. Consider an open network such that arrivals to node 0 occur from outside in accordance with a Poisson process with rate . After receiving

. Consider an open network such that arrivals to node 0 occur from outside in accordance with a Poisson process with rate À. After receiving service at node 0, the job (customer) may leave the system with probability q or may go one of the k nodes, the probability that it goes to node i being Pii= 1 ,..., k, Zi =, Pi = 1 - q. From node i, i = 1,2 ,..., k, it goes back to node 0. Each of the (k + 1) nodes has an exponential server, the rate at node i being ui, i = 0, 1,2 ,..., k.
Represent the network by a suitable diagram. Write down the routing matrix P and find Q' = (@, @], . .., "}). Assume that the system is at steady state. Show that the queues at the (k + 1) nodes behave as independent M/ M/1 queues. Denoting pi = i = 0,1, 2, ..., k, show that the system state ( no, n1, ..., n}) has probability k p(no, n) ...., nk) =(1-pop.
i=0 Show that the average total response time is given by k Pi E(R) = 1 - Pi i=0 E(B;)
1-XE(B;)
i=0 where E (B;) is the average service-time requirement at node i (Trivedi, 1982).