How to model count data with decay

I’m trying to understand how I might model count data where there’s diminishing marginal utility and a stochastic process.

So, let’s say we’re modeling the number of “useful intelligence tips” given to us by Russian agents dependent on how many shots of vodka we give them. I want to show that past a certain point, the number of useful tips starts to decay.

The way I’d think of this is by making the rate parameter lambda dynamic by time t.

So at time t = 1 we might say we get rpois(1,10) pieces of intel n. Poisson distributed n pieces of intel, with mean 10.

But let’s say we start to see decay so that by time t = 100 only 2 pieces of intel are returned. We can model time t = 100 with rpois(1,100).

My thinking on this is we build a vector so that lambda is dynamic and varies at each point t:

lambda_t[t] = t * rexp(1, rate = z).

We then just need to solve for z.

for (1:t) Y[t] ~ rpois(1, lambda_t[t]);

Does this approach make sense? I’ve written in pseudo-code using a bit of stan syntax and R syntax as I imagine I’ll mix the two. But I really am curious how count data is generally modeled when the counts vary over t.