Rectangle Coverage with Circles - Strategy 2 • Largest rectangle ( a × b ) with aspect ratio L ac W ac from the disc ! "# of radius r c ( a b = L ac W ac ) • Use rectangles to cover the $ "# rectangular region L ac × W ac . √ L 2 ac + W 2 • m � � × = ac 4 r o cos π √ n L 2 ac + W 2 � � ac 4 r o cos π n � L 2 ac + W 2 � 2 × n ac minimize mn = � (1) 4 r o cos π n � r 2 s − r 2 o ≥ H ac subject to : h c = (2) Region-based faults – p. 3/3
Effect of variation of r s on r o and n #$$ '$ ! "# $%$&''($) "# $%$&*'($+ "# $%$&' ! "# $%$&''($) "# $%$*'($+ "# $%$&' )*'+*,-./#/012,3*45, *+(,+-./0#0123-4+56- )*'+*,-./!/012,3*45, *+(,+-./0!0123-4+56- "$ !"$ ' 6 ( 7 8 7 &$ !$$ %$ #$ "$ !$ $ $ !" #$ #" %$ %" &$ ( ) !" #$ #" %$ %" &$ ' ( ($ ! "# $%$&''($) "# $%$&''($* "# $%$&' +,-./012. ) 3 '$ 4 &$ #$ $ !" #$ #" %$ %" &$ ) * Region-based faults – p. 6/19
Effect of variation of H ac on r o and n (# $!# , *+ -.-"##/-0 *+ -.-1#/-2 3 -.-$# ) '( *+*"##,*- '( *+*".#,*/ 0 *+*$# 452*5678-$-9:;6+5<=6 $## 12/'2345*$*6783(29:3 '# 452*5678-"-9:;6+5<=6 2 > 12/'2345*"*6783(29:3 "!# ? / ; &# < "## $# !# # # ! "# "! $# $! %# %! ! "# "! $# $! %# %! ) *+ & '( "$# , *+ -.-"##/-0 *+ -.-"##/-1 2 -.-$# "## -5678+9:;8 (# 1 3 4 '# &# $# # ! "# "! $# $! %# %! ) *+ Region-based faults – p. 7/19
Observation 1 • Increase in r s results in increase in r o and decrease in mn for both the strategies. • Intuitive - as the radius of the coverage sphere increases, the radius of the circular orbit of the ANPs will increase and the total number of ANPs needed to cover the entire air corridor will decrease. • When r s is too small compared to H ac , there may not be a feasible solution. Region-based faults – p. 2/5
Observation 2 • Increase in H ac results in decrease in r o and increase in mn for both the strategies. • Intuitive - as the height of the air corridor increases, the radius of the circular orbit of the ANPs has to decrease. • Consequently, the total number of ANPs needed to cover the entire air corridor must increase. Region-based faults – p. 3/5
Observation 3 • Number of ANPs in an orbit remains a constant ( n = 5 ) irrespective of changes in L ac , W ac , H ac and r s . • This is because the factor n/cos 2 ( π n ) , present in the objective function, reaches its minimum at n = 5 . !' !& ()*+,- ! . ! *(/ %' %& ' ! " # $ %& %! %" %# %$ !& ( Region-based faults – p. 4/5
Observation 4 • The cost of the solution (i.e., the number of ANPs needed to provide complete coverage of the air corridor) using strategy 1 is less than that of strategy 2 • Explanation - by comparison of the objective functions of the strategies. Region-based faults – p. 5/5
! Architecture and Algorithms for an Airborne Network � � How to solve the Connected Coverage Problem? � � A two phase approach � � Phase 1: Find out the minimum number of ANPs required to provide radio coverage in the air corridor at all times. � � Phase 2: Find out the minimum transmission range required by the ANPs to form a backbone network that remains connected at all times.
