Design of a Web-Based Decision Support System for Service Portfolios in Heterogeneous Radio Access Network Environments

Abstract

Newly emerging radio access technologies have produced a novel heterogeneous network environment. Wireless service operators should build the best service portfolio strategy for each user by focusing on the co-existence of multiple access networks and complex service combinations, while maximizing the overall network utilization. Web-based Decision Support System (web-based DSS) is one of the best ways of making service portfolios available to every user in a multiple access network environment. Service designers, customer relationship managers, and network engineers can build the best match relationship between services and networks to enhance user utilization. In addition, the easily accessible web-based DSS in an optimal heterogeneous network operation framework provides opportunities for designing new services. The network load and financial effect of newly designed services could also be analyzed and reshaped easily by testing the DSS functionality. Various mathematical tools have been developed for DSS to integrate different network domains. To demonstrate its applicability to the integration of network domains, we tested various service scenarios in a heterogeneous network environment and evaluated the versatile functions of web-based DSS.

Appendices

Appendix 1

The minimization problem is stated as follows:

Minimize the total cost $T(u_{1}, u_{2}, \ldots, u_{N} ) = \int\nolimits_{0}^{{u_{1} }} {f_{1} (u)du + \cdots + \int\nolimits_{0}^{uN} {f_{N} (u)du} }$ subject to the constraints $u_{1} \ge 0, \ldots, u_{N} \ge 0$ and $u_{1} + u_{2} + \cdots + u_{N} = C$. Because the domain is compact, the total cost function has the minimum in its domain. The Karush–Kuhn–Tucker conditions require that there are constants $\eta_{1}, \ldots, \eta_{N}, \lambda$ such that for $k = 1,2, \ldots, N$,

$$ \begin{gathered} \frac{\partial }{{\partial u_{k} }}T(u_{1}^{*}, \ldots, u_{N}^{*} ) - \eta_{k} + \lambda = 0, \hfill \\ u_{k}^{*} \ge 0, \hfill \\ \eta_{k} \ge 0, \hfill \\ \eta_{k} u_{k}^{*} = 0, \hfill \\ \end{gathered} $$

and $u_{1}^{*} + \cdots + u_{N}^{*} = C$. Therefore, if $u_{k}^{*} > 0$ for all $k = 1,2, \ldots, N$, then $f_{1} (u_{1}^{*} ) = f_{2} (u_{1}^{*} ) = \ldots = f_{N} (u_{N}^{*} )$.

Appendix 2

We will prove that V(t) converges to zero as the time t approaches infinity.

Let $A_{ij} : = c_{ij} (f_{i} (u_{i} ) - f_{j} (u_{j} ))$. The total traffic $u_{1} + u_{2} + \cdots + u_{N}$ is conserved over time because

$$ \frac{{d(u_{1} + u_{2} + \cdots + u_{N} )}}{dt} = - \lambda \sum\limits_{i,j} {A_{ij} } = 0\quad \because \;A_{ij} = - A_{ji} $$

Because A _ij  = A _ji and the marginal costs functions are increasing, we have

$$ \begin{aligned} \dot{V} = \frac{d}{dt}\sum\limits_{i,j} {c_{ij} (f_{i} (u_{i} ) - f_{j} (u_{j} ))^{2} } = & 2\sum\limits_{i,j} {A_{ij} (f^{\prime}_{i} (u_{i} )\dot{u}_{i} } - f^{\prime}_{j} (u_{j} )\dot{u}_{j} ) \\ = & -2\lambda \sum\limits_{i,j} {A_{ij} \left( {f^{\prime}_{i} (u_{i} )\sum\limits_{k} {A_{ik} - f^{\prime}_{j} (u_{j} )\sum\limits_{k} {A_{jk} } } } \right)} \\ = & -2\lambda \left( {\sum\limits_{i,j} {\sum\limits_{k} {f^{\prime}_{i} (u_{i} )A_{ij} A_{ik} - \sum\limits_{i,j} {\sum\limits_{k} {f^{\prime}_{j} (u_{j} )A_{ij} A_{jk} } } } } } \right) \\ = & -4\lambda \left( {\sum\limits_{i} {f^{\prime}_{i} (u_{i} )\sum\limits_{j,k} {A_{ij} A_{ik} } } } \right) \\ = & -4\lambda \sum\limits_{i} {f^{\prime}_{i} (u_{i} )\left( {\sum\limits_{k} {A_{ik} } } \right)^{2} } \\ = & -\frac{4}{\lambda }\sum\limits_{i} {f^{\prime}_{i} (u_{i} )(\dot{u}_{i} )^{2} } \le 0, \\ \end{aligned} $$

which implies that V(t) always decreases unless the traffic distribution (u ₁(t), …, u _N (t)) is not in a unique equilibrium state. Because V(t) is bounded below, we should have $\dot{V} \to 0$ as t → ∞. However, $\dot{V} = 0$ if and only if $\dot{u}_{i} = 0$ for all i = 1, …, N, which is also equivalent to the equation f ₁(u ₁) = f ₂(u ₂) = … = f _N (u _N ), leading to V = 0. Therefore, as the dynamics evolve over time, it follows that $\dot{u}_{i} \to 0$ (i = 1, …, N). This completes the proof.