Fork-Join Processing System. Consider an m-node fork-join network that processes jobs as follows. Jobs arrive every u time items (u is a constant) and each job splits into m tasks, which are...

Fork-Join Processing System. Consider an m-node fork-join network that processes jobs as follows. Jobs arrive every u time items (u is a constant) and each job splits into m tasks, which are simultaneously assigned to the m nodes for processing. The nodes operate independently, and each node processes jobs like a single-server G/M/1 system with independent exponential service times with rate μ. When all the m tasks for a job are finished, the job is complete and exits the system. The network is shown in Figure 1.4 in Chapter 1, where the operating rules were different. Assume the system is empty at time 0. Let X(t)=(X₁(t),...,X_m(t)) denote the numbers of tasks at the m nodes at time t., and find its limiting distribution. Let Wi n denote the time to complete the task at node i for the nth job. Then the sojourn time in the system for the nth job (i.e., the time to process the job) is Wn = max{W1 n,...,W m n }. Show that Wn d → W as n → ∞, and determine the distribution of W (which is a product of exponential distributions). Find the distribution of W when G is an exponential distribution.