## 4 Model Solution
### Solving the Separation-Aggregation Queueing Network Model Using the Parallel Strategy
#### 4.1.1 Constructing a Closed-Loop Queuing Network to Compute State-Based Service Rates  
A closed-loop queuing network model based on separation-aggregation is shown in Figure 2, where there are \(K_{r}\) pickups in the closed-loop network, \(Q_{l}\) is the number of customers waiting to be served in \(Q_{l}^{q}\) and \(Q_{s}\)D is the number of customers waiting to be served in \(Q_{s}^{q}\). The state variable \(s_{k}\) of this closed-loop queueing network can be given as:  
\[s_{k}=(Q_{s},Q_{L}),k=Q_{l}+Q_{s}+mQ_{l}=Q_{l}(m+1)+Q_{s} \tag{20}\]  
where \(\mathrm{m}=\mathrm{min}\ (K_{r},\mathrm{M})\) is the maximum number of customers waiting to be served in \(Q_{s}^{q}\) and \(Q_{l}+Q_{s}<m+1,\ Q_{l}=0,1,Q_{s}=0,1,\ldots,m\). If the system completes an aggregation using the parallel operation strategy, there are two scenarios:  
1. At \(Q_{l}=1\), the hoist completes its operation and reaches the convergence point. This process requires the shuttle to complete the pickup operation and move to the entrance of the aisle; then, the service rate of this process is \(\mu_{s}\). 2. At \(Q_{s}>1\), the shuttle completes the pickup operation and arrives at the hoistway entrance, and this process requires the hoistway to move to the entrance of the target level. Then, the service rate of this process is \(\mu_{11}\). Setting \(\pi(s_{k})\) as the state probability in state \(s_{k}\), the throughput rate can be expressed by the two cases of aggregation as:  
\[\lambda_{f}(K_{r})=\sum_{Q_{s}=0}^{m}\mu_{\mathrm{I}1}\pi(Q_{s},0)+\sum_{Q_{s}=0 }^{m-1}\mu_{s}\pi(Q_{s},1) \tag{21}\]  
The service time of shuttle service point \(\mu_{s}\) is independently and identically distributed, denoted as \((\alpha,\ \mathrm{T})\), where the initial probability vector \(\alpha= \begin{bmatrix}1&0&0&0&\dots\end{bmatrix} _{1\times p}\) is the initial probability vector of the p-order Coxian distribution and \(\mathrm{T}\) is the p-order square matrix. The service time of hoist service point \(\mu_{\mathrm{I}1}\) is independently and identically distributed, denoted as \((\beta,\ S)\), and the initial probability vector \(\beta= \begin{bmatrix}1&0&0&0&\dots\end{bmatrix} _{1\times p}\) is the initial probability vector of the q-order Coxian distribution and \(\mathrm{S}\) is the q-order square matrix. The state variables of the separation-aggregation closed-loop queuing network model are set to \((s_{k},\ i,j)\), where \(i,j\) are the states in which the shuttle and hoist are located, respectively, where \(i=1,2,3\dots p;\textit{and}\ j=1,2,3\dots q\). Let \(\pi_{t}\) be the steady-state probability distribution of \(s_{k}\). As the two cases complete the parallel operation strategy complete the aggregation, \(\pi_{t}=[\pi_{0},\pi_{1}]\). Therefore, the state transfer rate matrix \(\mathrm{Q}\) of the state variable \(s_{k}\) is:  
\[Q= \begin{bmatrix}B_{00}&B_{01}\\ B_{10}&B\end{bmatrix}  \tag{22}\]  
The transfer diagram of the generation and extinction process is shown in Figure 8.  
\(B_{01}\) represents the rate matrix for the transfer of the \(\pi_{0}=[\pi(s_{0}),\pi(s_{1}),\pi(s_{2})\dots\pi(s_{m})]\) state to the \(\pi_{1}=[\pi\Big{(}s_{m+1}),\pi(s_{m+2}),\pi(s_{m+1}),\pi(s_{m+3}),\dots,\pi(s_ {2m})\Big{]}\) state. \[B_{01}=\left[ \begin{array}{ccccc}(Q_{s},Q_{l})&(0,1)&(1,1)&(2,1)&\dots&(m-1,1 )\\ (0,0)&I_{T}\otimes S^{\circ}&&&\\ (1,0)&&&I_{T}\otimes S^{\circ}&&&\\ (2,0)&&&I_{T}\otimes S^{\circ}&&&\\ \vdots&&&\ddots&\\ (m-1,0)&&&I_{T}\otimes S^{\circ}\\ (m,0)&&&0\end{array} \right] \tag{23}\]  
\(B_{00}\) represents the rate matrix for the transfer of the \(\pi_{0}=[\pi(s_{0}),\pi(s_{1}),\pi(s_{2})\dots\pi(s_{m})]\) state to the \(\pi_{0}=[\pi(s_{0}),\pi(s_{1}),\pi(s_{2})\dots\pi(s_{m})]\) state. Figure 8: Transfer diagram of the generation and extinction process. \[B_{00}=\left[