Consider a k-node Jackson type network with the modification that each node i has s servers rather than one server. Each server at i has an exponentially distributed service time with rate μi. The exogenous input rate to node i is ρi = λ0Q0: and each output from i is switched to j with probability Qij and switched out of the system with probability Qi0 (as in the text). Let λi,1 ≤ i ≤ k, be the solution, for given λ0, to k λj = ) λiQij, i=0 1 ≤ j ≤ k, and assume that λi <>sμi; 1 ≤ i ≤ k. Show that the steady-state probability of state m is k Pr{m} = n pi(mi), i=1 where pi(mi) is the probability of state mi in an (M, M, s) queue. Hint: Simply extend the argument in the text to the multiple server case.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.