In a time-shared computer system M terminals share a central processor. Let µ be the processing rate at the CPU, with the processing time having an exponential distribution. If a terminal is free at...

In a time-shared computer system M terminals share a central processor. Let µ be the processing rate at the CPU, with the processing time having an exponential distribution. If a terminal is free at time t, the probability that it will initiate a job in the infinitesimal interval (t, t + ?t] is ??t + o(?t) and it will continue to be free at t + ?t has the probability 1 - [??t + o(?t)]. (a) Let {pn} be the probability distribution of the number of busy terminals as t ? 8. Determine pn, n = 0, 1, 2,...,M. (b) Show that in the long run, the arrival rate at the CPU is given by M?/1 + ?W where W is the mean response time ( = mean waiting time of a job arriving at the terminal.) (c) Equating the arrival rate with the departure rate from the processor show that the mean response time can be obtained as M/µ(1 - p0) = 1/?