Robust Network Design - Connectivity and Beyond Consider n nodes (SSying platforms) in an m -dimensional space � m (for AN scenario m = 3 ) x i ( t ) ∈ � m denotes the coordinates of the node i at time t , where by convention x i is considered an m × 1 column vector, and by n ( t )] T , the mn vector resulting from stacking the x ( t ) = [ x T 1 ( t ) , . . . , x T coordinates of the nodes in a single vector The dynamics of node i (for all i ∈ { 1 , 2 , . . . , n } ), is given by x i ( t ) = u i ( t ) , where u i ( t ) is the control vector taking values in some ˙ set U ⊆ � m Region-based faults – p. 2/5
Robust Network Design - Connectivity and Beyond In vector notation, the system dynamics become x ( t ) = u ( t ) ˙ n u ( t )] T are n ( t )] T , u ( t ) = [ u T x T x T 1 ( t ) , . . . , u T where ˙ x ( t ) = [ ˙ 1 ( t ) , . . . , ˙ mn × 1 vectors, respectively. The network of flying platforms described by the system dynamics, gives rise to a dynamic graph G ( x ( t )) G ( x ( t )) = ( V , E ( x ( t ))) is a dynamic graph consisting of • a set of nodes V = { 1 , . . . , n } indexed by the set of flying platforms, and • a set of edges E ( x ( t )) = { ( i, j ) | d ij ( x ( t )) < δ } with d ij ( x ( t )) = � x i ( t ) − x j ( t ) � as the Euclidean distance between the platforms i and j and δ > 0 is a constant. Region-based faults – p. 3/5
Robust Network Design - Connectivity and Beyond Can we control the motion of the ANPs so that G ( x ( t )) retains graph-theoretic properties of interest P for all time t > 0 ? Often times the property P will correspond to the requirement that the graph G remains connected at all times. Formally the problem can be stated as follows: Given C n,P , the set of all graphs on n nodes with property P , is it possible to � nd a control law u ( t ) such that if G ( x (0)) ∈ C n,P then G ( x ( t )) ∈ C n,P for all t ≥ 0 ? Alternately, given the movement pattern of the platforms (location, SSight path, velocity), we may want to know the minimum transmission range of the on-board transceivers so that the resulting network is always connected. Region-based faults – p. 4/5
Robust Network Design - Connectivity and Beyond Mobility Pattern for Connected Dynamic Graph (MPCDG): This problem has five controlling parameters: (i) a set of points { p 1 , p 2 , . . . , p n } on a two dimensional space (representing the centers of circular flight paths of the platforms), (ii) a set of radii { r 1 , r 2 , . . . , r n } representing the radii of circular flight paths, (iii) a set of points { l 1 , l 2 , . . . , l n } representing the initial locations (i.e., locations at time t = 0 ) of the platforms on the circular flight paths, (iv) a set of velocities { v 1 , v 2 , . . . , v n } representing the speeds of the platforms, and (v) transmission range T r of the transceivers on the airborne platforms. Region-based faults – p. 5/5
Robust Network Design - Connectivity and Beyond y c i • r i ( t ) � • c j s ij ( t ) � i • � r j ( t ) � R i ( t ) r c i � r c j � • j θ i ( t ) � R j ( t ) α c i θ j ( t ) α c j x O Region-based faults – p. 2/2
y c i • � r i ( t ) • c j s ij ( t ) � i • � r j ( t ) � R i ( t ) r c i � r c j � • j θ i ( t ) � R j ( t ) α c i θ j ( t ) α c j x O R j ( t )) 2 = R 2 ij ( t ) = ( � R i ( t ) − � j ( t ) − 2 � R i ( t ) · � s 2 i ( t ) + R 2 R j ( t ) Region-based faults – p. 16/19
y i (0) • β i � c i • R i (0) r c i � θ i (0) α c i x O R i (0) cos θ i (0) − r ci cos α ci tan β i = R i (0) sin θ i (0) − r ci sin α ci Region-based faults – p. 17/19
• Condition for having a link between nodes i and j s ij ( t ) ≤ D Region-based faults – p. 18/19
• Condition for having a link between nodes i and j s ij ( t ) ≤ D • Expression for s ij ( t ) s 2 ij ( t ) = r 2 c i + r 2 i + 2 r c i r i cos( β i + ω i t ) + r 2 c j + r 2 j + 2 r c j r j cos( β j + ω j t ) + r c i r c j cos α c i c j + r i r j cos( β ij + ( ω i − ω j ) t ) + r c i r j cos( α c i − β j − ω j t ) + r c j r i cos( α c j − β i − ω i t ) (4) Region-based faults – p. 18/19
Distance between i and j as a function of time - links being active and inactive 25 Edge exists between i and j No edge between i and j 25 Distance Between Nodes i and j D = 24 20 Distance between nodes i and j 20 D = 18 15 15 10 10 5 5 D = 4 0 0 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Time Time • For D = 4 : the link i − j never exists • For D = 24 : the link i − j always existss Region-based faults – p. 19/19
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$ Architecture and Algorithms for an Airborne Netw !"# Visualization of Flight Paths of Airborne Networking Platforms and its relationship with Topology Change of the Dynamic Graph formed by an Airborne Network
Mobile Network of Airborne Platforms � � Mobility Models � � Predictable Well-structured Flight Paths � � Predictable Ill-structured Flight Paths � � Unpredictable Flight Paths
