Cloud-Assisted Remote Sensor Network Virtualization for Distributed Consensus Estimation

0.23 MB·English

by  Sherif Abdelwahab

Cloud-Assisted Remote Sensor Network Virtualization for Distributed
Consensus Estimation
Sherif Abdelwahab∗, Bechir Hamdaoui∗, and Mohsen Guizani†
∗
Oregon StateUniversity,abdelwas,[email protected]
†
Qatar University,[email protected]
Abstract
remotesensingtaskrequestsfromcloudclients. Whenare-
5
questisgranted,theagentisalsoresponsibleforvirtualizing
1
0 We develop cloud-assisted remote sensing techniques for asensornetworktoperformthe requestedsensingtask. An
2 enabling distributed consensus estimation of unknown pa- exampleofsuchtasksisdistributedconsensusestimation,in
n rameters in a given geographic area. We first propose which an unknown set of parameters need to be estimated.
a a distributed sensor network virtualization algorithm that Imagine for example the virtual sensing task in which it is
J searchesfor,selects,andcoordinatesInternet-accessiblesen- requiredtotrackthelocationofanRFID-taggedpersonina

5 sors to perform a sensing task in a specific region. The large campus. A group of smart-phones connected through
1 algorithmconvergesinlinearithmic time forlarge-scalenet- machine-to-machinephysicallinkscanestimatethelocation

works, and requires exchanging a number of messages that of a person based on the RFID signal strength each smart-
]
I isatmostlinearinthenumberofsensors. Second,wedesign phone receives and measures. In this case, a virtual sens-
N
an uncoordinated, distributed algorithm that relies on the ing request is sent to the cloud agent which dispatches the
s. selected sensors to estimate a set of parameters without re- request to few smart-phones in the swarm. These smart-
c quiring synchronizationamong the sensors. Our simulation phones autonomously search for a group of smart-phones
[
results show that the proposed algorithm, when compared that can read RFID tags and are willing to participate in

1 to conventional ADMM (Alternating Direction Method of the sensing task to estimate the distance based on received
v Multipliers), reduces communication overhead significantly signalstrengths. Amongallthesesmart-phones,asubgroup
7 withoutcompromisingtheestimationerror. Inaddition,the ofthemisthenselectedtoformavirtualsensornetworkcon-
4
convergencetime, though increases slightly, is still linear as nectedaccordingtoatopologythatistobespecifiedbythe
5
3 in the case of conventional ADMM. cloud agent itself.Such virtual sensor network distributedly
0 and cooperatively estimates the location of the person us-
1. 1 Introduction ingdistributedlinearconsensusalgorithms(seeforexample
0 [20, 11]), and continuously sends an update of the location
5 of the person to the cloud agent, which in turn sends it to
As the Internet of Things (IoT) emerges, a rapid growth of
1
the cloud client.
: the number of sensor-equipped things (e.g., smart-phones,
v In this paper, we develop cloud-assisted remote sensing
tablets, etc.) is observed where it became important to re-
i algorithms that enable distributed consensus estimation of
X think the way conventionalsensor-basedmanagement tech-
unknown parameters. Specifically, we propose:
r niques are designed. Although sensor-based distributed
a
sensing has already been investigated in the past, cloud-
An efficient network virtualization algorithm that can
assisted remote sensing is a new paradigm that capitalizes •
searchandselectsensorsfromthe swarmtoformavir-
on the capabilities of IoT to enable what is called Sensing
tual sensor network that can perform a sensing task.
as a Service [1]. Cloud-assisted remote sensing based dis-
Thealgorithmconsistsofselectingasetofsensorsthat
tributed parameter estimation is one example of such ser-
are willing to participate in the requested remote sens-
vices.
ing task, and finding optimal one-to-one mappings be-
Traditional sensor networks depend primarily on the so-
tween virtual and participatory sensors.
phistication andaccuracyof the sensorydevices themselves
toperformsensingtasksandmeetqualityofservicerequire- Anefficientestimationalgorithmthatreliesonthe vir-
ments. In the Sensing as a Service model, cloud-based sen- • tual sensor network, formed by our proposed virtual-
sornetworksrelymainlyonswarmsofparticipatorysensors ization algorithm, to estimate a set of unknown pa-
to perform remote sensing tasks. Unlike traditional sen- rameters in a distributed way and without requiring
sory devices, participatory sensors, though come with new synchronizationamong participatory sensors.
opportunities, present key challenges, mainly pertaining to
their sporadic availability and unpredictable mobility. Ourresults show that givena virtualsensingtask requir-
In the cloud-based remote sensing paradigm, a cloud ing g sensors and a swarm of n sensors where a pair of
agent(ormanager)isresponsibleforreceivingandhandling sensors is considered connected if the sensors are within a
1
distance r from one another, our virtualization algorithm mum allowed processing power, maximum allowed memory
finds the capabilities of the sensors in O(r−1logn) with capacity, etc.).
