In Unmanned Manufacturing Factory (UMF), we consider a workstation consisting of a machine and several workbenches, each assigned an AGV to perform the parts transporting service. Every workpiece requires multiple processes at the same workbench, and the time for the AGV to transport a part is greater than the time for the machine to assemble it. This operating model is complex, as several workpieces corresponding to several workbenches are in service at the same time. We extract the production system into a Semi-Open Queuing Networks (SOQNs) model. We use four parameters to describe the states of the system and construct the transition rate matrix. We find that the matrix has a particular structure that enables us to solve it with Matrix Geometric Method (MGM). Following a First-Come-First-Served (FCFS) policy, the performance of this production system with unlimited queuing space is evaluated in terms of service intensity, queue length, sojourn time, and throughput. The numerical experiments demonstrate a significant reduction in the sojourn time of the workpiece in the system when the number of workbenches increases to a certain value. Our work can provide important suggestions for designing UMF.