Airline Route Maps Route Map – Continental United States Route Map – North and South Americas Transatlantic Route Map
Airline Route Maps Route Map – North and South Americas
Airborne Network Route Map – North and South Americas
A Day in the Life of Air Traffic over the United States
Predictable Ill Structured Flight Path
Unpredictable Flight Path
Network without Infrastructure - Unpredictable Flight Path Initial Distribution: The ANPs are distributed uniformly over the deployment region. Relationship between transmission range ( c n ) and number of nodes ( n ) can be calculated using theoretical results in the literature. Let us consider a random graph G n ( x ) , which is constructed on independent random points U 1 , . . . , U n distributed uniformly on [0 , 1] d , d ≥ 1 , in which two distinct such points are joined by an edge if the l ∞ distance between them is at most some prescribed value 0 < x < 1 . Region-based faults – p. 1/5
Appel and Russo’s Result Proposition: For each n and x , { c n ≤ x } = { G n ( x ) is connected } Theorem: n � � c d lim = 1 , if d = 1 n log n n →∞ 1 = 2 d, if d ≥ 2 Region-based faults – p. 2/5
Unpredictable Flight Path - Mo- bility Model Mobility Model: • Modified Random Way Point model. • Time domain is divided into equal intervals. • Select randomly a value within a specified range for displacement along x -axis and y -axis. • For each node randomly select the directions for both dimensions, i.e. East/West for x dimension and North/South for y dimension. • If the corresponding destination falls outside the deployment region for either dimension, bounce back with the excess amount. • Reach destination at the end of the time interval. Region-based faults – p. 3/5
Unpredictable Flight Path - Dis- tribution after Movements How does the mobility of the nodes effect the distribution? X 1 , X 2 - random variables representing the x and y co-ordinates of a point over a unit square before movement (uniform distribution). Y 1 , Y 2 - random variables representing the x and y co-ordinates of a point over a unit square after movement. It can be derived that: F Y 1 Y 2 ( y 1 , y 2 ) = y 1 y 2 ∂ 2 � � f Y 1 Y 2 ( y 1 , y 2 ) = F Y 1 Y 2 ( y 1 , y 2 ) = 1 ∂ y 1 ∂ y 2 Therefore, distribution of nodes remain uniform and results derived for uniformly distributed static nodes can be used for the mobile airborne networks as well. Region-based faults – p. 4/5
! Architecture and Algorithms for an Airborne Network Path Switching and Routing Problem � � Flying ANPs result in change in the backbone network topology � � A path established between a source-destination node pair may not last for the entire duration of communication � � In order to complete communication, paths may have to be switched
! Architecture and Algorithms for an Airborne Network Path Switching and Routing Problem � � Path switching has associated overhead and is undesirable � � Communication has to complete with as few path switching as possible � � Minimum Path Switch Routing Algorithm
! Architecture and Algorithms for an Airborne Network Path (Channel) Diversity � � Quality of links may depend on various factors (mobility, atmospheric condition, jamming) � � Multi-channel communication � � Two channel (high and low) communication model � � Possibilities � � High and Low quality channels exists on a link � � Low quality channel exists on a link � � No
! Architecture and Algorithms for an Airborne Network Routing performance in diverse networking conditions
! Architecture and Algorithms for an Airborne Network Routing performance in diverse networking conditions Option 1: P 2 � P 1 � P 2 � P 3 � P 4 � P 5 Option 2: P 5 � P 6 � P 4 � P 5 � P 6
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+ 5"164)+7$)8".9+:$4(/#+;+!"##$%&'()*+,#-+<$*"#- Further limitations of Connectivity as the metric of fault-tolerance � � It does not capture the number or the size of the connected components in which the network disintegrates when the number of failed nodes exceeds the connectivity of the network . Connectivity is 1 for both the graphs. However, in one case, after failure of one node, one large connected component is guaranteed to exist. No such guarantee can be given in the other case. � � Our solution: Region-based Component Decomposition Number
+ 5"164)+7$)8".9+:$4(/#+;+!"##$%&'()*+,#-+<$*"#- Objectives of our study � � How to design a network that will be k region-based fault tolerant under single/multi-region fault model? � � How to design a network whose RBCDN is at most k? � � How to design a network whose RBSCS (or RBLCS) is at least k? � � How to design a network that ensures that the key nodes retains a specified structural property (large connectivity, small diameter) under single/multi-region fault model? � � How to design a multi-layer network so that impact of failures in one layer i (say, layer 1) can be mitigated by resources in layer j?
