# Analysis and Optimization of the Robotic Mobile Fulfillment Systems Considering Congestion
## 3 Modeling and Solution  
### Semi-Open Loop Queuing Network Model (SOQN)  
According to the system analysis and scheduling rules, the semi-open loop queuing network model of the AGV system can be established, as shown in Figure 3. In this model, the AGV in the warehouse is regarded as a secondary resource when it performs the task of loading and unloading. It is assumed that the AGV is running at a constant speed without considering the acceleration and deceleration process. The assignment strategy of the AGV is the random assignment; that is, any idle AGV can be assigned to any task as soon as possible. It is assumed that the arrival rate of the external task arrival system obeys Poisson distribution. The task service in the system follows the first come, first serve rule (FCFS). In this SOQN model, the number of service nodes is equal to \(A+6\), in which the aisle nodes are parallel service desks. Each aisle belongs to a node, evenly distributed on the left and right sides of the cross-aisle, each with \(A/2\) nodes. So the network has \(A\) aisle nodes. There are four cross-aisle nodes in the system. That is the left cross-aisle node \(A+1\) and right cross-aisle node \(A+1\) for leaving the station correspond to \(CA_{l}^{D}\) and \(CA_{R}^{D}\) in the network model respectively, and the left cross-aisle node \(A+3\) and the right cross-aisle node \(A+4\) for arriving the station correspond to \(CA_{l}^{A}\) and \(CA_{R}^{A}\) in the network, respectively. The station node is identified with \(A+5\), and the virtual synchronization service node used by the system to match AGV car and external task queue is identified with \(A+6\).  
When a task arrives in the system and enters the external order queue \(Q_{o}\). The synchronous node \(A+6\) finds an idle AGV in the AGV queue \(Q_{R}\) and matches it with a task. Then, the material is transported by AGV through one of the cross-aisle nodes to leave the station. It is equally possible for AGV to pass the left cross-aisle node or the right cross-aisle node and enter the aisle nodes on the left and right sides, and the probability is \(1/2\). The AGV then enters the designated aisle node to unload and load the next material and leaves the aisle node to return to the operation station to process the material through the cross-aisle nodes. After completion of the picking work, the released AGV re-enters the inner AGV queue \(Q_{R}\) to wait for the next assignment. In the selection of the aisles, the uniform distribution strategy is adopted, and the probability of choosing each aisle is equal to \(1/A\). That is to say, any AGV can be assigned to any task in any aisle equally.  
This queueing network model has the characteristics of both closed network and open network, which is called semi-open queueing network model. Because the external tasks arrive at the system continuously and enter the external queue \(Q_{o}\), it has the property of an open network. At the same time, because the number of AGV in the system is limited and they always exist in the system, any task arriving in the system needs to wait for the  
Figure 2: Schematic of the congestion.  
matching with the idle vehicles to carry out the execution process. So the network also has the property of a closed network. In order to ensure the steady-state of the queuing network model, the arrival rate of the task should be less than the maximum throughput calculated by the closed network system when there are \(R\) vehicles in the system. That is, the number of tasks arriving at the system per hour should be less than the number of tasks that can be processed per hour when all AGVs are busy. The semi-open queueing network described above is a non-product form, so an approximate method is needed to solve the model [24]. Before solving, the service time expression of each node in the network is described.