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Optimization and Distributed Algorithms for Resource Allocation in Multi-hop Wireless Networks. R. Srikant Department of ECE and CSL University of Illinois at Urbana-Champaign. Motivation. Objective: Fair and Efficient Resource Allocation in Multi-hop Wireless Networks Questions:

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optimization and distributed algorithms for resource allocation in multi hop wireless networks

Optimization and Distributed Algorithms for Resource Allocation in Multi-hop Wireless Networks

R. Srikant

Department of ECE and CSL

University of Illinois at Urbana-Champaign

University of Illinois at Urbana-Champaign

motivation
Motivation
  • Objective: Fair and Efficient Resource Allocation in Multi-hop Wireless Networks
  • Questions:
    • What is the optimal network architecture? Does it naturally arise from the objective?
    • Are there distributed algorithms that implement the various layers of the protocol stack?
    • Where approximations are necessary for implementability, can we quantify the degree of approximation?
    • How easy is it to extend the model to accommodate other traffic models (multicast, network coding, etc.)?
    • Network designed for fixed number of flows. Stability with dynamic traffic? (Lin, Shroff, S.)

University of Illinois at Urbana-Champaign

closely related work
Closely Related Work
  • Scheduling/Routing:
    • Tassiulas-Ephremides; Tassiulas
  • Resource Allocation for the Internet:
    • Kelly et al; Low et al; S.
  • Resource Allocation in Wireless Networks
    • Stolyar; Neely, Modiano & Li; Lin&Shroff
  • Distributed Algorithms:
    • Lin & Rasool; Gupta, Lin & S., Joo &Shroff, Sarkar et al (slotted time)
    • Kar et al, Gupta-Stolyar (random access)
    • Xiao-Johansson-Boyd, Chiang, Huang-Berry-Honig (power control)
  • Extensions to network coding:
    • Eryilmaz & Lun, Ho et al, Chiang et al

University of Illinois at Urbana-Champaign

outline
Outline
  • A simple three-node example: Internet versus wireless networks
  • Joint scheduling, routing and congestion control for multi-hop wireless networks (Eryilmaz, S.)
  • Extensions to multicast traffic (Bui, Stolyar, S.)
  • Low-complexity distributed MAC algorithm (Sanghavi, Bui, S.)

University of Illinois at Urbana-Champaign

three node internet
Three-Node Internet

User 1

ca=1

cb=1

User 0

User 2

subject to

University of Illinois at Urbana-Champaign

solution
Solution

Solution:

University of Illinois at Urbana-Champaign

functional decomposition
Functional Decomposition

Lagrange Multipliers (nodes):

Congestion Control (sources):

  • Lagrange multipliers ≈ Queue lengths
  • But not true queue dynamics
  • Reasonable model for the Internet

University of Illinois at Urbana-Champaign

wireless network
Wireless Network

User 1

cA=1

cB=1

User 0

User 2

subject to

a is the fraction of

time link A is used

University of Illinois at Urbana-Champaign

lagrange multipliers
Lagrange Multipliers

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decomposition
Decomposition

Congestion control (sources and nodes):

Maxweight MAC or Scheduling (network):

Solution is an

extreme point

Earlier comment regarding queue lengths and Lagrange multipliers applies

University of Illinois at Urbana-Champaign

alternative formulation
Alternative Formulation

User 1

cA=1

cB=1

User 0

User 2

subject to

a0 is the fraction of

time link A is used for

user 0

University of Illinois at Urbana-Champaign

decomposition1
Decomposition

Congestion control (per-flow queues):

MAC or Scheduling (Backpressure):

University of Illinois at Urbana-Champaign

resource constraints and queueing dynamics
Resource Constraints and Queueing Dynamics

x1

μa1

x2

μb2

pa0

pb0

x0

μa0

μb0

subject to

  • Queue stability constraints:
  • Arrival rate into a queue is departure rate from previous queue
  • Still not precise: what happens if previous q=0?

University of Illinois at Urbana-Champaign

differences in the two formulations
Differences in the Two Formulations
  • Arrivals instantaneously arrive at all nodes in the route

versus

node-by-node queueing behavior

  • Sources react to sum of queue lengths

versus

Sources react to entry queue length

  • Why is it sufficient to react to only the entry queue length?
    • Back-pressure algorithm

University of Illinois at Urbana-Champaign

outline1
Outline
  • A simple three-node example: Internet versus wireless networks
  • Joint scheduling, routing and congestion control for multi-hop wireless networks (Eryilmaz, S.)
  • Extensions to multicast traffic (Bui, Stolyar, S.)
  • Low-complexity distributed MAC algorithm (Sanghavi, Bui, S.)

