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A (Cynic‟s) Recent History of
Wireless Network Capacity
Research
Stavros Toumpis (Informatics Dept., ΩΠΑ!)
TUC, Summer 2010
1
Organization
1. Introduction
2. A few basic questions
3. Asymptotic Capacity of Wireless
Networks with Immobile Nodes
4. Asymptotic Capacity of Wireless
Networks with Mobile Nodes
5. Capacity of Massive Wireless Networks
2
PR
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AB
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1. Introduction
3
Cellular Wireless Networks
• Mobile terminals communicate with others exclusively through base stations.
• Mobile terminals have very little responsibility.
• A wireless access network.
PSTN,
Internet,
etc.
Α Β
Truly Wireless Networks
• Mobile Terminals communicate through their neighbors.
• Mobile Terminals have many responsibilities.
– For example, they must forward other terminals‟ data.
• Much more challenging.
PSTN,
Internet,
etc.
Α
Β
Prehistory
• Research started in the 70‟s
– ARPA Project
– Some military communications systems came out of it (ΕΡΜΗ!)
• Interest cooled off in the 80‟s
• Renewed interest in the 90‟s
– Wireless communications very popular
– Technology became more powerful and could support algorithms.
• Currently, interest is still going strong.
6
Preprehistory : Naval
Communications at
the Turn of the
(Previous) Century
• Problem: Stop the German High Seas
Fleet going in/out of Denmark Strait
• Setting:
– You are in 1914, most of your ships have no wireless. Must
depend on visual communication.
– Fog, i.e., fading
• Solution: A Hierarchical, Mobile, Visual Sensor
Network.
• Many other examples in history, even in antiquity7
Many names for the same thing
1. Packet Radio Networks (70‟s)
2. Multihop Wireless Networks (80‟s)
3. Wireless ad hoc networks (90‟s)– Mostly EE people
4. Mobile Ad Hoc Networks - MANETs (90‟s)– Mostly CS people
5. Wireless Networks (future?)
Question: What do you think is the reason for this constant change of names?
8
Special Types of Wireless Networks
1. Wireless Sensor Networks
2. Vehicular Ad Hoc Networks
3. Next Generation Cellular Networks
4. Delay Tolerant Networks
5. Wireless Mesh Networks
• Others will come up sure enough
• Commercial products exist for most of them9
1. Wireless Sensor Networks
SINK
SINK
Task Manager
Internet,
Satellite,
UAV
• Applications:
– Sensing forest fires
– Monitoring earthquakes
– Military applications
– Agricultural Applications
• Challenges
– Integrate data collection / compression with
communications
– Low energy
– High lifetime requirement
11
2. Wireless Mesh Networks
INTERNET
Question: Why do you think AMWN is so large?
3. Next Generation Wireless Access
(Hybrid) Networks
Under current 3G technology, mobile
phones only communicate with the
base stations
In next generation wireless access
networks, mobile terminals will
exchange information directly with
each other, saving energy and
bandwidth.
With 3G technology, user A looses
connectivity
Α
Β
In next generation networks, user
B will forward user A‟s data
4. Vehicular Ad Hoc Networks
(VANETs)
• Applications
– Security (for example, car notifies when there is an
accident 300 meters ahead)
– Infotainment
• Challenges
– Topology changes fast
– Security, QoS
• Obvious interest from auto companies
15
5. Delay Tolerant Networks
• Basic Idea: Delays in the communication are so large, topology changes
• Some times, delays are unavoidable
– Partitioned Networks
• Some times, delays are acceptable
– Data is not time critical
• Applications
– Space telecommunication
– Wildlife tracking, for zebras, whales, etc.
16
Peculiarities of the wireless channel
1. Communication links are coupled
– i.e., wireless medium is inherently broadcast
2. Bandwidth is limited
– 106 times less bandwidth than optical networks
3. Communication is localized, because signals decay with distance
– The graphs that describes who can talk to whom directly is embedded in a 2D space.
Question: Which one do you think is the most important from a networking perspective?
17
Two Main Research Areas
• Protocols research: Trying to achieve the
capacity
– Media Access, Routing, Transport, etc.
– Also cross layer: Routing/MAC protocols, etc.
• Capacity research: trying to find the capacity
in the first place
18
Caveat: Capacity means different
things to different people
• Information Theoretic Capacity
– “What is the maximum rate of communication under all
modulation schemes in the AWGN channel?”
• Media Access Capacity
– “What is the maximum probability of success in Slotted
Aloha?”
• Cellular Capacity („When not to spread spectrum…‟)
• Erlang Capacity
– “How many restrooms do I need a cinema with 500 seats,
so that blockage probability is less than 1%”?
• Network Capacity19
Network Capacity (roughly)
• Consider a complete network model:
– Node placements (regular, arbitrary)
– Channel/Transceiver model (see next slides)
– Assume perfect coordination:
– Perfect medium access (we never have collisions)
– Optimal routing (no routing overhead)
– Perfect power control (no unnecessary interference anywhere)
• How to think about: GOD tells everyone when to transmit,
to whom, what data, with what power, rate, etc.
• Network Capacity is the traffic carrying capability of the
network under this condition.
20
• Network Capacity Research is necessarily cross-layer.
• We need to simplify! Otherwise the problem becomes
impossible to solve.
• Next, we see two common simplifications1 of the
PHY layer (Information Theorists, brace yourselves.)
