Optimization and Distributed Algorithms for Resource Allocation in Multihop Wireless Networks

1
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

2
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.)

3
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 LinShroff
• 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

4
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.)

5
Three-Node Internet
User 1
ca1
cb1
User 0
User 2
subject to
6
Solution
Solution
7
Functional Decomposition

• Lagrange Multipliers (nodes)
• Congestion Control (sources)

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

8
Wireless Network
User 1
cA1
cB1
User 0
User 2
subject to
?a is the fraction of time link A is used
9
Lagrange Multipliers
10
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
11
Alternative Formulation
User 1
cA1
cB1
User 0
User 2
subject to
?a0 is the fraction of time link A is used
for user 0
12
Decomposition

• Congestion control (per-flow queues)
• MAC or Scheduling (Backpressure)

13
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 q0?

14
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

15
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.)

16
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
?(w,m)
?(m,v)
?(n,v)
v
Slot 1
Slot 2
time
17
Traffic Model

• The set of flows that share the network.
• Each flow is described by a source-destination
  pair No predefined routes.

• Let xf denote the rate of flow f
• 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.

18
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.

19
Node Model

• Each node maintains a queue for each destination
  node.

i
qn,j
s(i,n)
s(n,m)
(j)
(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

20
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 by qb(f)t.
• Congestion Control

or

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

21
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 is W(n,m)maxt
  the maximum value among all destinations
• Then, choose the rate vector ?t 2 ? that
  satisfies

22
Node m
An example
1
2
5
W(n,m)max (max5-1,7-2,2-5)5 d(n,m) 2
5
Node n
7
W(n,k)max (max5-6,7-8,2-4)0 d(n,k) ?
2
6
Node k
8
4
23
Queue Stability

• Define the Lyapunov function

where q 2 K?. Drift analysis results in
Theorem 1 For some finite constant c, we have
24
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

25
Stochastic Models

• The set of allowable rates at each time instant
  can be time-varying
• Dont 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

26
Stochastic model ? Fluid model

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

K
K
K/2
K/q
K/(q-1)
. . .
. . .
0
1
2
q
?
?
?
?
?
The steady-state mean and the variance of the
above Markov chain are both T(K).
27
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.)

28
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)
29
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
30
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

31
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.)

32
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)

33
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)
34
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
35
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-Weig
ht Matching (with queues as edge weights) renders
the queues stable.
2
3
3
5
2
36
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
37
Communication Overheads
Scheduling
Scheduling
Service
Service
Capacity results only indicative of efficiency
in service part.
Resources wasted in scheduling not accounted for,
grow with n
Growing overheads gt what does capacity region
mean ?
38
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.)

39
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
40
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.

41
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
42
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
43
Simulations
44
Simulations
45
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.
46
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?