an average number of messages per sensor of Θ(1). Our Weassumethateachsensori S iscapableofestimating
results also show that our proposed algorithms achieve a a vector of unknown parameter∈s, θ RN, through noisy
virtualization benefit that is very close to an upper bound measurements, x RM. That is, ∈
i
∈
in O(max r−1nlogn,g3 ) time with a O(n) worst case av-
{ } x =H θ+u , i=1,...,n
erage number of messages per sensor. Finally, our results i i i
show that our proposed distributed parameter estimation where H RM×N is sensor i’s sensing model (typically
i
∈
algorithm has a linear convergence time and incurs com- known to i only) relating x to θ , and u is an additive
i i i
munication overhead that is at least an order of magnitude Gaussian noise with zero mean and variance σ2. We as-
i
lesser than that incurred by the conventional Alternating sume that u and u are independent from one another for
i j
Direction Method of Multipliers (ADMM). all i,j S. Because different sensors may have different
∈
In our design, the cloud agent does not need to have full sensing models and/or different measurement methods, it
knowledge ofthe underlying swarmof sensors,including its is very likely that different sensors have different estimates
topology, sensor capabilities, and sensor availability. We of θ. Also, we do not assume/require that the sensors are
only assume that the cloud agent has a direct communica- synchronized; that is, the consensus algorithms we develop
tionwithsome,butnotnecessarilyall,sensors. Eachsensor in this paper to estimate θ are asynchronous.
has to exchange small sized messages with only its direct
neighbors to diffuse information acrossthe whole or part of 2.2 Virtual Remote Sensing
the swarm. We also assume that the swarm is dynamic by
nature in that sensors’ availability, locations, connectivity, We assume that there exists a cloud manager/agent that
and capacities may change sporadically and unpredictably is responsible for managing the participatory sensors, han-
over time. dlingvirtualsensingtaskrequests(tobesubmittedbycloud
This paper is organized as follows: In Section 2, we clients), and ensuring that clients’ Service Level Agree-
overviewour algorithmicdesign, describe the requirements, ments (SLAs) are met once their requests are granted by
and state the related work. In Section 3, we propose and the cloud. Each virtual sensing task request is represented
presentoursensornetworkvirtualizationalgorithm. InSec- byaquadruple, g,θ,c,δ ,wheregisthenumberof(virtual)
tion 4, we propose and present our distributed parameter sensors requestehd to periform the sensing task, θ RN is a
∈
estimationalgorithm. Finally,weevaluateandcomparethe column vector of unknown parameters to estimate by the
performanceofourproposedalgorithmsnumericallyinSec- virtualsensors,andthe locationc and radiusδ indicate the
tion 5 and conclude the paper in Section 6. area of interest that needs to be sensed; i.e., all requested
sensorsmustbelocatedwithinδ distancefromthe centerc.
Generally speaking, SLAs consist of: (i) a maximum time
2 Framework Overview: System
within which the sensing task must be completed, (ii) an
Model, Problem Formulation, absolute tolerance ǫabs >0 of the estimation quality, (iii) a
relative tolerance ǫ > 0 of the estimation quality (maxi-
rel
and Related Work
mum gap between the g sensors’ local estimates of θ), and
(iv) a maximum rejection rate, defined as the ratio of the
In this section, we provide a brief overview of the different numberoffailedvirtualizationstothetotalnumberofsens-
frameworkcomponentsandalgorithmsthatareproposedin ing task requests.
this paper to perform cloud-assisted remote sensing based Upon receiving a sensing task request, the cloud agent’s
parameter estimation. We also talk about some previously job is then to define a set V of g virtual sensors to be real-
done works that are related to the algorithms proposed in izedbyg connectedparticipatorysensors,alllocatedwithin
this paper. distance δ from the center c, that can collaboratively and
distributively estimate θ. Depending on the SLAs of the
2.1 Participatory Sensors sensing task, the cloud agent needs to determine each of
the following three parameters before proceeding with sen-
We consider a swarm of a large number of participatory sor network virtualization needed to perform a requested
sensors that are managed by a cloud platform. Each sen- sensing task. First, it needs to choose a suitable virtual
sor of the swarm is cloud accessible via the Internet and topologythat connectsthe setofvirtualsensors,V,so that
is assumed to have some sensing capability. We model the theycanperformthesensingtaskcollaboratively. Although
swarmasanEuclideangeometricrandomgraph =(S,L), othertopologiescanbeused,wefocusinthispaperonthree
G
where S is a set of n sensors, and L is the set of all links types: complete, cyclic, and star. For a given topology, let
connecting the sensors,where two sensorsare consideredto E denote the set of virtuallinks connecting the virtual sen-
be connected if they are within a transmission radius, r, of sors and Υ = (V,E) be the graph representing the virtual
eachother. Letloc(i)denote the physicallocationofsensor sensor network. Note that the cloud agent needs to make
i, and C(i) denote its sensing capability or capacity (C(i) sure that the virtual sensor network is connected according
can for e.g. refer to maximum allowed sensing time, maxi- to the chosen topology.
