Download channels

Multi-Channel Wireless Networks
Nitin Vaidya
Illinois Center for Wireless Systems
University of Illinois at Urbana-Champaign
September 5, 2007
Capacity of
Multi-channel Wireless Networks
with Random (c, f) Assignment
Vartika Bhandari & Nitin H. Vaidya
Motivation: Switching Constraints
915 MHz
2.4 GHz
5 GHz
Increased push towards
opening up spectrum for
unlicensed use:
Total available spectrum may be
large, and divided into multiple
(possibly non-contiguous) channels
60 GHz
Devices may have only one or
few interfaces; may not be able
to switch over entire frequency
range
•Hardware limitations
•Policy Issues
Some Previously Proposed Models

Adjacent (c, f) assignment

Random (c, f) assignment
Adjacent (c, f) Assignment
 Each node assigned a block location i from 1, 2, …, c-f+1
with prob. 1/c-f+1 each; can then switch on channels:
i, i+1, .., i+f-1
For all channels i, Pr[ a node can switch on channel i]
= min{i, c-i+1, f, c-f+1}/c
Example: f=2, c=8
Random (c, f) Assignment
 Each node is assigned a random f-subset of channels
 Pr[ a node can switch on channel i] = f/c, for all i
Example: f=2, c=8
Network Model
1
s(1)
3
s(2)
…
…
4
s(f)
6
5
…
Each node has
one interface
…
…
n nodes randomly
deployed over a unit
area torus
No. of channels c=O(log(n))
c orthogonal channels
2
c
Each channel has bandwidth W/c
Interface can switch between f channels
2≤f≤c
Network Model

Protocol Model [Gupta&Kumar] for interference
X→Y transmission is successful if
• XY ≤ r
• ZY≥(1+Δ)r, for other concurrently transmitting Z

Each node is source of exactly one flow
Chooses its destination as node nearest to a randomly chosen
point (same as in [Gupta&Kumar])
Avg. src-dst distance is Θ(1)
Network Capacity

lim Pr[ can guarantee each flow a throughput
λ(n)=c1(f(n)) ] =1
but
lim Pr[ can guarantee to each flow a throughput
λ(n)=c2(f(n)) ]<1
Per flow capacity is Θ(f(n))
Factors Affecting Capacity
Connectivity [Gupta&Kumar]
D
S
D
S
Sufficient TX range: all
SD pairs connected
Small TX range: some node S isolated
Interference [Gupta&Kumar]
Each receiver “occupies”
circle of radius Δr/2
≤r
≥(1+Δ)r
Δr/2
Δr/2
≥(1+Δ)r
≤r
Factors Affecting Capacity
Interface Constraint [Kyasanur&Vaidya]
In multi-channel case, if not enough interfaces, some channels
unutilized
Example:
c=10; m=1; only 8 nodes in region;
can use only 4 channels at a time
Destination Bottleneck [Kyasanur&Vaidya]
A node can be destination of multiple
flows; restricts per-flow capacity
New Factors Affecting Capacity
(5, 7)
(1, 4)
(3, 4)
(4, 5) (5,6)
(4, 6)
(5, 6)
(3, 4)
(1, 2)
(4, 5) (6, 7)
(3, 6)
(6, 7)
(7, 8)
(1, 3)
(6, 7)
(2, 5)
(4, 5)
Connectivity
A device can communicate directly
with only a subset of the nodes
within TX range
Node isolated despite having other nodes in range
Bottleneck Formation
(1, 3)
(1, 2) A
B
(1, 2)
C (1, 2)
D
(1, 5) (1, 4)
Both flows forced to use channel 1;
cannot transmit concurrently
Some channels may be scarce in
certain network regions, leading
to possible overload on some
channels/nodes
Prior Results at a Glance
Unconstrained
assignment
Adjacent (c,f)
Random (c,f)
Use c channels
Use f common channels
Gap:
Bounds did
not match
Recap of Prior Result:
Sufficient Condition for Connectivity
Want to show that any pair of nodes
x, y are connected through some path
Divide network into square cells of area
y
Choose r(n)=√(8a(n))
x
For each node can construct a
connected backbone spanning all cells
Backbone(x): tree rooted at x
Show that w.h.p. backbones for all
nodes have a point of connection
Connectivity Results
No switching constraint
f=Ω(√c)
Adjacent (c,f)
Random (c,f)
f=c
[Gupta-Kumar]
:
Convergence of Comm. Probability
Random (c, f)
Adjacent (c, f)
Random (c, f) probability
converges to 1 much
faster than adjacent (c, f)
probability
Random (c, f) Assignment:
Capacity Lower Bound

