How can I simulate this Poisson process ?
We model the instants of arrival of a bus at a given stop with the Poisson process of intensity $\lambda$. And an instant T at which an individual reaches this stop. We want to calculate the average time between the previous bus, and the one that the individual will take.
We know that the cumulative probability distribution function (cdf) for a Poisson process is given by $F(t)=1-e^{-\lambda t}$.
Get a random number generated between 0 and 1 - let this be $P$.
Then $P=1-e^{-\lambda t}$
$e^{-\lambda t}=1-P$
$-\lambda t=\ln (1-P)$
$t=-\frac{\ln (1-P)}{\lambda }$
This value $t$ is the time until the next bus arrives.
Repeat.
Start your simulation at t=0. No buses have yet arrived.
Generate your first random number $P_1$ and find $t_1=-\frac{\ln (1-P)}{\lambda }$.
Generate your second random number $P_2$ and find $t_2=-\frac{\ln (1-P)}{\lambda }+t_1$.
Continue generating your random numbers $P_i$ and find $t_i=-\frac{\ln (1-P_i)}{\lambda }+t_{i-1}$.
From what you say, you have a fixed time $T$ in mind at which the person will arrive.
Continue until $t_n > T$. The waiting time is $T-t_n$.
Run the simulation again to find a large number of waiting times and find their means.
Alternatively you could choose for the passenger to arrive at a random point. In my simulation of this, I let 1000 buses arrive. I stored this set of arrival times. I then picked at random an arrival time for a passenger using a random number generator to give $T$ such that $0<T<t_{1000}$. I then worked out the waiting time.
This was easier to simulate because I only had to create one list of bus arrivals, but I can see that you might not believe the outcome of this.