2
Another parameter the cloud agent needs to fix and as- of the topology. Our proposed technique for performing
sociate with Υ is the maximum allowed path length, h¯, be- such aGtask is presented in Section 3.
tween any pair of virtual sensors. h¯ can be viewed as a
Related work. In a recent work, Perera et. al [18, 17] de-
waytolimitthenumberofsensors/hopsmessageexchanges
scribed a system of context-aware sensor search to address
amongvirtualsensorscangothrough. Itisaparameterthat
the research challenges of searching for sensors when large
can be used to impose an upper bound on end-to-end mes-
numbers of sensors with overlapping and redundant func-
sage delays, and thus, on the sensing task completion time.
tionality are available to the cloud. Such a sensor search
Note that a virtual link between two virtual sensors may
approach suffers from practical limitations as it relies on
be realized by more than two physical sensors, and some
centralized knowledge of all available sensors and requires
of these sensors may not necessarily realize a virtual sensor
continuous tracking of the sensors’ dynamics, such as the
in V by itself. That is, some sensors may only be used for
sensors’ availability, connectivity, and mobility.
forwarding traffic without participating in performing the
Sensorsearchalgorithmsneedtobesimple,havebounded
sensing task.
searchlatency,andincurminimumcommunicationoverhead
The thirdparameterthe cloudagentneeds to chooseand
betweenthe cloudplatformandthe largenumber ofpartic-
setisthesensingcapacitythresholdR(j)foravirtualsensor
ipatory sensors. Unlike centralized resource discovery al-
j V. The capacity C(i) of a participatory sensor i which
∈ gorithms [3], gossip-based search protocols are distributed
realizes j must then exceeds the capacity threshold R(j).
andtopology-independent, whichare more suitable for sen-
This threshold canfor e.g. representsthe minimum storage
sor search in the IoT context. Gossip protocols are origi-
capacity, the minimum CPU computing power, and/or the
nally designed for information dissemination [4], and have
minimum amount of time required by the sensing task.
been demonstrated to be effective in resource discovery in
The required values of the parameters associated with Υ
including E, R(j), and ¯h will be mainly determined by the Peer-to-Peer (P2P) networks [9]. Our proposed framework
relies on gossip techniques to determine the set of sensors
client’s SLA, and is beyond the scope of this paper. In this
amongallparticipatorysensorsthatarecapableofperform-
paper,wefocus insteadonthe designofefficientalgorithms
ing sensing and are willing to participate in the formation
thatmeetthesedesignrequirements. Ourultimateobjective
of a virtual sensor network.
in this work is to design a distributed consensus algorithm
that enables the estimation of the unknown parameter vec-
tor, θ, subject to the design parameter requirements speci-
fied by both the cloud client and the cloud agent. To this
2.2.2 Sensor Network Virtualization
end, the key tasks that need to be executed by the swarm
of participatory sensors to perform estimation are:
This virtualization task consists of finding (i) a set A S′
⊂
ofexactly g connectedsensorsselected amongall sensorsin
2.2.1 Sensor Search
S′ (the g selected sensors should be connected according to
It consists ofsearching for the sensors,among all participa- the virtual topology chosen by the cloud agent) and (ii) a
tory sensors, that meet the Υ requirements. More specifi- set (i,j) A V :j (i) of one-to-onemapped
A
M ⊂{ ∈ × ∈D }
cally, the swarm searches for a subset of participatory sen- pairs (each participatory sensor in A is mapped to one and
sors, S′ S, such that a sensor i S′ if: onlyonevirtualsensorinV)suchthatthelength,h(i,i′),of
⊂ ∈
any simple path connecting two distinct participatory sen-
i) itcansenseandestimateθthroughasensingmodelH ;
i sors i,i′ in A mapping a pair of directly connected virtual
i.e., it can observea vectorxi that can be expressedas sensors (j,j′) E is less than or equal to h¯. We refer to
x =H θ+u , ∈
i i i a possible A, pair as a feasible virtualization of the
A
{ M }
ii) it is geographically located within δ distance from c, requestedvirtualsensornetworkΥ. Notethatforanypossi-
and ble set A, there could exit multiple possible sets, A, each
M
can form a feasible virtulization when paired with A, and
iii) its capacity C(i) R(j) for at least one virtual sensor theobjectiveofasensornetworkvirtualizationalgorithmis
≥
j V. thentofindthe’optimal’feasiblevirtualization, A, ∗.
