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
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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
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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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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