Dynamic Provider Selection & Power Resource Management in Competitive Wireless Communication Markets

Abstract

In this paper the combined problem of Wireless Internet Service Provider (WISP) selection by the mobile customers and corresponding power allocation is treated, in order to meet user expectations and satisfaction in a competitive wireless communication market with several co-existing WISPs. Each WISP is characterized by a price and service-based reputation, formed based on its adopted pricing policy and its success to satisfy customers’ Quality of Service (QoS) prerequisites, the latter implicitly characterizing the specific WISP’s market penetration factor. The customers who act as learning automata selecting the most appropriate WISP adopt a machine learning based mechanism. The optimal power allocation is concluded from the maximization problem of each user’s utility function, which is confronted as a non-cooperative game among users and its Nash equilibrium is determined. The output of the resource allocation problem feeds the learning system in order to build knowledge and conclude to the optimal provider selection. A two-stage iterative algorithm is proposed in order to realize the machine learning provider selection and the distributed resource allocation. The performance of the proposed approach is evaluated via modeling and simulation and its superiority against other state of the art approaches is illustrated.

Appendix A – Proof of Theorem 1

Towards proving the quasi-concavity of mobile customer’s utility function, we examine the sign of its second order derivative with respect to p _i . Considering the real time users, we have:

$$ \frac{\partial^2{U}_i^{NET}}{\partial {p_i}^2}= g\left({\gamma}_i\right)+ h\left({\gamma}_i\right)-{ce}^{p_i} $$

(6)

where $g\left({\gamma}_i\right)=\frac{2 R}{{p_i}^3}{\left(1-{e}^{- A{\gamma}_i}\right)}^{M-1}\left[-{ M Ae}^{- A{\gamma}_i}{\gamma}_i+1-{e}^{- A{\gamma}_i}\right]$ and $h\left({\gamma}_i\right)=\frac{{ M A}^2 R}{{p_i}^3}{\left(1-{e}^{- A{\gamma}_i}\right)}^{M-2}{e}^{- A{\gamma}_i}{\gamma_i}^2\left({ M e}^{- A{\gamma}_i}-1\right)$. We examine the sign of the individual terms of eq. (6). Considering the function g(γ _i ), we apply the Bolzano theorem, which is an important specialization of the Intermediate Value Theorem [19]. For ${\gamma}_i=\frac{ \ln M}{A}$, we have: $g\left(\frac{ \ln M}{A}\right)=- \ln M-\frac{ \ln M}{A}+1<0$ and for ${\gamma}_i=\frac{ \ln {10}^4 M}{A}$, we have: $g\left(\frac{ \ln 10000 M}{A}\right)=-\frac{ \ln 10000 M}{A}-\frac{ \ln 10000 M}{A}+1>0$, ∀M ∈ (1, 1000). Hence, there exists a ${\gamma}_{RT}\in \left(\frac{ \ln M}{A},\frac{ \ln M}{A}\right)$ such that g(γ _i ) = 0. Moreover, since g(γ _i )is continuous, we conclude that

$$ g\left({\gamma}_i\right)<0,\forall {\gamma}_{\iota}\in \left(\frac{ \ln M}{A},{\gamma}_{RT}\right) $$

(7)

For the second term of (6), we have:

$$ h\left({\gamma}_i\right)<0\iff {\gamma}_i>\frac{ \ln M}{A} $$

(8)

The third term of (6) is always negative. Combining (7) and (8) we define the modified strategy space, where$\frac{\partial^2{U}_i^{NET}}{\partial {p_i}^2}<0$, for the real time users:

$$ {\gamma}_{\iota}\in \left(\frac{ \ln M}{A},{\gamma}_{RT}\right) $$

(9)

Considering the non-real time users, we have:

$$ \frac{\partial^2{U}_i^{NET}}{\partial {p_i}^2}=\frac{AMR_i}{{p_i}^3 \ln 10}\tau \left({\gamma}_{\iota}\right)-{ce}^{p_i} $$

(10)

where

$$ \begin{array}{l}\tau \left({\gamma}_i\right)=\frac{{\left(1-{e}^{- A{\gamma}_i}\right)}^{M-2}{e}^{- A{\gamma}_i}{\gamma}_i}{1+{\left(1-{e}^{- A{\gamma}_i}\right)}^M}\left\{\begin{array}{l}-2+2{e}^{- A{\gamma}_i}-\\ {} A{\gamma}_i\left(1-{ M e}^{- A{\gamma}_i}\right)\end{array}\right\}\\ {}- A M{\left(\frac{{\left(1-{e}^{- A{\gamma}_i}\right)}^{M-1}{e}^{- A{\gamma}_i}{\gamma}_i}{1+{\left(1-{e}^{- A{\gamma}_i}\right)}^M}\right)}^2+\frac{2 \ln \left(1+{\left(1-{e}^{- A{\gamma}_i}\right)}^M\right)}{AM}\end{array} $$

(11)

We apply again the Bolzano theorem for the function τ(γ _i ). For ${\gamma}_i=\frac{ \ln M}{A}$, we have:

$$ \begin{array}{l}\tau \left(\frac{ \ln M}{A}\right)=\frac{{\left(\frac{M-1}{M}\right)}^{M-2}\frac{1}{M}\frac{ \ln M}{A}}{1+{\left(\frac{M-1}{M}\right)}^M}\left(-2+\frac{2}{M}\right)\\ {}- AM{\left(\frac{{\left(\frac{M-1}{M}\right)}^{M-1}\frac{1}{M}\frac{ \ln M}{A}}{1+{\left(\frac{M-1}{M}\right)}^M}\right)}^2+\frac{2 \ln \left(1+{\left(\frac{M-1}{M}\right)}^M\right)}{A M}<0\end{array} $$

(12)

and for ${\gamma}_i=\frac{ \ln {10}^4 M}{A}$, we have:

$$ \begin{array}{l}\tau \left(\frac{ \ln 10000 M}{A}\right)=-\frac{1}{10000 M}\frac{ \ln 10000 M}{2}\left(2+ \ln 10000 M\right)\\ {}-\frac{1}{A M}\frac{1}{10000^2}{\left(\frac{ \ln 10000 M}{2}\right)}^2+\frac{2 \ln 2}{A M}>0\end{array} $$

(13)

∀M ∈ (1, 1000) and ∀A ∈ (0.1, 100). Hence, there exists a ${\gamma}_{\mathrm{N} RT}\in \left(\frac{ \ln M}{A},\frac{ \ln M}{A}\right)$ such that τ(γ _i ) = 0. Moreover, since τ(γ _i )is continuous, we conclude thatτ(γ _i ) < 0 and $\frac{\partial^2{U}_i^{NET}}{\partial {p_i}^2}<0$, for non-real time users:

$$ \forall {\gamma}_{\iota}\in \left(\frac{ \ln M}{A},{\gamma}_{\mathrm{N} RT}\right) $$

(14)

Hence, combining eqs. (9) and (14) we conclude that all utility functions are simultaneously concave (hence quasi concave) in the altered strategy space:

$$ {\gamma}_{\iota}\in \left(\frac{ \ln M}{A},{\gamma}_{final}\right) $$

(15)

where γ _final  = min {γ _RT , γ _NRT }. ■.