∈ { MA}
For each participatory sensor i, we define the virtual do- We now define and introduce what an ’optimal’ feasi-
main of i, (i), as the set of all virtual sensors that can be ble virtualization means. We consider that the cost (to
D
supported by sensor i; that is, the cloud) of virtualizing a virtual sensor network request,
Υ = (V,E), is determined by the amounts of requested
j V :C(i) R(j) if loc(i) c δ resources and given by Cost(Υ) = αV + β E , where α
(i)= { ∈ ≥ } k − k≤ (1) | | | |
D ( otherwise, denotes an incentive paid by the cloud to each participa-
∅ torysensor,andβ denotesanincentiveassociatedwitheach
and the objective of this task is to construct, with the min- physical path between each pair of participatory sensors in
imum possible communication overhead,eachparticipatory A. Anincentivecouldbemonetaryorcouldbeinanyother
sensor i’s virtual domain, (i), and to determine the set form (e.g., credits, services, etc.). On the other hand, the
D
S′ asfast aspossible, allwithout assuming priorknowledge totalbenefit (to the cloud/swarmof sensors)resultingfrom
3
a feasible virtulization, A, , can be expressed as measurementvector,x ,anditssensingmodel,H ,andthen
A i i
{ M }
solves the following Least Squares (LS) problem
C(i) R(j) h¯ h(i,i′)
Benefit= α − + β − , minimize 1 x Hθˆ 2 (3)
C(i) ¯h 2k − k
(i,jX)∈MA (i,Xi′)∈P (2) where θˆ is here the optimization variable. The unbiased
whereh(i,i′)isagainthepathlength(innumberofhops)of maximum-likelihood (ML) estimate of θ is simply θˆLS =
the path connecting the sensors of the pair (i,i′) mapping HTH −1HTx.
the virtual link between j and j′ and P = (i,i′) A This centralized LS approach, though simple, requires
{ ∈ × (cid:0) (cid:1)
A : (i,j),(i′,j′) ,(j,j′) E denotes the set of all thateachvirtualizedsensorexchangesitsmeasurementvec-
A
∈ M ∈ }
such pairs. Note that for a given sensing task request (i.e., tor and its sensing model with the cloud agent, which can
for a given virtualization cost), the lesser the used physical create significant communication overhead,especially when
resources, the higher the total benefit. The sensor network thenumberofmeasurements,M,andthenumberofvirtual
virtualization algorithm that we propose consists then of sensors, g, are large.
finding a feasible virtualization that maximizes the total We instead propose, in this paper, a decentralized ap-
benefit given in Eq. (2). We refer to the optimal solution proach that relies on the virtual sensor network to provide
as A, ∗. Clearly,finding A, ∗ is a hardproblem anestimationof the parametervectorθ. We rely onthe re-
A A
{ M } { M }
due to the factorial size of the solution space (in n). Our cent results presented in [21] to develop our distributed es-
firstcontributioninthisworkistodevelopanalgorithmthat timationalgorithm,whichreducescommunicationoverhead
solves this virtualizationproblem efficiently. The algorithm significantly when compared to the conventional ADMM
is presented in Section 3. approach [16] in addition to not requiring synchronization
Related work. Network virtualization techniques pro- amongsensors. TheproposedalgorithmispresentedinSec-
posed in the past decade consist mainly of virtual network tion 4.
embedding algorithms, which instantiate virtual networks Related work. Distributed parameter estimation ap-
on substrate infrastructures [10, 3]. Most of these virtual proacheshavebeenproposedin[16,20,11]. Estimationcan
networkembeddingalgorithmsarecentralized(e.g. [7])due fore.g. becarriedoutbyfirstcomputingalocalestimateat
totheeaseofdeploymentofcentralizedapproachesincloud eachvirtualsensorandthenperformadistributedweighted
platforms where the cloud providerdesires to have full con- averageof the localestimates [20]. This approachresults in
trol on the physical network resources. anMLestimate,butdoesnotlimit/boundthevariationbe-
Distributedvirtualnetworkembeddingandvirtualization tweenmean square errorsof localestimates. More recently,
algorithmshavealsobeenproposedinliterature[12,2]. One Paul et al. [16] propose a distributed estimation algorithm
ofthe limitation ofthe algorithmproposedin [12]lies inits basedon ADMM. Although this approachresults in an op-
unsuitability for swarm virtualization, as the authors as- timal mean square error when compared to LS, it exhibits
sume unlimited physical resources and consider an offline a significant in-network communication overhead that re-
resource virtualization approach. As for the more recent quires even more messages to be exchanged among sensors
work proposed in [2], although the virtualization phase of than that exchanged in the centralized LS. One approach
the physicalnetworkdoesnotrequirefull/globalknowledge also proposed in [16] to overcome this problem is to ap-
of the swarm, the cloud must initially partition the swarm proximate the computation of primal and dual variables at
intohierarchiesanddelegateeachvirtualnetworkrequestto eachstep of the algorithm by using earlier versions of these
a different hierarchy,which also requiresfull knowledgebut variables instead of sharing them at each iteration which
about the sensors in each hierarchy. Unlike this approach, marginally reduces the communication overhead. In addi-
our proposed virtualization algorithm is fully distributed. tiontotheincreasedcommunicationoverhead,conventional
In addition, it does not require any synchronization among ADMMrequiressynchronousoperationofthesensors. This
sensors. isverychallengingfromapracticalviewpoint,anddoesnot
scalewellespeciallywhenappliedintheIoTcontext. Ithas
been shown recently that an asynchronous implementation
2.2.3 Distributed Consensus Estimation
of ADMM has O(1/k) convergence [21].
