Saturday 22 April 2023 1
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
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Outline
Introduction and motivation Capacity properties of finite networks Application to experimental traces Asymptotic network capacity Application of asymptotic analysis
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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
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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
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Introduction
Node mobility can be exploited to carry data across the network Store-carry-forward communication
scheme
S DR
Drawback: large delays (minutes/hours)
Delay-tolerant networking
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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 ?
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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
S. Diggavi, M. Grossglauser, and D. Tse, Even One-Dimensional Mobility Increases Ad Hoc Wireless Capacity, IEEE Trans. on Information Theory, November 2005
Per-node throughput is still !
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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 ?
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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
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Outline
Introduction and motivation Capacity properties of finite networks Application to experimental traces Asymptotic network capacity Application of asymptotic analysis
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Arbitrary (finite) networksMain 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
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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
One possible transmission configuration
Another possible transmission configuration
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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:
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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
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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
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Outline
Introduction and motivation Capacity properties of finite networks Application to experimental traces Asymptotic network capacity Application of asymptotic analysis
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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
ij measured in bytes
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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
ij measured in seconds
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Experimental tracesVirtual 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)
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Experimental contact graphs
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
class 1class 2class 3class 4
x 10x 100x capacity
“minDC”flow assignment
“MaxDC”flow assignment
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Theoretical performance of experimental networks
Traffic scenario Maximum aggregate capacity
Average number of hops
Umass mDC 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
Pro
bab
ilit
y
Number of hops
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Outline
Introduction and motivation Capacity properties of finite networks Application to experimental traces Asymptotic network capacity Application of asymptotic analysis
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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)
Assumptions
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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 (1+Δ)RT
ij
k
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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) )
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Uniformly dense networks We define the local asymptotic node density ρ(XO) at
point XO as:
Where is the disk centered in XO , of radius
A network is uniformly dense if:
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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)
The scheduling policy S*
Theorem: S* is optimal, i.e., link capacities resulting from S* satisfy
i j
(1+Δ)RT(1+Δ)RTnote: it complies with the protocol interference model
Transmission from i to j is enabled when
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Outline
Introduction and motivation Capacity properties of finite networks Application to experimental traces Asymptotic network capacity Application of asymptotic analysis
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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: Sarafijanovic-Djukic, M. Piorkowski, and M. Grossglauser, Island Hopping: Efficient Mobility-Assisted Forwarding in Partitioned Networks,, IEEE SECON 2006
From: J.H.Kang, W.Welbourne, B. Stewart, G.Borriello, Extracting Places from Traces of Locations, ACM Mobile Computing andCommunications Review, July 2005.
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Heterogeneous nodes with restricted mobility
“home-points” of the nodes
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Restricted mobilityThe 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
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Anisotropic node density (clustering)
Achieved through the distribution of home-points
0
1
0 1
Uniform model: home-points randomly placed over the area according to uniform distribution
n = 10000
0
1
0 1
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
Scaling the network size
?
10 nodes……100 nodes…..1000 nodes
We assume that:
Moreover: node mobility process does not depend on network size
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Asymptotic capacity results
Recall:
0 1/2
per-
nod
e c
ap
aci
ty
0
-1/2
-1
logn [(n)] Uniform Model
Independently of the shape of s(d) !
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Asymptotic capacity results
Recall: #clusters
0 1/2
per-
nod
e c
ap
aci
ty
0
-1/2
-1
logn [(n)] Clustered Model
“Super-critical regime”: mobility helps
“Sub-critical regime”: mobility does not help
?Lower bound : in case s(d) has finite support
Lower bound : in case s(d) has finite support
?
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Super-critical regime
Let (m = n in the Uniform Model)
When we are in the super-critical regime
Theorem: in super-critical regime a random network realization is uniformely dense w.h.p.Transmission range is
optimalScheduling policy S* is optimal
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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 nodesedges are weighted by
Network capacity is obtained by solving the maximum concurrent flow problem over the constructed graph
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Upper bound : network cut
0 1
1
1/2
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Average/random network flows
0 1
1
1/2
The “average” flow through the cut is easily computed as
fundamental question:
Proof’s idea:
Consider regular tessellation where squarelets have area γ(n)
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
Answer: YES !
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An optimal routing scheme
The above routing strategy sustains per-node traffic
s
d
Routing strategy:
Consider a regular tessellation where squarelets have area
Create a logical route along sequence of horizontal/vertical squarelets, choosing any node whose home-point lie inside traversed squarelet as relay
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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
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Model variationsMobility 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)
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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
In general, network capacity can depend on the geometry of the space visited by the nodes
n1/2-β
n-1/2
Best solution:
network capacity is Θ( n3/2-β)
n-β/2
Worst solution:
network capacity is Θ( n1-β/2)
n-β/2
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ConclusionsOne 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
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Comments ?
Questions ?
