Providing Throughput and Fairness Guarantees in Virtualized WLANs Through Control Theory

Abstract

With the increasing demand for mobile Internet access, WLAN virtualization is becoming a promising solution for sharing wireless infrastructure among multiple service providers. Unfortunately, few mechanisms have been devised to tackle this problem and the existing approaches fail in optimizing the limited bandwidth and providing virtual networks with fairness guarantees. In this paper, we propose a novel algorithm based on control theory to configure the virtual WLANs with the goal of ensuring fairness in the resource distribution, while maximizing the total throughput. Our algorithm works by adapting the contention window configuration of each virtual WLAN to the channel activity in order to ensure optimal operation. We conduct a control-theoretic analysis of our system to appropriately design the parameters of the controller and prove system stability, and undertake an extensive simulation study to show that our proposal optimizes performance under different types of traffic. The results show that the mechanism provides a fair resource distribution independent of the number of stations and their level of activity, and is able to react promptly to changes in the network conditions while ensuring stable operation.

Appendix

Theorem 1

Given the definition of e _i in Eq.  14 , there exists an unique solution to the system defined by e _i  = 0 ∀ i that satisfies e _opt = 0 and e _fair,i  = 0 ∀ i.

Proof

By subtracting e _j from e _i we obtain

$$ \begin{array}{rll} e_i - e_j &=& (N-1)S_i - S_j - (N-1) S_j + S_i \\ &=& N (S_i - S_j), \end{array} $$

(38)

and therefore, given that e _i  = 0 ∀ i,j, the above results in S _i  = S _j ∀ i, j, and therefore we have that e _fair,i  = 0 ∀ i. Furthermore, this results in the following relation (as already expressed in Eq. 6),

$$ \frac{n_i \tau_i}{1-\tau_i} = \frac{n_j \tau_j}{1-\tau_j}, $$

(39)

which specifies, for a given (n _i , n _j ) pair, a one-to-one relationship between τ _j and τ _i ∀ i,j, and therefore we can take e.g. τ ₁ as reference. In this way, if we express e _opt = 0 as

$$ \prod (1-\tau_k)^{n_k} = P_e^*, $$

(40)

we have that the rhs of the above equation is a constant between 0 and 1, while the lhs is a decreasing function of τ ₁ from 1 to 0. Therefore there exists a unique solution that solves the above equation, thus ensuring also that e _opt = 0. □

Theorem 2

The K _P and K _I relationship specified by Eq.  34 guarantees stability.

Proof

According to [33], we need to check that the following transfer function is stable

$$ (I - z^{-1}CH)^{-1} C. $$

(41)

Computing the above yields

$$ (I - z^{-1}C K_H I)^{-1}C = \frac{K_P + \frac{K_I}{z-1}}{1 - z^{-1} \left(K_P + \frac{K_I}{z-1}\right)K_H} I, $$

(42)

which can be expressed as

$$ (I - z^{-1}C K_H I)^{-1}C = \frac{P(z)}{z^2 + z a_1 + a_2}I, $$

(43)

where P(z) is a polynomial and

$$ a_1 = -1(1+K_PK_H). $$

(44)

$$ a_2 = K_H(K_P-K_I). $$

(45)

According to [33], a sufficient condition for stability is that the zeros of the pole polynomial fall within the unit circle. This can be ensured by choosing the coefficients {a ₁,a ₂} that belong to the stability triangle [17]:

$$ a_2 < 1, $$

(46)

$$ a_1 < a_2+1, $$

(47)

$$ a_1 > -1 -a_2. $$

(48)

Equation 46 is satisfied as long as K _P  > K _I , while Eq. 48 is satisfied if K _I  > 0. By operating in Eq. 47 we obtain the relationship $K_P < -K_H^{-1} + K_I/2$, which combined with the previous relations results in the conditions expressed by Eq. 34. □