. M/ M/1 Queue with Bernoulli feedback.

Consider an M/ M/1 FCFS queue with Bernoulli feedback such that after completion of service, the job may leave the system with probability q or may be fed back into the system with probability p.

(a) Show that the effective average arrival rate to the queue is À/q.

(b) Show that the number of jobs N in the system has a geometric dis-

tribution Pr{N = n} =

1

-

n=0,1,2 ,....

qu qu

-

(c) Show that the time Y from the last input (to the server) to the next feedback has the distribution PM Ry(t) = Pr{Y ≥ t} =-
-e-(1-1) 91-2 H - 1 and that the interinput time I has hyperexponential distribution.

(d) Show that the number of jobs Na left behind by a customer departing from the system has the same distribution as N-that is, Pr[N] = n) = Pr[N = n), n = 0,1,2,

(e) Further, show that the departure process is Poisson with rate À (Burke, 1976).