Visualization of Airborne Networks Visualization of Airborne Networks ANPs in Reality Region-based faults – p. 2/2
Specification of ANPs • The STS-111 is an unmanned Mid Altitude Long Endurance UAV and is designed to carry crucial communications technology, provide persistent surveillance capabilities and work in tandem with other UAVs. • Multi-layered communications software package that enables multi-vehicle IP-based messaging (TCP/IP , UDP) over line-of-sight (LOS) or beyond-line-of-sight (BLOS*) communication channels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egion-based faults – p. 2/2
Backbone Network Design in Presence of Faults • Earlier, we presented algorithms to compute the minimum transmission range of the airborne transceivers so that the backbone network remains connected at all times. • This transmission range ensures a time-invariant property (connectivity) of a time-varying network topology. • Our earlier results did not consider impact of an attack on an airborne network. • Our current results provide an algorithm to compute the minimum transmission range of the airborne transceivers so that the backbone network remains connected at all times in spite of failure of several airborne nodes due to an attack. Region-based faults – p. 2/2
Visualization of Airborne Network with Fault Region-based faults – p. 2/2
Previous Result • We considered a similar scenario in an Infocom 2006 paper where the nodes are stationary • We extended these results to fit in a Airborne Network scenario where nodes are mobile • Mobility of the nodes increases the complexity of the problem significantly. Region-based faults – p. 2/16
Previous Result (cont’d) • Infinite number of points can potentially be centers of fault regions. • However, it is sufficient to consider only O ( n 2 ) points as centers of fault regions (Sen et al. - Infocom 2006). & %#$ ( # $ !"$ !"# ' %#$ Vulnerability Zone ( V Z i ): Circular area of radius R centered at node i I-points: The intersections of two vulnerability zones of two different nodes. If VZ of one node does not intersect with any other VZ, then the location of the node is considered as an I-point. Region-based faults – p. 3/16
Previous Result (cont’d) Computation of Region-Based Connectivity *%&&"*+",'*%-)%&"&+'. *%&&"*+",'*%-)%&"&+'/ !"#$%&'() *%&&"*+",'*%-)%&"&+'0 • Algorithm computes, for each region, the minimum number of nodes inside the region whose removal will disconnect the graph • The objective is to find the minimum transmission range so that graph remains connected even after the failure of all the nodes in any of the regions. • Required information: 1. Time-varying network topology without faults in every time slot (interval) 2. Nodes within each fault region in every time slot (interval) Region-based faults – p. 4/16
Computation of time-varying network topology without faults in every time slot (interval) Region-based faults – p. 5/16
Time-varying Distance Between a Pair of Nodes y c i • r i ( t ) � • c j s ij ( t ) � i • � r j ( t ) � R i ( t ) r c i � r c j � • j θ i ( t ) � R j ( t ) α c i θ j ( t ) α c j x O R j ( t )) 2 = R 2 ij ( t ) = ( � R i ( t ) − � j ( t ) − 2 � R i ( t ) · � s 2 i ( t ) + R 2 R j ( t ) Condition for having a link between nodes i and j : s ij ( t ) ≤ D Region-based faults – p. 6/16
