reservation mechanism with parameter r. be free on a link before it accepts an alternatively routed call. Then Consider a single link in such a network and let, n, be the number the process, n, modelling the utilization of a link is again a birth and of circuits in use. The process, n, can be modelled by a birth and death process with rates as given by Equations 5.25 and 5.26 but death process with transition rates as in Equations 5.25 and 5.26, now with the overflow rate (o) given by Equation 5.33 and the loss where o is the rate of overflow traffic. probability (L) given by Equation 5.34. (5.25) (5.33) q(n,n-1)n n1,.,C (5.34) (5.26) The independent blocking approximation allows o to be ex- Note that these expressions reduce to those in Equations 5.27 and pressed in terms by Equation 5.27. 5.28 with the choice M = 1. The repeated substitution method can again be used to determine a solution for B1 and B2 given some o = 2AB,(1-B2) (5.27) initial starting values. Figure 5.6 shows solutions to these fixed point equations when r The interpretation of Equation 5.27 is that blocking conditions of = 0 (i.e. no trunk reservation is applied) with M=1,5 and a range of different links are approximately independent, and that overflow values of a and C. Note that when r=0 we have that B1=B2=B, traffic arriving at a link is approximately Poisson. A call blocked say. Observe the possibility of multiple solutions for B for large 5/8 Teletraffic theory C=120 C=1000 C=infinite C=120 C=1000 C=infinite Figure 5.6 Blocking probability B with M = 1, 5 enough and for a narrow range of the ratio N/C and that these effects are magnified in the case of larger M. The upper and lower solutions 5.7). The optimal choices of r are also shown in Figure 5.8 together correspond to stable fixed point solutions while the middle solution with the limiting case where M tends to infinity. corresponds to an unstable fixed point. 5.3.2 Simulations also exhibit this bistable behaviour where the net- Implied costs work has two quasi-stable states. A high