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Low Complexity Resource Allocation Techniques for Symmetrical Services in OFDMA Systems

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Abstract

The popularity of symmetrical services like video conferencing, telemedicine etc. is in an upsurge. With the rapid increase in the number of customers for such services, the demand for reliable and computationally feasible resource allocation techniques satisfying all the users are also intensifying. This paper proposes two computationally feasible techniques for allocation of resources in OFDMA system specifically for services that demand similar quality in the uplink and downlink directions. The resource allocation problem is multiobjective in nature with the objectives to maximize the data rates in both directions meanwhile minimizing the difference in the bidirectional data rates for each user. Fairness has been incorporated as a major constraint in the formulation of the optimization problem. Equal Power allocation and weighted sum method are implemented for power allocation while subcarrier allocation is carried out using an evolutionary technique. The solution for subcarrier allocation is represented in a novel manner such that the complexity is reduced to a great extent at the same time attaining acceptable data rates.

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Authors and Affiliations

  1. Department of Electronics and Communication Engineering, Mar Baselios College of Engineering and Technology, Trivandrum, India

    P. S. Swapna & Sakuntala S. Pillai

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  1. P. S. Swapna
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  2. Sakuntala S. Pillai
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Appendix

Appendix

The objectives of the multiobjective optimization problem is to

The constraints considered are

figure c

The remaining constraints are considered in the implementation of the NSGA II algorithm. The Lagrangian of the problem can be expressed as:

$$ L\left( . \right) = \mathop \sum \limits_{i = 1}^{ K} \alpha_{i1} R_{i1} + \mathop \sum \limits_{i = 1}^{ K} \alpha_{i2} R_{i2} - \lambda_{1} \left[ {\mathop \sum \limits_{j = 1}^{N} \mathop \sum \limits_{i = 1}^{K} p_{ij1} w_{ij1} - p_{BS} } \right] - \mathop \sum \limits_{i = 1}^{K} \lambda_{i2} \left[ {\mathop \sum \limits_{j = 1}^{N} p_{ij2} - p_{t} } \right] $$

Substituting for the expression of data rates from Shannon’s theorem,

$$ L\left( . \right) = \mathop \sum \limits_{i = 1}^{ K} \alpha_{i1} \mathop \sum \limits_{j = 1}^{N} \omega_{ij1} \log_{2} \left[ {1 + \gamma_{ij1} \tilde{p}_{ij1} } \right] + \mathop \sum \limits_{i = 1}^{ K} \alpha_{i2} \mathop \sum \limits_{j = 1}^{N} \omega_{ij2} \log_{2} \left[ {1 + \gamma_{ij2} \tilde{p}_{ij2} } \right] - \lambda_{1} \left[ {\mathop \sum \limits_{j = 1}^{N} \mathop \sum \limits_{i = 1}^{K} p_{ij1} w_{ij1} - p_{BS} } \right] - \mathop \sum \limits_{i = 1}^{K} \lambda_{i2} \left[ {\mathop \sum \limits_{j = 1}^{N} p_{ij2} - p_{t} } \right] $$

Grouping the \( \tilde{P}_{ij1} \;{\text{and}}\;\tilde{P}_{ij2} \) terms,

$$ L\left( . \right) = \mathop \sum \limits_{i = 1}^{K} \alpha_{i1} \mathop \sum \limits_{j = 1}^{N} \omega_{ij1} \log_{2} \left[ {1 + \gamma_{ij1} \tilde{p}_{ij1} } \right] - \lambda_{1} \left[ {\mathop \sum \limits_{j = 1}^{N} \mathop \sum \limits_{i = 1}^{K} p_{ij1} w_{ij1} } \right] + \mathop \sum \limits_{i = 1}^{ K} \alpha_{i2} \mathop \sum \limits_{j = 1}^{N} \omega_{ij2} \log_{2} \left[ {1 + \gamma_{ij2} \tilde{p}_{ij2} } \right] - \mathop \sum \limits_{i = 1}^{K} \lambda_{i2} \left[ {\mathop \sum \limits_{j = 1}^{N} p_{ij2} } \right] + \lambda_{1} \left[ {p_{BS} } \right] + \mathop \sum \limits_{i = 1}^{K} \lambda_{i2} \left[ {\mathop \sum \limits_{j = 1}^{N} p_{t} } \right] $$
$$ L\left( . \right) = \left( {\mathop \sum \limits_{j = 1}^{N} \omega_{ij1} \log_{2} \left[ {1 + \gamma_{ij1} \tilde{p}_{ij1} } \right]} \right)\left\{ {\mathop \sum \limits_{i = 1}^{K} \alpha_{i1} } \right\} - \lambda_{1} \left[ {\mathop \sum \limits_{j = 1}^{N} \mathop \sum \limits_{i = 1}^{K} p_{ij1} w_{ij1} } \right] + \left( {\mathop \sum \limits_{j = 1}^{N} \omega_{ij2} \log_{2} \left[ {1 + \gamma_{ij2} \tilde{p}_{ij2} } \right]} \right)\left\{ {\mathop \sum \limits_{i = 1}^{ K} \alpha_{i2} } \right\} - \mathop \sum \limits_{i = 1}^{K} \lambda_{i2} \left[ {\mathop \sum \limits_{j = 1}^{N} p_{ij2} } \right] + \lambda_{1} \left[ {p_{BS} } \right] + \mathop \sum \limits_{i = 1}^{K} \lambda_{i2} \left[ {\mathop \sum \limits_{j = 1}^{N} p_{t} } \right] $$

