A packet-switched communication network is viewed as an open queueing network with each link modeled as an M/M/1 queueing system. Packets arrive at various nodes as independent Poisson processes, and the length of every packet is assumed to be exponentially distributed. Let lij be the rate of packets to be transmitted over link (j), and let Hij be the transmission rate (packets/s) of link (ij). (a) Show that the average delay of packets (measured from arrival at some node to departure at some other node) is given by E[S] 1 all (ij) Hij - dij where A is the total arrival rate (packets/s) into the network. (b) Consider the 5-node 7-link network shown in Fig. 8.31, where the transmission rate of all links is it = 3 packets/s. Packets arrive at nodes 1 and 2 with rates n = 1 packet/s and r2 = 3 packets/s respectively, and have to be transmitted to nodes 3,4,5. The following routing probabilities are used: P12 = P14 = 1/2 P23 = 3/5, P24 = P25 = 1/5. Find the average delay of packets in the network, and all the link utilizations. (C) Suppose that all packets arriving at node 1 are destined for node 5. Find the average delay of these packets. A packet-switched communication network is viewed as an open queueing network with each link modeled as an M/M/1 queueing system. Packets arrive at various nodes as independent Poisson processes, and the length of every packet is assumed to be exponentially distributed. Let lij be the rate of packets to be transmitted over link (j), and let Hij be the transmission rate (packets/s) of link (ij). (a) Show that the average delay of packets (measured from arrival at some node to departure at some other node) is given by E[S] 1 all (ij) Hij - dij where A is the total arrival rate (packets/s) into the network. (b) Consider the 5-node 7-link network shown in Fig. 8.31, where the transmission rate of all links is it = 3 packets/s. Packets arrive at nodes 1 and 2 with rates n = 1 packet/s and r2 = 3 packets/s respectively, and have to be transmitted to nodes 3,4,5. The following routing probabilities are used: P12 = P14 = 1/2 P23 = 3/5, P24 = P25 = 1/5. Find the average delay of packets in the network, and all the link utilizations. (C) Suppose that all packets arriving at node 1 are destined for node 5. Find the average delay of these packets