Notonlyisourproposedestimationalgorithmbothasyn-
Thistaskreliesonthevirtualsensornetwork(formedinthe
chronous and distributed, but also reduces communication
previous step) to provide an estimate of θ that is at most
overhead significantly when compared to the conventional
ǫ from the optimal, and that all the virtualized sensors
abs
ADMM approach [16].
consent to the same estimate value of θ with a tolerance of
ǫ .
rel
Without loss of generality, consider indexing the selected 3 Sensor Network Virtualization
g sensorsinthe virtualsensornetworkas1...g andletx=
T T T T T T T T T
x ,...,x , H = H ,...,H , and u= u ,...,u . We begin by presenting our proposed Randomized and
1 g 1 g 1 g
The aggregate measurements can then be written as x = Asynchronous Distributed Virtualization (RADV) algo-
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
Hθ +u. One simple approach of estimating θ is to have rithm, which consists of four phases: (I) searching for sen-
the cloud agent first collects from each virtual sensor i its sorsthat can supporta virtualsensing task request Υ,(II)
4
pruning of virtual domains (i) for all i S, (III) con- in Eq. (4), s contacts only one of its neighbors s′ at time
D ∈
struction of benefit matrices in a distributed manner, and k. Then, for all (i) D : i = s′, s pushes (i) to s′
s
(IV) solving assignment problems at virtual sensors. This only if s′ did notDrecei∈ve (i) bef6ore and h(i,s) <Dh¯. Also,
D
approach results in multiple solutions each evaluated by a for all (i) Ds′ : i = s, s pulls (i) from s′ only if s did
different sensor, and the cloud agent selects the solution not recDeive∈ (i) befor6 e and h(i,sD′) < h¯. If no information
D
with the maximum benefit. is exchangedbetween s and s′ at time k, s stops contacting
We design time-invariant gossip based algorithms for the any of its neighbors. However, s may restart contacting its
firstthreephasesinwhichsensorsexchangeinformationran- neighbors again if it updated D after time k+1.
s
domly [19, 4]. At the k-th time slot, let sensor i be active Whens constructsits D ,it startsbypruning (s). The
s
D
andcontactitsneighborsensorj (i.e.,(i,j) L)withprob- pruning is performed by deleting a virtual sensor j (s)
∈ ∈ D
ability T >0 only if j canbe contactedby morethan one (i.e., (s) (s) j ) if none of the virtual sensors that
i,j
D ←D \{ }
of its neighbors. The probability T denotes the probabil- are connected to j, j′ V : (j,j′) E , is not included
i,i
{ ∈ ∈ }
ity that i does not contact any other sensor. Let the n n in any received (i), i.e. j / (i) : (i) D . This prun-
s
× D ∈ D D ∈
matrix T = [T ] be a doubly stochastic transition matrix ing rule ensures that the virtualized sensors maintain the
i,j
of non-negative entries [5]. A natural choice of T is required topology E and the constructed benefit matrices
i,j
shall result in a feasible virtualization.
1
, if i=j or (i,j) L,
Ti,j =di+1 ∈ (4) Phase III—Construction of Benefit Matrices
0, otherwise,
As mentioned earlier, finding a feasible virtualization,
where di =|{j ∈S :(i,j)∈L}| is the degree of sensor i. A, A ∗,thatmaximizesthetotalbenefitgiveninEq.(2)
Wenowpresenteachofthefourphasesingreaterdetails. { M }
is a hard problem due to the large size of the solution
space. Therefore, this phase proposes an efficient way of
Phase I—Sensor Search
solvingthis virtualizationproblem. Specifically, wepropose
Theobjectiveofthisphaseistoconstruct (s)foralls S a method that solves this problem in a distributed manner
andS′ asfastaspossiblewithoutassumingDpriorknowle∈dge and without requiring any synchronization among sensors,
of the topology. First, the cloud initiates the search at as described next.
