Analysing multiprogramming queues by generating functions

The generating function approach for analysing queueing systems has a longstanding tradition. One of the highlights is the seminal paper by Kingman on the shortest queue problem, where the author shows that the equilibrium probabilities P_{m,n} of the queue lengths can be written as an infinite sum of products of powers. The same approach is used by Hofri to prove that for a multiprogramming model with two queues the boundary probability P_{0,n} can be expressed as an infinite sum of powers. The present paper shows that the latter representation does not always hold, which implies that the multiprogramming problem is essentially more complicated than the shortest queue problem. However, it appears that the generating function approach is very well suited to show when such a representation is available and when not.