# Energy-Aware Online Non-Clairvoyant Scheduling Using Speed Scaling with Arbitrary Power Function
## 4 A 2-Comptitive Scheduling Highest Scaled Importance First (HSIF)
### Scaled Importance-Based Flow Plus Energy
#### 4.1.2 Potential Function \(\Phi(t)\)
**Case 2: If \(pr_{o}=pr_{h}\) then, one can observe that \(\forall a\in[0,\ a_{o}]\) there is a decrement in \(pr_{o}(a)\) at the rate of \((s_{o}/a_{o})\), due to which the possible maximum rate of increment in \(\Phi\) is:**  
\[\frac{d\Phi}{dt}\leq 2\int_{0}^{a_{o}}P^{\prime}(Q(pr(a)))\bigg{(}\frac{s_{o}}{a _{o}}\bigg{)}\ da \tag{2}\]  
\(\forall a\in[0,\ a_{o}]\), \(pr(a)=(pr_{h}(a)-pr_{o}(a))_{+}\leq pr_{h}-pr_{o}=0\),  
\[\Rightarrow P^{\prime}(Q(pr(a)))=P^{\prime}(Q(0)) \tag{3}\]In Equation (1) substituting the values for \(f(x)=P^{\prime}(x)\), \(m=s_{o}\) and \(n=P^{\prime}(Q(0))\), it provides:  
\[\int_{0}^{s_{o}}f(x)\;dx+\;\;\;\int_{0}^{P^{\prime}(Q(0))}f^{-1}(x)\;dx\geq\;s_{ o}\cdot P^{\prime}(Q(0)) \tag{4}\]  
Using Equations (3) and (4) in (2), it provides  
\[\frac{d\Phi}{dt}\leq 2\int_{0}^{a_{o}}P^{\prime}(Q(pr(a)))\bigg{(}\frac{s_{o}} {a_{o}}\bigg{)}\;da=2\;s_{o}\cdot P^{\prime}(Q(0))\]  
\[\leq 2\bigg{(}\int_{0}^{s_{o}}f(x)\;dx+\;\int_{0}^{P^{\prime}(Q(0))}f^{-1}(x)\; dx\bigg{)}=2\;P(s_{o})\]  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]0  
Hence the running condition is satisfied for \(pr_{o}=pr_{h}\). **Case 3:** If \(pr_{o}<pr_{h}\) then, one can observe that a decrement in \(pr_{h}(a)\) creates a decrement in \(\Phi\) and a decrement in \(pr_{o}(a)\) creates an increment in \(\Phi\). \(\forall a\in[0,\;a_{h}]\), there is a decrement in \(pr_{h}(a)\) at the rate of \((s_{h}/a_{h})\), due to which the possible rate of change of \(\Phi\) is:  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]1  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]2  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]3  
\(\forall a\in[0,\;a_{o}]\), there is a decrement in \(pr_{o}(a)\) at the rate of \((s_{o}/a_{o})\), due to which the possible rate of change of \(\Phi\) is  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]4  
\(\forall a\in[0,\;a_{o}]\), \((a)=\left(pr_{h}(a)-pr_{o}(a)\right)_{+}\leq\left(pr_{h}-pr_{o}\right)\), thus  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]5  
Adding the Equations (6) and (8)  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]6  
Let i, \(s_{h}\) and \(s_{o}\geq 0\) be real numbers. Since P is strictly increasing and convex, \(P^{\prime}(0)\geq 0\) and \(P^{\prime}(x)\) is strictly increasing. Substituting the values of f(x) = P\({}^{\prime}\)(x), m = s\({}_{o}\) and n = P\({}^{\prime}\big{(}Q\big{(}pr_{h}-pr_{o}\big{)}\big{)}\) in Equation (1), it provides  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]7  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]8  
\[\frac{d\Phi(t)}{dt}=2\int_{0}^{\infty}P^{\prime}\big{(}Q\big{(}pr^{t}(a)\big{)} \big{)}\frac{d\big{(}pr^{t}(a)\big{)}}{dt}\ da\]9  
\[pr_{h}^{t}+P\big{(}s_{h}^{t}\big{)}+\frac{d\Phi(t)}{dt}\leq c\cdot\big{(}pr_{o }^{t}+P\big{(}s_{o}^{t}\big{)}\big{)},\textit{where }c=2\]0Substituting the values from Equation (11) to (9), it provides  
\[ \begin{array}{ll}\frac{d\Phi}{dt}\leq 2P(s_{o})&+2P^{\prime}(Q(pr_{h}-pr_{o})) \cdot Q(pr_{h}-pr_{o})-2(pr_{h}-pr_{o})\\ &-2P^{\prime}(Q(pr_{h}-pr_{o}))s_{h}\\ &=2(P(s_{o})-(pr_{h}-pr_{o}))+2P^{\prime}(Q(pr_{h}-pr_{o}))\cdot(Q(pr_{h}-pr_{o} )-s_{h})\end{array}