A lower bound construction that achieves capacity
Optimal Construction (1)
Divide torus into square
cells of area a(n)
Every cell has Θ(na(n))
nodes w.h.p.
r/√8
Cell Division based on [El Gamal]
Optimal Construction (2)
In each cell select
nodes as “backbone candidates”
The rest are deemed “transition facilitators”
Optimal Construction (3)


Notion of proper channels:
A channel i is said to be proper in cell D if the number of
backbone candidate nodes in cell D that can switch on
channel i is at least
By choice of cell-area a(n), the number of proper channels in
any cell is at least:
Optimal Construction (4)
Utilizes a notion of
“partial backbones”
y
x
backbone(x) grows into cells
traversed by flows for which x is
either src or dst
Flow’s packets proceed most of
the way on src-backbone;
when c2/f2 hops left to
destination, try to jump onto
dst-backbone
If straight-line route too short,
perform detour routing
Ensuring Load-Balance

Straight-line source-backbones grown in lock-step
One hop at a time

At each step, incremental load-balance ensured
Inductive procedure
Channel/Node assignment algorithm at each
cell in each step involves computing matchings on a
bipartite graph
Growing Backbones
Consider a backbone
that must enter cell
D in step i
?
D
Then its previous hop
node must lie in an
adjacent cell
Need to select:
1. Channel on which to
schedule incoming
backbone link
2. Node in D that will
act as next relay
Channel Allocation
D
Link entering cell D
Previous hop node u
Link will be allocated a channel from amongst the channels proper in
cell D that u can switch on; no. of such choices available to u is at least:
Bipartite Graph for Allocating Channels to Links
Entering D
Channel allocation
cast as problem of
computing a matching
that saturates all
proved
vertices in set L
Existence of such a matching
using Hall’s Marriage Theorem
Set L: A vertex for
each backbone-link
entering the cell at
this step
If such a matching
exists then each
incoming link gets a
channel and no
channel gets more
than k incoming links
Set P : k vertices
for each proper
channel in the cell
k=
Relay-Node Allocation
Suppose a link was allocated channel i for entering cell D in step i
These nodes switch on
channel i
D
Can again show via a matching argument that we can
entering cell D
allocate relayLink
nodes
ensuring that no node is assigned
channel i links in step i
more than 14onincoming
Link will be allocated a relay node from amongst the nodes in cell D that
can switch on channel i
Since i is proper in D, the no. of possible choices is at least Mu
Transmission Schedule



Summing over all steps, can show that no channel
or node gets overloaded in any cell of the network
Destination/detour backbones can be grown
without load-balance concerns
Thereafter obtain a 2-level transmission schedule
Coloring of cell-interference graph yields inter-cell
schedule
Within each cell-slot, obtain an intra-cell schedule via
coloring a conflict graph of links to be scheduled
The
New
Picture
Prior
Results
No switching constraint
Use c channels
[Kyasanur-Vaidya]
Adjacent (c,f)
f=c
Random (c,f)
Recent Result
Use f common channels
Factor of
improvement
capacity is:
√c-switchability yields
order-optimal capacity
√c-threshold

Random (c, f) assignment
At least in asymptotic sense: √c-switchability as good as
full-switchability

Interesting point of trade-off
 √c-switchability may cost less than full-switchability
Might want to design systems around this operating
point!

