# Entropy Analysis of a Flexible Markovian Queue with Server Breakdowns
## 2 Model Formulation and Previous Results
### Entropy Solution Using Both First and Second Moments  
Our objective here is to improve the probability vector obtained in the previous subsections. This is realized by including both first and second moments to the previous formulation. We will show that the use of the two moments as extra constraints leads to best approximation to the initial steady state probability vector obtained analytically. The nonlinear maximization problem to solve in this case is the following:  
\[\text{max} Z=-\sum_{i=0}^{\infty}p_{i}\ln p_{i}\] \[\text{s.t. } \sum_{n=0}^{c-1}a_{n}p_{n}=1+\frac{\alpha_{1}}{\beta_{1}} \tag{18}\] \[(EP3) \sum_{i=0}^{\infty}ip_{i}=E(X)\] (19) \[\sum_{i=0}^{\infty}i^{2}p_{i}=E(X^{2})\] (20) \[p_{i}\geq 0,\text{ for all }\,i.\]  
Similarly to the previous cases, we use the classical method of Lagrange. The following system of nonlinear equations, where the unknowns are the Lagrange multipliers \((\lambda_{1},\lambda_{2},\lambda_{3})\) is obtained:  
\[e^{a_{0}\lambda_{1}}+e^{a_{1}\lambda_{1}+\lambda_{2}+\lambda_{3 }}+e^{a\lambda_{1}}\sum_{k=2}^{c-1}e^{a\lambda_{1}+k\lambda_{2}+k^{2}\lambda_ {3}}+\sum_{k=c}^{\infty}e^{k\lambda_{2}+k^{2}\lambda_{3}}-e = 0\] \[e^{a_{1}\lambda_{1}+\lambda_{2}+\lambda_{3}}+e^{a\lambda_{1}} \sum_{k=2}^{c-1}ke^{a\lambda_{1}+k\lambda_{2}+k^{2}\lambda_{3}}+\sum_{k=c}^{ \infty}ke^{k\lambda_{2}+k^{2}\lambda_{3}}-eE(X) = 0\] \[e^{a_{1}\lambda_{1}+\lambda_{2}+\lambda_{3}}+e^{a\lambda_{1}} \sum_{k=2}^{c-1}k^{2}e^{a\lambda_{1}+k\lambda_{2}+k^{2}\lambda_{3}}+\sum_{k=c} ^{\infty}k^{2}e^{k\lambda_{2}+k^{2}\lambda_{3}}-eE(X^{2}) = 0\] \[.\]  
This system can be solved numerically. The values of \((\lambda_{1},\lambda_{2},\lambda_{3})\) obtained numerically will be replaced in the following formula to obtain the probabilities \(p_{k}\):  
\[p_{k}=e^{-1}e^{a_{k}\lambda_{1}}e^{k\lambda_{2}}e^{k^{2}\lambda_{3}},\quad k=0,1,2,\cdots. \tag{21}\]  
**Example 3**. : _Let us take the same data as in Examples 1 and 2 and calculate the initial probability vector using the analytical method (exact), entropy approach with first moment only (Entropy 1), entropy approach with second moment only (Entropy 2), and entropy approach with both first and second moments (Entropy 1&2). Tables 3 and 4 show the exact solution and the approximate solutions along with the corresponding percentage errors for \(\lambda=0.5\) and \(\lambda=5.5\), respectively._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Exact** & **Entropy 1** & **PE 1** & **Entropy 2** & **PE 2** & **Entropy 1\&2** & **PE 1\&2** \\ \hline \(p_{0}\) & 0.7519 & 0.7265 & 0.0338 & 0.6560 & 0.1275 & 0.7550 & 0.0040 \\ \(p_{1}\) & 0.1876 & 0.1794 & 0.0441 & 0.2892 & 0.5414 & 0.1795 & 0.0431 \\ \(p_{2}\) & 0.0465 & 0.0653 & 0.4049 & 0.0523 & 0.1237 & 0.0527 & 0.1324 \\ \(p_{3}\) & 0.0112 & 0.0206 & 0.8346 & 0.0025 & 0.7810 & 0.0108 & 0.0346 \\ \(p_{4}\) & 0.0024 & 0.0065 & 1.6986 & 0.0000 & 0.9858 & 0.0018 & 0.2421 \\ \hline Average & & & 0.6032 & & 0.5119 & & 0.0912 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial probability vectors comparison (\(\lambda=0.5\)). _We read from Table 6 three points: First, the approximation results obtained using Entropy 1 & 2 are clearly better than the ones obtained by the two other methods. Second, Entropy 1 is much better than Entropy 2, that is, if we are using a single moment, then better use the first moment than the second moment. Third, the efficiency of all three methods is directly proportional to the values of \(\mu_{1}\)._  
_We can see from Figure 4 that we always have PE \(1\&2<PE\ 1<PE\ 2\), which confirms our conclusions from Table 6 stated above._  
**Effect of \(\mu_{2}\) on the Percentage Error of \(p_{0}\). **  
_The results are summarized in Table 7 and Figure 5._  
_We read from Table 7 three points: First, the approximation results obtained using Entropy 1 & 2 are much better than the ones obtained by the two other methods. Second, if we are using a single moment, then better use Entropy 1 than Entropy 2.