the flow of packets through the node. It is possible to have a very is given by expression 3.81. good estimate of how the mean sojourn time depends on this Queuing models for multi-access channels 3/15 quantity but not have more detailed information about the distribu- be located at an interior point. This is therefore determined by the tion than this. zeros of the derivative and must be satisfied at only that point. In these circumstances detailed modelling of the sojourn time H has a derivative as in Equation 3.88. distribution appears inappropriate. As will now be shown it is possible to deduce a great deal about performance without having (3.88) to make further assumptions which would be difficult to justify and which may also complicate analysis (Kelly, 1989). These models are capable of considerable generalisation including having priority queues although this extension will not be discussed here. It can now be seen that Equation 3.86 must have just one solution The nodes within the network are labelled 1, 2, J. Each customer since the solution determines the unique maximum of H. class ,r, has N customers in it. For each such class there is a rule Repeated substitution is one of the most numerically efficient which determines how the customers in the class move around ways of solving the above sets of equations. One begins with an within the network. This rule is based on cycles with one being initial value of the for example as in Equation 3.89. The 9,'s completed every time certain sequences of visits are executed. and p;'s may then be solved recursively as in Equations 3.90 and A simple rule of this type is given if a route is determined by a list 3.91. of nodes which have to be visited in order. A cycle is completed once all the nodes in the list have been visited. Note that repeat visits to (3.89) nodes within the list are permitted. A more general rule is given if it is supposed that routing in each class is determined by an irreducible homogeneous Markov chain. Let denote the probability that the cus