Stability and Fairness of Service Networks Jean Walrand – U.C. Berkeley Joint work with A. Dimakis, R. Gupta, and J. Musacchio.

Stability and Fairness of Service Networks Jean Walrand – U.C. Berkeley Joint work with A. Dimakis, R. Gupta, and J. Musacchio

Outline Stability of Longest Queue First  Fluctuations can stabilize Fairness through flow control  Control of long term rates Fairness of multiple access  Impatience may help in a crowd

Outline Stability of Longest Queue First  Fluctuations can stabilize Fairness through flow control Control of long term rates Fairness of multiple access Impatience may help in a crowd

Stability of Longest Queue First with Antonis Dimakis (PhD 5/06) Motivation Easy Case Subtle Effect

LQF - Motivation Wireless:  Goals: Simple protocol, large throughput  Transmission priority increases with backlog 2345 16

LQF: Motivation Iterated Longest Queue First (iLQF) [McKeown’95] :  Queues are considered in decreasing queue size order. Maximum throughput? 11 12 22 21 11 22 input 1 input 2 output 2 output 1 Q 11 (t) Q 12 (t) Q 21 (t) Q 22 (t) Switch:

Stability of LQF: Easy Case Example: LQF: 12 – 9 – 8 7 – 9 – 8 9 – 9 – 8 w.p. 1/2 123 1 2 3 service vectors i.i.d. arrivals w.p. 1/2

Stability of LQF: Easy Case Necessary: 1 + 2 <1, 2 + 3 <1. Sufficient! Under LQF, longest queues tend to decrease:  Say, Q 1 ¼ Q 2 >>Q 3, for some time.  Then, Q 1 +Q 2 decreases, and so do Q 1,Q 2. Key: locally in time, service from common resource pool. 123 1 2 3

Stability of LQF: Easy Case Local Pooling: Assume {1, 2} are longest for some time Note that {1, 2} are served at constant rate (1) (We say that {1, 2} satisfies Local Pooling.)  {1, 2} must decrease (because 1 + 2 <1)  longest queue must decrease 123 1 2 3

Stability of LQF: Easy Case Local Pooling: Assume {1, 2, 3} are longest for some time Note that {1, 2} are served at constant rate (1) (We say that {1, 2, 3} satisfies Local Pooling.)  {1, 2} must decrease (because 1 + 2 <1)  longest queue must decrease 123 1 2 3

Stability of LQF: Easy Case Local Pooling: Assume {1, 3} are longest for some time Note that {1, 3} are served at constant rate (2) (We say that {1, 3} satisfies Local Pooling.)  {1, 3} must decrease (because 1 + 3 <2)  longest queue must decrease 123 1 2 3

Stability of LQF: Easy Case Local Pooling: Set L satisfies LP if it has a subset K that LQF serves at a constant rate Theorem: If every set L satisfies LP and if the rates are feasible, then LQF makes system stable Proof: Longest queue is a Lyapunov function (Consider fluid limit ….) 123 1 2 3

Stability of LQF: Easy Case Graphs that satisfy Local Pooling: Trees 3, 4, 5 Cycles Combinations

Stability of LQF: Subtle Effect Graph that does not satisfy Local Pooling: 3 65 14 2 {1, 2, 3, 4, 5, 6} has no subset served at constant rate  {1, 2, 3, 4, 5, 6} does not satisfy LP Every proper subset satisfies LP E.g., {1, 2, 3, 5} longest  serve {2, 3} at rate 1 Service Vectors: {1, 3, 5}, {2, 4, 6} {1, 4}, {2, 5}, {3, 6}

Stability of LQF: Subtle Effect Note: Deterministic inputs with rate close to 0.5  unstable (LQF serves 2/6 a positive fraction of time) 3 65 14 2 Theorem: LQF stable if i.i.d. arrivals with nonzero variance Key Idea: {1, 2, 3, 4, 5, 6} cannot be set of longest queues for a positive fraction of time!  LP holds most of the time  Longest queue decreases

Stability of LQF: Subtle Effect 3 65 14 2 Key Idea: {1, 2, 3, 4, 5, 6} cannot be set of longest queues for a positive fraction of time! Assume all queues are longest for a while  {2, 3} and {5, 6} served at same rate 

Stability of LQF: Subtle Effect

Max – Min large at k  (n)  A subset L of queues dominates the others during interval  This subset satisfies LP  Longest queue decreases.

Stability of LQF: Subtle Effect Theorem: Assume that whenever a set L does not satisfy LP, the corresponding service vectors have rank ≤ |L| - 2. Assume also the arrivals are i.i.d. with positive variance (and satisfy a large deviation bound). Then LQF is stable for any feasible arrival rates.

