. Consider Jackson's open-network model with a single server at each of the k nodes. Let T and S denote, respectively, the total time spent in the system and the total service time received by a unit. Assume that the system is in equilibrium and that the network is such that a unit can never visit any node more than once. If u; is the rate of service for the server (exponential) at node i and p; = x;/u ;, i = 1,2, .. ., k, then show that A

E(T) = A-1 2 Pi 1 - P i == 1 E(S)= A-1 [(1+p))

i=1 and A

p

,2 E(W) = E(T- S)= A -!

-

p?

where A = >", À ¡, À, being the rate of arrival from outside to node i.

Show that the LST of T is the transform of a mixture of exponential distributions and find the same (Lemoine, 1977).