# Buffer Allocation via Bottleneck-Based Variable Neighborhood Search
## 4 Numerical Example
### Small-Scale Topology Experiment
#### 4.1.2 Results  
Table 3 lists the results of the five methods at the computation time when buffer allocation is finished. Figure 9 shows the computation time versus throughput curves for the five methods in the case of \(\lambda=0.4\). The results show that API-VNS and API-BS can generate higher throughput than the other three methods in the first 2 h. However, API-VNS achieved the same throughput more rapidly than API-BS. Subsequently, ETS obtained a higher throughput than API-VNS because of its high solution quality for available computation time. Figure 10 shows the computation time versus throughput curves for the five methods in the case of \(\lambda=0.6\). The results show that API-VNS can achieve the highest throughput among the comparison methods, which shows the advantage of its solution quality in available computation time. Moreover, compared to other four methods, API-VNS obtained the same throughput with least computation time. Figures 11 and 12 show the scatter plots of the designed buffer size versus the solution throughput for the five methods in the cases of \(\lambda=0.4\) and \(\lambda=0.6\), respectively. In the case of \(\lambda=0.4\), ETS achieved the highest solution throughput, followed by API-VNS and API-BS. However, ETS allocated less designed buffers than API-VNS and API-BS. ETS adds several buffers in each buffer update, which test more candidate buffer allocation solutions. Therefore, it can achieve the highest solution quality in the experiment. Moreover, API-VNS and API-BS obtain the same solution throughputs as they both use API; Inv-BS and Inv-VNS also obtain the same solution throughput because both use the inventory indicator. Furthermore, API-VNS and API-BS have the highest solution quality because both achieve the highest solution throughput with the same designed buffer size. \begin{table}
\begin{tabular}{l c} \hline \hline
**Items** & **Parameters** \\ \hline Arrival rates & \(\lambda_{1}=\lambda\) (jobs/s) \\ Service rates of buffers & \(\mu_{1}=\mu_{4}=\mu_{7}=\mu_{11}=\mu_{13}=\mu_{15}=\mu_{nor}\) (jobs/s) \\ Service rates of buffers & \(\mu_{2}=\mu_{5}=\mu_{8}=\mu_{10}=\mu_{12}=\mu_{14}=\mu_{con}\) (jobs/s) \\ Service rates of servers 3, 6, 9 & \(\mu_{3}=\mu_{6}=\mu_{9}=\mu_{sor}\) (jobs/s) \\ Service discipline of merging & First come, first served: jobs arriving at the merging node enter it first preemptively. \\ Service discipline of splitting & Random: jobs at the splitting node move to ensuing nodes randomly. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Input parameters of experiment 1. Figure 8: Small-scale topology. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pattern (\(\lambda\), \(\mu_{\text{zero}}\),\(\mu_{\text{cc}}\),\(\mu_{\text{per}}\))** & **Algorithm** & \begin{tabular}{c} **ComputationTime** \\ \(t_{\text{sol}}\) \\ (**s**) \\ \end{tabular}  & \ \begin{tabular}{c} **Obtained Throughput** \\ \(T_{\text{sol}}\) \\ (**jobs/s**) \\ \end{tabular}  &
\begin{tabular}{c} **Designed Buffer Size \(N_{\text{sol}}\)** \\ \(B_{\text{sol}}\),\(\mu_{\text{b}}\),\(\mu_{\  
Figure 11: Scatter plot of designed buffer size vs. solution throughput for the five methods in the case of \(\lambda=0.4\) in experiment 1. Figure 10: Computation time vs. throughput curves for the five methods in the case of \(\lambda=0.6\) in experiment 1. Figure 9: Computation time vs. throughput curves for the five methods in the case of \(\lambda=0.4\) in experiment 1. Unlike that in the case of \(\lambda=0.4\), API-VNS and API-BS obtain different solution throughputs, and Inv-BS and Inv-VNS also obtain different solution throughputs. API-VNS and API-BS use API as the BI to detect bottlenecks. However, the real bottleneck is not detected in the initial neighborhood structure, and VNS is used to search for more candidate bottlenecks. Therefore, API-VNS can achieve a higher solution throughput. Similarly, Inv-VNS achieves a higher solution throughput than Inv-BS. Compared to the API, the inventory indicator is not sufficiently effective in the experiment.