On the capacity
of mobile ad hoc wireless
networks
Michele Garetto – Università di Torino, Italy
Paolo Giaccone - Politecnico di Torino, Italy
Emilio Leonardi – Politecnico di Torino, Italy
Wednesday, 16 March 2016
1
Outline





Introduction and motivation
Capacity properties of finite networks
Application to experimental traces
Asymptotic network capacity
Application of asymptotic analysis
2
Introduction
 The sad Gupta-Kumar result:
In static ad hoc wireless networks with n nodes,
the per-node throughput behaves as
P. Gupta, P.R. Kumar, The capacity of wireless networks, IEEE Trans. on
Information Theory, March 2000
3
Introduction
 The happy Grossglauser-Tse result:
In mobile ad hoc wireless networks with n nodes,
the per-node throughput remains constant
M. Grossglauser and D. Tse, Mobility Increases the Capacity of Ad Hoc
Wireless Networks, IEEE/ACM Trans. on Networking, August 2002
4
Introduction
Node mobility can be exploited to carry
data across the network
 Store-carry-forward communication scheme
S
R
D
 Drawback: large delays (minutes/hours)
 Delay-tolerant networking
5
Mobile Ad Hoc
(Delay Tolerant) Networks
 Have recently attracted a lot of attention
 Examples
pocket switched networks (e.g., iMotes)
vehicular networks (e.g., buses, taxi)
sensor networks (e.g., disaster-relief networks,
wildlife tracking)
Internet access to remote villages (e.g., IP over usb
over motorbike)
 Key issue: how does network capacity depend on
the node mobility pattern ?
6
The impact of mobility
 In their original paper, Grossglauser and Tse assume
that the mobility pattern of each node produces a
uniform distribution of node presence over the network
area
 results have later been extended to a restricted
mobility model in which each node moves over a random
great circle on the sphere
Per-node throughput
is still
!
S. Diggavi, M. Grossglauser, and D. Tse, Even One-Dimensional Mobility
Increases Ad Hoc Wireless Capacity, IEEE Trans. on Information Theory,
November 2005
7
The general (unanswered) problem
 What properties in the mobility pattern of nodes
allow to avoid the throughput decay of static
networks ?
 What are the sufficient conditions to obtain
per-node throughput ?
 Are there intermediate cases in between
extremes of static nodes (Gupta-Kumar ’00) and
fully mobile nodes (Grossglauser-Tse ’01) ? Under
which conditions ?
8
Our work
 We take one step forward in addressing the
general problem
 We obtain basic results for arbitrary (finite)
networks
 We provide a general framework to compute the
asymptotic transport capacity of mobile
networks with anisotropic mobility patterns
 We apply the proposed framework to a class of
mobile networks comprising heterogeneous
nodes and spatial inhomogeneities
9
Outline





Introduction and motivation
Capacity properties of finite networks
Application to experimental traces
Asymptotic network capacity
Application of asymptotic analysis
10
Arbitrary (finite) networks
Main result:
Under very general assumptions, the
capacity (in networking sense) of a
mobile wireless network depends on the
mobility process only through the joint
stationary distribution of nodes over
the area
The details on how nodes move (change
of speed, direction, group movements)
have no impact on network capacity
11
Arbitrary (finite) networks
 Assumptions:
 n nodes moving according to a stationary and ergodic mobility
process (possibly correlated among the nodes)
 A source node s generates traffic for destination d according
to a stationary and ergodic process with rate sd
 Transmissions between pairs of nodes occur at fixed rate r
 The set of transmissions that can be scheduled successfully at
time t depends only on the instantaneous node positions
Another
possible
One possible
transmission
transmission
configuration
configuration
12
Joint scheduling-routing formulation
 Scheduling policy S: selects π(t) among all feasible
transmission configurations at time t
 According to S, between any pair of nodes ( i, j ) there exist a
virtual link of capacity
 Routing policy R: described by
, denoting the
average fraction of the traffic flow from node s to node
d, which is routed through link ( i, j )
 Quantities
satisfy the classic flow-conservation constraints:
13
Traffic sustainability

