# Process Mining IPTV Customer Eye Gaze Movement Using Discrete-Time Markov Chains +
## 4 Results
### DTMC-\"Content Purchase\" Screens
#### 4.1.2 Expected Time to Payment-Screen A&B  
For screen A, the transition matrix can be taken and separated into subsections, as shown in Equation (15). Given there are 7 transient states and 1 absorbing state, \(m=7\) and \(n=1\), therefore we calculate the \(Q\) and \(R\) sections in the transition matrix, and the results are shown in Figure 15 below. \[Q= \begin{bmatrix}0.8216&0.0094&0.0235&0&0&0&0.1455\\ 0.005&0.9005&0.01&0.005&0&0&0.0796\\ 0.0014&0.0014&0.7968&0.009&0.0069&0&0.1832\\ 0.0016&0.0016&0.0126&0.739&0.0047&0.0016&0.2382\\ 0&0&0.0063&0.0013&0.7747&0.0025&0.2153\\ 0&0&0&0.0051&0.0153&0.7913&0.1883\\ 0.007&0.0033&0.0641&0.0305&0.0315&0.0167&0.8435\end{bmatrix} R= \begin{bmatrix}0 \\ 0\\ 0.0014\\ 0\\ 0\\ 0.0033\end{bmatrix} \]  
Figure 14: (**a**) Transition Matrix for Screen B; (**b**) DTMC Plot for Screen B.  
Figure 15: Screen A: (**a**) Section \(Q\) in the transition matrix; (**b**) Section \(R\) in the transition matrix. The Fundamental Matrix can then be calculated using Equation 16, where \(I_{n,}\) in this case, will be an identity matrix of size \(m\) =7, so we have \(N\) in Figure 16 below. Next, using Equation 17, the expected number of steps to absorption can be calculated, as given there are 7 transient states, \(c\) will be a 7 \(\times\) 1 column vector of 1's, and therefore \(t\) will also be a matrix of size 7 \(\times\) 1. Besides, given that \(t\) represents the expected number of steps to absorption for all states, then \(t_{i}\) represents the expected number of steps to absorption from a given state \(i\). In this case, the states are A, B, C, D, E, F and Z. Also, considering the created discrete Markov chain follows a time step of 0.165 s between states, the expected number of steps for each state will be multiplied by 0.165 to get the real expected time to absorption from each state. Table 2 shows the calculated expected time to absorption from each state. From these calculations, it can be inferred that despite the initial state (or AOI on screen A) being fixated on, it takes relatively the same time to reach the payment state, with less than 0.2 s of a difference between the 'quickest' initial state Z and the'slowest' initial state B. Likewise, the calculations by Equations (15)-(17) are applied to screen B. For this screen, there are 5 transient states and 1 absorbing state, so \(m\) = 5 and \(n\) = 1. The Q and R sections in the transition matrix and the results are shown in Figure 17 below. The Fundamental matrix, in this case, by Equation (16), is shown in Figure 18. \begin{table}
\begin{tabular}{c c} \hline \hline
**AOI/State Name** & **Expected Time to Absorption** \\ \hline A & 7.88 s \\ B & 7.96 s \\ C & 7.81 s \\ D & 7.84 s \\ E & 7.85 s \\ F & 7.86 s \\ Z & 7.77 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Expected time to absorption from each state–DTMC for screen A.  
Figure 16: Fundamental Matrix \(N\) for DTMC–Screen A.  
Figure 17: Screen B: (**a**) Section \(Q\) in the transition matrix; (**b**) Section \(R\) in the transition matrix. \[N= \begin{bmatrix}25.404&39.336&136.98&82.125&302.34\\ 15.719&48.831&136.73&82.173&302.35\\ 15.647&38.892&42.84&82.406&301.64\\ 15.574&38.866&136.23&87.382&300.6\\ 15.597&38.817&136.08&82.078&302.45\\ \end{bmatrix} \]  
Again, using Equation (17), the expected number of steps to absorption can be calculated, as given there are 5 transient states, \(c\) will be a \(5\times 1\) column vector of 1's, and therefore \(t\) will also be a matrix of size \(5\times 1\). For screen B, the states are A, B, C, D, and Z. Table 3 below shows the calculated expected time to absorption for the DTMC on screen B. According to Table 3, it can be concluded that irrespective of the initial state, it takes approximately the same expected time to the absorption state (Payment screen), with almost the same range between the 'quickest', again the Z state, and the'slowest', both A and B have the same expected time.