Efficient Wireless Scheduling Algorithms: A Message Passing Study

1. Message-Passing for Wireless Scheduling: an Experimental Study Paolo Giaccone (Politecnico di Torino) Devavrat Shah (MIT) ICCCN 2010 – Zurich August 2nd, 2010

2. Scheduling in wireless networks • schedule simultaneous transmissions • to avoid interference among neighboring nodes • needs coordination across the communication medium • simplified interference model • a transmission is successful if none of its neighbors are transmitting • neighbors defined simply by the transmission range RT

3. System model and notation • packet duration is fixed and time is slotted • i is the node • xi=1 if node is transmitting, 0 if not • X=[xi] is the transmission vector • N(i) is the set of neighboring nodes at a distance < RT from node i, i.e. the set of nodes that may eventually interfere • a interference-free X must be

4. Interference graph • G=(V,E) • V is the set of nodes • edge • an independent set (IS) on G corresponds to a subset of nodes that can transmit simultaneously without interference

5. Optimal scheduler • Optimal scheduling • for generic constrained resource allocation problem • Tassiulas and Ephremides, IEEE Trans. Automatic Control, 1992 • to maximize throughput, compute the maximum weight independent set (MWIS) at each timeslot • weight wi of a node i is the number of enqueued packets 10 5 5 10

6. Centralized algorithms for IS • IS is NP-complete • greedy approximations • Rnd-IS: S is a random permutation of nodes • MaxW-IS: S is a sequence of nodes in decreasing order of weights 10 10 9 9 9 9 1 1

7. Message passing approach • derived from belief propagation to perform inference on graphical models, such as Bayesian networks and Markov random fields • successfully employed in many fields: physics, computer vision, statistics, coding (Viterbi algorithm), generic combinatorial optimization • amenable to parallel implementation • network protocols are based on message passing algorithms

8. Message passing • update phase • each node sends a message to the neighbors based on the received messages • is the message from node i to j at iteration n • estimate phase • each node takes a local decision

9. Message Passing for MWIS Derived by Sanghavi, Shah, Willsky, IEEE Transactions in Information Theory, 2009

10. Computational tree interpretation 1 2 5 3 4

11. Contribution • for a generic graph with loops, messages may not converge, leading to unfeasible solutions • to improve converge we propose • use of memory • message averaging • we investigate their effects on the performance

12. Memory • exploit “continuity” in the system state • queue evolution is limited: |wi(t+1)-wi(t)|≤1 • Property: |MWIS(t+1)-MWIS(t)|≤ N • MWIS(t) is also a good candidate for time t+1 • idea: keep the most recent messages from the previous timeslot as the initial value • leads to reduced convergence time

13. Message averaging • observation: message may oscillate • idea: to average message values with a weighted moving average filtering • filter constant: α=1  no filtering

14. Asynchronous update • Earlier pseudocode of MP-IS assumes that all the nodes update synchronously their messages in parallel at each iteration • this assumption is not needed • We assume uncoordinated, asynchronous update • each node wakes at some random time • it updates the outgoing messages based the messages received so far • its sends the new updated messages to all its neighbors

15. Simulation results • given • interference graph • traffic pattern • the simulator estimates • throughput • packet delay • packet loss for the whole network and for each node

16. Noisy grid as interference graph • random geometric graph • place N nodes on a perfect grid • add some noise to the position (η parameter) • η=0 corresponds to a perfect grid • η very large corresponds to a bidimensional Poisson process • all the nodes with distance < RT are connected η=0.0 η=0.5 η=1.0

17. Admissible traffic pattern • given G, finding the admissibility rate region is NP-hard • ri is the normalized arrival rate at node i • ρ is the load factor • ρ=1 is such that Rnd-IS will obtain 100% throughput • K is a traffic parameter • K=1  unbalanced traffic • large K  balanced traffic

18. Perfect grid • N=100 nodes • ρ=1.35 • Conclusions • memory boosts performance of MP-IS • one iteration is enough for MP-IS to be optimal

19. Noisy grid • ρ=1.0 • Conclusions • very little throughput degradation in irregular graphs

20. Conclusions • MP-IS with just 1 iteration + memory + averaging performs comparable with centralized algorithms • similar result for Tree-Reweighted Message Passing algorithm • promising approach for the limited protocol overhead • belief propagation is taking care of • longer queues -> messages are proportional to wi • graph structure -> messages depend on the graph