P(U_k \leq t) = 1 - e^{-\lambda t} - \lambda t e^{-\lambda t}, t \geq 0$$

We are interested in the distribution $M_t$ of vehicles crossing this fixed point during $[0, t]$. First we observe that the distribution of $U_k$ given is the Erlang-2 distribution. Thus, we may think of each $U_k$ as being the sum of two inter arrival times in a Poisson process with rate $\lambda$. That is, the times the vehicles cross the given point may be thought of as $U_1 = T_2, U_1 + U_2 = T_4, U_1 + U_2 + U_3 = T_6, ...$, where $T_1, T_2, ...$ are the times of arrivals in a Poisson process $N_t$ with rate $\lambda$. [I don't understand this. It seems like if $U_1$ is an inter arrival time then $U_1 < T_2$. I'm confused about what is meant by inter arrival time.] Then the number $M_t(\omega)$ of vehicles crossing is equal to 6 if and only if the Poisson process $N$ has, for that realization $\omega$, 12 or 13 arrivals in $[0, t]$ [Why would it be 12 or 13?]. So for any $t \geq 0$, $$P(M_t = k) = P(N_t = 2k) + P(N_t = 2k+1) = \frac{e^{-\lambda t}(\lambda t)^{2k}}{2k!} + \frac{e^{-\lambda t}(\lambda t)^{2k+1}}{(2k+1)!}$$

Please clarify, especially with an explanation of what is meant by inter arrival times. Please provide specific examples of arrival times and inter arrival times and why they have that relationship. And please explain $M_t$ with an example.


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