1. A set of four host computers A', B', C' and D' are interconnected with each other using a packet switched network consisting of an array of four switching nodes A, B, C, D (see Fig. 1 below). The routes chosen by the packets is always the shortest path between the source and destination computer. The service rates are independent on the number of packets in their respective queues and the arrival rates from the exterior of the packet switched network are independent of the number of packets in the system.

(a) How many classes of customers are defined by the pathways and topology of Fig.1 ?

(b) A decision is made to utilize the FIFO service discipline with similar mean service rates for all classes of customers. Using the information provided below, where r_i is the utilization of rate of the server of queue i, evaluate the probability that k = (1, 4, 5, 3, 2, 1,3,1) where k_i is the number of customers in queue i.

i	ri
1	0.356
2	0.789
3	0.146
4	0.459
5	0.682
6	0.5
7	0.419
8	0.872

Table 1: Utilization rates

2. Now consider a case where the number of servers at the switching nodes is always sufficient for there always to be at least one free server for the incoming packets, what now will the probability that k = (1, 4, 5, 3, 2, 1,3,1)?