# Analysis and Optimization of the Robotic Mobile Fulfillment Systems Considering Congestion
## 2 System Introduction and Scheduling Rules
### Congestion Factors and Scheduling Rules  
There will be vehicle congestion in the RMFS with fixed shelves; that is, multiple AGVs occupy the same space resource at the same time, resulting in vehicle delay. In this system, the cross-aisle is a one-way driving path (the departure and arrival directions belong to two paths), and the AGV driving on the same path is in the same direction with the constant speed, so the cross-aisle conflict is not considered. Therefore, the congestion phenomenon in the system can be divided into two kinds of routing conflicts, namely, aisle conflict and cross-aisle intersection conflict. The specific congestion situation is shown in Figure 2.  
In the aisle, the path setting is two-way driving. Therefore, when an AGV is already in the aisle and another AGV tries to enter the aisle, there is a conflict phenomenon. In the same way, when one AGV tries to leave the aisle to enter the cross-aisle and another AGV is driving or preparing to enter the aisle from the cross-aisle, the intersection conflict will occur. In order to solve this conflict, the rule called LCFS-PR (last come, first serve preemptive resume) is proposed to be used in the aisle. When the service rule is used, the priority of the existing AGV in the aisle is lower than that of the later arriving AGV. That is, when the latter arrival AGV tries to request access to the aisle, the AGV working in the same aisle will drive to the end port of the aisle, and then it will continue to work after the latter arrival AGV finishes the operation. By considering the use of the service rules in the modeling process, the focus of this paper is to study the impact of congestion factors on system performance.  
When the AGVs conflict in the intersection of the cross-aisle and the aisle, the system will appear as a congestion phenomenon. In order to solve this problem, a rule is formulated, that is, after each operation in the aisle, the AGV leaves the aisle immediately and waits to enter the cross-aisle at the shelf port where the aisle connects with the cross-aisle.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbols** & **Meanings** & **Symbols** & **Meanings** \\ \hline \(A\) & Number of aisles & \(w_{ca}\) & Width of cross-aisle \((m)\) \\ \(C\) & Number of cross-aisle & \(L_{pqz}\) & Distance between station and storage edge in the \\ \(R\) & Number of AGV & \(L_{pqz}\) & Distance between station and storage edge in the \\ \(W\) & Width of storage area \((m)\) & \(L_{pqz}\) & Distance between the adjacent station \\ \(L\) & Length of storage area \((m)\) & \(l\) & The side length of each storage location \((m)\) \\ \(K\) & The ratio of length to width of storage area & \(w_{ca}\) & The side length of the station area \((m)\) \\ \(v_{R}\) & The average running speed of AGV & \(\lambda\) & The arrival rate of picking order entering system \((\)per hour\()\) \\  & in the aisle \((m/s)\) & \(l_{m}\) & Time for an AGV to lift or drop the shelf \((s)\) \\ \(l_{\text{CR}}\) & The average running speed of AGV in & \(l_{p}\) & Time for a picker at the station to process a shelf \((s)\) \\  & across-aisle \((m/s)\) & \(l_{s,d}\) & Time for AGV to return to stand by position from the station \((s)\) \\ \(w_{m}\) & Width of shelf module \((m)\) & \(l_{s,d}\) & Time for AGV to target shelf from stand by position \((s)\) \\ \(L_{m}\) & Length of shelf module \((m)\) & \(l_{s,s}\) & Time for AGV to move the target shelf to the station \((s)\) \\ \(w_{a}\) & Width of the aisle \((m)\) & \(l_{s,st}\) & Time for AGV to move the target shelf to the station \((s)\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main system parameters and their meanings.