1. “Protocol Model”
2. “Physical Model”
– Also known as SINR model and Interference Model
• Others exist. They all exhibit tradeoff
between accuracy and simplicity
21
1 P. Gupta and P. R. Kumar, „The Capacity of Wireless Networks,‟
IEEE Trans. On Information Theory, Vol. 46, No. 2, Mar. 2000, pp 388-404.
Simplified PHY
1. Protocol Model
• A transmission from node Xi to node Xj is
successful iff
– | Xi - Xj |<r
– For all other simultaneously transmitting nodes Xk
we have | Xk - Xj |>(1+Δ)r (First Option)
– For all other simultaneously transmitting nodes Xk
we have | Xk - Xj |>(1+Δ) | Xi - Xj | (Second Option)
• All transmissions are with rate W
22
Critique
• According to the model, the transmission on
the left is a success, and that on the right leads
to a collision.
• In reality, the inverse is more probable.23
2. Physical Model
• When node Xi transmits with power Pi, node Xj
receives a signal with power Pr,j=GijPi.
– Commonly, Gij=K(dij)-a, a>2
• Reception in node Xj is successful provided that
Rj ≤fR(γj) ≡Wlog2(1+γj/Γ)
where
γj = GijPi /(η+Σk ϵT, k ≠i Gkj Pk)
and T is the set of transmitting nodes.
• Obviously more realistic than protocol model. Effects
of interference are now additive.
24
2. A few basic questions
25
Basic Question 1
• Assume a source, a sink, and many nodes in between that can act as relays.
• Basic Question 1: Is it better to transmit with one long hop, or with many small ones?
• An obvious tradeoff exists:– With one long hops, you use the channel once, but
need to transmit with high power, so you interfere more.
– With many short hops, you transmit many times, but with less power.
– Is there a “sweet spot”?
26
Answer
• Many small hops are better,
because they occupy less area.
• Unfortunately, this complicates
both the design protocols and the
capacity calculations
tremendously.
• Explanation carries through
under most reasonable models.
• Two basic resources: Bandwidth
and Area!!
27
Basic Question 2
• Basic Question 2: So, given we use short hops (see Basic Question 1) how many hops on the average is the distance between two nodes in a network of n nodes?
• If all nodes were on a square grid, we would need on the order of n1/2 hops
• But this is too restricting.
• What happens when nodes are placed randomly?
28
Quick Lemma
• n nodes
• n/(k1log n) cells C1,C2,…
• mj nodes in cell Cj
• On the average we have
k1log n nodes per cell
• In fact, something stronger holds:
P{k1log n /2 ≤mj ≤2k1 log n for all j}→1, n→∞.
29
Overview of Proof
• Bound probability that one cell is empty.
• Use union bound to bound probability that
there is an empty cell among all:
• Problem related to Coupon Collector’s
Problem
n
i
i
n
i
i EPEP11
)()(
30
Basic Question 3
• Basic Question 3: Given we use short hops (see Basic
Question 1), how many transmitter receiver pairs can
be active simultaneously?
• Easy answer with protocol model when we have a grid of
m2 nodes
• But happens in random case and more realistic models?31
Useful Lemma1
• Let a large number n of transmitters be placed on
an area
• Let a receiver at the center of the area
• Physical protocol
• Assume Rician/Rayleigh/log-normal/etc. fading
• Then power coming from closest transmitter is
comparable to sum of powers of all else.
32
1 B. Hajek and A. Krishna and R. O LaMaire, „On the Capture Probability for a
Large Number of Stations,‟ IEEE Trans. Communications, Vol. 45, No. 2, Feb.
1997.
Explanation
• Random placement of nodes forces more
variation in the signal than fading!
• Proof: by use of order statistics.
33
A direct corollary1
• Let n nodes
• Let everyone transmit to closest neighbor with
probability θ
– Obviously, 0<θ<1.
• Then on the average φn successful
transmissions (with SINR>γ), with 0<φ<θ !!!!
34
1 M. Grossglauser, D.N.C. Tse, „Mobility increases the capacity of ad hoc
wireless networks,‟ IEEE/ACM Transactions on Networking, Vol. 10, No. 4, pp.
477-486, Aug. 2002.
Alternative, less smart proof1
• Divide area in lattice of
n cells
• Divide cells in 9 regular
sub-lattices
• Divide time frame in 9
slots
• Use law of small
numbers
35
1 S. Toumpis, „Capacity Bounds for Three Classes of Wireless Networks:
Asymmetric, Cluster, and Hybrid,‟ in Proc. ACM Mobihoc, May 2004,
Roppongi, Japan.
• In each slot, a single receiver in each cell of
the corresponding sub-lattice receives, from a
single transmitter placed in a nearby cell
(assuming they exist).
• Trivial to bound useful signal.
• Easy to bound interference.
• So we have O(n) transmissions at any given
time.
36
So what do we know so far?
1. To achieve capacity, always transmit across
small distances.
2. In a network of n nodes, two random nodes
are roughly n1/2 hops away.
3. In a network of n nodes, there can be O(n)
simultaneous transmissions (with finite, non-
decreasing SINR), provided they are between
nearest neighbors.
37
3. Asymptotic Capacity of
Networks with Immobile
Nodes1
38
1 P. Gupta and P. R. Kumar, „The Capacity of Wireless Networks,‟ IEEE Trans.
On Information Theory, Vol. 46, No. 2, Mar. 2000, pp 388-404.
