I am having trouble with a summation notation in a paper I am reading (talking about Semi Markov Processes), I am not how to use it. 


The equation is as follow: 


$$MTTSF\_{\phi}=\sum\_{i - 1}\frac{1}{\xi\_{i}}\sum\_{j}x\_i.\theta\_{ij}$$


Which give you the Mean Time to Security Failure of your system. 
Here are the definition of the variables: 


$x\_i \in \mathbb{N}\_{0}^{+}$. $x\_i$ is expressed in time units. $\theta\_{ij}$ is an element of a matrix, that represents the existence of a transition in a Semi Markov Process, between a state $i$ and a state $j$. 


Finally, $\xi\_i$ is computed as follow: $\sum\_{l \in \{j\in J | \theta\_{ij} > 0\}} 1$. This mean that for a given state $i$ and for each $\theta\_{ij} > 0$ for this $i$ in the matrix, you add $1$ tp $\xi\_i$. 


$x\_i$ is computed using the following formula: $\frac{TTDV\_i}{PoCoB\_i}$. These are annotations on the SMP states. $TTDV\_i$ is the Time to Discover Vulnerability, expressed in time units, and $PoCoB\_i$ is the Probability of Component Break. These two values are considered as inputs and expressed like this: $(TTDV\_i, PoCoB\_i))$


As an example, authors provide the following example: 
SMP States are named as follow: $S=\{Access, Web, Pars, DA, DB, \Omega\}$
They are annotated as follow: $X=\{(200, 0.1264), (250, 0.1896), (125, 0.2528), (150, 0.1264), (300, 0.1264)\}$, in order of states apparition.


As you can see, $\Omega$ is the absorbing state of the SPM, hence it is not annotated as other states.


The matrix $\Theta$ is as follow: 
$$\begin{pmatrix}
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 2 & 0 \\
0 & 0 & 0 & 0 & 0 & 2 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}$$


Based on that, you can compute every $\xi\_i$ for each component, wich give you the vector: $\Xi=\{1,1,1,1,1,0\}$


Using these information, authors obtain $MTTSF\_{\phi} = 5256,78~time~units$ 


However, doing it manually I obtain the following (I am not writing the $\theta\_{ij} = 0$): 
$1 \* \frac{200}{0.1264} + 1\*\frac{250}{0.1896} + 1 \* \frac{125}{0.2528} + 2\*\frac{150}{0.1264} + 2\*\frac{300}{0.1264} = 10515,55907~time~units$.


What I would like to ask is the following: 


does the first sum of the equation $\sum\_{i-1}\frac{1}{\xi\_i}$ means that, for each state $i$ you compute $\frac{1}{\xi\_i}$, and then you compute the second sum, for each $j$. Then, you repeat for each $i$ until you reach $i-1$ (which is, I believe, the state space length minus 1)? I am not so sure as I am actually more strict on my notations, and always write a sum properly, with the starting element below and the bound on top. 
If anyone could maybe verify, it would be very appreciated. Thanks a lot.



Summation unknown notation
Actually I have identified the problem. In the paper, authors example used an SMP where the attacker can chose between two paths, both with the same probability $\frac{1}{2}$. Because the start state is not taken into account in the system fromalized control flow, I belive that they made a mistake which resides here. 


