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Equilibrium Price and Dynamic Virtual Resource Allocation for Wireless Network Virtualization

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Abstract

Economic and technical features are equally important to radio resource allocation in wireless network virtualization (WNV). Regarding virtual resource (VR) as commodity, this paper proposes an effective VR allocation scheme for WNV from the perspective of the market-equilibrium theory. First, physical meaning clear utility functions are defined to characterize the network benefits of user equipments (UEs), infrastructure providers (InPs) and virtual network operators (VNOs) in WNV. Then, the VR allocation problem between one InP and multiple VNOs is formulated as a multi-objective optimization problem. To reduce the algorithm complexity, the multiple-objective problem is first decoupled into two single-objective sub-problems. The supplier-layer sub-problem aims to maximize the benefit of the unique InP, while the customer-layer sub-problem aims to maximize the benefits of the multiple VNOs. Both of the separated sub-problems are solved by using standard convex optimization method, and are combined by searching for the equilibrium-price (EP) of the VR market. As a result, the Pareto optimal solution of the original multi-objective problem is found, at which no one (the InP or anyone of the VNOs) can increase its benefit by deviating the EP without hurting others’ benefits. The effectiveness of the proposed VR allocation scheme is testified through extensive experiments.

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Notes

  1. The proof for the uniqueness is given in Proposition 5.

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Acknowledgments

This work was supported in part by the Shenzhen S&T Innovation Project under Grant JCYJ20140610151856732, in part by the National Natural Science Foundation of China under Grant 61471361, 61572389, 61572191, and 61371127, in part by the National High-Tech R&D Program (863 Program) under Grant 2015AA01A705, and in part by the Natural Science Foundation of Jiangsu Province, China under Grant BK20140202.

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Correspondence to Lianming Zhang.

Appendices

Appendix A

1.1 Proof of proposition 1

Taking the first- and second- order derivatives of π0(α) with respect to c m , we have

$$ \frac{\partial {\uppi}_0\left(\alpha \right)}{\partial {c}_m}=\frac{\partial {U}_m}{\partial {c}_m}-\alpha, \kern0.36em m=1,\dots, M $$
(26)

and

$$ \frac{\partial^2{\uppi}_0\left(\alpha \right)}{\partial {c}_m^2}=\frac{\partial^2{U}_m}{\partial {c}_m^2}<0,\kern0.36em m=1,\dots, M $$
(27)

In order to guarantee the concavity of π0(α), it is ordered to derive the condition under which ∂π0(α)/∂c m  > 0. By substituting Eq. (4) into Eq. (26), the condition is derived as shown in Eqs. (12) and (13).

Appendix B

2.1 Proof of proposition 2

To solve problem (10), we exploit the Karush-Kuhn-Tucker (KKT) optimality condition [21]. The Lagrangian function of problem (10) is written as

$$ {L}_0\left(\alpha \right)={\displaystyle \sum_{m=1}^M{U}_m}+\alpha \left(S-{\displaystyle \sum_{m=1}^M{c}_m}\right)+\mu \left({\displaystyle \sum_{m=1}^M{c}_m}-S\right)-{\displaystyle \sum_{m=1}^M{\nu}_m{c}_m}+{\displaystyle \sum_{m=1}^M{\rho}_m\left({c}_m-S\right)} $$
(28)

where μ, v m and ρ m , ∀ m = 1, …, M, are the non-negative Lagrangian multipliers. The KKT optimality conditions for this problem are:

$$ \frac{\partial {L}_0\left(\alpha \right)}{\partial {c}_m}=\frac{ \ln \left(1+\frac{G_m}{c_m}\right)-1}{c_m{G}_m \ln \left(1+\frac{G_m}{c_m}\right)}-\alpha +\mu -{\nu}_m+{\rho}_m=0,\kern0.36em m=1,\dots, M $$
(29)

with the additional complementary slackness conditions as:

$$ \mu \left({\displaystyle \sum_{m=1}^M{c}_m}-S\right)=0 $$
(30)
$$ {\rho}_m\left({c}_m-S\right),\kern0.36em m=1,\dots, M $$
(31)
$$ {\nu}_m{c}_m=0,\kern0.36em m=1,\dots, M $$
(32)

Since 0 < c m  < S and ∑ M m = 1 c m  < S for ∀ m = 1, …, M, constraints (10.1), (10.2) and (10.3) are all satisfied without equality. So we get v m  = 0, ρ m  = 0 and μ = 0. Then, by solving Eq. (29), we can obtain the optimal solution to problem (10) as shown in Eq. (14).

Appendix C

3.1 Proof of proposition 3

First, we assume that the unique optimal solutionFootnote 1 to problem (16) is

$$ {\tilde{C}}_n=\left({\tilde{c}}_{n,1},{\tilde{c}}_{n,2},\dots, {\tilde{c}}_{n,{K}_n}\right),\kern0.36em n=1,\dots, N $$
(33)

The resultant network benefit for the nth VNO is

$$ {\tilde{\uppi}}_n={\displaystyle \sum_{k=1}^{K_n} \ln \left({\tilde{c}}_{n,k}{w}_0{ \log}_2\left(1+{b}_{n,k}\frac{p_{n,k}\cdot {g}_{n,k}}{{\tilde{c}}_{n,k}{w}_0{n}_0}\right)\right)}-\alpha \cdot {\displaystyle \sum_{k=1}^{K_n}{\tilde{c}}_{n,k}},\kern0.36em n=1,\dots, N $$
(34)

We need to prove that Eq. (33) is also a solution (but not necessarily the unique one) to problem (15). This can be proved by following contradiction.

Assume that Eq. (33) is not a solution to problem (15). According to the Pareto optimality, there must exists another VR allocation vector

$$ {\widehat{C}}_n=\left({\widehat{c}}_{n,1},{\widehat{c}}_{n,2},\dots, {\widehat{c}}_{n,{K}_n}\right),\kern0.36em n=1,\dots, N $$
(35)

different from Eq. (33) which can increases the benefits of part of the VNOs without decreasing the benefits for the other VNOs. Without loss of generality, we assume that, with the new VR allocation vector (35), the benefits of the N − 1 VNOs remain unchanged except for the \( \overline{n} \)th VNO. That means

$$ {\widehat{\uppi}}_n={\tilde{\uppi}}_n\kern0.36em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.36em n=1,\dots, N\kern0.36em \mathrm{and}\kern0.36em n\ne \overline{n} $$
(36)

and

$$ {\widehat{\uppi}}_{\overline{n}}>{\tilde{\uppi}}_{\overline{n}}\kern0.36em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.36em n=\overline{n} $$
(37)

On condition that Proposition 4 is satisfied, π n would increase monotonically with c n,k , for ∀ k ∈ {1, …, K n }. Thus, to increase \( {\tilde{\uppi}}_{\overline{n}} \), we have to allocate more VR to the \( \overline{n} \)th VNO. As the total amount of VR available for the N VNOs is fixed, one or some of the N − 1 VNOs (other than the \( \overline{n} \)th VNO) would receive reduced amount of VR. This contradicts with the assumption that the VR allocation vector (35) is Pareto optimality.

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Zhang, G., Yang, K., Jiang, H. et al. Equilibrium Price and Dynamic Virtual Resource Allocation for Wireless Network Virtualization. Mobile Netw Appl 22, 564–576 (2017). https://doi.org/10.1007/s11036-016-0766-9

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