Capacity of MultiChannel Wireless Networks with Random c, f Assignment

1
Capacity of Multi-Channel Wireless Networks
with Random (c, f) Assignment

• Vartika Bhandari Nitin H. Vaidya

ACM MobiHoc 2007
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Motivation Heterogeneous Multi-Channel Wireless
Networks
915 MHz
2.4 GHz
5GHz

• Multiple available spectral bands many channels
• Individual interfaces may not operate on all
  channels
• Heterogeneity in operational frequency range
• Possible scenarios low-cost sensor nodes with
  limited capability transceivers, mesh networks
  with different device types
• Channels may have different characteristics

Much multi-channel wireless research based on a
homogeneity assumption Need to develop an
understanding of the impact of heterogeneity
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Interface Heterogeneity Building an Initial
Understanding

• A first step towards understanding the
  implications
• What is the asymptotic scaling behavior under
  switching constraints? How is it different from
  the case of unconstrained switching?
• In prior work Infocom07, proposed some models
  to capture some possible switching constraints
• Each interface can switch on f channels out of c
  these f channels are a priori assigned
• How are these f channels assigned?
• Adjacent (c, f) assignment
• Random (c, f) assignment

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Adjacent (c, f) Assignment

• Each node assigned a block location i from 1,
  2, , c-f1 with prob. 1/c-f1 each can then
  switch on channels
• i, i1, .., if-1
• For all channels i, Pr a node can switch on
  channel i
• mini, c-i1, f, c-f1/c

Example f2, c8
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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 f2, c8
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Network Model
s(1)
s(2)

• c orthogonal channels

s(f)
Each node has one interface
No. of channels cO(log(n))
n nodes randomly deployed over a unit area torus
Each channel has bandwidth W/c
Interface can switch between f channels 2 f c
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Network Model

• Protocol Model GuptaKumar for interference
• X?Y transmission is successful if
• XY r
• ZY(1?)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 GuptaKumar)
• Avg. src-dst distance is T(1)

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Network Capacity

• lim Pr can guarantee each flow a throughput
  ?(n)c1(f(n)) 1
• but
• lim Pr can guarantee to each flow a throughput
  ?(n)c2(f(n)) lt1

Per flow capacity is T(f(n))
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Factors Affecting Capacity
Connectivity GuptaKumar
D
D
S
S
Sufficient TX range all SD pairs connected
Small TX range some node S isolated
Interference GuptaKumar
?r/2
r
(1?)r
Each receiver occupies circle of radius ?r/2
?r/2
(1?)r
r
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Factors Affecting Capacity
Interface Constraint KyasanurVaidya
In multi-channel case, if not enough interfaces,
some channels unutilized
Example c10 m1 only 8 nodes in region can
use only 4 channels at a time
Destination Bottleneck KyasanurVaidya
A node can be destination of multiple flows
restricts per-flow capacity
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New Factors Affecting Capacity
(1, 4)
(5, 7)
Connectivity
(3, 4)
(5,6)
(4, 5)
(4, 6)
(5, 6)
(3, 4)
(1, 2)
A device can communicate directly with only a
subset of the nodes within TX range
(6, 7)
(4, 5)
(3, 6)
(6, 7)
(7, 8)
(1, 3)
(6, 7)
(2, 5)
(4, 5)
Node isolated despite having other nodes in range
Bottleneck Formation
Some channels may be scarce in certain network
regions, leading to possible overload on some
channels/nodes
Both flows forced to use channel 1 cannot
transmit concurrently
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Prior Work Connectivity Results
Random (c,f)
fO(vc)
No switching constraint
Adjacent (c,f)

fc

Gupta-Kumar
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Recap of Prior ResultSufficient 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)v(8a(n))
For each node can construct a connected backbone
spanning all cells Backbone(x) tree rooted at x
x
Show that w.h.p. backbones for all nodes have a
point of connection
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Convergence of Comm. Probability
Random (c, f)
Adjacent (c, f)
Random (c, f) probability converges to 1 much
faster than adjacent (c, f) probability
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Prior Results at a Glance
Kyasanur-VaidyaMobicom05
Gap Bounds did not match
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Random (c, f) AssignmentCapacity Lower Bound

