# Throughput of Buffer with Dependent Service Times
## 6 Numeric Illustrations  
In the numerical examples, rather than using fixed arrival and service processes, we use a class of arrival processes and a class of service processes, both dependent on parameters. By manipulating these parameters, we can obtain throughput-related and loss-related characteristics for various standard deviations of the interval between arrivals and different correlations between transmission times. Specifically, the Markovian service process has here the following matrices \(\mathbf{L}_{0}\) and \(\mathbf{L}_{1}\):  
\[\mathbf{L}_{1}=\left[ \begin{array}{ccc}275.1850&0.0368&0.0169\\ 0.0103&30.8598&0.1123\\ 0.0113&0.1708&3.3982\end{array} \right], \tag{71}\]  
\[\mathbf{L}_{0}=\frac{1}{a}\cdot\left[ \begin{array}{ccc}-0.3549&0.1037&0.2512\\ 0.1813&-0.3431&0.1618\\ 0.1556&0.2317&-0.3873\end{array} \right]-\mathbf{L}_{1}, \tag{72}\]  
where \(a\) denotes a parameter. These matrices were generated randomly in such a way that they produce a coefficient of correlation of about 0.4 (exactly 0.433), without the parameter \(a\). Introducing the parameter \(a\) in (72) makes it possible to obtain any correlation in the range 0-0.433. Specifically, by decreasing \(a\), we also decrease the correlation. In Figure 2, the dependence of \(\rho_{1}\) on \(a\) is depicted. In what follows, when manipulating \(a\), rather than giving \(a\) in the plots, the value of \(\rho_{1}\) is presented, which is more interesting. Regardless of the value of \(a\), the service rate for these matrices remains unaltered and equal to \(100\) pkts/s. The packet interarrival interval distribution is  
\[G(t)=1-\frac{0.9b}{b-10}e^{-bt}-\frac{0.1b-10}{b-10}e^{-10t}, \tag{73}\]  
where \(b\) is a parameter. No matter what \(b\) is, the expected arrival rate is \(100\)pkts/s. However, the coefficient of variation for the interarrival interval, \(C\), changes with \(b\). In Figure 3, this relationship is shown. When manipulating \(b\), rather than giving \(b\) in the plots, the value of \(C\) is shown. If not declared otherwise, the buffer volume is \(K=20\). This will be altered in Sections 6.1 and 6.2, where much larger buffers will be used as well. In Figure 4, the output traffic intensity in time is depicted for \(C=4\) and various values of \(\rho_{1}\). In every case, the buffer is initially empty, \(N(0)=0\). In Figure 5, the output traffic intensity is displayed for the same \(C\) and various \(\rho_{1}\) but with an initially full buffer, \(N(0)=K\). In Figure 6, the same curves as in Figure 5 are depicted, but over a much longer time scale. Therefore, in Figures 4-6, we can observe the effect of increasing \(\rho_{1}\) on the time-dependent output traffic intensity in the case of initially empty buffer versus full buffer (Figure 4 vs. Figure 5), as well as in the short versus long time scale (Figure 5 vs. Figure 6). In Figure 4, we observe a high peak in the output traffic intensity, approaching \(200\), shortly after the start. This peak is not related to \(\rho_{1}\) but is solely caused by the high value of \(C\) (this will also be discussed next to Figure 7). The initial intensity in Figure 5 is even higher, being nearly \(300\). Such high intensity is a reaction to the initially full buffer, meaning the service process can depart several packets from the buffer without waiting for new arrivals. Figure 3: Coefficient of variation, \(C\), for the time between packets versus parameter \(b\). Figure 2: Correlation of two consecutive transmission times, \(\rho_{1}\), versus parameter \(a\). Growing service correlation has at least two noticeable effects in Figures 4-6. Firstly, a high \(\rho_{1}\) may increase the output traffic intensity over a short time scale, as seen in both Figures 4 and 5. This can be attributed to the fact that a high positive correlation induces short periods of higher-than-average service intensity.