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Connectivity and Capacity of Multichannel Wireless Networks with Channel Switching Constraints

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Title: Connectivity and Capacity of Multichannel Wireless Networks with Channel Switching Constraints


1
Connectivity and Capacity of Multi-channel
Wireless Networks with Channel Switching
Constraints
  • Vartika Bhandari
  • Joint Work with Nitin H. Vaidya

Communications Seminar, CSL-UIUC October 30, 2006
2
Multi-channel Research in the Wireless Networking
Group
  • Capacity
  • Theoretical limits on performance
  • Insights useful for protocol design
  • Protocol Design
  • Channel-assignment
  • Routing
  • Implementation
  • Net-X project
  • OS support for protocols that utilize multi-
    capabilities
  • Currently supports multi-channel protocols
    easily extensible to other multi-
  • Testbed of Soekris nodes
  • Uses Net-X architecture
  • Multi-channel, multi-radio

Soekris net4521
3
Switching Constraints Motivation
  • Earlier work on multi-channel networks often
    assumes unconstrained switching ability
  • Why switching could be constrained
  • Hardware limitations
  • Policy issues

Total available spectrum may be large, and
divided into multiple channels
Devices may have only one or few interfaces may
not be able to switch over entire frequency range
4
Hardware limitations
  • Low cost, low power transceivers, e.g., for
    sensor nodes
  • Limited tunability of oscillator
  • May tune within some sub-band of total band
  • Some proposed designs have a bank of filters
  • Possible that bank size may be smaller than
    available frequencies

5
Policy Issues
  • Dynamic Spectrum Access via Cognitive radio
  • Secondary users may use a band when primary is
    inactive
  • Must relinquish medium on detecting primary
    activity

6
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
7
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)

8
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))
9
Previous Results
  • GuptaKumar If one channel or dedicated
    interface per channel at each node
  • Per flow capacity in random networks scales as

What if c channels and mltc interfaces per node?
10
Multi-Channel Capacity (mltc)
  • KyasanurVaidya no restriction on channel
    switching

c channels of bandwidth W/c
1
1
m
m
c
11
What if switching constraints?
  • Results of KyasanurVaidya assume no
    restriction on switching
  • What if each node can only switch on a subset of
    f channels?

Need to study connectivity and capacity under
these switching constraints
12
Capacity The Original Issues
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
13
Capacity The Original Issues
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
14
Capacity The New Issues
Connectivity
(4, 5)
A device can communicate directly with only a
subset of the nodes within TX range
(2, 3)
(5, 6)
(1, 2)
(1, 3)
(6, 7)
(3, 6)
Bottleneck Formation
(7, 8)
(2, 5)
Some channels may be scarce in certain network
regions, leading to possible overload on some
channels/nodes
15
Proposed Models
  • Adjacent (c, f) assignment
  • Models limited tunability
  • Also encompasses untuned radio model
  • Random (c, f) assignment
  • Spatially correlated assignment

16
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
  • Forall channels i, Pr a node can switch on
    channel i
  • mini, c-i1, f, c-f1/c

Example f2, c8
17
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
18
Spatially Correlated Assignment
  • n randomly deployed nodes
  • N randomly located pseudo-nodes
  • Each pseudo-node assigned a channel
  • Models presence of active primary user
  • Nearby secondary users denied channel access
  • Also models external noise sources rendering
    channels effectively unusable

R
1
R
2
Area-blocking models for CR considered by other
work too, e.g., Sahai et al , but have not
considered quantifying impact of varying channel
availability on multi-hop secondary networks
capacity
19
Results at a Glance
whenever
always achievable
20
The Method
Obtain Necessary Condition for Connectivity Min
TX range r(n)
Yields Upper Bound on per-flow Capacity O(W/(n
r(n)))
Obtain Sufficient Condition for
Connectivity What TX range r(n) suffices
Does it match necessary condition?
If Upper Lower Bounds match Optimal capacity
determined
Provide Lower Bound Construction
21
Adjacent (c, f) AssignmentCapacity Upper Bound
  • Necessary condition for connectivity
  • Tx range r(n) must be at least
  • Restricts capacity to

equiv. to
22
Lower Bound Construction
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
23
Lower Bound Construction
  • Notion of preferred channels
  • Prob. a node has that channel is at least f/2c
  • Includes most channels except a few at the
    fringes of band
  • Each node has at least f/2 preferred channels
  • By choice of a(n)
  • Every cell has T(log(n)) nodes capable of
    switching on each preferred channel w.h.p.

24
Routing of Flows
  • Earlier work GuptaKumar, ElGamal,
    KyasanurVaidya
  • straight-line routing
  • May not always work when switching is constrained
  • What if no common src-dst channel
  • Require minimum number of hops to guarantee
    transition from a source channel to a destination
    channel w.h.p.

