Hi. I'm having trouble with renewal processes. I want to obtain a long run cost, and to do that, I want to prove that a process is a renewal process, so I can estimate the long run cost by dividing the expected renewal cycle cost and the expected renewal cycle length.

I'm going to describe the process now. Suppose a random time T (for this case, the life of a bulb), and fixed parameters p0 (fixed probability threshold), k0 (fixed time). With this in mind:

A bulb is installed at time 0. A time t1 is defined as the time when the probability of T being less than t1 is over p0. Then a time t2 is defined as t2=t1+E( g(T) | T>t1 ), where "|" is for conditional expectation, and g(T) is a function of T (a function to scale T). The next bulb is installed at time t3=max(t2+k0, T) [so if the bulb fails before t2+k0, the next bulb is installed at t2+k0, but if it fails after t2+k0, then a new bulb is installed when the failure occurs].

I'm not sure if this is a renewal process where I can define a cycle between bulb intallations. I can see that the mechanism to install a new bulb is always the same, but the fact that t1 and t2 depend on T in some way makes me wonder if I can consider them as fixed or not.

Thanks for any help