Storage allocation under processor sharing and infinite server models

We consider storage allocation models, which have m primary holding spaces and infinitely many secondary ones. All of the spaces are numbered and ordered. An arriving customer takes the lowest available space. We define the traffic intensity rho to be lambda0/mu where lambda0 is the customers' arrival rate and mu is the service rate. We also define N1 to be the number of occupied primary spaces and N2 to be the number of occupied secondary spaces. Then we define S to be the set of the indices of the occupied spaces, and the "wasted spaces" W are defined as the difference between the largest index of the occupied spaces (Max S) and the total number of occupied spaces (|S| = N1 + N2).;First we study an M/M/infinity queue model where all the customers are served at the same rate mu, i.e. there are infinitely many servers. We study the probability distribution of the wasted spaces asymptotically for rho → infinity. We also give some numerical results, and the tail behavior for rho = O(1).;For a processor sharing model, which has only one processor servicing the stored items (customers), we study the joint probability distribution of the numbers of occupied primary and secondary spaces. For 0 < rho < 1, we obtain the exact solutions for m = 1 and m = 2. For arbitrary m we study the problem in three asymptotic limits: (1) rho ↑ 1 with m fixed, (2) m → infinity with a fixed rho < 1, and (3) rho ↑ 1, m → infinity with m(1 - rho) = O(1).