Overall Methodology• Capacity is a random quantity if we assume that:
– Placement of nodes is random.
– Each node has random traffic requirements.
• So we only derive bounds on the capacity that hold
with high probability (w. h. p.), i.e., with probability
going to 1 as the number of nodes goes to infinity.
• This allows the use of tools like the law of large
numbers.
• Therefore, these bounds do not hold only for a given
network realization, but for whole classes of
networks.
• This approach also sheds light to the capabilities of
networks with a modest number of nodes. 39
Setting
• n immobile nodes X1,…Xn
uniformly and independently
placed in unit square.
• Each node has a single
destination node, chosen
randomly.
• All nodes require common
end-to-end rate λ(n).
• We define the capacity C(n) as the maximum achievable
λ(n), multiplied by the number of nodes n. 40
Back-of-the-envelope calculation
• There can be O(n) simultaneous transmissions,
so aggregate throughput is O(n).
• Each packet needs n1/2 hops so aggregate
thoughput without counting retransmissions is
O(n) /n1/2=O(n1/2)
• Per node throughput is λ(n)= O(n1/2) /n=O(n-1/2)
• Gupta and Kumar were (almost1) the first to
formalize this train of thought.
41
1 A. Silvester and L. Kleinrick, „On the Capacity of Multihop Slotted ALOHA
Networks with Regular Structure,‟ in IEEE Trans. On Communications,
Vol. COM-31, No. 8, Aug. 1983, pp. 974-982.
Basic Result
• Theorem: There are two constants K1, K2, such
that
• Definition: A sequence of events En occurs with
high probability (w.h.p.) iff
limn→∞ P(En)=1.
• Therefore, theorem states that w.h.p.,
42
1log
)(log
lim 21
n
nKnC
n
nKP
n
n
nKnC
n
nK
log)(
log21
Discussion
• As the number of nodes n→∞, the per-node throughput will necessarily go
to zero, roughly as (n log n )-1/2.
• However, the aggregate throughput will go to infinity as (n log n )-1/2.
• Result is very good or very bad, depending on your point of view.
• With high probability, capacity lies in short interval!
Question: why is there a logarithm there? Do you think it is
fundamental?43
Sketch of Proof
of Upper Bound:
44
1 P. Gupta and P. R. Kumar, „The Capacity of Wireless Networks,‟ IEEE Trans.
On Information Theory, Vol. 46, No. 2, Mar. 2000, pp 388-404.
.1log
)(lim 2
n
nKnCPn
• Assume protocol model with
max distance r(n)
• Each transmission consumes
an area around the receiver:
no other receiver can be in
the shaded disk.
• Number of simultaneous
transmissions T(n) multiplied by π(Δr(n)/2)2 must be less
than area of networks (equal to 1):
T(n) ≤4/[πΔ2r(n)2].
• Therefore, at most 4W/[πΔ2(r(n))2] bps are transmitted by
the network at any given time. 45
Offered Traffic
• If every node creates traffic with rate λ(n),
aggregate created traffic is nλ(n).
• Each packet must be transmitted L/r(n) times,
where L is the average distance between
source and destination nodes.
• Offered traffic: nλ(n)L/r(n).
• Offered traffic must be less than achievable
traffic, so
46)(
4)(
))((
4
)()(
222 nnrL
Wn
nr
W
nr
Lnn
Connectivity Requirement
• We must have r(n)>[log n /n]1/2, otherwise
some of the nodes will be out of the range of
everyone else!
• Sketch of Proof:
• Setting the minimum value for r(n), we get the
upper bound.47
n
i nji
ii
n
i
i
XPXP
XPP
1 1
j
1
})isolated are X,({})isolated is ({
}isolated is {[]node isolatedan is there[
Sketch of Proof
of Lower Bound:
48
1 S. Toumpis and A. J. Goldsmith, Large Wireless Networks under Fading,
Mobility, and Delay Constraints,‟ in IEEE Infocom 2004, Hong Kong, China,
mar. 2004, vol.1, pp 609-619.
.1)(log
lim 1
nC
n
nKPn
Proof Overview• We know that many transmissions over small
distances are better than a few transmissions over
a large distance.
• We will construct a scheme that uses the
principle.
– The basic complication is that node positions are
random.
• The per-node throughput of the scheme will be
better than K1/(n log n )1/2.
• The capacity is the supremum of all achievable
aggregate throughputs, so necessarily
C(n)>nK1/(n log n)1/2. 49
Step 1: Cell
Lattice
• n nodes
• n/(k1log n) cells C1,C2,…
• mj nodes in cell Cj
• On the average we have
k1log n nodes per cell
• In fact, as we saw:
P{k1log n /2 ≤mj ≤2k1 log n for all j}→1, n→∞.
50
Step 2: Routing
• n nodes
• n/(k1 log n) cells C1, C2,…
• Nodes only transmit
to neighboring cells
• Hops per route
< 2(n/k1 log n)1/2
• lj routes through cell Cj
• W.h.p., for large enough k2,
lj < (k2 n log n)1/2 for all j
• Proof: again, Union Bound51
Step 3: Time
Division
• Divide area in lattice of
cells
• Divide cells in 9 regular
sub-lattices
• Divide time frame in 9
slots
• In each slot, a single
receiver in each cell of
the corresponding
sublattice receives data
from a single
transmitter. 52
Throughput Calculation
• Guaranteed throughput per cell: T(n)=W/D.
