Calculation of Steady-State Probabilities for Content of Buffer with Correlated Inputs

Mathematical models for the behavior of a switching node t h a t receives data from a (large) number of terminals over low-speed access lines have been considered by Gopinath and Morrison, 1,2 and some particular examples have been investigated by Fraser, Gopinath, and Morrison. 3 In this paper, we consider one of the models and give the details of an alternate procedure, which was alluded to by Gopinath and Morrison, 1 for calculating certain steady-state probabilities. We first describe the model which we will consider. It is assumed that the data are received at the switching node in the form of packets of fixed size. As the packets arrive, they are placed in a buffer, which is a firstin-first-out queue. The buffer processes packets at a uniform rate, provided that it is not empty. In an actual computer network, the buffer capacity is finite, and a packet is lost if the buffer is full when it attempts to enter it. In our mathematical model, it is assumed that the buffer has 3097 infinite capacity, so t h a t no overflow is possible, and we are interested in calculating the steady-state probability that the buffer content (i.e., the number of packets in the buffer) exceeds the proposed capacity of the buffer. We let the time that it takes for the buffer to process a packet through the node be our unit of time. We suppose that £ n is the number of packets which enter the buffer in the time interval (n,n + 1]. If b n denotes the buffer content at time n, then the buffer content at time n + 1 is given by the equation bn+1 = (bn-l)+ + tn, +