Stability of LQF: Subtle Effect Examples: 1 2 3 4 5 6 7 8 3 65 14 2

Outline Stability of Longest Queue First Fluctuations can stabilize Fairness through flow control  Control of long term rates Fairness of multiple access Impatience may help in a crowd

Fairness Through Flow Control (with John Musacchio, UCSC) Motivation Analysis

Motivation Example Intuitively: h large enough  max – min fair Long-term average rates  max – min for h >> 1 h = discard threshold

Motivation h = discard threshold

Motivation

Fairness Through Flow Control (with John Musacchio, UCSC) Motivation Analysis

Q n (nt): Scale thresholds and speed up

Analysis Q n (nt)/n: Scale space Q n (nt)/n  fluid limit Q(t) with suitable rates….

Analysis Roughly, x(n; t) := Q n (nt)/n  uoc fluid limit Q(t) For t ≥ t 0, Q(t) = q* with suitable rates. This implies Key argument: Most of the time t ≥ 0, x(n; t) ≈ q* However, we want

Motivation q*:

Analysis Key argument: Average throughput close to max min

Outline Stability of Longest Queue First Fluctuations can stabilize Fairness through flow control Control of long term rates Fairness of multiple access  Impatience may help in a crowd

Fairness of Multiple Access with Rajarshi Gupta (PhD 5/05) Motivation Protocol Analysis Simulations

Motivation: Exponential Backoff is Unfair Exponential backoff scheme (e.g. 802.11b)  Nodes pick backoff uniformly in a backoff range  If collision, double the backoff range Multiple interference domains  Node in center sees more contention and collision  It backs off more  Gets lesser share of bandwidth Unfair towards middle nodes in network Active Link Rcvd on A Rcvd on B Rcvd on X A6 A,B66 A,X33 A,B,X442 All rates in Mbps

Protocol: Impatient Backoff Algorithm Approach: Nodes that face more contention should get higher priority Key Mechanism  Upon collision, nodes decrease their backoff Need to worry about  Stability  Fairness  Throughput

Protocol: Backoff Update If collision or quiet  Decrease the mean backoff delay  b := b/m, where m>1 If successful transmission  Increase the mean backoff delay  b := bm Note: Distributed reset mechanism When a node’s mean delay falls below threshold, node broadcasts “multiply by K” ….

Protocol: Simplified MAC Model All packet lengths are same Transmissions occur slot by slot Local synchronization is assumed  Similar to any slotted protocol No RTS/CTS

Protocol: IBA Mechanism Backoff Contention Phase  Each node has mean backoff b  Picks backoff delay B using exponential variable with mean b  Sends out Slot Capture Message after B backoff mini-slots  If a node carrier senses another message sooner – it keeps quiet Packet Transmission Phase  Starts after completion of Backoff Contention Phase  Nodes with successful Slot Capture Messages transmit  Constant packet length  Transmission confirmed by ack Collision occurs if two neighbors pick same backoff Neither hears slot capture Both try to transmit Packet transmission wasted

Markov Chain Models Two extreme topologies  Star Topology (unfair)  Triangle Clique Topology (symmetric) Model ratio between mean backoffs Prove stability, fairness Throughput-fairness tradeoff (in Star)  Max throughput = 0 + 4  1 = 4  But fair throughput = 0.5 + 4  0.5 = 2.5 interferenc e

Star Topology: Birth-Death Chain Stable: Positive recurrent for m>1 Strong drifts towards stable state S 0 Fair: Expected transmission rate for all nodes is 0.5 interference

Star Topology: Varying Neighbors Model sleeping nodes Every 100 slots, some nodes go to sleep Fairness = 1 Average success probability  Middle Node(s X )= 0.473  Outer Nodes(s Z )= 0.470

Triangle Topology Markov Chain interference m=2 Prove positive recurrence using Lyapunov function Chain drifts towards bottom-left stable states Fairness is due to symmetry

Simulations on Random Topology Exponential Backoff Min Throughput = 9% of mean Jain’s Fairness Index = 0.58 Mean Throughput = 0.101 Min Throughput = 49% of mean Jain’s Fairness Index = 0.68 Mean Throughput = 0.102 Circle = Node : Center = Location, Area = Throughput Impatient Backoff

Variations in Simulation Nodes execute random walk Initial bias against selected nodes Nodes switch between active and sleep cycles Similar comparisons with exponential backoff  Comparable throughput  Significantly better fairness

Conclusions Stability of LQF if either  Local Pooling, or   L not LP, rank(Service) ≤ |L| - 2 (  L “splits”) Fairness through flow control  Key idea is long term rates through 2 scalings: Impatient backoff for fair MAC in ad hoc networks

Stability and Fairness of Service Networks Jean Walrand – U.C. Berkeley Joint work with A. Dimakis, R. Gupta, and J. Musacchio.

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Stability and Fairness of Service Networks Jean Walrand – U.C. Berkeley Joint work with A. Dimakis, R. Gupta, and J. Musacchio.

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Presentation on theme: "Stability and Fairness of Service Networks Jean Walrand – U.C. Berkeley Joint work with A. Dimakis, R. Gupta, and J. Musacchio."— Presentation transcript:

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