Link Lifetime Distance between i and j as a function of time - links being active and inactive 25 Edge exists between i and j No edge between i and j 25 Distance Between Nodes i and j D = 24 20 Distance between nodes i and j 20 D = 18 15 15 10 10 5 5 D = 4 0 0 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 Time Time • For D = 4 : the link i − j never exists • For D = 24 : the link i − j always exists Region-based faults – p. 7/16
Link Active/Inactive Time Intervals !"#$5%678"9, !"#$5%:,;< !"#$%&' !"#$%(' !"#$%)' *"+,-"#, & ( ) . / 0 1 2 3 &4 &( &) &. &/ &0 &1 && • Check for the graph-connectivity in each of these intervals between time t = t 1 and t = t 2 (time of operation) • Binary search over the range [0 , T max ] to find the minimum transmission range to make the graph connected Region-based faults – p. 8/16
Computation of nodes within each fault region in every time slot (interval) Region-based faults – p. 9/16
Vulnerability Zone for an ANP • Vulnerability Zone of an ANP moves along with the ANP • At time t , it is denoted by V Z i ( t ) • For a pair of ANPs, ANP i and ANP j , the I-points are denoted as I 1 ij ( t ) and I 2 ij ( t ) &'()#*+,-.,#*%/-!"#-0*1-!"$- 1)-*)+-#*+/2,/%+3 *+,#- & # %#$ ( !"$ )$ !"# %$ $ )./-.+0*10*+,#- )# # %# ' $ !"# %#$ !"$ I-points when VZs intersect I-points when VZs do not intersect Region-based faults – p. 10/16
Lifetime of I-points • Distance s ij between ANP i and ANP j at time t is given as: R j ( t )) 2 = R 2 ij ( t ) = ( � R i ( t ) − � j ( t ) − 2 � R i ( t ) · � s 2 i ( t ) + R 2 R j ( t ) • The existence of the I-points ( I 1 ij ( t ) and I 2 ij ( t ) ) at time t , is denoted by indicator variables II 1 ij ( t ) and II 2 ij ( t ) defined as follows: 1 , if s ij < 2 R II 1 ij ( t ) = 0 , otherwise $%&'+,+!!"# (%&'+,- $%&'+,+!!"# (%&'+,$ !!"# !!"# & %#$ # !"#$%$&' (%&' !"#$%$&' !"$ !"# !"# %$ $ # %# $%&' !"# $ ' !"# %#$ &")* !"$ vvehvuberviue II 1 ij = II 2 II 1 ij = II 2 ij = 1 ij = 0 Region-based faults – p. 11/16
Nodes within Fault Region • F = { f 1 , f 2 , . . . , f l } : the set of regions centered at I-points, where l = n ( n − 1) + n • D pk ( t ) = distance between center of region f p and ANP k at time t • For every region f p and every node ANP k , if at time t the distance D pk ( t ) ≤ R , then ANP k is within the fault region f p 2345+6"!7"0+%&+-+8(9/90: 2345+6"!7"0+%&+-+8(9/: )!*+,-. '&0 1 1 1 '&/ )!*+,-. '&()!*+,-. %& !"#$ Region-based faults – p. 12/16
Putting it Altogether • Step 1: Computation of time-varying network topology without faults in every time slot (interval) • Step 2: Computation of nodes within each fault region in every time slot (interval) Combining Step 1 and Step 2: • Step 3: Division of the time intervals computed in step 1 into sub-intervals, to identify the set of vulnerable nodes in each one of them, due to occurrence of an attack (fault) Region-based faults – p. 13/16
Computation of the Minimum Transmission Range to keep the Residual Graph Connected when Some or All Nodes in a Fault Region f p have Failed ;6;6; :(6< :( :) :0 %"&'( ( + %"&', %"&'+ %"&') %"&'* ) * %"&'* %"&') %"&'( =!6!(6>6126&?!6@A@"%@B%$ ) * + !( ( , - . !) !* / 0 !"#$ :(4!)56<6 :(4!*56<6 4!56789 32+ %"&'( ( %"&'( 32* 4!56789 ( + %"&') + %"&') 32) 4!56789 ) 32( 4!56789 * ) %"&'* %"&'* 12 * !"#$ • From link lifetime information, the graph G i corresponding to time interval i is computed • From the fault region ( f p ) information, we compute the subintervals, during which a subset of nodes of G i is vulnerable due to fault f p Region-based faults – p. 14/16
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