Z(t): network backlog, i.e., the amount of traffic (in
bits) already generated by sources that has not yet
been delivered to destinations at time
t
 traffic matrix
is sustainable if there
exists a scheduling policy S and a routing strategy R,
such that:
 Capacity region: the set of all sustainable traffic
matrices
14
Main theorem
 Maximum throughput is achieved by the “simple” class
of scheduling policies based only on position information
(stateless and memoryless policy)
 No gain in terms of throughput can be achieved by
complex scheduling policies considering:
 Speed and direction of nodes
 History of encounters
 Queue lengths
 Age of stored information
 Proof’s idea: if a given traffic matrix can be sustained
by some arbitrarily complex scheduling, there exist a
simple scheduling policy that sustains it
 Corollary: the capacity region of a mobile wireless
network is convex
15
Outline
 Introduction and motivation
 Capacity properties of finite networks
 Application to experimental traces
 Asymptotic network capacity
 Application of asymptotic analysis
16
DieselNet-Umass experiment
 30 buses running the campus transport
service in Spring 2005 (60 days)
 traces provide the amount of data
transferred through TCP connections
using WLAN 802.11 MAC access
 mij measured in bytes
17
Infocom2005-iMotes experiment
iMotes carried by 41 volunteers attending
the conference (5 days)
traces provide the radio contact duration
between any two iMotes using Bluetooth
MAC 802.15 access
 mij measured in seconds
18
Experimental traces
 Virtual capacities
are directly measured
for each pair of nodes (i,j)
 contact graph: a capacitated graph in which
 each vertex corresponds to a mobile node
 edge (i,j) is weighted by
 The maximum throughput is computed by solving
a standard multi-commodity flow problem (for a
given flow assignment)
19
Experimental contact graphs
“minDC” class 1 class 2 class 3 class 4
“MaxDC”
capacity
10x
100x
flow assignment x
flow assignment
 significant asymmetries and inhomogeneous capacities
 few edges contribute most of the capacity: class 3-4 contribute for 80%
(Umass) and 60% (iMotes) of the overall transport capacity
20
Theoretical performance of
experimental networks
Traffic scenario
Maximum aggregate
capacity
Average number of
hops
21.131 Gbytes
2.31
Umass MDC
28.578 Gbytes
1.85
iMotes mDC
1.756 Msec
2.60
iMotes MDC
3.008 Msec
1.69
Probability
Umass mDC
Number of hops
21
Outline
 Introduction and motivation
 Capacity properties of finite networks
 Application to experimental traces
 Asymptotic network capacity
 Application of asymptotic analysis
22
Assumptions
 n nodes moving over unit-area closed connected region
 independent, stationary and ergodic mobility processes
 uniform permutation traffic matrix: each node is origin
and destination of a single traffic flow with rate  (n)
bits/sec






 all transmissions employ the same nominal range or power
 all transmissions occur at common rate r
 single-radio, omni-directional antennas
 interference described by protocol model (next slide)
23
Protocol Model
 Let dij denote the distance between node i and node j,
and RT the common transmission range
 A transmission from i to j at rate r is successful if:
for every other node k simultaneously transmitting
RT
i
(1+Δ)RT
j
k
24
Asymptotic analysis
 We say that the per-node capacity is Θ( (n) ) if there
exist two constants c and c’ such that
 Equivalently, we say that the network capacity in this
case is Θ( n (n) )
25
Uniformly dense networks
 We define the local asymptotic node density ρ(XO) at
point XO as:
Where
radius
is the disk centered in XO , of
 A network is uniformly dense if:
26
Properties of uniformly dense networks
 Theorem: the maximum network capacity is
achieved by scheduling policies forcing the
transmission range to be
 Corollary: simple scheduling policies leading to
link capacities
are asymptotically optimal, i.e., allow to achieve
the maximum network capacity (in order sense)
27
The scheduling policy S*
 Transmission from i to j is enabled when
(1+Δ)RT
(1+Δ)RT
i
j
note: it complies with
the protocol
interference model
 Theorem: S* is optimal, i.e., link capacities
resulting from S* satisfy
Outline





