The distribution of wasted spaces in the M/M/. queue with ranked servers

We consider the M/M/. queue with m primary servers and infinitely many secondary servers. All the servers are numbered and ordered. An arriving customer takes the lowest available server. We define the wasted spaces as the difference between the highest numbered occupied server and the total number of occupied servers. Letting . = .0/. be the ratio of arrival to service rates, we study the probability distribution of the wasted spaces asymptotically for . . .. We also give some numerical results and the tail behavior for . = O(1).