time kG= 0 by sending Υ to one or more arbitrary sensors. Duringthisphase,eachsensorslocallyconstructsitsown
Atanylatertimek,sanditsneighbors′ exchangeinforma- set,A(s),ofgsensorsthatschoosesasvirtualizedsensorsto
tion as follows. s pushes Υ to s′ only if s′ does not have Υ, assignto virtual sensorsin V. Eachsensors also maintains
orpulls itfroms′ onlyif s does nothaveΥ. If s contactss′ g row vectors, B(s) R1×gand i A(s), that we define
i ∈ ∈
andbothsands′ havereceivedΥbefore,sstopscontacting as the benefit vector of sensor i seen by s, where the j-th
any other sensor. Upon receipt of Υ, a sensor s constructs element,B(s),denotesthebenefitofassigningparticipatory
i,j
(s) according to Eq.(1) and S′ as sensori A(s) tothe virtualsensorj V as seenby s,and
D ∈ ∈
is given by
S′ = s S : (s) >0 .
{ ∈ |D | }
A virtual domain of sensor i, (i), evaluated during the C(i) R(j) h¯ h(j,s)
sensorsearchphaseisnotsufficienDttotellwhetherafeasible B(s) = α C−(i) +β − ¯h if j ∈D(i),
i,j 
embedding can be found if sensor i virtualizes (is mapped 0 otherwise.
to) j. For example, the virtualization in which sensor i
virtualizes j and sensor i′ virtualizes j′ where there is a Our objective is then to construct, for each s S, the
virtual link (j,j′) is not feasible if i′ is not reachable from benefit matrix B(s) =[B(s) ] as fast as possible,∈and find
i∈A(s)
i and vice versa. When this situation occurs, we say that afeasiblevirtualization, A, ,thatmaximizesthetotal
A
sensoriisincapableofsupportingthetopologyrequirement benefit, { M }
specified by E, initiating thus a virtual domain pruning. B(s),
i,j
(i,jX)∈MA
Phase II—Virtual Domain Pruning
among all s S without knowing the structure. More-
∈ G
During this phase, we ensure that all virtualized sensors over,the pathlengthbetweenasensors andanyother sen-
maintain the topology E by allowing a sensor to receive sor i that s includes in its benefit matrix must not exceed
the virtual domains of other sensors and delete a virtual ¯h. Finally, a sensor s shall include only the benefit vectors
sensor j from its domain if there exists a virtual link (j,j′) of the g sensors with the largest possible benefit.
such that j′ is not included in any other received domains. Each sensor s initially sets A(s) =A(s) s if (s) / ,
∪{ } D ∈∅
Let Ds (i) : i S denotes the set of domains that sets h(i,s)=0for alli S, and sets
⊂ {D ∈ } ∈
sensor s has at time k. Initially D = (s) and h(i,s)=
s
{D }
0for alli S1. Usingthesametransitionmatrix,T,defined C(s) R(j)
∈ α − +β, j (s),
1Knowledge about other sensors existence is not needed, and h is Bs(s,j) = C(s) ∈D
typicallyevaluated dynamically. 0, otherwise.

5
Also,smaintainsascalar,bmin,definedastheminimumto- ensurethatthe lengthofthe paths betweeni ands andbe-
s
talbenefitithasreceivedfromanyothersensorandwritten tween i′ and s are the shortest possible ones to ensure that
as the length of the path between i and i′ is also the short-
bmin =min B(s), est, as in this case, s, i, and i′ reside in the same clique
s i,j
i
j∈V with high probability. We evaluate the effectiveness of this
X
relaxation in Section 5 and show that our virtualization al-
and the corresponding sensor,
gorithm performs well even when the condition g n does
≪
imin =argmin B(s). not hold.
s i,j
i
j∈V
X
Phase IV—Solving Local Assignment Problem
Initially, bmin =0 and remains so until A(s) =g.
s | | After reception of the g benefit vectors, s proceeds to this
Using the same transition matrix, T, defined in Eq. (4),
phase of the algorithm only if it stops communicating and
s contacts its neighbor s′ only once at each time k. Then,
A(s) = g. Each sensor s S with A(s) =g solves locally
for alli A(s) : i = s′, s pushes the benefit vector B(s) to | | ∈ | |
s′ only i∈f h(i,s)<6h¯ and i the following assignment problem:
maximize B(s)m
i,j ij
Bi(,sj)− βh¯ >bms′in. subject to i∈PA(s) mj∈PiDj(=i) 1, i A(s),
jX∈V (cid:18) (cid:19) j∈D(i) ∈ (5)
P m =1, j V,
Also,for alli A(s′) : i=s,spullsthebenefitvectorB(s′) {i:j∈D(i)} ij ∈
from s′ only i∈f h(i,s′)<6h¯ and i mijP 0,1 ,
∈{ }
where m are binary optimization variables indicating
B(s′) β >bmin. whether tihje participatorysensori is assignedto the virtual
i,j − h¯ s
j∈V (cid:18) (cid:19) sensorj. Theproblemformulatedin(5)isequivalenttothe
X
maximum weight matching perfect problem in a bipartite
If no information is exchanged between s and s′ at time k, graph, and hence, we propose to use the classical Hungar-
s stops contacting its neighbors at time k+1. However, s ian method to solve it (the worst case time complexity is
mayrestartcontactingitsneighborsagainifB(s) isupdated O(g3) [13, 14]).
