Message-Passing for Wireless Scheduling: an Experimental Study

Paolo Giaccone (Politecnico di Torino)Devavrat Shah (MIT)

ICCCN 2010 – Zurich

August 2nd, 2010

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

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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

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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

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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

510

10 5

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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

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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

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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

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Message Passing for MWIS

9Derived by Sanghavi, Shah, Willsky, IEEE Transactions in Information Theory, 2009

Computational tree interpretation

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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

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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 time12

Message averaging

• observation: message may oscillate• idea: to average message values with a

weighted moving average filtering– – filter constant: α=1 no filtering

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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 update1.each node wakes at some random time2.it updates the outgoing messages based the messages

received so far3.its sends the new updated messages to all its neighbors

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Simulation results

• given– interference graph– traffic pattern

• the simulator estimates – throughput– packet delay– packet loss

for the whole network and for each node

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Noisy grid as interference graph

• random geometric graph1. place N nodes on a perfect grid2. add some noise to the position (η parameter)

• η=0 corresponds to a perfect grid• η very large corresponds to a bidimensional Poisson process

3. all the nodes with distance < RT are connected

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η=0.0 η=0.5 η=1.0

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

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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

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Noisy grid

• ρ=1.0

• Conclusions– very little throughput degradation in irregular graphs

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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

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