## 3 Verification of the Proposed Statistical Method  
The proposed method is verified in this section. First, the proposed method is directly compared with HSPICE transient simulation for cases with small number of SSO buffers. In this case, the occurrence probabilities of some dramatic cases such as all aggressors in the same transitioning state are not extremely low. This means that those cases are likely to appear during the transient simulation. In these cases, accuracy of the proposed method can be directly compared with full transient simulation. Then, the proposed method is compared to more complex cases. Lastly, the proposed method is compared with previous works to emphasize the accuracy. In Figure 8, case for four buffers sharing the same hierarchical PDN with a data rate of 1 Gb/s is verified by comparing the estimated statistical eye-diagram with the transient eye-diagram simulated using HSPICE. Since the lowest occurrence probability of worst-cases (all aggressor buffers in pull-up or pull-down transition) is around 0.01, which is likely appear in the transient simulation, direct comparison between two eye-diagrams is possible. In case with four buffers, the estimated eye-diagram showed good correlation with that obtained from the direct transient simulation. However, as shown in Figure 8a, the voltage variations which have occurrence PDFs below \(10^{-4}\) order, the transient simulation starts to lose accuracy or requires longer simulation time. To mitigate this issue in the transient simulation, simulation with longer bit patterns and longer simulation time should be conducted. In Figure 9, the case for the four buffers with a data rate of 2 Gb/s is verified by comparing the estimated statistical eye-diagram with transient eye-diagram simulated in HSPICE. With a higher data rate, the eye-opening voltage is decreased and timing jitter is increased, since more UIs are considered in the recursive convolution when defining the ISI-PDFs. In this case, transient simulation starts to lose accuracy with voltage variations, with occurrence probability below \(10^{-4}\) order. If the number of aggressor buffers increase, allowed aggressor combinations will dramatically increase with lower occurrence probabilities. Therefore, if the number of the aggressor buffer increases, transient simulation will lose accuracy or even fail to estimate some responses or noise characteristics. Figure 8: Total of four buffers operating at a data rate of 1 Gb/s. (**a**) HSPICE simulation result. (**b**) Estimated statistical eye-diagram using the proposed method. In Table 2, accuracy and simulation time (estimation time for the proposed method) comparisons between the HSPICE transient simulation and the proposed method are summarized. Conventional transient simulation is limited when the number of buffers increases. As the number of buffers increases, the transient simulation loses its accuracy to achieve precise eye-diagrams. Recently, BER analysis has become even more important, and this analysis begins from the obtained statistical eye-diagram. The BER obtained using the transient simulation is losing accuracy below \(10^{-4}\) order. The proposed method is capable of estimating BER below \(10^{-16}\) order or even lower with a higher data rate. The proposed method is compared with previous works. First, the proposed method is compared with the peak distortion analysis (PDA) method [12]. In Figure 10, estimated eye-diagrams obtained using [12] are plotted for the comparison. For both Figure 10a,b, four buffers sharing the hierarchical PDN are set, but they operate at different data rates: 1 Gb/s and 2 Gb/s. Compared to Figures 8 and 9, the PDA method is not capable of estimating an accurate eye-diagram under the influence of power/ground noise.