• Routes per cell: l(n)<(k2 n log n)1/2
• Guaranteed throughput per route (i.e., node):
λ(n)=T(n)/l(n)
• The capacity is the supremum of all achievable
throughputs, so
53
.log)(
)()( 1
n
nK
nl
nTnnC
Next: Some Extensions
1. Getting rid of the logarithm
2. Getting rid of the square root
3. Tougher traffic patterns
54
1. Getting Rid of the Logarithm1
• Question: do you think the logarithm is due
to the problem, or due to the solution?
• If we use the more realistic physical model, we
can dispense with the logarithm!
• But we need more advance mathematics.
• In fact, we need…
55
1 M. Franceschetti and O. Dousse and D. N. C. Tse and P. Thiran, „Closing
the Gap in the Capacity of Wireless Networks Via Percolation Theory,‟ in
IEEE Trans. On Information Theory, Vol. 53, No. 3, Mar. 2
Percolation Theory
• Motivation: what is the rate with which water
trickles through the porous material?
56
Typical
Result
• Each node is connected with each of its neighbors, with
probability p.
• If p>0.5, an infinite connected cluster exists almost surely.
• If p<0.5, an infinite connected cluster does not exist,
almost surely. 57
Highway
System
• By reducing the size of the cells, we can have enough
of them full, so that a „highway system‟ is formed,
consisting of around n1/2 horizontal and n1/2 vertical
highways.
• Using this construction, and the physical model, we
can get rid of the logarithm. 58
2. Getting Rid of the Square Root1,2
• Basic implicit assumption so far: nodes are not
allowed to cooperate in the coding/decoding
phase.
• If we use distributed MIMO, gains can be
impressive: Capacity increases faster than n1-ϵ,
for any ϵ>0.
59
1 S. Aeron and V. Saligramma, „Wireless ad hoc networks: Strategies and scaling
laws for the fixed SNR regime,‟ IEEE Trans. Inf. Theory, vol. 53, no. 6, pp. 2044-
2059, Jun. 20072 A. Özgür, O. Lévêque, and David N. C. Tse, ‘Hierarchical Cooperation Achieves
Optimal Capacity Scaling in Ad Hoc networks,’ in IEEE Trans. on Information
Theory, Vol. 53, No. 10, Oct. 2007, pp. 3549-3572.
Trick: Hierarchical Cooperation
• We divide nodes in hierarchical set of clusters.
• Within same hierarchical level, nodes
communicate as follows:
1. Phase 1: Within each cluster, nodes distribute
their information to other nodes
2. Phase 2: Long Range MIMO transmissions
across many clusters
3. Phase 3: Within each cluster, nodes distribute
information to recipients.
60
3. Asymmetric Traffic • n source nodes, placed
uniformly and
independently.
• nd destination nodes
placed uniformly and
independently, where
the destination
exponent dϵ(0,1).
• Each source node
chooses a random
destination node, so
there are around n1-d
sources for each
destination
• All sources require
end-to-end rate λ(n).61
Capacity of Asymmetric Networks• Let the capacity C(n) =n sup λ(n)
• With probability going to 1 as the number of source nodes
n→∞:
• To avoid the formation of bottlenecks in a network with n
nodes, we need at least n1/2 destinations.
• If destinations are costly and we want to minimize their
number, the network has a sweet spot:
– More than n1/2 will not improve the capacity significantly
– Less than that, and bottlenecks start to form. 62
.2
1d0 ,
log
,12
1 ,
)(log)(log
3
2/3
2/1
2
1
n
nK
dn
nK
nCnnKd
d
More Extensions• What happens if the network is three dimensional?
– Capacity increases like (n /log n)^2/3
– Gupta and P. R. Kumar, „Internets in the Sky: The Capacity of Three
Dimensional Wireless Networks,‟ Communications in Information and
Systems, vol. 1, issue 1, pp. 33-49, Jan. 2001.
• What happens if there is fading?
– Performance of scheme that achieves lower bound is not reduced more
than a factor of log n, for many types of fading models
– S. Toumpis and A. J. Goldsmith, Large Wireless Networks under
Fading, Mobility, and Delay Constraints,‟ in IEEE Infocom 2004, Hong
Kong, China, Mar. 2004, vol.1, pp 609-619.
• What happens if the bandwidth goes to infinity?
– Capacity increases like (n /log n)^2/3
– R. Negi and A. Rajeswaran, „Capacity of power constrained ad-hoc
networks,‟ in Proc. IEEE Infocom, Hong Kong, China, Mar. 2004.
63
More Extensions
• What happens when we have multicast traffic?
• What happens when traffic is localized?
• What happens when movement is constrained
or localized?
• Currently, Gupta/Kumar has 4602 citations, so
obviously no stone has been left unturned.
64
4. Asymptotic Capacity of
Networks with Mobile Nodes1
65
1 M. Grossglauser, D.N.C. Tse, „Mobility increases the capacity of ad hoc
wireless networks,‟ IEEE/ACM Transactions on Networking, Vol. 10, No. 4, pp.
477-486, Aug. 2002.
Basic Idea
• Traditional thinking is that mobility has an adverse
effect on the capacity of networks.
– Overhead of routing protocols increases roughly linearly
with level of mobility.
