# Analysis and Optimization of the Robotic Mobile Fulfillment Systems Considering Congestion
## 4 Simulation and Result Analysis  
### Simulation  
The simulation model is used to verify the theoretical model. In order to ensure the validity and reliability of the model, different scene experiments are obtained by changing the system design parameters. In this paper, the discrete event simulation module of Matlab's Simulink function was used to simulate the queuing network model. The simulation model was built according to the process of the queuing network. The first was random task generation. The random arrival process was generated by an exponential function, which obeys the Poisson distribution. The generated task was bound to AGV through a resource acquisition device. If there was no idle AGV at that time, the task would wait in the external queue. Finally, after binding with the idle AGV, the task entered the shelves system to receive service. The process of receiving service after entering the shelves system is as follows: First, enter the cross-aisle service, and choose the cross-aisle node on both sides with equal possibility. There is no waiting in this cross-aisle node because it is assumed that the cross-aisle node is an infinite capacity service node. Then the aisle node service is carried out. If there is already AGV working in the selected aisle, the AGV that are working in the aisle will enter the waiting queue. Later AGVs are served first, the queuing rule is Last In, First Out (LIFO). After the AGV completes the service in the aisle, it returns to the end port of the aisle and enters the cross-aisle service node again. Similar to the previous cross-aisle node service, it directly enters the cross-aisle service without waiting. Finally, the task with the AGV enters the queue of the picking station. If the picking station is free, the task will be served. If there are vehicles in operation, the task will wait in the queue. After completing the service, the AGV is released by the resource releaser. The task leaves the system, and the AGV returns to the resource pool to wait for matching the next task. The parameters of each node can be calculated according to the service time calculation expression in Section 3.2 and the given warehouse layout design parameters. The system parameters used in the simulation model are in Table 2.  
In order to fully verify the effectiveness and reliability of the theoretical model, we change the number of storage spaces \(N\), the aspect ratio of storage area \(K\) and the task arrival rate of the system \(\lambda\) to generate multiple scenes. For each scenario, the simulation model first runs 100 h of warm-up time to eliminate the system error as far as possible and then runs 100 times for 1000 h to achieve no less than 95% confidence. The collected index includes the average tasks queue waiting length of the synchronization node \(L_{Q_{o}}\), the AGV utilization \(\rho_{R}\), order throughput time \(E[CT]\), the average congestion delay time \(E[ABD]\) and the congestion time proportion \(\frac{E[ABD]}{E[CT]}\). During the running of the simulation model, the waiting time index of the task is tracked. We draw the curve of the tracking result at the synchronous matching node, as shown in Figure 6 above. It can be seen from the trend of the curve in Figure 6 that the waiting time of the task queue in the first 100 h was oscillatory. After the system reached a stable state, the change of the delay time was very small. Similarly, we tracked the congestion time of any two aisles and drew a graph, as shown in Figure 7. The system vibrated severely in the first 150 h, and then the fluctuation of the congestion time curve became very small as the system gradually ran to a stable state. We took the average delay time of all aisles as the simulation result of the congestion time to reduce the error.