Im trying to describe some events with the Poisson distribution and overall its a really good fit. There is however a slight error in the model since there is a dead time when every event is about to happen.

Imagine for instance taking a picture with an oldfashioned polaroid camera and the success criteria is whenever i receive the actual picture as output from the camera. The time from when i press the release to the picture is in my hand is dead- there cannot be another event in this time- the camera will simply ignore my action.

What Im trying to describe works similarly

Does anyone know how to do this? I think it has to look something like this but I am not sure.. :

π = P(0|event) + P(1|event) + P(2|event)… P(n|event) + P(D)

P(0) = Poisson(0, λ(t))

P(1) = Poisson(1, λ(t-d1))

P(2) = Poisson(2, λ(t-d2))

.

.

.

P(n) = Poisson(n, λ(t-dn))

t = time left , d = dead time

In order to sum to 1 I add P(D) which Im not sure how to quantify- this is my best bid:

P(D) = P(d1|P(1)) + P(d2|P(2)) + … P(dn|P(n))

P(dn|P(x)) = dn/t

Hope somebody is great at this and willing to help me! Thank you!