• On the other hand, if we are willing to tolerate very
large delays, then we can take advantage of the
mobility to deliver our packets through physical
transport, rather than over the wireless channel.
• Idea currently very hot (Delay Tolerant Networks).
66
Wishful Thinking (?)
• We know order O(n) simultaneous
transmissions are possible
• With immobile nodes, it is necessary to have
O(n1/2) such transmissions to reach destination
• With mobile nodes, if I find a way to only need
K, then total throughput will be O(n)/K=O(n)
• Grossglauser/Tse managed this, with K=2.
67
Network Model
• Nodes placed in a disk of unit
area A = 1.
• Nodes move independently
of each other, according to
a stationary and ergodic
random process.
– Brownian motion, random walk are both acceptable.
• Each node has a random destination node (who is also
moving).
• Nodes create traffic with a common rate λ(n) bps.
• Physical Transceiver Model 68
First Try (K=1)• Since nodes are mobile, sources should wait
for destinations to come close, then transmit.
– No relaying is needed.
• This idea is very simple, but does not perform
very well. In fact, any scheme that does not
use relaying can not do better than:
lim n →∞ P{λ(n)=cn-1/(1+a/2) }=0 (1)
• Intuition:
– It is best to transmit over small distances.
– At any given time, very few nodes are close to
their final destination. 69
Sketch of Proof
• The following inequality holds:
• Where Xj(i) is the destination of node Xi, C is a constant,
and S is the set of successful transmissions.
• Intuition: we cannot have too many transmissions over
too large distances, or SINR will be violated
somewhere.
• To exceed (1), we need more than cn-1/(1+a/2)
simultaneously successful transmissions.
• But then it is impossible to satisfy (2).70
)2()(
Si
a
iji CXX
Second try: Scheduling policy π (K=2)
• We slot time, and index slots by t.
• In each slot, each node transmits with
probability θ.
• Each transmits to its closest neighbor a packet
intended for its destination.
• There will be a lot of collisions, but by
previous result of Hajek et al., on the average
there will be φn successful transmissions.
71
But what do we transmit?
• In even slots, we transmit packets (to our
nearest neighbor) intended for our destination,
that he will give to our destination later on.
• In odd slots, we transmit packets intended for
our nearest neighbor, that we received some
time in the past.
72
The book analogy
• Imagine a large number of people moving around in a
city.
• Each one carries a stack of books for a friend of his. The
stack is very high.
• Whenever I bump on any other person on the street:
– I either give him a book for him to give to my buddy,
– or I give him a book that his buddy gave to me some time in
the past.
• Chances that I bump on my own buddy are negligible.
• Question: What is the average number of people that
their destinations are also nearest neighbors? This is
related to the famous hat (or wife) problem. 73
But Delay is Terrible!
• Each node has to wait at the queue of the source
before it gets transmitted.– This delay is not very large.
• Each packet will also have to wait at the queue of
the relay:– With n nodes, the probability that any the destination will be
the closest neighbor of the receiver is only around 1/n.
– Therefore, on the average a packet will have to wait for n slots.
– The average delay per packet E[d]~ n.
• To summarize:– The aggregate throughput is great: T(n) ~n packets at any time.
– The packet delay is terrible: E[d]~ n slots. 74
Throughput-Delay Tradeoff
• The scheme performs very well in terms of
throughput but very bad in terms of delay.
• Can we exchange the two?
• One way to reduce delay, is to have more nodes act as
relays.– In the original scheme, only one node acts as the relay of the packet.
– If many nodes act as potential relays, statistically the packet will arrive
– faster.
– But for more nodes to act as relays, some sort of redundancy must be
used. This redundancy will inescapably reduce the throughput.
• Another way is to allow not two, but many hops from
the source to the destination.
• Next, we take a look at the various schemes that exist.75
First Scheme1
• Unrealistic mobility model
• Instead of a node giving the packet only to a single relay, it
successively gives na copies of the packet to na consecutive
relays, where the parameter a lies in (0, 1).
• All of the relays must be nearest neighbors.
• Delay increases like n1−a.
• Aggregate throughput increases like n1−a.
• We affect the delay-throughput tradeoff by modifying a.
76
1 M. Neely and E. Modiano, „Capacity and Delay Tradeoffs for Ad-Hoc
Mobile Networks,‟ in IEEE Trans. On Information Theory, Vol. 51, No. 6,
June 2005.
Second scheme1
• Packets are duplicated, and at any given time, multiple relays
exist for the same packet.
• Instead of transmitting the packet multiple times (as in the first
scheme), nodes transmit their packets only once.
• But fewer transmissions are allowed at the same time, so
transmissions reach further away!
• The fewer the simultaneously allowed transmissions, the
smaller the aggregate throughput, but the smaller the delay.
• Aggregate throughput increases like na, 1/2 < a < 1.
• Delay increases like n2a−1.
77
1S. Toumpis and A. J. Goldsmith, Large Wireless Networks under Fading, Mobility,
and Delay Constraints,‟ in IEEE Infocom 2004, Hong Kong, China, mar. 2004,
vol.1, pp 609-619.
Third Scheme2
• Packets are not duplicated: at any given time, there is only one
copy of the packet in the network.
• But: packets are allowed to make multiple hops to reach their
destination, once they are close enough.
• So this scheme is a combination of the schemes of Gupta/Kumar
and Grossglauser/Tse.
