# Throughput of Buffer with Dependent Service Times
## 4 Buffer Throughput  
In this section, we deal with three throughput-related characteristics: \(D_{i}(n,t)\), \(D_{i}^{\prime}(n,t)\), and \(T\). Specifically, \(D_{i}(n,t)\) denotes the expected count of packets departed from the buffer in \((0,t)\), given that \(N(0)=n\), \(S^{*}(0)=i\). Its derivative, \(D_{i}^{\prime}(n,t)=\frac{\mathrm{d}}{\mathrm{d}t}D_{i}(n,t)\), is the output traffic intensity at time \(t\). Finally, \(T\) denotes the throughput of the buffer:  
\[T=\lim_{t\to\infty}D_{1}^{\prime}(0,t), \tag{7}\]  
i.e., the expected count of packets departed in a time unit in the stationary regime. Obviously, the initial values \(N(0)=0\) and \(S^{*}(0)=1\), chosen in (7), can be replaced with any other initial values because the stationary throughput does not depend on them. We initiate with the derivation of \(D_{i}(n,u)\). Assuming \(0<n<K\), we can assemble an equation for \(D_{i}(n,t)\) as follows:  
\[D_{i}(n,t)= \sum_{k=0}^{n-1}\sum_{j=1}^{m}\int_{0}^{t}p_{ij}(k,v)\big{[}k{+}D_ {j}(n-k+1,t-v)\big{]}\mathrm{d}G(v)\] \[+\sum_{j=1}^{m}\int_{0}^{t}r_{ij}(n,v)\big{[}n+D_{j}(1,t-v)\big{]} \mathrm{d}G(v)\] \[+\sum_{k=0}^{n}\sum_{j=1}^{m}kp_{ij}(k,t)(1-G(t)). \tag{8}\]  
In (8), \(p_{ij}(k,v)\) is the probability that \(k\) packets are departed from the buffer in \((0,v)\), while the service phase is \(j\) at time \(v\), provided that \(N(0)\geq k\) and \(S^{*}(0)=i\). On the other hand, \(r_{ij}(k,v)\) is probability that \(k\) packets are departed from the buffer in \((0,v)\), while the service phase is \(j\) at the close of \(k\)-th transmission, provided that \(N(0)\geq k\) and \(S^{*}(0)=i\). System (8) is obtained via the total probability formula, employed with respect to the first arrival epoch, \(v\). Specifically, the first component of (8) responds to the case, where \(k<n\) packets are departed by \(v<t\), and the buffer is not emptied by \(v\). In such cases, we know at the time \(v\) that the new expected count of packets departing by \(t\) must be \(\sum_{j}\big{[}k+D_{j}(n-k+1,t-v)\big{]}\) because the buffer occupancy at \(v\) is \(n-k+1\). The second component of (8) responds to the case where \(n\) packets are departed by \(v<t\), so the buffer empties at some time \(u<v\). Moreover, the service phase stops changing at \(u\). In such cases, we know at the time \(v\) that the new expected count of packets departing by \(t\) must be \(\sum_{j}\big{[}n+D_{j}(1,t-v)\big{]}\) because the buffer occupancy at \(v\) is \(1\). The third component of (8) responds to the case where \(v\geq t\) and \(k\) packets are departed by \(t\) of the probability \(\sum_{j}p_{ij}(k,t)(1-G(t))\). Assuming \(n=K\), we find the equation for \(D_{i}(K,t)\) as follows:  
\[D_{i}(K,t)= \sum_{j=1}^{m}\int_{0}^{t}p_{ij}(0,v)D_{j}(K,t-v)\mathrm{d}G(v)\] \[+\sum_{k=1}^{K-1}\sum_{j=1}^{m}\int_{0}^{t}p_{ij}(k,v)\big{[}k{+}D _{j}(K{-}k{+}1,t{-}v)\big{]}\mathrm{d}G(v)\] \[+\sum_{j=1}^{m}\int_{0}^{t}r_{ij}(K,v)\big{[}K+D_{j}(1,t-v)\big{]} \mathrm{d}G(v)\] \[+\sum_{k=0}^{K}\sum_{j=1}^{m}kp_{ij}(k,t)(1-G(t)). \tag{9}\]  
Equation (9) is similar to (8) with the exception that the first component of (8) is split into the first two components in (9). Namely, for \(k=0\), the new buffer content is \(K\) packets, not \(K+1\), as it would be in (8). Obviously, the buffer design does not allow the occupancy of \(K+1\) packets. The rest of (9) is indistinguishable from (8). Assuming \(n=0\), we have  
\[D_{i}(0,t)=\int_{0}^{t}D_{i}(1,t-v)\mathrm{d}G(v). \tag{10}\]  
This is obtained again via the total probability formula, employed with respect to the first arrival epoch, \(v\). If \(v<t\), then the the new expected count of packets departing by \(t\) must be \(D_{i}(1,t-v)\). If \(v\geq t\), then the count of packets departing by \(t\) is 0, so the component responding to such case vanishes.