# Time dependent Poisson arrival rate in wait-time theory

A typical Poisson process assumes an average rate $\lambda$ over time which is ignorant to previous events. To calculate how many events occurs after some time period could be done by simply calculating $\lambda*t$ in which $t$ is a particular amount of time intervals.

Now I would like to model the amount of arrival of passengers for a plane flight, assuming they will arrive between $t-3$ and $t-0.5$ hours before plain-departure time $t$. If I would take an average rate $\lambda$, on every point in time, the exact same amount of people would likely arrive. However, I believe it is more realistic that people arrive in a more centered way. That is, few people arrive at $t-3$, most people arrive around $t-2$ and again few arrive around $t-0.5$ (normal- or poisson distribution-like).

I approached this as follows. Say, I expect $136$ passengers. Then I calculated the elapsed time $dt_{i=1}$ after $t-3$ and multiplied it by the average $\lambda_{i} = \frac{136}{(t-3)-(t-0.5)}$ at that point in time ($i=1$). This results in, say, 6 passengers arriving. This leaves me with 130 passengers on $i=2$. I then repeat the same step, by calculating the new $\lambda_{i} = \frac{136-6}{(t-3)-(t-0.5)}$ and multiply it by $dt_{i=2} = 2*dt_{i=1}$, etc. ($dt_{i=3} = 3*dt_{i=1}$) etc. etc..

Note: when calculating $\lambda_i$, the divisor is still the entire time period, i.e. $t-3$ to $t-0.5$.

In short, on each iteration, I increment $dt_{i}$ and decrease the total amount of passengers and, in turn, $\lambda_i$. This results in a figure which has the shape I expected and wanted. However, I feel like I should incorporate the probabilities of a Poisson distribution.

Is this, mathematically, a sound solution for modelling the arrivals, or should I indeed take probabilities into account in some manner?

• Do you have any measurements of when people arrive that you could investigate before creating a model? Guessing a model could be okay, but actual measurements of when people arrive could often be better. – mathreadler May 22 '17 at 9:15
• Unfortunately I do not. At the moment, an approximation would suffice. I would, however, like my approximation to be valid and supported by theory/common practice. In wait-time theory, the amount of passengers, in this case, is just the input of the function so it is not covered that much. – Robin Kramer May 22 '17 at 9:17