Star-Shaped Network. Let X(t) denote a closed Jackson process with ν items for the star-shaped or central-processor network shown in Figure 4.5. Node 1 is the center node and nodes 2,...,m are points...

Star-Shaped Network. Let X(t) denote a closed Jackson process with ν items for the star-shaped or central-processor network shown in Figure 4.5. Node 1 is the center node and nodes 2,...,m are points of the star such that the routing rates λ1j and λi1 are positive. All the other routing rates are 0. Suppose node 1 operates like an M/M/∞ node with service rate μ1, and the other nodes operate like M/M/1 nodes with service rate μi for node i. Find a solution wi of the traffic equations, and then find the stationary distribution for X(t). Is this process reversible? Find the equilibrium probability that there are no items at node 1.