This paper considers optimal design of queueing networks in which each node consists of a single-server FIFO queue and an infinite-server queue, which is referred to as incubation queue. Upon service completion at a FIFO queue, a job splits (forks) into two parts: the first part is routed to the next node on its route, and the second part is placed in the incubation queue. Routing of the jobs of multiple types is governed by a central decision maker that decides on the routes for each job type and aims to minimize the mean turnaround time of the jobs, i.e., the time spent in the system until service completion at the FIFO queue in the last node, and at all incubation queues on the job's route, which may be viewed as a join operation. We provide explicit results for the turnaround time when all service and inter-arrival time distributions are exponential and invoke the Queueing Network Analyzer when these distributions are general. We then develop a Simulated Annealing approach to find the optimal routing configuration. We apply our approach to determine the optimal routing configuration in a chemistry analyzer line.
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