16. There are three jobs that need to be processed, with the processing time of job i being exponential with rate . There are two processors available, so processing on two of the jobs can immediately start, with processing on the final job to start when one of the initial ones is finished.

(a) Let Image denote the time at which the processing of job i is completed. If the objective is to minimize

, which jobs should be initially processed if ?

(b) Let M, called the makespan, be the time until all three jobs have been processed. With S equal to the time that there is only a single processor working, show that For the rest of this problem, suppose that . Also, let be the probability that the last job to finish is either job 1 or job 2, and let be the probability that the last job to finish is job 3.

(c) Express in terms of and .
Let be the value of when i and j are the jobs that are initially started.

(d) Show that .

(e) If show that is minimized when job 3 is one of the jobs that is initially started.

(f) If show that is minimized when processing is initially started on jobs 1 and 2.