## 3 System Outbound Process Modelling
### Average Service Time Model for a Four-Way Shuttle System Server
#### 3.2.1 Server Service Time Model Using Parallel Operation Strategy  
First, the shuttle and the hoist need to carry out horizontal pickup operations and vertical target-level seating operations, respectively, constituting a parallel operation strategy in which the shuttle and the hoist complete their tasks at the same time. Subsequently, the shuttle carries the goods to the hoist, the hoist accepts the goods to complete the convergence, and finally, the shuttle carries the goods to level 1. The probability in this case \(P_{1}\) is:  
\[P_{1}=\frac{C_{T-1}^{M-1}}{C_{T}^{M}} \tag{3}\]  
Therefore, the average service time of the server in vertical jobs, horizontal jobs and remaining jobs are calculated as follows:  
1. Operation service time of vertical operations  
In accordance with assumption 4, this paper adopts the endpoint stay strategy. The hoist stays at the endpoint of the previous task, which is the initial position of system level 1; therefore, the movement distance of the vertical job is the vertical distance from the target level to level 1. Thus, the service time of vertical operation \(T_{I1}(t)\) is:  
\[T_{I1}(t)=\frac{(t-1)H}{V_{t}}+2(\varepsilon_{l}+\gamma_{l}) \tag{4}\]  
1. Operating service time of horizontal operations  
When the four-way shuttle is at the target t level, the movement distance can be divided into (1) moving from the hoistway entrance (buffer zone) to the target cargo location; and (2) returning to the hoistway entrance. The distance of movement includes the horizontal distance across the lane and the vertical distance along with the lane movement, which involves the cross-aisle movement of the shuttle. According to expert consultation, the number of cross-aisles \(x\) is determined by the number of aisles served by hoist \(N\). If \(N\) is an even number, the location of the hoist is the \(\frac{N}{2}\)th aisle from left to right; if \(N\) is an odd number, the location of the hoist is the \(\frac{N+1}{2}\) th aisle from left to right, then the number of aisles \(x\) spanned takes a range of values \(0\leq x\leq\left|\frac{N}{2}\right|\) and the probability of crossing aisle \(x\) is:  
\[P(x)=\left\{\begin{array}{l}\frac{1}{N},\,x=0\\ \frac{N}{N},N\,\,is\,even\,\,and\,\,x=\frac{2}{N}\\ \frac{2}{N},\,N\,is\,even\,\,and\,\,0<x<\left\lfloor\frac{N}{2}\right\rfloor \\ \frac{2}{N},N\,is\,odd\,\,and\,\,0<x<\left\lfloor\frac{N}{2}\right\rfloor\end{array}\right. \tag{5}\]The value of \(x\) is: \[x=\left|\left\lceil\frac{N}{2}\right\rceil-a\right|\] (6) Then, the service time of the shuttle during this phase \(T_{S}(M_{ac})\) is: \[T_{s}(M_{ac})=\frac{2\big{(}x+\left\lceil\frac{c}{4}\right\rceil\big{)}W_{A}+2 Lx+cW}{V_{s}}+\varepsilon_{s}+4\gamma_{s}\] (7)
9. Operating service time of the remaining operations The hoist moves from the target t level with cargo to level \(1\), and the operating service time \(T_{ls}(t)\) is: \[T_{ls}(t)=\frac{(t-1)H}{V_{l}}+2(\varepsilon_{l}+\gamma_{l})\] (8)  
In this case, there is a parallel operation between the hoist movement in \(1\) and the shuttle movement in \(2\). While the shuttle receives the outbound task, it applies for the hoist to wait at the target level; if one side arrives first, it will wait for the other side until it completes the rendezvous at the hoistway entrance picking up the goods and unloading them with the hoist to level \(1\).