# Analysis and Optimization of the Robotic Mobile Fulfillment Systems Considering Congestion
## 3 Modeling and Solution
### Service Time Expression of Each Node
#### 3.2.1 The Service Rate and the Parameter Estimation in the Aisle  
The service time of AGV in the aisle includes the whole running time after entering the aisle from the aisle port, selecting a location to store materials, and taking out materials from another location to return to the aisle port. Set \(t_{rt}\) is the running time in the aisle, \(t_{s}\) is the time to store material, \(t_{r}\) is the time to take out a material. In addition, the service time also includes the time from the cross-aisle to the aisle port. Therefore, the aisle service time expression is obtained.  
\[t_{A}=t_{rt}+t_{s}+t_{r}+\frac{2d_{arr}^{loc}}{v_{R}} \tag{1}\]  
In this expression, only \(t_{rt}\) is unknown, and its expected value \(E[t_{rt}]\) needs to be estimated according to the storage strategy of the storage area. Next, we estimate the solution of \(E[t_{rt}]\) under the random storage strategy and the classified storage strategy. Let \(i_{s}\) is the location number of the storage shelf, \(i_{r}\) is the location number of the rack, \(D(i_{s},i_{r})\) represents the running distance from the storage location to the pickup location and return to the starting aisle port.  
\[D(i_{s},i_{r})=2l\cdot\max(i_{s},i_{r})-l \tag{2}\]  
Figure 3: Queuing network model of the AGV system.  
It is set that there are \(N\) material spaces on each floor of the aisile. Under the random storage strategy, The values of \(i_{s}\) and \(i_{r}\) are relatively independent. So the joint probability of \(i_{s}\) and \(i_{r}\) can be expressed as  
\[P(i_{s}=k,i_{r}=q)=\frac{1}{N(N-1)}\qquad\forall\;k,q\in\{1,2,\ldots\ldots N\} \;k\
eq q \tag{3}\]  
Let \(i_{m}=\max(i_{s},i_{r})\), then the probability function of \(i_{m}\) is expressed as  
\[P(i_{m}=k)=\frac{2(k-1)}{N(N-1)}\qquad\forall\;k\in\{2,\ldots\ldots N\} \tag{4}\]  
The first and second moments of \(i_{m}\) can be expressed as  
\[E[i_{m}]=\frac{2(N+1)}{3} \tag{5}\]  
\[E\Big{[}i_{m}{}^{2}\Big{]}=2(N+1)\Big{(}\frac{N}{4}+\frac{1}{6}\Big{)} \tag{6}\]  
According to Formulas (2), (5) and (6), the first and second moments of the round trip distance can be obtained as  
\[E[D(i_{s},i_{r})]=\frac{l\cdot(4N+1)}{3} \tag{7}\]  
\[E\Big{[}D(i_{s},i_{r})^{2}\Big{]}=\frac{l^{2}\cdot(6N^{2}+2N-1)}{3} \tag{8}\]  
Therefore, both the time expectation of \(t_{tr}\) and the second moment of it can be obtained as  
\[E[t_{tr}]=\frac{E[D(i_{s},i_{r})]}{v_{R}}=\frac{l\cdot(4N+1)}{3v_{R}} \tag{9}\]  
\[E\Big{[}t_{tr}{}^{2}\Big{]}=\frac{E\Big{[}D(i_{s},i_{r})^{2}\Big{]}}{v_{R}{}^ {2}}=\frac{l^{2}\cdot(6N^{2}+2N-1)}{3v_{R}{}^{2}} \tag{10}\]  
According to Formulas (1), (9) and (10), the expected service time and the second moment in the aisile can be obtained as  
\[E[t_{A}]=\frac{l\cdot(4N+1)}{3v_{R}}+t_{s}+t_{r}+\frac{2d_{arr}^{loc}}{v_{R}} \tag{11}\]  
\[E\Big{[}t_{A}{}^{2}\Big{]}=\frac{l^{2}\cdot\big{(}6N^{2}+2N-1\big{)}}{3v_{R}{ }^{2}}+2\Bigg{(}t_{s}+t_{r}+\frac{2d_{arr}^{loc}}{v_{R}}\Bigg{)}\frac{l\cdot(4 N+1)}{3v_{R}}+\frac{l^{2}\cdot\big{(}6N^{2}+2N-1\big{)}}{3v_{R}{}^{2}} \tag{12}\]  
The aisile service rate, the variance of expected service time, and the variance coefficient of the square formula are as follows  
\[\mu_{A}=\frac{1}{E[t_{A}]} \tag{13}\]  
\[Var(t_{A})=E\Big{[}t_{A}{}^{2}\Big{]}-E[t_{A}]^{2} \tag{14}\]  
\[cv_{A}{}^{2}=\frac{Var(t_{A})}{E[t_{A}]^{2}} \tag{15}\]