Let

$$ \begin{aligned} f\left( {p_{ij1} } \right) = \log_{2} (1 + \gamma_{ij1} p_{ij1} )\left[ {\mathop \sum \limits_{i = 1}^{K} \alpha_{i1} } \right] - \lambda_{1} \mathop \sum \limits_{i = 1}^{K} \mathop \sum \limits_{j = 1}^{N} p_{ij1} \hfill \\ {\text{and}} \hfill \\ f\left( {p_{ij2} } \right) = \log_{2} (1 + \gamma_{ij2} p_{ij2} )\left[ {\mathop \sum \limits_{i = 1}^{K} \alpha_{i2} } \right] - \mathop \sum \limits_{i = 1}^{K} \lambda_{i2} \mathop \sum \limits_{j = 1}^{N} p_{ij2} \hfill \\ \end{aligned} $$

Substituting the above assumption in the previous derived equation

$$ L\left( . \right) = \left( {\mathop \sum \limits_{j = 1}^{K} \omega_{ij1} f\left( {p_{ij1} } \right)} \right) + \left( {\mathop \sum \limits_{j = 1}^{K} \omega_{ij2} f\left( {p_{ij1} } \right)} \right) + \lambda_{1} \left[ {p_{BS} } \right] + \mathop \sum \limits_{i = 1}^{K} \lambda_{i2} \left[ {\mathop \sum \limits_{j = 1}^{N} p_{t} } \right] $$
$$ L\left( . \right) = \left( {\mathop \sum \limits_{j = 1}^{K} \omega_{ij1} f\left( {p_{ij1} } \right)} \right) + \left( {\mathop \sum \limits_{j = 1}^{K} \omega_{ij2} f\left( {p_{ij2} } \right)} \right) + \lambda_{1} \left[ {p_{BS} } \right] + \mathop \sum \limits_{i = 1}^{K} \lambda_{i2} \left[ {\mathop \sum \limits_{j = 1}^{N} p_{t} } \right] + \mathop \sum \limits_{i = 1}^{K} \mu_{i1} \left[ { \delta } \right] + \mathop \sum \limits_{i = 1}^{K} \eta_{i1} \left[ { \delta } \right] - \mathop \sum \limits_{i = 1}^{K} (\upsilon_{i1} + \tau_{i1} )\left[ {R_{min} } \right] $$

The constant terms are neglected since it doesn’t affect the maximization of the function. Taking the derivatives of \( f\left( {p_{ij1} } \right) \) and \( f\left( {p_{ij2} } \right) \) with respect to power and setting it to zero yields,

$$ \frac{{df\left( {p_{ij1} } \right)}}{{dp_{ij1} }} = \left( {\mathop \sum \limits_{i = 1}^{K} \alpha_{i1} } \right)\left[ {\frac{{\gamma_{ij1} }}{{\ln 2 (1 + \gamma_{ij1} p_{ij1} }}} \right] + \lambda_{1} $$

Equating to zero yields,

$$ \left( {\mathop \sum \limits_{i = 1}^{K} \alpha_{i1} } \right)\left[ {\frac{{\gamma_{ij1} }}{{\ln 2 (1 + \gamma_{ij1} p_{ij1} }}} \right] + \lambda_{1} = 0 $$
$$ p_{ij1} = \left\{ {\left( {\mathop \sum \limits_{i = 1}^{K} \alpha_{i1} } \right)\frac{1}{{ - \lambda_{1} ln2}}} \right\} - \frac{1}{{\gamma_{ij1} }} $$

Similarly,

$$ p_{ij2} = \left\{ {\left( {\mathop \sum \limits_{i = 1}^{K} \alpha_{i2} } \right)\frac{1}{{\mathop \sum \nolimits_{i = 1}^{K} \lambda_{i} ln2}}} \right\} - \frac{1}{{\gamma_{ij2} }} $$

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Swapna, P.S., Pillai, S.S. Low Complexity Resource Allocation Techniques for Symmetrical Services in OFDMA Systems. Wireless Pers Commun 116, 3217–3234 (2021). https://doi.org/10.1007/s11277-020-07844-8

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