- freely available
Energies 2019, 12(20), 3819; https://doi.org/10.3390/en12203819
- Battery lifetime modeling is performed for battery-limited MTC nodes and the notions of an individual as well as network lifetimes are introduced and proven to equate to energy efficiency and power control in the discussed coexisting network.
- The uplink power control design in a coexisting HTC/MTC network is formulated as a stochastic differential game (SDG) by selecting appropriate state dynamics.
- A mean field game (MFG) for the MTC network is formulated which can accommodate a massive number of devices. State space of remaining battery energy and interference is considered to find an optimal power control policy for MTC transmit power allocations.
- A modified utility function is proposed for the formulated MFG depending on the signal-to-interference-plus-noise-ratio (SINR) thresholding and experienced interference power. A mean field approximation (MFA) technique is utilized to modify the formulated utility function to more appropriately adapt to the converted MFG design.
- The proposed MFG is solved using finite difference scheme with Lax-Freidrichs technique being utilized to solve the coupled FPK and HJB system of equations.
- Equilibrium analysis for the SDG as well as MFG design is performed by considering the formulated cost function.
2. Literature Review
3. Architecture and Problem Formulation
3.1. Network Model
3.2. System Model
3.2.1. Hybrid Interference Model
3.2.2. Individual Battery Lifetime
3.2.3. Battery Lifetime and Energy Efficiency
3.2.4. Network Battery Lifetime
3.3. Problem Formulation
4. Stochastic Differential Power Control Game
4.1. State Space Model
4.2. Utility Function
4.3. Optimal Control Policy
4.4. Stability and Equilibrium Analysis
5. Mean Field Power Control Game
5.1. Mean Field Game
5.1.1. Mean Field and Player Interactions
- Individual player makes a small contribution towards mass of all the other players and the MFG. This property ensures that the players act in a more rational and independent manner while choosing the optimal control policy.
- The action selection by each player depends on its own interests and only interacts with the mean field of players instead of participating in individual interactions.
- The continuity property of the mean field model is ensured by the provision of the massive number of MTC devices for power control.
- When analyzing a K-player game, the MFA technique is used to model the actions of the players and prove the exchangeability property.
5.1.2. Mean Field Utility Function
5.1.3. Mean Field Game Formulation and Equilibrium Analysis
5.2. Finite Difference-Based MFG Solution
5.2.1. Solving HJB and FPK Equations
5.2.2. Power Control Scheme
|Algorithm 1: Finite difference method for optimal mean field, Lagrangian and transmission power evaluation|
|Initialization:M(0, 0, 0), P(X + 1, 0, 0), λ(X + 1, 0, 0), Iter = 1|
Step 1: Mean field evaluation
for alli = 1: 1: Xdo
for allj = 1: 1: Ydo
for allk = 1: 1: Kdo
Calculate mean field M(i + 1, j, k) using Equation (31)
if P(i, j + 1, k) = 0
M(i + 1, j + 1, k + 1) = M(i, j, k)
else: M(i + 1, j + 1, k + 1) = 0
Step 2: Lagrangian evaluation
for all i = X + 1: −1: 1 do
| for allj = 1: 1: Y+1do|
for all k = 1: 1: K do
Calculate λ(i−1,j,k) using Equation (39)
Step 3: Transmission power evaluation
for all i = 1: 1: X + 1 do
for all j = 1: 1: Y + 1 do
for all k = 1: 1: K do
Calculate P(i−1,j,k) using Equation (38)
until Iter ≥ Itermax
6. Performance Evaluation and Discussion
6.1. Simulation Setup
6.2. Simulation Results
6.2.1. Mean Field Power Control Policy
6.2.2. MTC Power Allocation and Battery Energy State
6.2.3. Energy Efficiency and Network Lifetime
Conflicts of Interest
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