after time k+1. We can also tolerate an error ǫ > 0 of the resulting total
When s receives Bi(s′), s updates Bi(,sj) as benefit and relax the restriction of finding a perfect match-
ingforlargeg. Thisrelaxationisreasonablewhenthereare
β
B(s) if j (i), enough sensors involved in solving these local optimization
Bi(,sj) = i,j − h¯ ∈D problems, as in this case we can pick the best solution and
0 otherwise. discard those without a perfect matching. In such a sce-
nario, we can also use a linear time (1 ǫ)-approximation
If i / A(s), then we have two scenarios. In the first sce- algorithmtosolve(5)[8]. Inthispaper,−weusetheHungar-
∈
nario, s still has not received g benefit vectors, so bmsin = 0 ian method to solve our formulated optimization problems.
and A(s) <g,thensupdatesitssetofcandidatesensorsas Detailsofthealgorithmareomittedduetospacelimitation;
| |
A(s) =A(s) i . In the other scenario in which A(s) =g, readers are referred to [13, 14, 8] for detailed information.
∪{ } | |
s replaces the sensor corresponding to the minimum total Eachsensorsolveslocallytheoptimizationproblemgiven
benefit, imsin, with i so that A(s) = A(s) \ {imsin} ∪ {i}. in (5) and sends its obtained solution to the cloud agent.
On the other hand, if i A(s), then s updates B(s) if This is done asynchronously. The cloud agent then selects
∈ i,j
B(s′) > B(s). Finally, s updates bmin, imin, and the solution that leads to the maximum total benefit, and
i,j i,j s s
keeps all other solutions for later use in the event that the
j∈V j∈V
hP(i,s) as h(i,sP)=h(i,s′)+1. networkdynamicsinvalidatetheselectedsolutionbeforethe
Finding a feasible virtualization that maximizes the ben- virtual sensing task completes.
efitB(s) =[ B(s) ]insteadofthebenefitgiveninEq.(2) Complexity and message overhead. The time required
i∈A(s)
makes the problem easier because every sensor has a differ- tospreadΥacrossthe networkisO(r−1logn)[19]. Ittakes
entvalueforthebenefitB thatdependsonlyonthelength O(g) worst case time to evaluate (i) locally at sensor i.
i,j
D
of the physical path between i and s instead of the path Also, the time required to spread information in the prun-
lengthsofallpossiblecombinationsofsensorpairs(i,i′)that ing and benefit construction phases is O(r−1nlogn). The
can virtualize a virtual link. Intuitively, this relaxationstill pruning of the virtual domain (i) requires node i to ex-
D
leads to an optimal or near optimal virtualization, because amine g received virtual domains, each having at most g
for a connected swarm , the number of sensors that are entries. The worst case local running time of pruning is
G
directly connected by a single physical link (clique) grows then O(g2). Finally, the local running time of the Hungar-
logarithmically in n and hence this number is larger than g ian method is O(g3). Hence, the overall complexity of is
almost surely as g n. In such a case, it is sufficient to O(max r−1nlogn,g3 ).
≪ { }
6
The average number of messages communicated per sen- where θ ,i A are the optimization variables.
i
{ ∈ }
sorduringthesensorsearchphaseisΘ(1)andeachmessage By introducing an auxiliary variable, z, we decouple the
isO(g)insize. Duringpruningofvirtualdomains,sinceev- constraints in (6), so that θ z = 0 for all i A [15].
i
− ∈
ery sensor exchanges a maximum of n domains each of size However, this requires that z be shared among all g sen-
thatisalsoO(g),theaveragenumberofmessagescommuni- sors. Instead, we introduce g auxiliary variables, z , and
i
catedper sensoris O(n). However,because we restrictthat equivalently write the optimization problem as
messages be communicated up to h¯ hops for only a group
minimize 1 x H θ 2
of sensors that support the requirements of Υ, the average 2i∈Ak i− i ik (7)
number of messages per sensor is typically small. Figure 1
subject to θjPzi =0for all(i,j) P.
shows the total time and the average number of messages − ∈
per sensor required during both the domain pruning and Let λ = λ RN×1 : (i,j) P and ρ = ρ R :
i,j i,j
the benefit construction phases. The total time growth is (i,j) P{ den∈ote respectively∈the }set of Lagra{ngian∈mul-
linearithmic in n when Υ is sent to exactly one sensor and tiplier∈s an}d the set of penalty parameters. The augmented
when is connected. This time can, in practice, be de- Lagrangianis
G
creasedsignificantlyifΥisinitiallysenttomultiplesensors.
