# Holistic Utility Satisfaction in Cloud Data Centre Network Using Reinforcement Learning
## 3 Modelling of the CDCN
### The Utility Function Model for Cloud Users
One of the best functional forms that can meet the mentioned requirements is the product logistic function. The product logistic form for modelling energy-efficient data centres has been used in previous work such as [58]. Furthermore, the proposed product logistic model is in fact a non-linear extension of the utility model previously proposed in [59]). Therefore, we selected this function for modelling the user utility/satisfaction for each cloud user \(i=1,2,\cdots,M(t)\) in Equation (2):  
\[\mathcal{U}^{(i)}(t)=1+\mathcal{U}_{max}\Bigg{\{}\bigg{(}1+k_{1}^{(i)}\text{ exp}\Big{(}-\alpha_{1}^{(i)}\mathcal{C}^{(i)}(t)\Big{)}\bigg{)}\times\bigg{(}1+k_{2}^ {(i)}\text{exp}\Big{(}-\alpha_{2}^{(i)}\mathcal{D}^{(i)}(t)\Big{)}\bigg{)} \times\\ \bigg{(}1+k_{3}^{(i)}\text{exp}\Big{(}-\alpha_{3}^{(i)}\mathcal{G} ^{(i)}(t)\Big{)}\bigg{)}\times\bigg{(}1+k_{4}^{(i)}\text{exp}\Big{(}-\alpha_{ 4}^{(i)}\mathcal{B}^{(i)}(t)\Big{)}\bigg{)}\Bigg{\}}^{-1}\\ \times\text{exp}\Big{(}-\alpha_{5}^{(i)}\mathbb{E}[\mathcal{L}^{ (i)}(t)]-\alpha_{6}^{(i)}\mathbb{E}[\mathcal{H}^{(i)}(t)]\Big{)} \tag{2}\]  
where \(\mathcal{U}_{max}\) is the maximal level of SLA satisfaction (based on [60], \(\mathcal{U}_{max}\) was selected to be four). \(k^{(i)}>0\) and \(\alpha^{(i)}>0\) are the parameters associated with the logistic function. These parameters represent the steepness and horizontal drift associated with the logistic function. Therefore, these parameters must be selected based on the application types and characteristics for each user \(i\). \(\mathbb{E}[\mathcal{L}^{(i)}(t)]\) and \(\mathbb{E}[\mathcal{H}^{(i)}(t)]\) are the expected total packet loss ratio and round-trip time (RTT) associated with user \(i\) at time slot \(t\) and can be described as Equation (3) [23]:  
\[\mathcal{L}^{(i)}(t)=1-\prod_{j\in\mathcal{N}\setminus\mathcal{N}^{-i}(t)}(1- \mathcal{L}^{(i)}_{j}(t)) \tag{3}\]  
where \(\mathcal{L}^{(i)}_{j}(t)\) is the packet loss ratio associated with the communication path between user \(i\) and data centre \(j\) at time slot \(t\). \(\mathcal{N}\) and \(\mathcal{N}^{-i}(t)\) are the set of all data centres and the subset of those data centres that do not assign any resource to user \(i\) at each time frame \(t\), respectively. If we assume a _Poisson_ distribution of known mean rate parameter \(\lambda^{(i)}_{j}\) for the packet loss process of each communication link between user \(i\) and data centre \(j\)[61] and if we assume that the loss processes in each path are statistically independent, we can write the expected total packet loss for user \(i\) as Equation (4):\[\mathbb{E}[\mathcal{L}^{(i)}(t)]=1-\prod_{j\in\mathcal{N}\setminus\mathcal{N}^{-i}(t) }(1-\mathbb{E}[\mathcal{L}^{(i)}_{j}(t)])=1-\prod_{j\in\mathcal{N}\setminus \mathcal{N}^{-i}(t)}(1-\lambda^{(i)}_{j}) \tag{4}\]  
We assumed that the round-trip times associated with multiple communication paths between user \(i\) and the data centres is dominated by the largest one. Therefore, we can write, according to Equation (5):  
\[\mathcal{H}^{(i)}(t)=\sup_{j\in\mathcal{N}\setminus\mathcal{N}^{-i}(t)}\{ \mathcal{H}^{(i)}_{j}(t)\} \tag{5}\]  
where \(\mathcal{H}^{(i)}_{j}(t)\) is the measured round-trip time associated with the communication path between user \(i\) and data centre \(j\) at each time frame \(t\).