Elsevier

Information Sciences

Volume 514, April 2020, Pages 512-522
Information Sciences

A congestion game framework for service chain composition in NFV with function benefit

https://doi.org/10.1016/j.ins.2019.11.015Get rights and content

Abstract

The network function virtualization (NFV) is a network paradigm that aims at softwarizing network functions that usually have been implemented on specific hardware devices. In this paper, the service chain composition problem in NFV with function benefit is investigated. Using the non-cooperative game theory, the service chain configuration is converted to the seeking for the Nash equilibrium of the formulated game. The semi-tensor product (STP) of matrices is used as the algebraic tool for the game formulation and the Nash equilibrium calculation. The results on the service chain composition of the NFV are elaborated with a numerical example.

Introduction

The network function virtualization (NFV) [16], [1] is a network paradigm that uses virtualization related technologies to virtualize network function into virtual building blocks that are connected together to create communication services. Traditionally, these network functions are implemented to handle heavy traffic loads on dedicated network devices, which are expensive and inflexible to implement. NFV, a new networking paradigm, can transform the ossified internet in a more agile, flexible and dynamic ecosystem. Therefore, NFV is developing rapidly among the network market.

In the case of NFV network, a significant problem is service chain composition [5], which determines the required network functions and paths between them for the network service requests. Considerable work has been performed concerning the service chain configuration problem. For instance, integer linear programming formulation has been widely applied to optimize the network performance and provide solutions to the service chain composition problem. An optimization approach [8] has been proposed to provide a efficient orchestration scheme to mitigate congestion. In fact, the complex tasks of management, orchestration and resource allocation are in charge of only one entity, Telco Operator, which makes the deployment of the NFV paradigm unfeasible.

Compared with the centralized algorithms, game-theoretic control [7] brings a natural scheme to investigate the strategic interactions between the consuming users of the network system. It not only complements the insights gained by viewing distributed systems from a control theoretic perspective, but the outcomes predicted by the game paradigm can also serve as a benchmark for any control strategies that may be applied. As each network user selfishly aims to maximize their own benefits or minimize costs, congestion game [20] in which users share a common set of limited resources is a natural framework to seek the best operational point of the system.

The drawbacks of centralized placement solutions and a distributed service chain composition solution by exploiting congestion games are investigated in [6], [12]. D’oro et al., [6] studied a scenario where traffic flows generated by users contributes the congestion differently on the servers. [12] considered a situation where the traffic flow suffered different levels of influence of facility congestion. Considering the real-time operation of the network [17], [19], the cost such as bandwidth rate can fluctuate due to disturbances from environment. To feed distributed management entities the necessary information to perform dynamic configuration changes, efficient composition mechanism need to be implemented. As pointed in [19], the practical environment disturbs the real rates transferred by the network users, and the bandwidth remaining rate of nominal rate is of crucial importance. However, the researches discussed above did not consider this issue. So the extension of the service chain composition to include different bandwidths and sensitivities is necessary. Considering the player-specific character, this paper addresses the service chain composition problem in NFV networks from the congestion games with player-specific utility Function (CGPS) [12] game prospective, a particular class of congestion game. Compared with [12], this paper not only considers the incurred costs to the Virtual Network Functions (VNFs) servers providing the VNFs, also takes into account the actual perceived rates to fit well with the practical network system. In order to accommodate the requirements in actual application, appropriate parameters are introduced to weigh the contributions to the utility function of the different cost factors encountered when reaching the VNF servers. Finally in the game framework, this paper gives the explicit scheme to seek the desired solution.

When the players set is finite and each player has finite strategies to choose, the finite game is actually a finite-valued logic system. To analyze logical control networks [3], [9], [13], [24], the semi-tensor product of matrices (STP) [4] has been originally proposed as a generalization of conventional matrix multiplication, and its applications have been extended to various formulations such as stabilization, observer design of Boolean control networks [11], [26], [31], [32] and practical engineering problems [15], [27], [33]. After successful applications to the analysis and control of logical systems, the STP method can also be used to study finite games including potential equation [2], [14], control of evolutionary games [10] and Nash equilibrium seeking for dynamic games [28] since the matrix expression in the fashion of the STP is suitable for the implementation of the numerical algorithms. Whereas the effective algorithms have been addressed in the existing reports, the general game formulation based on the weighted game and its application including NFV has not been considered and still a challenging problem.