Are there other assignment models that yield
order-optimality with additional desirable
properties?
√c: reminiscent of distributed quorums [Maekawa]
If willing to allow Θ(√c)-switchability, can leverage
quorum ideas to get deterministic connectivity
Some further work on this theme
Insights


Insight #1: There is a strong coupling between
interface-selection at hop i and channel-selection
at hop i+1, leading to a coupling across hops
Insight #2: Two spatially proximate interfaces
that both switch on channel i are not necessarily
interchangeable; in smaller-scale networks, care is
needed in both channel and interface selection
Joint channel and interface selection problem at each link
Practical Relevance
802.11a
802.11a
Interface heterogeneity an increasingly likely
scenario
802.11a/b/g

802.11a/b/g
802.11b/g

802.11b
Similar issues will arise; insights from asymptotic
constructions useful
Ongoing/Future Work

Heterogeneous multichannel wireless networks
Non-asymptotic regime
Forms of
Heterogeneity
Different
operational
channel sets:
“Switching
Constraints”
Different
number of
interfaces
at each node
Channels with
different
data-rate/
propagation
characteristics
Time-variations
in channel
quality
What kind of routing and link-layer protocols
are needed to handle such scenarios?
Alternative Interpretations

Results can be viewed in context of random key
predistribution
Each node pre-assigned a random-set of keys
Neighboring nodes can communicate only if they have a
common preloaded key
Network Utility Optimization in
Multi-Channel Multi-Interface Networks
Simone Merlin
Nitin Vaidya
Problem statement


Resource allocation (Possibly joint/cross layer
solutions) :
Channel loading
Interface to channel binding
Transmission scheuling
Congestion control
Different approaches:
Theoretical bounds (no algorithms)
Practical heuristic protocols (no analytical properties)
Practical algorithms with provable performance
System model
Utility maximization (network flow problem):
Utiliy
Flow
conservation
Rate region
Algorithm overview

The optimization problem is decomposed in subproblems:
• Rate control
• Channel allocation  NEW CONTRIBUTION
• Routing & Scheduling (interfaces to channel binding /
trnsmission scheduling)
A dynamic algorithm is used to solve each sub-problem
Algorithm overview
[Motivated by work of Srikant, Shroff]

Rate control:. Queues length represent a ‘price’ for the rate
allocation.

Channel allocation: (vitual links allocation) local at each node

Routing/Scheduling: Based on Backpressure algo [Tassiulas]
Proposal: Algorithm
Load the channel with the
shortest queue,
provided the difference beween
input queue and channel
queue is positive
Back pressure algorithm
Algorithm evaluation

Scenario:
grid/random topologies
exclusive interference model
• Scheduling becomes an NP-Hard problem. A greedy
heuristic is used as scheduler in order to show the
algorithm behavior
» A lower bound on the value of the solution can
be proved,
» which depends on the interference degree.
Logarithmic utility function: proportional fairness
Performance are shown as a function of:
Number of channels
Number of interfaces
Number of multihop traffic flows
•
•
•
Algorithm evaluation: Utility
Grid totpology
The algorithm shows
the theoretically
predicted
behavior
2 interfaces
are enough
4 interfaces
are enough
Algorithm evaluation: Rate
Similar to the
utility behavior
Algorithm evaluation: Delay
All this cases reach
the maximum throughput
More channels allow for
a reduced delay
Algorithm evaluation: Capacity
Simulations results
Asymptotic result
Algorithm evaluation: Comparison
Bounds
Comparison with
an upper bound
in
M. Kodialam and T. Nandagopal.
Characterizing the capacity region in
multi-raadio multi-channel wireless
Networks, MobiCom, 2005.
Our
Algorithm
Conclusions


Algorithms/protocols with analytical basis need to
be developed.
The proposed algorithm a step toward this direction.
Many other issues remain open:
Reliable broadcast
Efficient routing mechanism
Practical protocol desing / overhead estimation
Fully distributed scheduled
More realistic interference models
Rate and Carrier Sense Adaptation in
Wireless Networks
Zhongning Chen
Nitin Vaidya
Adaptive Rate and Carrier Sense Adaptation




Dynamic adaptation can help improve spatial reuse
Our prior work developed such an adaptive
algorithm
Continuing on this work: we have improved the
adaptation algorithm to achieve better
performance in fading environments
Details in Zhongning Chen’s thesis
Thanks!
[email protected]