University of Illinois at Urbana-Champaign

slide16

Wireless Network Model

  • The network is represented by a graph:
  •  = set of link rates that are allowable in a time slot, i.e., we have:
  •  [t] 2, 8 t.

i

j

(m,j)

(i,n)

(n,m)

(m,w)

n

m

(v,n)

(w,m)

w

(m,v)

(n,v)

v

Slot 1

Slot 2

time

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slide17

Traffic Model

  • : The set of flows that share the network.
  • Each flow is described by a source-destination pair: No predefined routes.
  • Letxfdenote the rate of flowf
  • Letdenote the set of flow rates for which the corresponding link rates lie in.

e(f)=j

b(f)=i

i

flow f

j

n

m

w

flow h

flow g

v

  • Uf ( xf ) is a (strictly) concave function that measures the utility of flow f as a function of xf.

University of Illinois at Urbana-Champaign

slide18

Problem Statement

  • Design a mechanism that
    • guarantees stability of the queues,
    • allocates flow rates,{ xf }, that satisfy:
  • x*denotes the optimizer of the above problem, call it the fair allocation.

University of Illinois at Urbana-Champaign

slide19

Node Model

  • Each node maintains a queue for each destination node.

i

qn,j

s(i,n)

(j)

s(n,m)

(j)

m

s(i,n)

(k)

qn,k

s(n,v)

(k)

Node n

v

  • In general, the evolution of a queue length is described by

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slide20

Primal-Dual Congestion Controller

  • At the beginning of each time slot t, each flow, say f, has access to the queue length of its first node, denoted byqb(f)[t].
  • Congestion Control:

or

  • Increase rate when queue length is small
  • Decrease rate when queue length is large
  • K is a fixed parameter

University of Illinois at Urbana-Champaign

slide21

Back-pressure Scheduler

  • Assign a weight to each edge; find a feasible set of edges with the maximum sum weight
  • The differential backlog of link(n,m) for destination d is given by
  • Differential backlog of the link isW(n,m)max[t]: the maximum value among all destinations
  • Then, choose the rate vector [t] 2that satisfies:

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slide22

Node m

An example:

1

2

5

W(n,m)max = (max{5-1,7-2,2-5})+=5

d(n,m) = 2

5

Node n

7

W(n,k)max = (max{5-6,7-8,2-4})+=0

d(n,k) = 

2

6

Node k

8

4

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slide23

Queue Stability

  • Define the Lyapunov function

where q*2 K*.Drift analysis results in

Theorem 1:For some finite constant c, we have

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slide24

Fair Allocation

Theorem 2:There exists a finite B, such that for all f

  • For large K, the average rate allocation is fair
  • Tradeoff between delays and fairness

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stochastic models
Stochastic Models
  • The set of allowable rates at each time instant can be time-varying
  • Don’t need to know the statistics of the channel
    • The capacity region is unknown, but instantaneous capacity region is known
  • Can model randomness in the arrival processes
  • Proof: conditional mean drift of the Lyapunov function has the form shown in the previous page
  • Result:

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slide26

Stochastic model  Fluid model

  • Intuition: M/M/1 queue where the arrival rate decreases with the queue length.

K

K

K/2

K/(q-1)

K/q

. . .

. . .

0

1

2

q

The steady-state mean and the variance of the above Markov chain are both Θ(K).

University of Illinois at Urbana-Champaign

outline2
Outline
  • A simple three-node example: Internet versus wireless networks
  • Joint scheduling, routing and congestion control for multi-hop wireless networks (Eryilmaz, S.)
  • Extensions to multicast traffic (Bui, Stolyar, S.)
  • Low-complexity distributed MAC algorithm (Sanghavi, Bui, S.)

University of Illinois at Urbana-Champaign

multi rate multicast
Multi-rate multicast

x1, U1(x1)

μB

  • One sender, four receivers
  • Example of constraint:
  • Receivers can receive at different rates
    • Very important in wireless networks; otherwise, all rates will become zero frequently

μA

x2, U2(x2)

x

x3, U3(x3)

μC

x4, U4(x4)

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solution multi rate multicast
Solution: Multi-rate multicast

μB

  • Constraint:
  • A fictitious queueing network sending fictitious packets in the opposite direction enforces the constraints
  • The departures from the fictitious queues serves as tokens (credits) for the generation of real packets

μA

x

μC

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qos control delays
QoS Control: Delays

μA

x

μC

  • Source can send a packet for every token, or
  • Source can generate 9 packets for every 10 tokens received
  • Tokens inform the source of the amount of resources reserved for it
  • Source can use this information, but sends at a smaller rate to reduce delays

University of Illinois at Urbana-Champaign

outline3
Outline
  • A simple three-node example: Internet versus wireless networks
  • Joint scheduling, routing and congestion control for multi-hop wireless networks (Eryilmaz, S.)
  • Extensions to multicast traffic (Bui, Stolyar, S.)
  • Low-complexity distributed MAC algorithm (Sanghavi, Bui, S.)

