EE 685 presentation

1. Making Distributed Rate Control using Lyapunov Drifts a Reality in Wireless Sensor Networks By Avinash Sridrahan, Scott Moeller and Bhaskar Krishnamachari. EE 685 presentation

2. Objective of the paper • Formulating therate control problem, over a collection tree, in a wireless sensornetwork as a generic convex optimization problem • The paper proposes a distributed back pressure algorithm using Lyapunov driftbased optimization techniques. • First step is to show thatstochastic network optimizationcan be directly applied to a CSMA based WSN using the novel ans simpistic receiver capacity model developed by authors • The algorithm has been by implemented inTmote sky class devices in order to experimentally show that back-pressure algorithm on a real sensor network gives traffic rates close to the analytically predicted values

3. Motivation and basic approach • Stochastic networkoptimization approach that yields simple distributed algorithmsfor dynamic scenarios is based on theuse of Lyapunov drifts • Lyapunov drift based techniques described provide for distributed rate control based purely on localobservations of neighborhood queues. • The stability of the system is guaranteed and mechanisms to achieveoptimization with respect to given utility functions has been provided. • Until this paper, Lyapunov drift based techniques has only been applied to TDMA MAC based networks that operate under slotted-time assumption • TDMA based system difficult to have time-synchronized => so asynchronous CSMA based approach is more preferable • Primary contribution of this paper is to show that ratecontrol algorithms based on the Lyapunov drift frameworkcan be built over asynchronous CSMA-based MAC protocolsin a wirelesssensor network setting • This is achievable thanks to linear constraints presented by our receiver capacity model, transformedto a set of virtual queues

4. Problem Framework The problem is formulated for • Wireless sensor networks where the dominant topology is acollection tree where multiple sources are forwarding data toa single sink. • The optimization problem is defined as maximization of aggregate rate-dependent objective function • Where ri is the time average source rate for each source i,gi(ri) isassumed to be convex and Λis the capacity region for the collection tree. • To solve the above optimization problem, we need to the know thecapacity region Λ which constrainsthe optimization problem.

5. Problem Framework(capacity region problem) • Asynchronous CSMA based MAC creates capacity region problem due to difficult to predict and analytically untractable capacity utilization behavior • The authors utilize a simplistic and linearized receiver capacity approximation approach previously proposed by Sridrahan et al. • Thecore idea is to associate a constant bandwidth capacity witheach receiver in the network. • This capacity must be shared byall transmitters within interference range of that receiver. • Thismodel corresponds to a linear approximation of the capacityregion for each receiver. • Any linear combination ofneighborhood transmission rates is feasible so long as thenet overheard rate does not exceed the receiver bandwidth

6. Capacity region problem :an example network

7. Lyapunov Optimization formulation • By modeling optimization problem constraints as virtualqueues,stochastic network optimization can minimize the driftof a linear combination of the physical and virtual queues ofthe whole system. • Forwarding queue stability has been ensured while obeying constraints. • The objective function may beincorporated as a penalty or reward function included in thedrift boundwhich sets up the trade-off between systemqueue size/latency and utility optimality. • The modularityof the algorithms resulting from this approach makes them a promising and attractive option • This is achieved by the additionalconstraints to the optimization problem which use the receivercapacity model, and by relaxation of the exact channel capacityassumption inoptimization over Xi(t).

8. Lyapunov Optimization with Receiver Capacity VirtualQueues • The strength of proposed technique lies indecoupling the physical channel capacity region from thetransmission rate decisions (Xi(t)s). • The channel capacity is abstracted as follows • It is assumed that all nodes can transmit simultaneously withoutinterference, and support only two transmission values. • In agiven slot t, each Xi(t) is set to one of {0,Bmax}, with Bmaxset to a valuemarginally greater than the maximum receiver bandwidth ofany node in the network. • This way, nodes toggle between onand off modes of operation independently, with no concern forneighboring node’s activities. • The approach on the paper relies on the receiver capacitymodel constraints to enforce stability over the CSMA channel.

9. Lyapunov Optimization with Receiver Capacity VirtualQueues • Using the Lyapunov drift approach, eachof the constraints in the problem P1 is converted to a virtual queue. • Avirtual queue Zi is associated with each node. The queuing dynamics for each of the virtual queues Zi(t)is given as follows: • Each time slot, the queue is first serviced (perhaps emptied),then arrivals are received. • Each Zi queue therefore receives thesum of transmissions within the neighborhood of node i, thenis serviced by an amount equal to the receiver capacity ofnode i. • Therefore, for every timeslot in which neighborhoodtransmissions outstrip the receiver capacity of the node, thisvirtual queue will grow.

