# Throughput of Buffer with Dependent Service Times
## 6 Numeric Illustrations
Over a longer time scale, the effect is reversed, i.e., a higher \(\rho_{1}\) leads to lower output traffic intensity, as evident in Figure 6. Secondly, high service correlation can cause a complex evolution of the intensity, especially visible in Figures 5 and 6. Specifically, several local minima and maxima may occur over time, and their magnitudes are clearly related to the strength of the correlation--the higher \(\rho_{1}\), the more pronounced these extrema become. Figure 4: Output traffic intensity in time for four different values of \(\rho_{1}\). \(N(0)=0\) and \(C=4\) in every case. Figure 5: Output traffic intensity in time for four different values of \(\rho_{1}\). \(N(0)=K\) and \(C=4\) in every case. Figure 6: Output traffic intensity over a long time for four different values of \(\rho_{1}\). \(N(0)=K\) and \(C=4\) in every case. Another consequence of high service correlation is the convergence time to a stable throughput, as seen in Figure 6. Namely, the higher \(\rho_{1}\) is, the more time is required for the throughput to stabilize at the stationary value. In Figure 7, the output traffic intensity over time is illustrated for \(\rho_{1}=0.4\) and various \(C\) values. In each case, the buffer is initially empty, i.e., \(N(0)=0\). In Figure 8, the output traffic intensity is depicted for the same \(\rho_{1}\) and various \(C\) values but with an initially full buffer, i.e., \(N(0)=K\). In Figure 9, the same curves as in Figure 8 are depicted, but over a longer time scale. Thus, Figures 7-9 aim to demonstrate the effect of increasing \(C\) on the time-dependent output traffic intensity, comparing the case of an initially empty buffer versus an initially full buffer (Figure 7 vs. Figure 8) and short-term versus long-term behavior (Figure 8 vs. Figure 9). As seen in Figure 7, for the initially empty buffer, a high initial peak in output traffic intensity is caused solely by high values of \(C\). As \(C\) decreases, this peak diminishes accordingly. Conversely, the high initial intensity in Figure 8 is due to the strong correlation \(\rho_{1}=0.4\) and the full buffer, meaning several packets from the buffer can be processed quickly without waiting for new arrivals. In Figure 9, we may examine the convergence to the stable throughput depending on \(C\). As seen, \(C\) impacts the variability in output traffic intensity during the initial short time but has a minor effect on the total convergence time. For every \(C\), the convergence time is more or less the same. The difference is that for a small \(C\), the convergence is more regular and monotonic, while for a high \(C\), it is more variable, with local extrema. Figure 8: Output traffic intensity in time for four different values of \(C\). \(N(0)=K\) and \(\rho_{1}=0.4\) in every case. Figure 7: Output traffic intensity in time for four different values of \(C\). \(N(0)=0\) and \(\rho_{1}=0.4\) in every case. As seen in Figures 12-14, increasing \(C\) and \(\rho_{1}\) has a similar impact on the degradation of throughput, at least when \(C\) varies up to 4 and \(\rho_{1}\) varies up to 0.4. The impact of \(\rho_{1}\) is slightly stronger than that of \(C\), meaning the throughput decreases faster with increasing \(\rho_{1}\) than with increasing \(C\). Finally, in Figure 15, the dependence of the stationary loss probability, \(P\), on both \(C\) and \(\rho_{1}\) is depicted. As seen, increasing \(C\) and \(\rho_{1}\) increases the loss probability, while the impact of \(\rho_{1}\) is slightly stronger than that of \(C\), meaning \(P\) increases faster with \(\rho_{1}\) than with \(C\). Figure 3:0 Stationary throughput versus \(\rho_{1}\) for four different values of \(C\). Figure 3:1 Stationary throughput versus \(C\) and \(\rho_{1}\). Figure 3:2 Stationary throughput versus \(C\) for four different values of \(\rho_{1}\). Note that in the stationary case, throughput and loss probability are complementary; if one of them is high, the other must be low, and vice versa.