Optimal Control in Multi-hop Wireless Networks: Distributed Cross-layer Algorithms Presentation

1. Distributed Cross-layer Algorithms for the Optimal Control of Multi-hop Wireless Networks By Atilla Eryılmaz, Asuman Özdağlar, Devavrat Shah and Eytan Modiano EE 685 presentation

2. Objective of the paper • Proposing a general frameworkthat facilitates the development of distributed mechanismsto achieve full utilization of multi-hop wireless networks. • Describe generic randomized routing, scheduling andflow control scheme that allows for a set of imperfections in theoperation of the randomized scheduler to account for potentialerrors in its operation. • Studying the effect of suchimperfections on the stability and fairness characteristics of thesystem, and • Explicitly characterize the degree of fairness achievedas a function of the level of imperfections.

3. Motivation and basic approach • The results reveal therelative importance of different types of errors on the overallsystem performance, and provide valuable insight to the designof distributed controllers with favorable fairness characteristics. • A specificinterference model, namely the secondary interference model, have emphasized and distributed algorithms with polynomial communicationand computation complexity in the network size have been developed. • This isan important result given that earlier centralized throughput- optimalalgorithmsrelies on thesolution to an NP-hard problem at every decision. • This resultsin a polynomial complexity cross-layer algorithm that achievesthroughput optimality and fair allocation of network resourcesamongst the users. • It has also been shown that the algorithmic approach proposed by the authors enables efficientapproximation of the capacity regionof a multi-hop wireless network.

4. Contributions • A scheduling-routing algorithmcombined with a congestion controller for a general systemmodel whereby multi-hop flows are considered. • Explicit characterization of the effect ofdifferent types of errors on the overall performance. • A generic cross-layer mechanism withthree components: • A randomized scheduling component • A routing component (implemented by the network nodes) aimedat allocating resources to the flows efficiently; and a • A dualcongestion control component (implemented at the sources)aimed at regulating the flow rates to achieve fairness. • Studyof the proximity of the achieved rate allocationwith generic cross-layer scheme to the fair allocation, andcharacterization of the performance loss as a function ofthe imperfections of the underlying scheduler. • Distributed algorithm for the cross-layer mechanism with secondary interference model

5. System Model The problem is formulated for • A wireless network which is modeled as a undirected graph G = (N;L) where N is the set of nodes and L isthe set of links. Let N and L be the number of nodes andlinks in the network, respectively. • A time-slotted system with synchronized nodes in which each time-slot is long enough to accommodate a single packet transmission. • General interference model specified by aset of link-pairs that interfere with each other. It has been assumed that if twointerfering links are activated in a slot, both transmissions fail. • Link allocation and scheduling is made based on feasible allocations principle where no two active links will interfere with each other. • At each node, a buffer (queue) is maintained for each destination.

6. System Model • The flow that enters the network at node n and leaves it at node d as is referred as Flow-(n, d) • X[t] =X(d)n[t] denotes the vector of arrivals to the network in slot t corresponding to the arrivals for Flow-(n, d) • x(d)n [t] denotes the mean flow rate of Flow-(n, d) in slot t, i.e., x(d)n [t] = E[ X(d)n[t]]. • Then, the mean flow rate of Flow-(n, d)is defined as • S(d)(n,m)[t] ∈ {0, 1} is 1 if link(n,m) serves a packet destined for node d in that slot, and 0 otherwise. • This implies that for all (n,m) ϵN

7. Queue Evolution for nodes • At each node, a buffer (queue) is maintained for each destination. • We let Q(d)n[t] denote the length of the queue atnode ndestined for node d at the beginning of slot t. • Evolution of Q(d)n[t] when n ≠ d satisfies • We also have

8. Queue stability and Capacity region

9. Achieveable goals • Given the general model described above, the system goal is to designdistributed algorithms that achieve • Throughput-optimality • Fair allocation of the network resources amongst the flows. • A policy is called as throughput-optimal if it can supportany mean flow rate in the capacity region without violating the network stability. • To define fairness ,“utility maximization” frameworkof economics has been used: • With each flow, say Flow-(n, d), weassociate a utility function Un,d(·), of the mean flow rates whereby Un,d(x(d)n) is a measure of the utility gained by Flow-(n, d) for the mean flow rate x(d)n . • Based on thelaw of diminishing returns, the function Un,d(·) is concaveand non-decreasing for all flows. So throughput optimal fair allocation is achieved by mean flow rate vector

10. Generic Cross-layer Scheme • Genericcongestion control-routing-scheduling scheme aiming to achieve aforementioned throughput-optimality and fairness goals • The scheme combines ideas from state-of-art congestion controllersdesigned for wireless networks and the randomized scheduling strategy introducedby Tassiulas . • The proposed algorithm not onlyextends the use of randomized scheme of Tassiulas et al to multi-hopnetworks with general interference models, but also utilizes theparallel use of a dual congestion controller to achieve fairness.

