Consider a system with a single CPU server in which jobs arrive at rate A jobs per unit time. Each job requires average S units of CPU time. Assume that the jobs arrive at random points in time. Moreover, assume that jobs are processed using the FCFS scheduling policy. There are no I/O operations involved in processing a job.

a) What is the CPU utilization, denoted by U, in this system? Express U in terms of A and S.

b) Let the random variable X denote the number of jobs already present in the system when a new job arrives.

(i) What is Probability[X=0] ? This is the probability that a new arriving job finds the system idle.

(ii) What is the Probability[X>0] ? This is the probability that a new arriving job finds the system busy.

c) Let N denote the average number of jobs in this system and let T be the average turnaround time, i.e. the average amount of time a job spends in the system, starting from the instant when it enters the system to the point in time when it departs. (N is also the average value of the random variable X. Given that jobs arrive at random points in time, a newly arriving job would find the average number of jobs in the system equal to N. Also, by Little's Law, N = A T.) In the case when a newly arriving job finds the system busy, let W denote the average (expected) number of jobs it finds already present in the system. What is the relationship between W, N and U?

d) With FCFS job scheduling discipline, an arriving job has to wait for the completion of all of the jobs before it in the system. Ignoring the amount of service a job in the system may have already received, and assuming that each of these jobs would still require S units of service-time, give an expression for the average turnaround time T in terms of S, U and N.

e) Using Little's Law, which says N = A T, and the results from the part (d) above, derive an expression for N in terms of the server utilization U.