## 3 Consistent Path Congestion Model
### Consistent Path Congestion Model and Its Related Algebra
#### 3.1.3 Cumulative Differential Path Congestion Consistency Indicator  
To predict a stable high-volume traffic period, first we try to capture evidence that the _CDPCI \(W_{ij}(\theta,T,\Delta t_{n})\) increases_, which denotes \"_the situation that the congestion of the current path is worse than the backup path to a certain degree happens more/less often.\"  
We use the indicator \(s_{W,+/-}(\Delta t_{n})\) to denote the process that \(W_{ij}(\theta,T,\Delta t_{n})\) increases\(/\) decreases comparing to the last value \(W_{ij}(\theta,T,\Delta t_{n-1})\).  
\[s_{W,+}(\Delta t_{n})=\begin{cases}1,&\text{if }W_{ij}(\theta,T,\Delta t_{n})>W_{ ij}(\theta,T,\Delta t_{n-1})\\ 0,&\text{otherwise.}\end{cases} \tag{6}\]  
\[s_{W,-}(\Delta t_{n})=\begin{cases}1,&\text{if }W_{ij}(\theta,T,\Delta t_{n})<W_{ ij}(\theta,T,\Delta t_{n-1})\\ 0,&\text{otherwise.}\end{cases} \tag{7}\]  
The Cumulative Differential Path Congestion Consistency Indicator (CDPCCI) \(S_{+/-}(\theta_{8+/-},T_{s+/-},\Delta t_{n})\) is used to describe the situation that \"_the congestion of the current path is worse than the backup path to a certain degree more/less often, and the trend of increase/decrease has continued for more than a certain number of times during a recent time period_\", denoted by \"\(s_{W,+/-}(\Delta t_{n})\) keepsincreasing/decreasing for more than a certain number of times during a recent certain time period\"_, which in turn can be described with Equations (8) and (9) as shown below, in which \(\theta_{s+/-}\) is the Latency Consistency Sensitivity, which represents the \"_number of times that \(s_{W,+/-}(\Delta t_{n})\) has increased/decreased\"_, and \(T_{s+/-}\) is the Consistency Cumulative Window, which is the time period that is taken into consideration when measuring the CDPCCI.  
\[S_{W,+}(\theta_{s+},T_{s+},\Delta t_{n}) =\begin{cases}1,&\text{if }\sum_{n=0}^{T_{s+}/\Delta t}s_{W,+}( \Delta t_{n})>\theta_{s+}\\ 0,&\text{otherwise.}\end{cases} \tag{8}\] \[S_{W,-}(\theta_{s-},T_{s-},\Delta t_{n}) =\begin{cases}1,&\text{if }\sum_{n=0}^{T_{s-}/\Delta t}s_{W,-}( \Delta t_{n})>\theta_{s-}\\ 0,&\text{otherwise.}\end{cases} \tag{9}\]  
The meanings of \(S_{W,+}(\theta_{s+},T_{s+},\Delta t_{n})\) and \(S_{W,-}(\theta_{s-},T_{s-},\Delta t_{n})\) are explained as follows. When \(S_{W,+}(\theta_{s+},T_{s+},\Delta t_{n})=1\), it means path \(j\)'s situation will be better than path \(i\) for the foreseeable future, and this situation will last longer than the route switching time consumption, which is less than 1 s in our experiment. The values of \(\theta_{s+}\) and \(\theta_{s-}\) have a strong impact on the performance of the indicator's accuracy, and thus should not be too large or too small. If the \(\theta_{s+}\) value is too large, it means that the indicator \(S_{W,+}\) stays equal to 0 (which means \"stay in the current path\") even when the current channel's situation is worse than the backup one, thus causing unnecessary congestion, delay, and further QoS deterioration. When \(\theta_{s+}\) is too small, by contrast, it may make the indicator \(S_{W,+}\) too sensitive, jumping between the two paths with slight path latency jittering and causing routing oscillations. Similarly, when the \(\theta_{s-}\) value is set too high, the algorithm may fail to switch back to the original path when the backup path's situation deteriorates and becomes more congested than the original one, causing QoS deterioration. When \(\theta_{s-}\) is too small, indicators will be too sensitive to the path's latency changes and cause router flapping and network oscillation. The ideal threshold values are empirical facts, and may not be constant values. They depend on the real-time traffic on the paths in the network.  
In our evaluation experiment, where the traffic data used were from real-life Abilene peak time traffic data, we found that the ideal values \(\theta_{s+}=6\) and \(\theta_{s-}=-7\) or \(-6\) best indicate the situation of our experimental setup, as will be discussed in detail in Section 4.