A Multi-criteria Decision Making Approach for Scaling and Placement of Virtual Network Functions

Abstract

This paper investigates the joint scaling and placement problem of network services made up of virtual network functions (VNFs) that can be provided inside a cluster managing multiple points of presence (PoPs). Aiming at increasing the VNF service satisfaction rates and minimizing the deployment cost, we use both transport and cloud-aware VNF scaling as well as multi-attribute decision making (MADM) algorithms for VNF placement inside the cluster. The original joint scaling and placement problem is known to be NP-hard and hence the problem is solved by separating scaling and placement problems and solving them individually. The experiments are done using a dataset containing the information of a deployed digital-twin network service. These experiments show that considering transport and cloud parameters during scaling and placement algorithms perform more efficiently than the only cloud based or transport based scaling followed by placement algorithms. One of the MADM algorithms, Total Order Preference by Similarity to the Ideal Solution (TOPSIS), has shown to yield the lowest deployment cost and highest VNF request satisfaction rates compared to only transport or cloud scaling and other investigated MADM algorithms. Our simulation results indicate that considering both transport and cloud parameters in various availability scenarios of cloud and transport resources has significant potential to provide increased request satisfaction rates when VNF scaling and placement using the TOPSIS scheme is performed.

Appendix

1.1 TOPSIS

TOPSIS algorithm consists of easy implementation steps. After a decision matrix $\mathbf{A }_{s}=\left[ a_{ij}\right] _{K \times M}$ is created in the first step for a given request-$s \in {\mathcal {S}}$, a normalized decision matrix $\mathbf{N }_{s}=\left[ n_{ij}\right] _{K \times M}$ is formed in the second step (dropping the subscript s here on) as:

$$ n_{ij}=\frac{a_{ij}}{\sqrt{\sum \nolimits _{k=1}^{K}{a_{kj}^{2}}} }. $$

(8)

In the third step, a weighted normalized decision matrix $\mathbf{V }=\left[ v_{ij}\right] _{K \times M}$ is created by multiplying each column of the matrix $\mathbf{N }$ by a corresponding weight $w_{i}$ as:

$$ \mathbf{v }_{i} = w_{i} \cdot \mathbf{n }_{i}, $$

(9)

where

$$ \mathbf{n }_{i}=\left[ n_{1i},\ldots ,n_{pi}\right] ^{\text {T}},\quad \mathbf{v }_{i}=\left[ v_{1i},\ldots ,v_{pi} \right] ^{\text {T}},\quad \text {for}~i=1,2,\ldots ,M $$

In the fourth step, the positive $\text {V}^{+}$ and negative $\text {V}^{-}$ solution points are formed as:

$$\begin{aligned}&\text {V}^{+}=\left\{ v_{1}^{+},v_{2}^{+},\ldots ,v_m^{+} \right\} = \left\{ \max \limits _{i} v_{ij} | j \in \left\{ 1,2,\ldots ,M \right\} \right\} , \end{aligned}$$

(10)

$$\begin{aligned}&\text {V}^{-}=\left\{ v_{1}^{-},v_{2}^{-},\ldots ,v_{m}^{-} \right\} = \left\{ \min \limits _{i} v_{ij} | j \in \left\{ 1,2,\ldots ,M \right\} \right\} . \end{aligned}$$

(11)

In the fifth step, the Euclidean distance $L_{i}^{+}$ of each multiple decision point from the positive point $\text {V}^{+}$ and the Euclidean distance $L_{i}^{-}$ of each multiple decision point from the negative point $\text {V}^{+}$ are calculated as:

$$\begin{aligned}&L_{i}^{+} = \sqrt{\sum \limits _{j=1}^{p} \left( v_{ij}-v_{j}^{+} \right) ^{2} },\quad i=1,2,\ldots ,K, \end{aligned}$$

(12)

$$\begin{aligned}&L_{i}^{-} = \sqrt{\sum \limits _{j=1}^{p} \left( v_{ij}-v_{j}^{-} \right) ^{2} },\quad i=1,2,\ldots ,K. \end{aligned}$$

(13)

In the next step, the relative similarity of the alternatives from the positive and negative point is calculated as:

$$ T_{i} = \frac{L_{i}^{-}}{L_{i}^{-} + L_{i}^{+}},\quad i=1,2,\ldots ,K. $$

(14)

Then, the final solution $e^*$ (the best node selection for the placement of each containers that are scaled) is selected as:

$$ e^{*} = e_{i^{*}}\quad \text {where}\,i^{*}=\arg \max \limits _{i} T_{i},\quad i=1,2,\ldots ,K. $$

(15)

1.2 SAW

SAW algorithm is simple and is the most popular scoring method. In SAW, the score of each candidate node i is obtained by adding the contributions from each attribute $a_{i,j}$ multiplied by the weight factors $w_{j}$. Then, the selected node is

$$ e^{*} = e_{i^{*}}\quad \text {where}\,i^{*}=\arg \max \limits _{i} \sum \limits _{j} w_{j} a_{i,j}, $$

(16)

where $a_{i,j}=a_{i,j}/a_{j}^{+}$ for benefit parameters and $a_{i,j}=a_{j}^{-}/a_{i,j}$ for cost parameters and $a_{j}^{+} = \max \nolimits _{i} a_{i,j}$ and $a_{j}^{-} = \min \nolimits _{i} a_{i,j}$.