• A lower bound construction that achieves capacity

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Optimal Construction (1)
Divide torus into square cells of area a(n)
Every cell has T(na(n)) nodes w.h.p.
r/v8
Cell Division based on El Gamal
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Optimal Construction (2)
In each cell select nodes as backbone
candidates
The rest are deemed transition facilitators
At least
such nodes
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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

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Optimal Construction (4)
Consider cell D
Amongst all adjacent cells, consider any subset
of nodes all having channel i, such that
cardinality of this subset is
D
Then with high probability, the union of the
channel-sets of these nodes has cardinality at
least
This holds with high probability for all channels
i and all cells D
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Routing (1)

• Need c2/f2 intermediate hops on each route
• If straight-line route too short, perform detour
  routing

D
D
Detour routing does not increase asymptotic
routing load on any cell
P
S
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Routing (2)
Utilizes a notion of partial backbones backbon
e(x) grows into cells traversed by flows for
which x is either src or dst
y
Flows packets proceed most of the way on
src-backbone when c2/f2 hops left to
destination, try to jump onto dst-backbone
x
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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

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Growing Backbones

• Consider a backbone that must enter cell D in
  step i
• Then its previous hop node must lie in an
  adjacent cell
• Need to select
• Channel on which to schedule incoming backbone
  link
• Node in D that will act as next relay

?
D
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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
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Bipartite Graph for Allocating Channels to Links
Entering Cell D
Channel allocation cast as problem of computing a
matching that saturates all vertices in set L
Existence of such a matching proved using Halls
Marriage Theorem
If such a matching exists then each incoming link
gets a channel and no channel gets more than k
incoming links
Set L A vertex for each backbone-link entering
the cell at this step
Set P k vertices for each proper channel in the
cell
k
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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 allocate relay nodes ensuring that no node is
assigned more than 14 incoming links in step i
Link entering cell D on channel i
Link will be allocated a relay node from amongst
the backbone candidate 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
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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

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Capacity Lower Bound Obtained

• Capacity with random (c, f) assignment is

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Prior Results
The New Picture
fvc
No switching constraint
Adjacent (c,f)
Random (c,f)
fc
Use c channels Use f common channels
Kyasanur-Vaidya
Factor of
improvement
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vc-threshold

• Random (c, f) assignment
• At least in asymptotic sense vc-switchability as
  good as full-switchability
• Interesting point of trade-off
• vc-switchability may cost less than
  full-switchability
• May want to design systems around this operating
  point?
• Are there other assignment models that yield
  order-optimality with additional desirable
  properties?
• vc reminiscent of distributed quorums Maekawa
• If willing to allow T(vc)-switchability, can
  leverage quorum ideas to get deterministic
  connectivity
• Some further work on this theme

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Major Insights

• 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
• Coupling between channel and interface selection
  leads to a joint channel-interface selection
  problem
• Resultantly there is also a strong coupling
  across hops
• Choices at hop i have a major impact on available
  choices at hop i1, and resultantly on all future
  hops

Accentuates the need for suitable
routing/scheduling strategies
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Practical Relevance

• Interface heterogeneity an increasingly likely
  scenario

802.11a
802.11a/b/g
802.11a
802.11a/b/g
802.11b/g
802.11b

• Similar issues will arise insights from
  asymptotic constructions useful

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Ongoing/Future Work

• Heterogeneous multi-channel wireless networks
• Non-asymptotic regime

What kind of routing and link-layer protocols are
needed to handle such scenarios?
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Alternative Interpretations

• Results can be viewed in context of random key
  pre-distribution
• Each node pre-assigned a random-set of keys
• Neighboring nodes can communicate only if they
  have a common preloaded key

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Thank You Technical Reports available
at http//www.crhc.uiuc.edu/wireless