• The more hops packets are allowed to make, the smaller the
aggregate throughput becomes, but the smaller the average delay
also becomes.
• Aggregate throughput increases like na, where 1/2 < a < 1.
• Delay increases like na−1/2
78
2A. El Gamal, J. Mamen, B. Prabhakar, D. Shah, „Throughput-Delay
Trade-off in Wireless Networks,‟ in Proc. ACM/IEEE Infocom, Hong Kong,
China, Mar. 2004.
Tradeoff of
Exponents
• Per node throughput λ(n) ~nt, packet delay d(n) ~nd.
• Differences in the curves are mostly due to different
assumptions on mobility models, and not so much on
any inherent advantage of any of the schemes. 79
The 2nd scheme,
in greater detail
• Nodes placed in square area.
• Node movements are
independent and uniform.
• For simplicity:
– Within B secs, nodes do not move.
– Every S = NB secs, nodes get
perfectly reshuffled.
• (But results also hold for Brownian motion, various random walks,
etc.)
• Experiment lasts for 2NnD frames of duration B, where D is
integer, greater than 1.
• Physical Channel Model 80
Cell Lattice• n nodes.
• ( n1+d/k1 log n)1/2 cells
C1,C2, . . .
• d is a design parameter,
with 0 < d < 1.
• Let mij be the number of
nodes in cell Ci, in frame j.
Then
E[mij] =(k1n1−d log n)1/2.
• Lemma: With high probability, for all i, j,
k2(n1−d log n)1/2 < mij < k3(n
1−d log n)1/2
• Proof: Union bound.81
Time Division
• Divide cells in 4 regular
sub-lattices.
• Divide frames in 4 slots
of duration B/4 .
• In each slot, a single
receiver in each cell of the
corresponding sub-lattice
receives, from a single transmitter.
• Lemma: With high probability, the SINRs of at least 50%
of the links at all slots are greater than k4 log n.
• Proof: Straightforward.82
Tentative Frame Format
• By assumption, nodes are not moving for the
duration of a frame.
• After a frame passes, no nodes transmit
anything until nodes get perfectly reshuffled
again.83
Packet Transmissions
• Odd frames: Source-Relay Communication.– Each node transmits a single packet, intended for its destination.
– Nodes that receive it will act as relays in subsequent even frames.
– Data rate used: R(n) = fR( k5 log n).
– Packet duration: D(n) = k6n(d−1)/2 (log n)−3/2
– W. h. p., all nodes will get their chance to transmit a packet.
• Even frames: Relay-Destination Communication.
• Packets in the same cell with their destination are
delivered.
• W. h. p., all packets will have the time to be
transmitted.84
Final Frame Format
• Instead of waiting for node to be reshuffled,
we execute the same algorithm in parallel, N
times.85
Throughput and Delay Calculation
• Throughput calculation:
– Every 2B seconds, each node creates a packet.
– Each packet has a size of R(n) × D(n) bits.
– Per-node throughput is λ(n) = R(n)×D(n)/2Β=k6n(d−1)/2 (log n)−5/2
• Delay calculation:
– Each message is carried by around r(n) = (k1n1−d log n)1/2 relays.
– These relays spread out in c(n) = ( n1+d k1 log n)1/2 cells
– Probability that a packet will make it in a frame is only
r(n)/c(n) << 1.
– We need around c(n)/r(n) frames, or 2Nsc(n)/r(n) seconds.
– Lemma: W. h. p., all packets delivered with a delay smaller than
dmax = (4Ns)nd.
86
Sanity Check
• All previous results involve serious
idealizations:
– Number of nodes goes to zero
– Delay goes to infinity
– Throughput per node goes to zero
– Size of buffers goes to infinity
– Node movements are independent, etc.
• So is it worth it? What do we get out of it?
• Justification: these are all capacity results, which
express ultimate bounds, and abstract ways of
achieving them. 87
5. Capacity of Massive
Networks1
88
1 S. Toumpis, „Mother Nature knows Best: A survey of recent results on
wireless networks based on Analogies with Physics,‟ Computer Networks, Vol
52, Feb. 2008, pp. 360-383.
Underlying Theme
• In the modeling phase we frequently arrive
at equations / tradeoffs / concepts
occurring in nature
• Analogies with Physics should be
exploited
1. We gain intuition
2. We end up with problems beaten to death!
• Especially true in wireless networks
– Spatial component
In this PartA. “Packetoptics”
– Optimal route design using analogies with Optics
B. “Packetostatics”
– Optimal placement of nodes in wireless sensor
networks using analogies with Electrostatics
C. Cooperative Transmissions
D. Energy Efficient Routing
E. Load Balancing
A. “Packetoptics”1,2
1 P. Jacquet, „Geometry of Information Propagation in Massively Dense Ad
Hoc Networks,‟ in Proc. ACM Mobihoc, May 2004, Roppongi, Japan, 157-
162.2 R. Catanuto, S. Toumpis, and Giacomo Morabito, “On Asymptotically
Optimal Routing in Large Wireless Networks and Geometrical Optics
Analogy,” Computer Networks, vol. 53, no. 11, pp. 1939-1955, July 2009.
Appetizer
05.01030
1),( 24 xyx
Problem: Find route
between (0,0) and (0,200)
with minimum cost.
