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One would not ordinarily challenge these assumptions, because the diffusion equation generally tends to smear irregularities. 8) along the surfaces but these conditions applied to each of two intersecting surfaces generates some non-analytic behavior of f at the edges. We will have some specific illustrations of this later, but one can see where in the above arguments the difficulties originate. (t),j#k J these other Dj(t) are not also near boundaries. If Dk(t) were near Dk+l(t) + c k + l and also near Dk_l(t) simultaneously, then server k would be interrupted both because of the lack of storage and lack of customers.

We could also allow the initial state to be close to anyone boundary so that all boundary terms vanish except those associated with the one boundary of interest. Thus we can investigate the meaning of each boundary term individually. (t) J is near a boundary. Dk(t) to be near Dk+ l + c k+ l ' but no other The consequence of this is that the kth server might be interrupted for lack of storage in ck + l This, in itself, will have no immediate effect upon the behavior of any server except the kth.

F de- to to The above scribes the transition probabilities from (Yo' ••• , Yn) at time (x O' ••• , xn ) at time t which, along with the Markov property, gives a complete description of the stochastic behavior of the system. Our goal is to obtain some explicit solutions for relatively simple special cases or at least some properties of solutions, not to settle convergence questions. If one must resort to numerical methods to solve the diffusion equation, one might as well have solved the original queueing problem by numerical methods.