25
Routing of Flows
If straight-line route not long enough
Detour Routing
D
D
Ensure O(c/f) hops
P
S
Detour routing does not increase asymptotic
routing load on any cell
26
Channel Transition Strategy
  • Assign flow a source channel uniformly at random
    out of all preferred channels available at
    source also choose a preferred destination
    channel
  • Initially flow is in progress-on-source-channel
    mode
  • When it is only T(c/f) hops from destination, it
    enters transition mode

27
Channel Transition Strategy
  • Can transition from source channel l to
    destination channel r in at most T(c/f) hops

(1, 2, 3)
In progress on src-channel mode
( 3, 4, 5)
5
D
(4, 5, 6)
(4, 5, 6)
5
(1, 2, 3)
(2, 3, 4)
S
(3, 4, 5)
2
Enters transition mode
(1, 2, 3)
4
(4, 5, 6)
2
(2, 3, 4 )
(2, 3, 4)
3
(2, 3, 4)
(1, 2, 3)
(3, 4, 5)
(1, 2, 3)
Example Adjacent (6, 3) assignment Channels 2,
3, 4, 5 are preferred channels
28
Transmission Schedule
  • Can prove that in each cell, no channel gets
    overloaded
  • Can prove that in each cell T(c) channels can be
    scheduled simultaneously no node bottleneck

29
Conclusion
  • With adjacent (c, f) assignment, when
    cO(log(n)), the per-flow capacity is

Even when f2, get capacity of the order of vc
channels When fc, get optimal capacity for this
region
30
Untuned Radio Model
Proposed by Petrovic, Ramachandran and Rabaey
  • Possible to manufacture low-cost low-power
    radios
  • Large process variation leads to untuned nature
  • Model
  • Each radios carrier frequency uniformly
    distributed over F1, F2
  • Admits bandwidth B
  • Max. disjoint channels possiblec(F2-F1)/B

31
Untuned Radios Capacity
Model of Petrovic et al.
Random network of untuned radio devices, with
random src-dst pairs?
Relay backplane of untuned radio devices
Per flow capacity with store-and-forward approach?
S
D
S-D communication S, D can use all
channels Backplane of untuned radio devices to
relay data cmax. disjoint channels possible
Their Results With random coding, get T(c) S-D
capacity
Result obtainable by mapping to adjacent (c, f)
assignment
32
Untuned Radio Upper Bound
Strategy Create mapping with a (2c2, 3)
assignment
Perform virtual channelization of frequency band
into 2c disjoint sub-bands/channels
Add two extra virtual channels one at each end
Disconnection event in virtual (2c2, 3) assigned
network are preserved in the actual untuned radio
network
Thus necessary condition for connectivity with
adjacent (2c2, 3) assignment holds
33
Untuned Radios Lower Bound
Strategy Create mapping with a (4c1, 2)
assignment
Perform virtual channelization of frequency band
into 4c1 disjoint channels
Virtual schedule can be used almost as-is for
actual untuned radio devices
Partial overlap characteristic of untuned radios
allows for frequency transition
34
Untuned Radios Conclusion
When cO(log(n)), per-flow capacity in a random
network of untuned radio devices is
35
Random (c, f) AssignmentCapacity Upper Bound
  • Tx range r(n) must be at least
  • Restricts capacity to

36
Random (c, f) Assignment 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)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
37
Backbone Connections
Node common to both backbones
Direct connection
Indirect connection via transition facilitator
38
Interesting Observations
  • Critical range smaller than that for adjacent (c,
    f) assignment
  • At critical connectivity range for random (c, f)
    assignment
  • Sub-network induced by a single channel i may not
    necessarily be connected
  • Overall network is connected w.h.p.

39
Random (c, f) AssignmentCapacity Lower Bound
  • A lower bound construction that achieves capacity

whenever c, f satisfy
40
Optimal Capacity Construction
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 If
straight-line route too short, perform detour
routing
x
41
Ensuring Load-Balance
  • 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 a
    matching on a bipartite graph

42
Random (c, f) AssignmentSimpler Construction
  • A lower bound construction that achieves capacity
  • Similar to construction for adjacent (c, f)
    assignment

for all cO(log(n)), 2fc
43
Results A Glance Again
fc
always achievable
44
Summary
  • Better to use all channels instead of a common
    f-subset
  • Even with f2, get capacity of the order of vc
    channels

When fc, these cases reduce to case of
unconstrained switching and yield capacity of
same asymptotic order But we can achieve it with
a simpler technique of assigning channels to flows
45
Interpreting Results in Time Domain
  • Results can also be interpreted in time-domain
  • Can yield some insights about power-save protocols

46
Open Issues
  • Closing the remaining small gap for random (c, f)
    assignment
  • Can optimal random (c, f) assignment capacity be
    achieved using asynchronous procedures?
  • Capacity of spatially correlated assignment
  • Extensions to models/hybrid models
  • Efficient protocol design for networks of
    constrained devices

47
Thank You Technical Reports available
at http//www.crhc.uiuc.edu/wireless
48
Random (c, f) Assignment Necessary Condition for
Connectivity
With a random (c, f) assignment (when
cO(log(n)), if
where
and
then
49
Simpler StrategyRouting of Flows
  • Source and destination may have different
    channels
  • Need to ensure transition from a source channel
    to a destination channel along the route
  • If O(c/f) hops in route, then transition happens
    w.h.p.
  • Ensure that every route is long enough

50
Simpler StrategyRouting of Flows
If straight-line route isnt long enough, do
detour routing
D
D
P
Ensure O(c/f) hops
S
Detour routing does not increase asymptotic
routing load on any cell
51
Simpler Strategy Channel Transition
  • Assign flow a source channel l at random out of
    preferred channels at source similarly for
    destination channels, choose channel r
  • Initially flow is in progress-on-source-channel
    mode
  • When it is only T(c/f) hops from destination, it
    enters ready-for-transition mode
  • Look for a node having channel pair (l, r)
  • Makes a transition as soon as first such node
    found
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