1.3 MEW

MEW or weighted product model (WPM) is another scoring method where the node scores are determined based on weighted product of the attributes. The selected node is

$$ e^{*} = e_{i^{*}}\quad \text {where}\,i^{*}=\arg \max \limits _{i} \prod \limits _{j} a_{i,j}^{w_{j}}. $$

(17)

1.4 GRA

GRA algorithm is basically based on building grey relationships between elements of two series in order to compare each member quantitatively. One of the series consists of best-quality entities and the other series contains comparative series. If the difference between two series of the comparative series is low, then it is more preferable. A Grey relational coefficient (GRC) is defined to be used for describing the relationships between two series and is calculated based on the level of similarity and variability. GRA algorithm is generally implemented in six steps:

• 1. Apply classification of the series based on three conditions: larger-the-better, smaller-the-better and nominal-the-best. In this paper, larger-the-better is selected as the comparison condition. It is assumed that D series $({\mathbf {A}}^{1},{\mathbf {A}}^{2}, \ldots , {\mathbf {A}}^{D})$ are compared where each series ${\mathbf {A}}^{i}=[a_{i,1},a_{i,2},\ldots ,a_{i,M}]$ has M entities.

• 2. The upper, lower and moderate bounds of series elements are defined. The upper bound and lower bound are defined as $u_{j}=max\{a_{1,j}, a_{2,j},\ldots ,a_{D,j}\}$ and $l_{j}=min\{a_{1,j}, a_{2,j},\ldots ,a_{D,j}\}$ respectively where $j=1,2,\ldots ,M$.

• 3. Normalize the individual entities: before calculating GRC, the decision matrix $\mathbf{A }$ needs to be normalized. For normalization, following equation is used for larger-the-better condition.

  $$ a_{ij}=\frac{a_{ij}-l_{j}}{u_{j}-l_{j}}, $$

  (18)

  where $i=1,2,\ldots ,D$.

• 4. Define the ideal series: a reference series ${\mathbf {A}}^{0} = [a_{0,1}^{*},a_{0,2}^{*},\ldots ,a_{0,M}^{*}] = [u_{1},u_{2},\ldots ,u_{M}]$ is formed which corresponds to ideal solution.

• 5. Calculate the GRCs: the GRC can be calculated from

  $$ \varGamma _{0,i}=\frac{1}{M} \sum \limits _{j=1}^{M} w_{j} \frac{\varDelta _{min}+\varDelta _{max}}{\varDelta _{i}+\varDelta _{max}}, \quad i=1,2,\ldots ,D, $$

  (19)

  where $\varDelta _{i}=|a_{0,j}^{*}-a_{i,j}|$, $\varDelta _{max} = \max \nolimits _{i,j} (\varDelta _{i})$ and $\varDelta _{min} = \max \nolimits _{i,j} (\varDelta _{i})$. $\max \nolimits _{i,j} ()$ and $\max \limits _{i,j} ()$ are the functions of the maximum and minimum value of a set of numbers with varying i and j respectively.

• 6. Select the alternative that has the highest GRC

  $$ e^{*} = e_{i^*}\quad \text {where}\,i^{*}=\arg \max \limits _{i} \varGamma _{0,i}. $$

  (20)

1.5 ELECTRE

ELECTRE method is based on outranking relation theory and is another method that analyzes data over the decision metric. When making a decision, the concordance and discordance indexes are used to measure the amount of dissatisfaction during the decision making process. After obtaining normalization decision matrix $\mathbf{N }=\left[ n_{ij}\right] _{K \times M}$ as in (8) and weighted normalized decision matrix $\mathbf{V }$ as in (9), the concordance and discordance sets are applied. The set of criteria is divided into two different subsets. Denote ${\mathcal {K}} = \{k_{1}, k_{2}, k_{3}, \ldots , k_{K}\}$ a finite set of alternatives. In the following formulation, the data is divided into two different sets of concordance and discordance. If the alternative $k_{1}$ is preferred over alternative $k_{2}$ for all the criteria, then the concordance set is composed as follows:

$$ C(k_{1},k_{2}) = \{j | v_{k_{1}j} > v_{k_{2}j} \}, \quad (k_{1},k_{2} = 1, \ldots , M \;\; \text {and} \;\; k_{1} \ne k_{2}), $$

(21)

where $C(k_{1},k_{2})$ denotes the collection of attributes in which $k_{1}$ is better than, or equal, to $k_{2}$. Then the concordance index of $(k_{1}, k_{2})$ is defined as:

$$ C_{k_{1},k_{2}} = \sum _{j^{*}} w_{j}, $$

(22)

where $j^{*}$ are the attributes contained in the concordance set $C(k_{1},k_{2})$. Similarly the discordance set is defined as:

$$ D(k_{1},k_{2}) = \{j | v_{k_{1}j} < v_{k_{2}j} \}, \quad (k_{1},k_{2} = 1, \ldots , M \;\; \text {and} \;\; k_{1} \ne k_{2}). $$

(23)

The discordance index $D_{k_{1},k_{2}}$ represents the degree of disagreement in $k_{1} -> k_{2}$ in the following way:

$$ D_{k_{1},k_{2}} = \frac{\sum _{j^{+})} | v_{k_{1}j^{+}} - v_{k_{2}j^{+}} | }{ \sum _{j)} | v_{k_{1}j} - v_{k_{2}j} | } , $$

(24)

where $j^{+}$ are the attributes contained in the discordance set $D(k_{1},k_{2})$ and $v_{ij}$ is the weighted normalized evaluation of the alternative i on criterion j. This method implies that $k_{1}$ outranks $k_{2}$ when $C_{k_{1},k_{2}} \ge {\hat{C}}$ and, $D_{k_{1},k_{2}} \le {\hat{D}}$ where ${\hat{C}}$ is the averages of $C_{k_{1},k_{2}}$ and ${\hat{D}}$ is the averages of $D_{k_{1},k_{2}}$.