The main contributions of the paper include:

  • 1.

    A more realistic NFV model is presented for the actual perceived rates that may fluctuate due to disturbances from environment. Moreover, appropriate parameters are introduced to balance the contributions to the utility function of the different cost factors.

  • 2.

    The problem of service chain configuration in NFV networks is cast as a congestion game and its corresponding optimization problem is converted to the seeking of the Nash equilibrium under the game framework.

  • 3.

    By leveraging the STP framework, the algebraic representation and potential function of the formulated game are derived. The service chain composition is achieved by exploiting properties of the resulting potential games.

The reminder of this paper is organized as follows. In Section 2, the considered network scenario is described. Section 3 gives some necessary model notations and descriptions. In Section 4, the game-theoretic model for the service chain composition problem is proposed, the algebraic representation of the formulated game and the potential function are presented. A numerical example is carried out to illustrate the proposed results in Section 5. Section 6 is a concise conclusion.

First, the notations which can be used in the following is introduced here.

  • 1.

    R,, ∏: the set of real number, Kronecker product, Cartesian product.

  • 2.

    Rm×n: the set of real matrices of order m × n.

  • 3.

    Col(A) (Row(A)): the set of columns (rows) of matrix A, and δni:=Coli(In), and n:={δni|i=1,2,,n}.

  • 4.

    Dk:={1,2,,k}, and Dkn:=Dk××Dkn.

  • 5.

    A matrix MRm×n is called a logical matrix if the column of M is in the form of δnk.

  • 6.

    In is the n × n identity matrix.

  • 7.

    1n:=(1,1,,1)nT, |s|: the cardinality of s.

  • 8.

    si: the vector of all entries of s except the ith one.

  • 9.

    A: the transpose of matrix A.

Section snippets

Model description

Network function virtualization (NFV) [1] is a network paradigm that aims at softwarizing network functions that usually have been implemented on specific hardware devices. Due to the energy and space benefits of virtualization, network service providers rapidly deploy NFV in their carrier networks in the recent years.

A NFV network is composed of network users, virtual network function (VNF) servers and Telco Operator. User generated traffic. VNF servers are NFV-compatible nodes providing VNF

Congestion game formulation

Each potential game Γ [18] admits a potential function, into which the incentives of all players are mapped.

Definition 1

[18]

Consider a normal game G={N,(Ai)iN,(ui)iN}, where N={1,2,,n} is the players set, Ai is the action set of player i, and ai ∈ Σi is player i’s action, and ui is the utility function. If there exist a function P:ΣR and a set of positive numbers wi>0,i=1,,n satisfiesui(ai,ai)ui(ai,ai)=wi(P(ai,ai)P(ai,ai)),for every i ∈ N, aiAi and ai,aiAi. Then G is a weighted potential

Numerical example

To validate the proposed framework, we consider the scenario based shown by Fig. 2. There exists two network users in the network. The bandwidth-unit benefit β=240, the bandwidth remaining rate rv1bw=0.97,rv2bw=0.96,rv3bw=0.98,rv4bw=0.95, the bandwidths of users are b1=2,b2=1, and the data synchronization rates are w1=2,w2=1, respectively. The congestion costs of different NFV servers are given as follows: cv1(nv1(s))=(nv1(s))3, cv2(nv2(s))=5nv2(s), cv3(nv3(s))=3(nv3(s))2, cv4(nv4(s))=4(nv1(s))

Conclusion

In this paper, the service chain configuration problem in NFV networks with function benefit is investigated. The service chain composition problem is converted to the seeking of the Nash equilibrium under the congestion game framework. Based on STP, the potential function of the formulated NFV game is derived and the service chain composition is achieved. The numerical example shows the effectiveness of the approach. Further research includes the consideration of delayed neural networks [21],

Declaration of Competing Interest

None.

Acknowledgments

This work was supported the National Natural Science Foundation of China under Grant 61773090 and the JSPS KAKENHI Grant Number JP18H05899 and JP19K15019.

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