We begin by presenting a few notations that will be used in the rest of the document. This notation is used below to construct our optimization model.

C: The switches in the network where i with ( $i\in C$) represents switch i.

L: The links in the network where ( $\left(i,j\right)\in L;i\ne j$), represents the link between switches i and j.

n: Total number of switches in the network.

$C_{ij}$: The bandwidth or capacity of the link connecting switches i and j with $i\ne j$.

$E_i$: Power consumption by switch i when it is activated.

$E_{ij}$: Energy consumption of the link between switch i and j with ( $i\ne j$).

F: All network flows $f\in F$.

$f_{ij}=\left(sr,ds,{\lambda }_f\right)$: flows between switches i and j.

${\omega }_i$: Binary variable. If ${\omega }_i=1$ then switch i is enabled; otherwise ${\omega }_i=0$.

${\sigma }_{ij}$: Binary variable, if ${\sigma }_{ij}=1$ then a link between switches i and j is activated, 0 otherwise.

In this paper, we consider the FAT TREE data centre network as a weighted directed graph $G=\left(C,L\right)$ where $C=\left\{1,\cdots ,n\right\}$ represents a finite set of nodes (switches) in the graph; ( $\left(i,j\right)\in L;i\ne j$) represents a finite set of arcs (links) between switches, where each link is used for communication in both directions. Figure 1 shows two switches i and j interconnected by an $\left(i,j\right)$ link.

The network topology is modelled using a weighted directed graph $G=\left(C,L\right)$ with a set of vertices C, and a set of edges L. Each node is an OpenFlow switch and a switch is represented by the letter i. The role of each switch is to transmit information according to the path chosen by the network controller. Each edge of the graph is a link. The transmission of packets between two ports forms a link. The link between switches i and j is called $\left(i,j\right)$. The links and switches in the network can be in ON or OFF mode.

We define the binary variables ${\omega }_i$ and ${\sigma }_{ij}$: ${\omega }_i$ to specify the current mode of the switch. is 1 if switch i is ON, 0 otherwise. Furthermore, the variable ${\sigma }_{ij}$ specifies the mode state of the link between two connected ports. ${\sigma }_{ij}=1$ if the link between switches i and j is active and 0 otherwise.

We define another binary variable $f_{ij}$ to identify flows on the DCN topology. A flow represents a group of packets with the same source and destination addresses, using the same link to reach the destination.

The variable $f_{ij}$ takes the value 1 if the flow f crosses the link $\left(i;j\right)$ with $\left(i\ne j\right)$ and the value 0 otherwise. The link utilisation matrix, U, is constructed by considering the utilisation of each link. DCN traffic is represented by the set of flows $\mathbb{F}$, where each $f\in \mathbb{F}$ is defined as $f_{ij}=\left(sr,ds,{\lambda }_f\right)$. Source and destination switches are represented respectively by sr and ds. ${\lambda }_f$, transmission rate of f packets measured in bits per second.

In this model, we consider links and switches as the main energy-consuming components of the network. Our deployment model allows them to be dynamically enabled and disabled to save energy. To support distributed traffic, the model uses the smallest possible set of active links and the fewest possible switches. Our optimisation model considers the following features:

1) For the sake of simplicity, the model parameters are based on the state of the network at a given time.

2) The model is defined as a problem of flows of various commodities.

3) the problem is defined as allocating links and switches to each flow while minimising the total number of active links and switches.

4) Given the model formulation and parameters, the result of the optimizer is a list of active links and switches for each flow $f\in F$.

5) In our model, we assume that all links have the same energy consumption and that this is a fixed constant. Switches also have a similar constant energy consumption. In other words, and are constants in our problem.

We formulate the SD-DCN energy optimisation problem as an integer linear programme (ILP). To better characterise the problem, we define binary variables for each switch and link in the network.

${\omega }_i$: Binary variable indicating whether switch i is active (1) or on standby (0).

${\omega }_i=\{\begin{array}{l}1,\text{if}\text{the}\text{Switches}i\text{is}\text{activated}\\ 0,\text{otherwise}\end{array}$

${\sigma }_{ij}$: Binary variable indicating whether the link between switches i and j is active (1) or not (0).