Nodes distributed according
to spatial Poisson process
Cost per hop increases
quadratically with hop
length:.)( 2addc
05.01010
1),( 24 xyx
05.0105
1),( 24 xyx
05.0102
1),( 24 xyx
Question: what
happens in the limit?
Limiting case predicted by
Optics!
Macroscopic formulation
• Cost Function:
• Cost of route C that starts at A and ends at B:
• Problem: Find route from A to B that minimizes cost.
.),(
lim)(0
rr
dcc
.)(][ B
A
C dcAB rr
Relation to Optics
• Fermat‟s Principle: To travel from A to B, light will take
the route that locally minimizes the integral:
• Therefore we have the following analogy:
– Index of refraction n(r) becomes the cost function c(r)
– Rays of light become minimum-cost routes.
.)(11
B
A
B
A
B
A
dsnc
dsu
dt r
Advantages of Optical Routing
• We can use the rich body of math that
already exists in Optics for our setting.
– For example, we know that light satisfies the
following equations:
• We can use the intuition that already exists.
– For example, we know that rays of light bend
toward optically denser materials.
.||,)( nSnds
dn
ds
d
r
Various Choices for the Cost
Function
1. Promoting long hops
2. Promoting short hops
3. Promoting energy efficiency
4. Etc.
)()( rr c
)(
1)(
rr
c
.,)(,0,)(
)),(()(
xconstxfxxf
fc rr
Choice of cost function very important!
R1: Jacquet, R2: Constant cost,
R3: Energy limited, R4: Bandwidth limited
Broadcast RoutingThe optimal propagation of a packet resembles the
propagation of light emanating from a light source
Any practical gain by knowing the limit?
• With finite but many nodes, the optimum route is hard to find
• So let us find the optimum route in the macroscopic limit, and use it to create a near optimum route
What the Optics-Networking Analogy
does not tell us
• How does the source know the initial angle
with which the packet/ray should be launched?
• In some nonhomogeneous environments, there
are multiple rays connecting two points
– All of them local minimums
– One of them global minimum
Route
Discovery
• Basic idea: Nodes launch multiple rays
• Intersection points notify pairs of node
B. “Packetostatics”1,2
1 M. Kalantari, M. Shayman, „Energy Efficient Routing in Wireless Sensor
Networks,‟ in Proc. Conference on Information Sciences and Systems,
Princeton University, NJ, Mar. 2004.2 S. Toumpis and L. Tassiulas, “Optimal Deployment of Large Wireless
Sensor Networks,” IEEE Trans. on Inform.Theory, vol. 52, no. 7, pp. 2935-
2953, July 2006.
Setting
• Wireless Sensor Network:
1. Sense the data at the source
2. Transport the data from the sources to the sinks.
3. Deliver the data to the sinks.
• Problem: Minimize number of nodes needed
• What is the best placement for the wireless nodes? What
is the traffic flow it induces?
Macroscopic View
• This problem is way too complicated to be
solved without proper abstractions
• Standard approach is based on microscopic
quantities: individual node placement,
individual link properties, etc.
• We can take a novel macroscopic approach,
using macroscopic quantities: node density,
data creation density, etc.
The Program1. Macroscopic quantities are connected with
each other through ‘constitutive laws’
– Microscopic considerations enter only through the formulation of these laws.
2. Approach opens gateway to new (or old, depending on how you look at it) Math:
– Calculus of Variations, Partial Differential Equations, Optics, Electrostatics, etc.
• Results are not as detailed as with standard approach, but detailed enough to remain useful
Macroscopic Quantities• Node Density Function d(x,y), measured in nodes/m2.
– In area of size dA centered at (x,y) there are d(x,y)dA nodes
• Information Density Function ρ(x,y), measured in
bps/m2.
– If ρ(x,y)>0 (<0), information is created (absorbed) with rate ρdA
over an area of size dA, centered at (x,y).
• Traffic flow function T(x,y),
measured in bps/m.
– Traffic through incremental
line segment is |T(x,y)|dl.
What goes in, must come out• The net amount of information leaving a
surface A0 through its boundary B(A0), must
be equal to the net amount of information
created in that surface:
• Taking |A0|→0, we get the requirement:
)( 0 0
),()(ˆ
AB A
dSyxdss nT
(1)
yx
yxTT
T
Special Case1. Nodes only need to transfer data from
sources to sinks
1. They do not need to sense them at the sources
2. They do not need to deliver them to the sinks once
their location is reached
2. The traffic flow function and the node
density function are related by:
(2) ),(|),(| max yxdcyx T
Traffic must be irrotational
• We must minimize the number of nodes
• If (2) is satisfied, then the traffic must be
irrotational:
• Easy proof by contradiction.
.),( dAyxdN
.0
yx
xy TTT
„Packetostatics‟• The traffic flow T and information density ρ must
satisfy:
• In free space, the electric field E and the charge
density ρ are uniquely determined by:
• Therefore, the optimal traffic distribution is the same
with the electric field when we substitute the sources
and sinks with positive and negative charges!
.0 , TT
.0 , EE
Example: A point source and a linear
sink
Analogy is uncanny!