11. Generic Cross-layer Scheme • The scheduling component builds on two algorithms: • PICK, which randomly picks a feasible allocation satisfying aspecific condition • UPDATE, which contains a network-wide comparison operation

12. Generic Cross-layer Scheme • In the operation of PICK and UPDATEalgorithms, various types of imperfections andrelaxations have been allowed to accommodate errors and to facilitate distributedimplementations. • The routing component determines whichpackets to be served over which links so as to optimize theirroutes. • Finally, the congestion controller component adjuststhe rate of injected traffic into the network to fully utilize theresources, i.e., to solve (2). • The scheme operates in stages, each stage containing afinite number of time slots where the number of slots is adesign choice. • The scheduling-routing and congestion controldecision is updated at the beginning of each stage, and is kept unmodified throughout the stage.

13. Scheduling Component

14. Scheduling component imperfections • δrelaxes the constraint of picking the optimum feasible allocationin each iteration, hence significantly reduces thecomplexity of this operation; • γcaptures the potential errors inthe computation of the total weight of the randomly selectedschedule; • ψcaptures the potential errors in the comparisonof the weights of the previous and the random scheduler • The imperfections included in the scheduling component arelikely to occur when randomized or distributed methods areemployed to perform these operations..

15. Routing Component

16. Congestion control Component

17. Analysis : Theorem 1

18. Analysis : Theorem 1 : Proof

19. Analysis : Theorem 1 : Proof

20. Analysis : Theorem 1 : Proof

21. Analysis : Theorem 1 : Proof

22. Analysis : Theorem 1 : Proof

23. Analysis : Theorem 1 : Proof

24. Analysis : Theorem 1 : Proof

25. Analysis : Theorem 1 : Proof • Now we are aiming to bound (21)

26. Analysis : Theorem 1 : Proof • Thus, T0 is the first slot after t when the randomly pickedschedule R according to (5) is equal to the optimum schedule,and (6) is satisfied • T1 is the first slot after T0 when thecondition in (6) is violated. • Note that in the interval betweenT0 and T1, the system is well-behaved, and no undesired eventsuch as that in (6) occurs. • Finally, let us define T2 := T − min (T, T1) as the remaining time after T1 until the end of T slots, if any. • The idea is to show that if T is sufficiently large,the duration between T0 and T1 will dominate the interval ofduration T. • Next, we make this argument rigorous.

27. Analysis : Theorem 1 : Proof

28. Analysis : Theorem 1 : Proof

29. Analysis : Theorem 1 : Proof

30. Analysis : Theorem 2

31. Analysis : Theorem 2 : Proof

32. Analysis : Theorem 2 : Proof

33. Analysis : Theorem 2 : Proof

34. Analysis : Theorem 2 : Proof

35. Analysis : Theorem 2 : Proof • Noting thatV (Y) ≥ 0 for all feasible Y, and re-arranging the terms inthis expression, we can obtain

36. Analysis : Theorem 2 : Proof

37. Analysis : Theorem 2 : Proof

38. Theorem 2 Implications • Theorem 2 reveals the effect of the errors and relaxationin the operation of the scheduler. • In particular, we see that,when γ=ψ=0, and δ > 0, the cross-layerscheme achievesoptimal performance. • Also, we observe that the effect of ψcan be detrimental unless it is significantly smaller than δ. • In comparison, the effect of γ appears to be milder if it canbe made small. • Ideally, we would like to design schedulerswith γ=ψ=0 in which case optimal performance can beguaranteed. • One such schedulerthat is applicable to second order interference model will be proposed

39. Algorithm DesignConstantly Backlogged Sources • We have two sequential algorithms PICK and COMPARE • The PICK algorithmis a randomized, distributed algorithm thatyields a feasibleschedule R[t] satisfying (5) in finite time. • The COMPAREalgorithm compares the total weights of the old schedule S[t]with the new schedule R[t] according to (6) in a distributedmanner. • An important feature of the COMPARE algorithm isthe use of the conflict graph of the two schedules. • On theconflict graph, a spanning tree can be constructed in a distributedmanner and used for comparison of the weights of thetwo schedules in polynomial time. • The conflict graph enablesa natural partitioning of the network, whereby decisions canbe made independently in different partitions in a distributedmanner. • The operations on the conflict graphcan be mapped to the actual network operations owing to thespecial structure of the problem.

40. PICK ALGORITHM

41. PICK ALGORITHM

42. COMPARE ALGORITHM • We have two sequential algorithms PICK and COMPARE • TheCOMPARE Algorithm is composed of two proceduresthat are implemented consecutively: • FIND SPANNING TREEand COMMUNICATE & DECIDE. • The FIND SPANNING TREEprocedure finds a spanning tree for each connected componentof G′ in a distributed fashion. • Then, the COMMUNICATE &DECIDE procedure exploits the constructed tree structure tocommunicate and compare the weights of the two schedulesin a distributed manner.