Asymptotic Analysis of a Storage Allocation Model with Finite Capacity: Joint Distribution

We consider a storage allocation model with a finite number of storage spaces. There are m primary spaces and R secondary spaces. All of them are numbered and ranked. Customers arrive according to a Poisson process and occupy a space for an exponentially distributed time period, and a new arrival takes the lowest ranked available space. We let N-1 and N-2 denote the numbers of occupied primary and secondary spaces and study the joint distribution Prob[N-1 = k, N-2 = r] in the steady state. The joint process (N-1, N-2) behaves as a random walk in a lattice rectangle. We study the problem asymptotically as the Poisson arrival rate lambda becomes large, and the storage capacities m and R are scaled to be commensurably large. We use a singular perturbation analysis to approximate the forward Kolmogorov equation(s) satisfied by the joint distribution.