## 3 System Outbound Process Modelling
### Modelling the Outbound Process of a Four-Way Shuttle System Based on Two Operation Strategies
#### 3.3.1 Parallel Part Modelling  
Construct the separation-aggregation queuing network model and analyse the parallel operation strategy by decomposing the complex network method in the following steps:  
1. Constructing the separation-aggregation queuing network model  
In this section, we have shown that the whole parallel operation strategy is divided into vertical operations, horizontal operations and remaining operations. First, the pickup order arrives at the \"separation\" point at arrival rate \(\lambda_{r}\), at which time the pickup task is divided into a vertical operation dominated by a hoist and a horizontal operation dominated by a shuttle, where the \(T\) shelves are divided into T different types of customers, arriving at arrival rate \(\lambda_{r}^{1},\lambda_{r}^{2},\ldots,\lambda_{r}^{T}\). Next, the hoist and the shuttle wait in the vertical queue \(Q_{l}\) and the horizontal queue \(Q_{s}\), respectively. After both the vertical and horizontal operations are completed, they converge at the \"convergence\" point; finally, the hoist carries the goods to complete the last remaining operations. As assumption 6 obeys the single-command cycle, the lifters are not released until the task is completed; therefore, the lifters under node \(\mu_{I2}\) can provide an unlimited service capacity. The separation aggregation queuing network model is shown in Figure 3.  
Figure 2: Four-way shuttle system outbound process queuing network model.  
2. Constructing a closed-loop queueing network  
As the shuttle and hoist obey a general distribution, the parallel operation strategy described in Section 3.3.1 is a nonproduct solution and cannot be solved directly; thus, the Coxian distribution solution model, which approximates the general distribution, is applied in this paper. Figure 4 shows a schematic diagram of a closed-loop queuing network under a parallel strategy, where there are \(\mathbf{K}_{r}\) pick-up operations in the closed-loop network and the arrival rate of each tier of jobs is \(\lambda_{r}^{i}=\frac{\lambda_{r}}{T}\). After entering this network, the shuttles are already bound to the order, the maximum number of accesses that can be accepted in the system cannot exceed the number of shuttles and the time used in the jobs to receive service at the shuttles and hoists is equal to their own service time.  
The importance of building this closed-loop queuing network model is to create an equivalent service node to the open-loop queuing network model, which can simplify the complex queuing network.  
3. Constructing an open-loop queuing network  
First, the service point formed by the closed-loop queuing network is denoted as node 1, and the service rate of this service point obeys a negative exponential distribution, which leads to an open-loop queuing network with two servers in series, where the service rate of service point 1 is replaced by the throughput rate sought in 2, which is load dependent. In the open-loop queuing network, the pickup task arrives with an arrival rate of \(\lambda_{r}\), node 1 enters node 2 for the remaining operations after equivalence and as node 1 is composed of a closed-loop queuing network, it compensates for the lack of trolley waiting time for the orders in the open-loop network. Finally, the maximum entropy method is used to solve the metrics of this queuing network model. The open-loop queuing network formed under the parallel strategy is shown in Figure 5.  
Figure 4: Closed-loop queuing network using the parallel strategy.  
Figure 3: Separation-aggregation queuing network model.