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

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

Outline

R. Srikant

Department of ECE and CSL

University of Illinois at Urbana-Champaign

University of Illinois at Urbana-Champaign

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

- 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

- 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

Functional Decomposition

Lagrange Multipliers (nodes):

Congestion Control (sources):

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

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Wireless Network

User 1

cA=1

cB=1

User 0

User 2

subject to

a is the fraction of

time link A is used

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Lagrange Multipliers

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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

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

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Decomposition

Congestion control (per-flow queues):

MAC or Scheduling (Backpressure):

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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?

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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

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

- 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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- : 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
- Letdenote 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.

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- 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.

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- 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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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

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- 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] 2that satisfies:

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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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- 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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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

- 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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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

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

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

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

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

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- 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

- 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

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)

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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

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

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

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

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

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

- 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

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 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

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Simulations

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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

- 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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