${\sigma }_{ij}=\{\begin{array}{l}1,\text{if}\text{a}\text{link}\text{between}\text{switches}i\text{and}j\text{is}\text{activated}\\ 0,\text{otherwise}\end{array}$

$f_{ij}$: Packet flow over the link between switches i et j.

The optimisation problem described at the beginning of this section can be modelled as follows:

$\min \left(\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\omega }_i\cdot E_i+\underset{i=1}{\overset{n}{{\displaystyle \sum }}}\underset{j=1;i\ne j}{\overset{n}{{\displaystyle \sum }}}{\sigma }_{ij}\cdot E_{ij}\right)$(1)

At the same time as relation (1), certain constraints are necessary to meet the limitations of the problem.

$\forall i;j$ j among the switches, with, $i\ne j$,

${\omega }_i+{\omega }_j-1\le {\sigma }_{ij}\le {\omega }_i;{\sigma }_{ij}\le {\omega }_j$(2)

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\omega }_i\le K$(3)

$f_{ij}\le C_{ij}{\sigma }_{ij},\forall i;j\text{et}i\ne j$(4)

$\underset{i\ne k}{{\displaystyle \sum }}{f}_{ik}=\underset{j\ne k}{{\displaystyle \sum }}{k}_j,\forall k\notin S\cup T$(5)

$\underset{j:\left(s,j\right)\in \text{link}}{{\displaystyle \sum }}{f}_{sj}-\underset{i:\left(i,s\right)\in \text{link}}{{\displaystyle \sum }}{f}_{is}\ge D_{st},\forall \left(s,t\right)\notin S\times T$(6)

$\underset{i:\left(i,t\right)\in \text{link}}{{\displaystyle \sum }}{f}_{it}-\underset{j:\left(t,j\right)\in \text{link}}{{\displaystyle \sum }}{f}_{tj}\ge D_{st},\forall \left(s,t\right)\notin S\times T$(7)

${\omega }_j\le {\displaystyle {\sum }_{i\ne j}{\sigma }_{ij}}+{\displaystyle {\sum }_{i\ne j}{\sigma }_{ji}},\forall j$(8)

${\omega }_j\le {\displaystyle {\sum }_{i\ne j}\left(f_{ij}+f_{ji}\right)}$, $\forall j$ is neither source nor destination(9)

Relation (1) is made up of two terms; the first term represents the energy consumption of the links traversed by a network flow, and the second term concerns the energy consumption of all the active switches in the network. Constraint (2) concerns network connectivity. Constraint (3) stipulates that the controller cannot simultaneously manage a certain number of switches more than the maximum capacity. The constraint is given in Equation (4), which represents the flow constraint and states that the total packet throughput between two switches cannot exceed the capacity of the link. Constraint (5) represents the flow conservation constraint and states that the number of incoming and outgoing packets for switches that are neither the destination nor the source of the flow must be equal. Constraints (6) and (7) are path constraints for the source and destination and indicate that the flow from the source switch must reach the destination switch. Constraint (8) ensures that no flow passes through an idle switch. Constraint (9) states that if no flows pass through the links connected to a given switch, the switch will be disabled.

Relation (1) is made up of two terms; the first term represents the energy consumption of the links traversed by a network flow, and the second term concerns the energy consumption of all the active switches in the network.

Constraint (2) concerns network connectivity. Constraint (3) states that the controller cannot simultaneously manage a certain number of switches in excess of the maximum capacity. The constraint is given in Equation (4), which represents the flow constraint and states that the total packet throughput between two switches cannot exceed the link capacity. Constraint (5) represents the flow conservation constraint and states that the number of incoming and outgoing packets for switches that are neither the destination nor the source of the flow must be equal. Constraints (6) and (7) are path constraints for the source and destination and indicate that the flow from the source switch must reach the destination switch. Constraint (8) ensures that no flow passes through an idle switch. Constraint (9) states that if no flows pass through the links connected to a given switch, the switch will be disabled.