University of Illinois at Urbana-Champaign

limitations of the approach
Limitations of the Approach
  • Each source needs to know only its ingress queue length to perform congestion control (decentralized)
  • Routing, MAC, power control, etc. are done using the backpressure algorithm: centralized, infeasible
  • Question: Are there decentralized approximations to the backpressure algorithm that achieve a large fraction of the capacity region?
    • Fix power levels
    • Fix routing
    • Focus only on scheduling (which links should be turned ON or OFF)

University of Illinois at Urbana-Champaign

primary interference model
Primary Interference Model

Wireless Network == graph with nodes and edges

Nodes == wireless devices

Communication only between neighbors

At any given time, a link can be “ON” or “OFF”

Constraint: no two adjacent links can

be “ON” at same time

(ON links form a matching in the graph)

(Corresponds to fixed power levels, orthogonalization, pairwise-only

Communication: Hajek and Sasaki)

University of Illinois at Urbana-Champaign

scheduling problem
Scheduling Problem

To decide what edges to turn ON at each time

- so as to “maximize data rates”

- abiding by interference constraints

- assume one-hop flows (easy extension)

Each edge has an associated queue

Stochastic packet arrivals to each

queue (not controlled, easy

extension to controlled)

OFF == no service for the queue

ON == one packet served

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capacity region
Capacity Region

Average arrival rate vector

( one for each edge, length of vector = |E| )

(capacity region) if and only if

is in convex closure of all matchings.

Max-Weight Matching (with queues as edge weights) renders the queues stable.

2

3

3

5

2

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existing algorithms
Existing Algorithms

Max-Weight Matching takes time to find new schedule.

Maximal Matching achieves ; communication overhead scales with n

Randomized Algorithm

1) In each time, generate random new matching s.t.

2) Switch if new better than

This achieves . Needs random generator, network-wide compare

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communication overheads
Communication Overheads

Scheduling

Service

Scheduling

Service

Resources wasted in scheduling

not accounted for, grow with n

“Capacity” results only indicative of

efficiency in service part.

Growing overheads => what does “capacity region” mean ?

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main result
Main Result

A constant-overhead algorithm that can achieve any fixed fraction of the capacity region.

  • In particular, given any we have an algorithm that
  • Achieves
  • Forms new schedule in handshake times.
  • (one handshake time = time for exchanging a control packet between neighbors.)

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algorithm idea
Algorithm: Idea

Make local improvements to existing schedule.

  • A node that is not part of the matching initiates a “query” to possibly increase the weight of the previous matching
  • The query is propagated on a path where links in the matching and links not in the matching alternate
  • Query stops after steps
  • Compare weight of links not in the matching with weight of links in the matching
  • Flip the status of the links on the path if weight can increase

2

1

3

1

2

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algorithm randomization
Algorithm: Randomization
  • Initially, each node randomly becomes “active”, i.e., initiates a query. So, multiple simultaneous requests in network.
  • If a request reaches an active or dead node, request
  • “fails”: no new active node, edge not special.
  • If two requests collide at a node, both fail.
  • This process makes disjoint alternating paths and edges.
  • Net queue length info. propagated along till the end.
  • Decision of switch/no switch made at end, relayed back.
  • All selected edges implement switching decision.

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proof sketch
Proof Sketch

Recall randomized Algorithm:

1) In each time, generate random new s.t.

2) Switch if new better than

Our Algorithm: a technique to generate this new , and

switch if it is better.

Theorem 1: The new generated by our algorithm satisfies

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proof sketch1
Proof Sketch

So, we approximately meet the criterion of Tassiulas

This implies corresponding rate region.

Theorem 2: Given any , if there is an algo. that generates

such that

and switches if gain, then that algo achieves

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simulations
Simulations

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simulations1
Simulations

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implications
Implications

Theoretical:

- Constant-time algorithms that can achieve any

a-priori intended fraction of capacity region.

- Precise accounting of overheads.

Practical:

- Allows protocol to be designed independent of

network size.

- = tunable parameter that allows selection of

best protocol given channel coherence times,

data type, etc.

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open problems
Open Problems
  • Approximating back-pressure routing (packet-by-packet routing is complicated to implement)
  • Distributed algorithms for more complicated interference models
  • Distributed power control and scheduling
  • Admission control and routing for inelastic flows
  • Where are the biggest gains compared to the existing protocol stack?

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