I’m a graduate student in ops research, but my research is in the optimization side of things more so than the stochastic processes. I need a suggestion about how to solve this inventory-motivated problem, let me explain it briefly.

Suppose products are sold from a warehouse according to a Poisson process with rate lambda. But, an empirically determined percentage of all sales, say pi, will be returned to the warehouse. We’ll assume for simplicity that we can instantly resell the products and that they’re non-perishable (i.e. a customer would view it to be as good as a non-used item).

Here’s the catch: the amount of time it takes for a return to come back is exponentially distributed with parameter beta. So, even if I viewed the potential returns leaving the warehouse as being a Poisson process with rate pi*lambda, which I’ve been trying to do, we still have to account for the fact that their ‘return time’ is also stochastic.

How do I determine the return distribution in the long run? That is, from the warehouse’s perspective, with what sort of interarrival distribution should I expect to see the returns arrive? My gut tells me it’s exponential with parameter pi*lambda, since in the long run, we’re just `shifting’ the Poisson process by a mean of 1/beta. But, my gut also tells me that since we’re adding so much variance to the interarrivals, this might not be the case.

Any ideas on what kind of models exist to explain this sort of situation? Again, I apologize for any naivete, but I don’t work in stochastic processes.