AbstractWe study max-min fair rate allocation and routing in energy harvesting networks where fairness is required among both the nodes and the time slots. Unlike most previous work on fairness, we focus on multihop topologies and consider different routing methods. We assume a predictable energy profile and focus on the design of efficient and optimal algorithms. Our analysis provides insights into the problem structure and can be applied to other related fairness problems..

Algorithms for Max-min Fair Rate Assignment and Routing in Energy Harvesting NetworksJelena Marašević1, Cliff Stein2, Gil Zussman1

1Department of Electrical Engineering, 2Department of Industrial Engineering and Operations Research,Columbia University, New York, NY

We are grateful to Professor Mihalis Yannakakis for useful discussions. This research was supported in part by NSF grants CCF-1349602, CCF-09-64497, and CNS-10-54856.Poster based on: J. Marašević, C. Stein, G. Zussman. Max-min Fair Rate Allocation and Routing in Energy Harvesting Networks: Algorithmic Analysis. ACM MobiHoc’14, 2014.

Importance of Fairness

Finding Single-path Routes An approximation to a “good” routing tree is NP-hard

to determine Finding a “good” single-path routing is NP-hard • However, we have designed an algorithm that

determines (in polynomial time) a time-invariable routing that maximizes the minimum sensing rate

Nodes:• 2 gets 0 rate• gets 0 rate

Total throughput maximization:

References[1] B. Radunovic and J.-Y. L. Boudec. A unified framework for max-min and min-max fairness with applications. IEEE/ACM Trans. Netw.,15(5):1073-1083, Oct. 2007.[2] R. Srivastava and C. E. Koksal. Basic performance limits and tradeoffs in energy- harvesting sensor nodes with finite data and energy storage, IEEE/ACM Trans. Netw., 21(4):1049-1062, 2013.[3] B. Gurakan, O. Ozel, J. Yang, and S. Ulukus. Energy cooperation in energy harvesting two-way communications. In Proc. IEEE ICC, 2013.[4] S. Sarkar, M. Khouzani, and K. Kar. Optimal routing and scheduling in multihop wireless renewable energy networks. IEEE Trans. Autom. Control, 58(7):1792-1798, 2013.[5] S. Plotkin., D. Shmoys, and É. Tardos. Fast approximation algorithms for fractional packing and covering problems. Math. of OR, 20.2 (1995): 257-301.

System Model

• A sink• nodes • links• time slots• ’s known• Battery capacity: • Rates:

The sink

1 2 3 T-1 T… …𝑡𝐵𝑖 𝑏𝑖 ,𝑡

𝑒𝑖 ,𝑡

𝝀𝒊 ,𝒕

=1 =2

1 2

Routing Topologies

a) Routing Tree

b) Single-path Routing

c) Multi-path Routing

Routing can be time-variable (change from one time slot to another ), or time-invariable (remain fixed over time slots )

≤ ≤

Rate Assignment and Routing Algorithms Water-filling framework; max-min fairness lexicographic maximization

¿

1 2 3 40

200

400

I (

W/c

m2 )

DaysTime:• High rates at the

peak, zero over night

Motivation: IoT..

Academia

• Networks of devices that traditionally have not been networked.

Maximizing Fixing Total Rates Routing

Single-path Fixed fractional Time-variable fractional

Rate Assignment in a Single-path Routing

descendants

Maximization:

1.For each node , find the maximum supported rate, assuming ’s descendants can support the same rate

2.Return the minimum rate from 1.

Fixing:

1.Fix all ’s for which 2.Fix the rates in all the slots with no extra

energy preceding

3.Fix the rates off all the descendants of ’s fixed in 1. and 2., in the same time slots

𝑏𝑖 ,𝑡

𝑩

11… 10987654𝑡=10

Fractional (Multi-path) Routing Restructuring constraints get a packing problem

Feasible rates: at least as hard as feasible 2-commodity flow• Unlikely to be solved optimally without linear programming

PTAS design:• Maximization: packing algorithm [5] + structural properties • Fixing: 1 LP over current solution’s -neighborhood

, for

, for

Future Work Fairness guarantees with a:• Distributed algorithm? (Challenge: communication overhead)• Online algorithm? (Challenge: small prediction window)

Different types of fairness:• Proportional fairness?• General -fairness?

Challenge: guaranteed convergence time