Poisson/exponential process with step-wise decreasing rate

[NotEuler]

So I've been stuck on trying to figure out the following problem. This is basically just for my own interest, not for a school problem or anything like that. It came out of trying to get my head around Poisson and exponential distributions.

Imagine you have a process where you have a fixed time T during which events happen. Now if the events all happened at a fixed rate λ, then the number of events that have taken place by time T would be Poisson distributed.

But what if the rate parameter decreases after each event? That is, we start with λ1, but after the first event, the rate parameter decreases to λ2<λ1. And after the second event, it decreases to λ3<λ2, and so on.

What, if anything can we now say about the average number of events that take place in the fixed time interval T? Or about the variance, or other features of the distribution of this number of events?

If I understand correctly, a non-homogeneous Poisson distribution would describe the process if the rate parameter decreased as a function of time. But in my example, it decreases as a function of the number of events. And the time intervals between the events are random.

 

[NotEuler]

A brief addition: It seems that it's quite easy to find the average time until n events have taken place.
Each time interval is exponentially distributed, with mean duration 1/λ1, 1/λ2 and so on.

Because of additivity of the mean, the mean until n events have taken place is then simply sum of the means: 1/λ1+1/λ2+...+1/λn.
Similarly, variance in the time until n events have taken place is 1/λ1²+1/λ2²+...+1/λn².

Does this seem correct?

Still, my original intention was to figure out the mean number of events if the period of time is considered fixed. I'm curious if this can be done in a simple way.