ies service at a rate which depends on the total Table 3.4 Queuing network transition rates number of customers present 4; (nj) taken to be zero if there are no customers present. Each customer within the queue receives a Transition part of this effort according to the service allocation vector Arrivals y(k,nj),k nj The amounts of service a customer requires at Departures Qg(ng)y(k,ng) each queue are independent of one another and distributed exponen- tial with unit mean. Some examples of how this vector might be set, Moving between queues y(e,ni)qi(ni)o(k,nh+1 along with the service rate, to obtain various kinds of server, are as follows: departure of a customer on route r from position k at its last queue L'(C). The state of the queue immediately after a customer 1. ./M/1 queue with first come first served scheduling as in Equ- following route r moves from queue i position e to queue h position ation 3.60. 2. ./M/m queue with first come first served scheduling as in k is M'ie.hk (C). Denote by j the first queue on route r and g the last Equation 3.61 and Equation 3.62. queue on that route, then these transitions take place at the rates 3. Processor sharing as in Equation 3.63. given in Table 3.4. The equilibrium queue distribution is of product form, as in = () otherwise (3.60) Equations 3.64 to 3.66, where G. is a normalisation constant, a is the total traffic arriving at node j, and is given by Equation 3.67, and plat=mqv(j.n)=`.jsnsm (3.61) a; is the number of visits that customers on route r make to node j and (with a little abuse of notation) "r(k) is the arrival rate of the p(n)=mqn>m class of the customer who is currently in position k. Queuing networks 3/13 (3.64) In order to make use of this result we must find suitable transition rates q'(C,D). This is done by constructing the time reversed que- uing network which has transition rates given by Table 3.5. (3.65) In the reversed network customers start from their last queue and follow their route backwards, and also the roles of Y and d are swapped. Thu