10. Lyapunov Optimization with Receiver Capacity VirtualQueues • Every node also hasa physical forwarding queue Ui. • The queuing dynamics of the physical queue Ui(t) is similarto that of the virtual queues and is given by: • Each node i first attempts to transmit Xi(t) unitof data to its parent, then receives units of data fromeach child node j. • Attemptedtransmissions (Xi(t)) are differentiated from true transmissions ( ). • Thedifference being that while it may be most optimal to transmita complete Xi(t) units of data in this timeslot, the queue maynot contain sufficient data to operate optimally, so ˆXi(t) ≤Xi(t).

11. Lyapunov Optimization :control decision and admission decision • Combining the objective function with the queueingdynamics presented in equations (7) and (8), Lyapunov drift optimization will result in analgorithm that has two components: • A control decision • An admission decision. • Each decision will be performed by everynode in the network at each time step. • A node performs acontrol decision to determine whether it is optimal to forwardpackets up the collection tree. • The admission decision isperformed in order to determine if a local application layerpacket should be admitted to the forwarding queue.

12. Lyapunov Optimization :Control decision • The control decision for a node iwith a parent k is the following: . • If condition (9) is true, maximize Xi(t) by setting it to Bmax. • Anode transmits data to the parent if and only if the differentialbacklog between the node and its parent exceeds the sum ofvirtual queues within the local node’s neighborhood. .

13. Lyapunov Optimization :Admission decision • The local admission decision fora node i is based on selecting Ri(t) so as to maximize the following . • Node i then selects a volume of localadmissions in timeslot t equal to Ri(t) such that expression(10) is maximized. • Note that Vopt, the tuning parameter that determines howclosely we achieve optimal utility, appears only in theadmissiondecisions. • As Vopt grows, so does the acceptable backlogfor which admissions are allowable (Ui(t)). .

14. Lyapunov Optimization :Vopt as tuning parameter • An intuition for this behavior of Vopt can be obtained bylooking at the feasible solutions of the optimizationproblemP1. • In the optimal solution of P1, all the constraints in P1need to be tight. • This implies that the system needs to beat the boundary of the capacity region => system will be unstable (queue sizes will be unbounded). • For a stable system, the constraints should be loose • This requires that the system to achieve a suboptimal solution with respect to the objective function while ensuring stability. • Thus, Vopt tunes how closely the algorithm operates to the boundary of the capacity region. .

15. Variables used in Lyapunov Formulation

16. Lyapunov Optimization :Derivation of admission and control decisions • Let the discrete time queueing equations for forwarding queues (Ui(t)) and virtual queues (Zi(t)) be thosedefined by update equations (8) and (7) respectively. • We definethe Lyapunov function as follows: • Then Lyapunov drift could be written as :

17. Lyapunov Optimization :Derivation of admission and control decisions • Squaring the forwarding queue discrete time queueing equationyields the following: • In typical systems, there exists a bound to themaximum values Xi(t) and Ri(t). • We know that Xi(t) < Bmax. Letthe bound on admissions per timeslot be Rmaxi for node i.. We’ll define constant Gi as follows:

18. Lyapunov Optimization :Derivation of admission and control decisions • Similar manipulation can be carried out for the virtual queues. • Define constant Ki in a manner similar to Gi:

19. Lyapunov Optimization :Derivation of admission and control decisions • Substitution of Gi and Ki into equations (13) and (14), thensumming over all nodes i, and finally taking the expectationwith respect to (Ui(t),Zi(t)), yields the following Lyapunovdrift bound:

20. Lyapunov Optimization :Miniziming Lyapunov Drift • Prior work shows that minimizing Lyapunov drift provides guaranteedstability over system inputs lying within the capacity region. • As was demonstrated in [6],an utilityfunction can be incorporated into the drift bound. • Let Y (t) = ∑Gi(Ri(t)) be thesystem utility,we subtract • from both sides of(15), yielding:

21. Lyapunov Optimization :RHS minimization • In order to minimize RHS, minimize the righthand side for every system state • Constant terms involving Ki and Gi.neglected • The remaining terms canbe separated into coefficients multiplying Xi(t) and Ri(t). • The goal is to minimize these termsthrough intelligent selection of per-timeslot decision variablesXi(t) and Ri(t).

22. Lyapunov Optimization :Optimal stable control decision involving Xi(t) • Consider with node i with parent node k • The coefficient associated with transmission variable Xi(t) is • If transmission rates Xi(t) and Xj(t) are independent∀i, j, then in order to minimize the RHS of(16), we maximize Xi(t) ∀i such that (17) is negative. • A nodetherefore transmits data to the parent whenever the differentialbacklog between the node and its parent exceeds the sum ofvirtual queues within the local node’s neighborhood.

23. Lyapunov Optimization :Optimal stable admission decision involving Ri(t) • The coefficientassociated with admission variable Ri(t) is: • In order to minimize the RHS of (16), wemaximize Ri(t) ∀i such that (18) is negative. • This equates toa simple admission control scheme. • If the forwarding queuesize scaled by admission rate exceeds (Vopt/2) times the utilityfor all admission rates, then the admission request is rejected. • Otherwise, a rate is chosen which maximizes g(Ri(t))−Ri(t).