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Resource allocation in future HetRAT networks: a general framework

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Aggregation of resources in space, spectrum, and so on, is the fundamental idea behind many technology building blocks of 5G networks, such as massive multi-input multi-output and carrier aggregation, among others. In this respect, another important dimension for parallelism is the aggregation of diverse radio access technologies (RATs), such as WiFi, LTE, and 5G, in heterogeneous networks (HetRAT networks). Simultaneous multiple RAT connectivity facilitates both users and network operators in achieving even higher data rates and in balancing the traffic load, respectively. In this paper, we review the technological supports for the realization of the multi-RAT connectivity and focus on a general framework for resource allocation in HetRAT systems. Specifically, a distributed algorithm is proposed to allocate, with minimal overhead signaling, the bandwidths of available RATs among users considering both the system operator’s objective and the users’ demands and constraints. The optimality of the proposed distributed algorithm is further proved, and its convergence is investigated. Finally a test case is introduced in which users are power constrained and the system operator enforces capacity limits and motivates users for a fair load balancing over RATs. Numerical results shows the efficiency of the proposed algorithm and its fast convergence in few iterations.

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Correspondence to Mohammad Hossein Manshaei.

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Appendix 1: Proof of Proposition 1

A feasible strategy profile \(\varvec{s}^*=\{\varvec{s}^*_i\}_{i=1}^{Q}\) is an NE of the \(\mathcal {G}\) if

$$\begin{aligned} \Gamma _i(\varvec{s}^{*}_i,\varvec{s}^{*}_{-i})\le \Gamma _i(\varvec{s}_i,\varvec{s}^*_{-i})\quad \forall \varvec{s}_i\in \mathcal {S}_i, i=1,\ldots , Q, \end{aligned}$$

This means that, for \(i=1,\ldots ,Q\), \(({\varvec{s}}^*_i,s^*_{-i})\) solves the optimization problem of the ith UE given in (4). As a result, the strategy profile \(({\varvec{s}}^*_i,{\varvec{s}}^*_{-i})\) satisfies in the Karush-Kuhn-Tucker (KKT) conditions in the form of

$$\begin{aligned}&0\le s^*_{i,n} {{\perp }} \left( \frac{\partial ( u_i({\varvec{s}}^*_i)+ f({\varvec{s}}^*,{\varvec{r}}))}{\partial s_{i,n}} - \sum _{m=1}^{c_i}\lambda _{i,m}\frac{\partial g_{i,m}({\varvec{s}}^*_i)}{\partial s_{i,n}}\right) \ge 0;\nonumber \\&\quad n=1,\ldots ,N, \end{aligned}$$
$$\begin{aligned}&0\le \lambda _{i,m} {{\perp }} g_{i,m}({\varvec{s}}^*_i)\ge 0;\quad m=1,\ldots ,c_i, \end{aligned}$$

where \(\{\lambda _{i,m}\}_{m=1}^{c_i}\) are the Lagrange multipliers and the operator \({{\perp }}\) means the orthogonality of the operands. On the other hand, if the KKT optimality conditions of every UE’s problem in (17)–(18) (\(i=1,\ldots ,Q\)) are put together, it can be readily verified that they are identical to the first-order optimality conditions of the global problem in (5).

Appendix 2: Proof of Proposition 2

The first order method is given by

$$\begin{aligned} \bar{\varvec{s}}\leftarrow \bar{\varvec{s}}+\alpha \nabla _{\varvec{s}}\mathcal {L}(\bar{\varvec{s}},\bar{\varvec{\lambda }}),\quad \bar{\varvec{\lambda }}\leftarrow \bar{\varvec{\lambda }}+\alpha \varvec{g(\bar{\varvec{s}}),} \end{aligned}$$

where the operator \(\nabla _{\varvec{s}}\) gives the vector of first derivatives (gradient) of its operand with respect to \(\varvec{s}\), \(\varvec{g(\bar{\varvec{s}})}=[g_{1,1}(\bar{\varvec{s}}_1),\ldots ,g_{Q,c_Q}(\bar{\varvec{s}}_Q)]\) and \(\alpha >0\) is a scalar stepsize. On the other hand, (19) can be re-written as

$$\begin{aligned} \bar{\varvec{s}}_i\leftarrow \bar{\varvec{s}}_i+\alpha \nabla _{\varvec{s}_i}\mathcal {L}(\bar{{\varvec{s}}}, \bar{\varvec{\lambda }}),\quad \bar{\varvec{\lambda }}_i\leftarrow \bar{\varvec{\lambda }}_i+\alpha \varvec{g_i(\bar{\varvec{s}}),} \end{aligned}$$

for \(i=1,\ldots ,Q,\) where \(\varvec{g_i(\bar{\varvec{s}})}=[g_{i,1}(\bar{\varvec{s}}_1),\ldots ,g_{i,c_i}(\bar{\varvec{s}}_Q)].\) Equations in (20) show that distributed implementation of the first order method is equivalent to the centralized implementation.