Introduction and motivation
Capacity properties of finite networks
Application to experimental traces
Asymptotic network capacity
Application of asymptotic analysis
29
A realistic mobility model for DTNs
 Realistic mobility processes are characterized by :
Restricted mobility of
individual nodes:
Non-uniform density due to
concentration points
From: J.H.Kang, W.Welbourne, B. Stewart,
G.Borriello, Extracting Places from Traces
of Locations, ACM Mobile Computing and
Communications Review, July 2005.
From: Sarafijanovic-Djukic, M. Piorkowski, and
M. Grossglauser, Island Hopping: Efficient
Mobility-Assisted Forwarding in Partitioned
Networks,, IEEE SECON 2006
30
Heterogeneous nodes with
restricted mobility
“home-points” of
the nodes
31
Restricted mobility
 The shape of the spatial distribution of each
node is according to a generic, decreasing
function s(d) of the distance from the home-point
s(d)
d
32
Anisotropic node density (clustering)
 Achieved through the distribution of home-points
Uniform model: home-points
randomly placed over the area
according to uniform distribution
Clustered model: nodes randomly
assigned to m = nν clusters
uniformly placed over the area.
Home-points within disk of radius r
from the cluster middle point
1
1
n = 10000
0
0
0
1
0
1
33
Scaling the network size
10 nodes……100 nodes…..1000 nodes
?
We assume that:
Moreover: node mobility process does not depend on network size
Asymptotic capacity results
logn [(n)]
Uniform Model
per-node capacity
0
-1/2
Independently of the shape of s(d) !
-1
0
Recall:
1/2
35
Asymptotic capacity results
logn [(n)]
Clustered Model
per-node capacity
0
-1/2
??
Lower bound
: in case
s(d) has finite
support
“Super-critical
regime”:
mobility
helps
-1
Lower bound : in case s(d) has finite support
“Sub-critical regime”: mobility does not help
0
Recall: #clusters
1/2
36
Super-critical regime
Let
(m = n in the Uniform Model)
When
we are in the supercritical regime
 Theorem: in super-critical regime a
random network realization is uniformely
dense w.h.p.
Transmission range
Scheduling policy S* is optimal
is optimal
37
Mapping over Generalized Random
Geometric Graph (GRGG)
 Link capacities can be evaluated in terms of contact
probabilities:
which depend only on the distance dij between the
homepoints of i and j
 We can construct a random geometric graph in which
 Vertices stand for homepoints of the nodes
 edges are weighted by
 Network capacity is obtained by solving the maximum
concurrent flow problem over the constructed graph
38
Upper bound : network cut
1
0
1/2
1
39
Average/random network flows
 The “average” flow through the cut is easily computed as
 fundamental question:
Answer: YES !
1
Proof’s idea:
Consider regular tessellation
where squarelets have area γ(n)
0
1/2
1
Take upper and lower bounds for
number of homepoints falling in each
squarelets, combined, respectively,
with lower and upper bounds of
distances between homepoints
belonging to different squarelets
40
An optimal routing scheme
Routing strategy:
s
Consider a regular tessellation
where squarelets have area
d
Create a logical route along
sequence of horizontal/vertical
squarelets, choosing any node
whose home-point lie inside
traversed squarelet as relay
The above routing strategy sustains
per-node traffic
41
Sub-critical regime
 Restricted mobility does not allow nodes whose
home-points belong to different clusters to
communicate using
Nodes have to use
System behaves as network of m static node
42
Model variations
Mobility of nodes dependent on network size
Spatial distributions not symmetric around
home-point. E.g.: mobility restricted on lines
with random orientation
Different classes of nodes (e.g.: fully mobile
nodes + nodes with restricted mobility)
Can be studied using the same techniques
(e.g. computation of network flow through
physical cut)
43
An interesting case
 Nodes contrained to move uniformly over rectangles of
area n-β (1/2 < β < 1), with minimum edge n-1/2 and random
orientation
Best solution:
Worst
solution:
n1/2-β
n-β/2
n-1/2
n-β/2
network capacity
3/2-β)
is
Θ(
n
network capacity
is Θ( n1-β/2)
 In
general, network capacity can depend on the
geometry of the space visited by the nodes
44
Conclusions
One step forward in the capacity analysis
of mobile wireless networks with general
node mobility processes
Some results of general validity for finite
and infinite number of nodes
Mapping over maximum concurrent flow
problem over geometric random graphs
Application to a general class of mobile
networks with heterogenous nodes and
clustering behavior
45
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