. M/ M/c queue with servers' vacations (Levy and Yechiali, 1976).

Consider an M/ M/c system in which a server proceeds on vacation when he has no unit to serve (the length of time he is on vacation being given by an exponential random variable with parameter 0) and in which the server proceeds on another vacation if he finds the queue empty on return. Suppose that the system is in steady state.

Find the joint distribution of the number of busy servers B and the number of customers N in the system Pr(N= k,₿ =r), r = 0,1,2 ,..., k ≥ r.

Show that the average number of busy servers is A/u (which is the same as the average number of busy servers in an M/ M/ c queue).

Show that the number of customers N in the system when all servers are on vacation has a geometric distribution given by co Pr(N= k) =

% + ce () + c)

Show that for c = 2, the average number of customers in the system is given by

@x[X(1 - z1) +0]
p L = +
p= 1 -7 where and x = [x(1 - 2}) + 20z]
(2 +u+0) - (x+u+0)2 - 42/11/2 22 (For queues with vacation, see Section 8.3.)