Appendix 3: Proof of Proposition 3

Newton’s method for solving the Lagrangian system is

$$\begin{aligned} \bar{\varvec{s}}\leftarrow \bar{\varvec{s}}+\Delta \bar{\varvec{s}},\quad \bar{\varvec{\lambda }}\leftarrow \bar{\varvec{\lambda }}+\Delta \bar{\varvec{\lambda }}, \end{aligned}$$

where \(\Delta \bar{\varvec{s}}\) and \(\Delta \bar{\varvec{\lambda }}\) are given by solving the system of equations

$$\begin{aligned} \underbrace{\nabla ^2\mathcal {L}(\bar{{\varvec{s}}},\bar{\varvec{\lambda }})}_{\mathbf {K}}\left( \begin{array}{c} \Delta \bar{\varvec{s}} \\ \Delta \bar{\varvec{\lambda }} \end{array}\right) =-\nabla \mathcal {L}(\bar{{\varvec{s}}},\bar{\varvec{\lambda }}). \end{aligned}$$

Without loss of generality, consider the case in which two single constrained UEs compete for resource allocation. Under this assumption, we have

$$\begin{aligned}&\mathbf {K} =\left( \begin{array}{l l} \mathbf {K}_1&{} \mathbf {K}_{1,2}\\ \mathbf {K}_{2,1}&{}\mathbf {K}_2 \end{array}\right) , \quad \nabla \mathcal {L}(\bar{{\varvec{s}}},\bar{\varvec{\lambda }})= \left( \begin{array}{c} \nabla _{\varvec{s}_1}\mathcal {L}(\bar{{\varvec{s}}},\bar{\varvec{\lambda }})\\ g_{1,1}(\bar{\varvec{s}}_1) \\ \nabla _{\varvec{s}_2}\mathcal {L}(\bar{{\varvec{s}}},\bar{\varvec{\lambda }})\\ g_{2,1}(\bar{\varvec{s}}_2) \end{array}\right) , \end{aligned}$$


$$\begin{aligned}&\mathbf {K}_1=\left( \begin{array}{l l} \nabla ^2_{\varvec{s}_1,\varvec{s}_1}\mathcal {L}(\bar{{\varvec{s}}},\bar{\varvec{\lambda }})&{} \nabla _{\bar{{\varvec{s}}}_1} g_{1,1}(\bar{{\varvec{s}}}_1)\\ {\nabla _{\bar{{\varvec{s}}}_1} g_{1,1}(\bar{{\varvec{s}}}_1)}^T&{}\mathbf {0} \end{array}\right) ,\nonumber \\&\mathbf {K}_2=\left( \begin{array}{l l} \nabla ^2_{\varvec{s}_2,\varvec{s}_2}\mathcal {L}(\bar{{\varvec{s}}},\bar{\varvec{\lambda }})&{} \nabla _{\bar{{\varvec{s}}}_2} g_{2,1}(\bar{{\varvec{s}}}_2)\\ {\nabla _{\bar{{\varvec{s}}}_2} g_{2,1}(\bar{{\varvec{s}}}_2)}^T&{}\mathbf {0} \end{array}\right) ,\nonumber \\&\mathbf {K}_{1,2}=\left( \begin{array}{l l} \nabla ^2_{\varvec{s}_1,\varvec{s}_2}\mathcal {L}(\bar{{\varvec{s}}},\bar{\varvec{\lambda }})&{} \mathbf {0}\\ \mathbf {0}&{}\mathbf {0} \end{array}\right) ,\nonumber \\&\mathbf {K}_{2,1}=\mathbf {K}_{1,2}^T. \end{aligned}$$

The coupling submatrices \(\mathbf {K}_{1,2}\) and \(\mathbf {K}_{2,1}\) hinder the distributed implementation of the Newton step. This problem is resolved if the matrix \(\mathbf {K}\) is replaced by \(\mathbf {K}_0\) given by

$$\begin{aligned} \mathbf {K}_0 =\left( \begin{array}{l l} \mathbf {K}_1&{} \mathbf {0}\\ \mathbf {0}&{}\mathbf {K}_2 \end{array}\right) . \end{aligned}$$

On the other hand, based on [38], we can conclude that using \(\mathbf {K}_0\) instead of \(\mathbf {K}\) does not affect the convergence to the optimal solution if coupling is below a specific threshold, mathematically described as \(\rho (\mathbf {I}-\mathbf {K}_0^{*^{-1}}\mathbf {K}^*)<1\).

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Heidarpour, M.R., Manshaei, M.H. Resource allocation in future HetRAT networks: a general framework. Wireless Netw 26, 2695–2706 (2020).

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