Switch dimension (N) Switching Systems: Architecture and Performances The same conclusion can be obtained by noting that there exists a value of p maximizing Equation 7.12. Deriving with respect to p and substituting in Equation 7.12 the abscissa of the maximum, it is found that the maximum efficiency is attained for = 1/N and it is (7.15) SO that, for large values of N the throughput tends to 1/e=36.8% as shown in Figure 7.20. The information regarding the saturated throughput is quite important since in a func- tional system the offered load has to be smaller, and in general much smaller, than the saturated throughput. As a matter of fact, the queues of a realistic system are not infinite and an offered load greater than the saturated throughput will cause the average number of stored packets to increase continuously till it generates system congestion due to the inability of the switch fabric to accept new input packets. In the example of Figure 7.20, fixing the bus bit rate at 20 Gbit/s and consequently N to 16, assuming that the realistic traffic that the switch fabric can manage is half the saturated throughput and that the traffic coming from different inputs is statistically similar and independent, it is obtained that the maximum offered load that the switch can manage with a minimal packet loss probability is on the order of magnitude of 445,000 packets per second per input port that is an average capacity of 216 Mbit/s per port, even if the line rate on the input line cards is 1.2 Gbit/s and the bus bit rate is as high as 20 Gbit/s. 7.3.2.2 Switch Fabric Based on an Unslotted Random Access Bus In the case of the unslotted random access, the bus access policy is the same as in the prec- edent case, but no frame exists on the bus and every access point transmits the packets on the shared media in an asynchronous way. We will maintain the hypothesis that the packets are all of the same length and are transmitted on input lines with the same speed. The hypothesis of heavy load is also maintained. Let us fi