Low complexity and distributed energy minimization

1. Low complexity and distributed energy minimization L. Lin, X. Lin and N. B. Shroff Presented by: SrikanthHariharan

2. Introduction • Goal: To minimize total power consumption in a multi-hop wireless network subject to a given load. • Total power consumption can be reduced by jointly optimizing power control, scheduling and routing. • Known optimal solution is centralized with high computational complexity. • Approximate optimal solution is also centralized and it yields a 3-approximation ratio. • Here, a low complexity and distributed (2+ε) - approximate solution is developed.

3. Notations and Model • G(V,E) – Network with nodes V and links E. • No(v) – Outgoing links of node v. • Ni(v) – Incoming links of node v. • Node-exclusive interference model and time slotted system. • h(Re) – Power to support a data rate Re of link e – assumed to be convex, non-decreasing with h(0)=0. • D – set of destinations. • Tvd – Long-term average data rate of the flow that needs to be supported from source node v to destination d. • fed – Average amount of data rate on link e allocated for destination d.

4. Problem Formulation Problem: Scheduling component in (*) has a high computational complexity.

5. Approximation • For all time-slots mwhen link e is activated, Re(m) is independent of m in any power-optimal scheme. • So, the objective function of (*) is now • From existing results on low – complexity scheduling, we have and if , where ηis arbitrarily small, then a maximal schedule can be computed such that each link is activated for fraction of time-slots. • Replace scheduling constraints by • β = 1 – Lower bound. • β = ½ - η– Upper bound on optimum.

6. Handling Non-convexity • Objective function and constraints non-convex. • Solution – Do change of variable • te – Fraction of time-slots link e is activated. • te = 0 => fed = 0 for all d. • Then, the long term average power consumption from link e can be expressed as

7. Convex optimization problem

8. Lagrange Duality and Energy Minimization Algorithm Strong duality holds.

9. Interpretation • Power cost – • Scheduling cost – • Utility – • can be interpreted as an approximation of the scaled version of the queue length at node v for destination d if constant step sizes are used. le is minimum when all the data rates are assigned to the destination with the maximum positive backlog difference.

10. Interpretation • What is the optimal te? • When is te = 0? • All backlog differences are negative => Utility does not increase by transporting data to next hop on this link. • Energy saving and/or the need to gain more flexibility in scheduling is more important than the utility of transporting data to next hop.

11. Components • Routing: Choose the flow with maximum positive backlog difference. • Power control: Choose Re to minimize le(Re). • Link Assignment: Choose te to minimize tele(Re) – This determines the amount of time link e should be on. • Maximal Scheduling: Need to determine exact time slots for link e given the fraction of up-time • Distributed algorithms exist for the node-exclusive interference model when .

12. Convergence Results • Constant step sizes: When step sizes are small, • dual variables converge to within a small neighborhood of the optimal dual solution • primal variables may not converge, however, the long-term average of the resultant power consumption is arbitrarily close to the optimal power consumption. • Diminishing step sizes: For step sizes that satisfy , the algorithm converges to the optimal value. • If the step sizes are chosen as , if f and t converge, then . • Power efficiency ratio in an AWGN channel is at most (2+ε) that of the optimal value with β=1.

13. Numerical Results t= 4000: σ(1,7) (gain) decreases. t=8000: Data rate of flow 2 decreases to 250 kbps.

14. Conclusion • Joint power control, link scheduling and routing algorithm proposed to minimize energy. • Distributed algorithm with low computational complexity. • Power efficiency ratio tighter than existing solution. • Solution adapts to changes in channel conditions and flow rates. • Methodology can be extended to a general interference model.