There are three jobs that need to be processed, with the processing time of job i being exponential with rate ui. 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 T, denote the time at which the processing of job i is completed. If the objective is to minimize EIT1 + T2 + T3], which jobs should be initially processed if μ1 < μ2 < μ3?

(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 141 112 μ, μ3 λ. Also, let P(u) be the probability that the last job to finish is either job 1 or job 2, and let P(a) 1- P) be the probability that the last job to finish is job 3

(c) Express E[S] in terms of P(u) and P() Let Pi, j(u) be the value of P() when i and j are the jobs that are initially started

(d) Show that P1,2(a) < Pista)

(e) If μ > λ show that E[M] is minimized when job 3 is one of the jobs that is initially started (f) If μ < λ show that E[M] is minimized when processing is initially started on jobs 1 and 2