Electrostatics NetworksPotential differences Number of hops
Non-homogeneous
dielectrics
Non-homogeneous
propagation environments
Conductors Mobile sources and sinks
Thomson‟s theorem Source/Sink placement
optimization
Intersection of electric field
lines and equipotential
lines
Node locations
Generalized Problem• Let
be the density of nodes needed to support the sensing/transport/delivery
• Optimization Problem:
• Minimization over all possible traffic flows T(x,y) that satisfy the constraint
• Standard tool for such problems: Calculus of Variations
|)),(|,,(),( yxyxGyxd T
).,(),( :subject to
)|),(,|,( :minimize 2
yxyx
dSyxyxGN
T
T
Result• The traffic flow is given by:
• where the potential function φ is given by the scalar non-
linear partial differential equation:
• together with appropriate boundary conditions, and G’,
H, properly defined functions
,),,(,,(2
1),(
yxHyxGyxT
),,(,,(2 yxHyxG
Example: Gupta/Kumar physical
layer
2
1
1max),(),( yxdcyx T
Example: Super Gupta/Kumar
3
2
1max),(),( yxdcyx T
Example: Sub Gupta/Kumar
83
1max),(),( yxdcyx T
Example:
Mixed
case
below),(
above),(),(
8
3
1
3
2
1
max
yxdc
yxdcyxT
A final look at the optimization
problem
).,(),( :subject to
,)|),(,|,( :minimize 2
yxyx
dSyxyxGN
T
T
The integrant can have alternative
interpretations: delay, energy, etc.
This is a problem in optimal
transportation
C. Cooperative
Transmissions1
1 B. Sirkeci-Mergen, A. Scaglione, and G. Mergen, “Asymptotic analysis of
multistage cooperative broadcast in wireless networks,” IEEE/ACM
Transactions on Networking, Vol. 14, Issue SI, June 2006, pp. 2531-2550
Setting• Topology: source placed on left side of strip,
destination placed on right side of strip, relays
are placed in strip, Poisson distributed.
• Reception model: nodes susceptible to
thermal noise, power decays with distance as
pr(d)=kd-2, reception successful if SINR>γ.
• Protocol: We slot time. In first slot, source
transmits. In i-th slot, everyone transmits if he
received for first time in previous slot.
Transmission powers add up at potential
receivers.
What the simulations say
• For sufficiently low threshold, a wave is
formed that propagates along the strip. After a
while, wave achieves fixed width and goes on
for ever.
• For high threshold, wave eventually dies out,
irrespective of how many nodes initially had
the packet.
• Position of initial relays critical.
The massively dense
assumption
• Analysis very hard because of random
placement of nodes.
• Assumption: We have so many nodes, that
there is a node practically everywhere.
• Not interested in which node receives in i-th
slot.
• Interested in which region of space receives
in i-th slot.
The result
• Region that receives successfully in i-th slot is
vertical strip of width di=h(di-1).
D. Energy Efficient
Routing1
1 M. Kalantari, M. Shayman, „Energy Efficient Routing in Wireless Sensor
Networks,‟ in Proc. Conference on Information Sciences and Systems,
Princeton University, NJ, Mar. 2004.
Setting
• A Wireless Sensor Network with multiple
sources and a central sink.
• A very large number of nodes
– Modeled by node density function.
– Not subject to optimization.
• Problem: Find routes from sources to the sink
that are energy efficient.
• Intuition: Avoid concentration of traffic in any
given location.
Solution• Minimize following integral:
• Then traffic satisfies Maxwell‟s equations:
• Only intuitive justification
– Works very well in some environments, not so well in other
environments.
• Extensions:
– Multiple types of traffic (handled by SVD)
– Traffic and Network inhomogeneities
.|),(| 2dxdyyx T
.0 , TT
Example
E. Load Balancing1
1 E. Hyytia and J. Virtamo, „On traffic load distribution and load balancing in
dense wireless multihop networks,‟ in EURASIP Journal on Wireless
Communications and Networking, Vol. 2007, No. 1, Jan. 2007.
Setting
• Until now, we supposed only one type of
traffic, or at most a few.
• In general case, if there are n nodes, there will
be n(n-1) distinct traffics (and that ignoring
multicasting!)
• Macroscopic approach: Location r1 creates
traffic for location r2 with rate λ(r1,r2),
measured in bps/m4.
Problem Formulation
• Set of all paths is P.
• Traffic through location r with direction θ has angular flux Φ(P,r,θ), measured in bps/m/rad.
• Total volume that passes through location r is given by scalar flux Φ(P,r):
• Problem: Find optimal distribution of paths, so that
maximum traffic load is minimized:
.),,),2
0
drr PP Φ(Φ(
).,maxargminopt rr PP P Φ(
Results
• Problem still very hard, even for highly
symmetric networks.
• Clever, sharp upper and lower bounds can be
found.
• Insightful closed form expressions for angular
flux.
• Methods borrowed from the modeling of
particle fluxes in Physics.
Conclusions
• New framework for studying problems, based on macroscopic approach.
• Many optimization problems with a pronounced spatial aspect can be handled.
• Some detail is sacrificed, but solutions are insightful.
• Math borrowed from Physics.
• Elephant in the room: we do not have convergence rates!
Parting Comments• Analogies with Physics are well worth
investigating
• The field is particularly promising in wireless
networks due to their spatial aspect
• I showed you two examples of such analogies
• Many more exit1
• Can you come up with others, in your own
research?
1A. Silva, E. Altman, P. Bernhard, M. Debbah, „Continuum Equilibria and
Global Optimization for Routing in Dense Static Ad Hoc Networks,‟
Computer Networks, In-Press, DOI: 10.1016/j.comnet.2009.10.019
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