Consider the job sharing computer system illustrated below. Incom- ing jobs arrive from the left in a Poisson stream. Each job, independently of other jobs, requires preprocessing in system 1 with probability Q. Jobs in system 1 are served FCFS and the service times for successive jobs entering system 1 are IID with an exponential distribution of mean 1/μ1. The jobs entering system 2 are also served FCFS and succes- sive service times are IID with an exponential distribution of mean 1/μ2. The service times in the two systems are independent of each other and of the arrival times. Assume that μ1  > λQ and that μ2  > λ. Assume that the combined system is in steady state.

(a) Is the input to system 1 Poisson? Explain.

(b) Are each of the two input processes coming into system 2 Poisson? Explain.

(c) Give the joint steady-state PMF of the number of jobs in the two systems. Explain briefly.

(d) What is the probability that the first job to leave system 1 after time t is the same as the first job that entered the entire system after time t?

(e) What is the probability that the first job to leave system 2 after time t both passed through system 1 and arrived at system 1 after time t?

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.