Additionally, the average number of messages per sensor is
(θ,z,λ)= 1 x H θ 2
shownto scale linearly with n, and is typically a very small Lρ i∈A "2k i− i ik
fraction of n. P λT (θ z )
− i,j i− j (8)
j∈A:(i,j)∈P
24 2.4 P
23 2.2 +j∈A:(i,j)∈P ρi2,jkθi−zjk2#.
22 2 es P
g
21 1.8 essa Bysettingthegradientw.r.tθiofEq.(8)tozeroandsolving
Time 20 MessTagimees 11..46 No. of M for θi, we get
19 1.2 age −1
18 1 Aver θi = HiTHi+ ρi,jI
j∈A:(i,j)∈P !
17 0.8 P
T
16 400 800 1200 1600 200 00.6 . Hi xi+ (λi,j +ρi,jzj) .
j∈A:(i,j)∈P !
n
P
Figure1: Time(innumberofiterations)andmessageoverhead(innumber Similarly, we solve for zi by setting the gradient w.r.t to
ofcommunicatedmessages)resultingfromconstructingthebenefitmatri- z to zero and rearranging the indices of the Lagrangian
ces. i
multipliers and the penalty parameters. It follows that
1 1
4 Distributed Estimation of θ zi = θj λj,i .
g − ρ
j∈AX:(i,j)∈P(cid:18) j,i (cid:19)
After completing the sensor virtualization task, using the
The former analysis leads to the conventional ADMM-
proposed RADV algorithm that we described in Section 3,
based distributed consensus estimation algorithm given by
the virtual sensors run an in-network parameter estimation
algorithm to compute θˆ distributedly. In this section, we −1
present our proposed Randomized and Asynchronous Dis- θ(k+1) = HTH + ρ I
i i i i,j
tributed Estimation(RADE) algorithm. We firstfollowthe j∈A:(i,j)∈P !
P
standard ADMM approach to derive primal, dual and La-
. HTx + λ(k)+ρ z(k) ,
grangian variable update equations, then we describe the i i i,j i,j j (9)
j∈A:(i,j)∈P !
proposedRADE algorithm. For clarityofnotation,inwhat z(k+1) = 1 Pθ(k) (cid:16) 1 λ(k) , (cid:17)
follows,werefertothesetofgselectedparticipatorysensors, i g j − ρj,i j,i
j∈A:(i,j)∈P
determined by means of the proposed RADV algorithm, λ(k+1) =λ(k) Pρ θ(cid:16)(k+1) z(k+1) (cid:17),
simply as A. j,i j,i − j,i j − i
The centralized estimation approach given in (3) is first where the superscript k(cid:16)denotes the val(cid:17)ue of the variable
decomposed into g local estimates of θ (one θˆi for each i at the k-th iteration. This conventional ADMM algorithm,
∈
A) while constraining the local estimates with the coupling given in (9), requires synchronization and variable update
constraints θi = θjfor all(i,j) P. This results in the among the sensors [6, 22]. Moreover, at each iteration k,
∈
following optimization problem: each sensor i must send z(k) and θ(k) to all other sensors it
i i
is connected to, so as to evaluate their k+1 primal, dual,
minimize 1 x H θ 2
2i∈Ak i− i ik (6) and Lagrangian multipliers. When M is small, this algo-
subject to θiPθj =0 for all (i,j) P, rithm incurs communication overhead that can be shown
− ∈
7
to be worse than the communication overhead incurred by 1
centralizedestimationmethods. However,when M is large,
the conventional ADMM algorithm incurs lesser communi-
cation overhead than what centralized estimation methods
incur, but it still remains practically unattractive due to
E
other weaknesses, detailed later in Section 5. S 0.1
M
Given the absolute and relative tolerances, ǫ and ǫ ,
abs rel
specifiedbytheSLAsbetweenthecloudclientandthecloud
LS
agent, we define the primal and dual tolerances,controlling ADMM-Complete
RADE-Complete
the convergence of the algorithm at iteration k, as RADE-Cyclic
RADE-Star
0.01
-10 -8 -6 -4 -2 0 2 4
ǫpri(k)=√gǫ +ǫ max( θ(k) , z(k) ), Noise Power (dB)
i abs rel k i k k− i k
Figure2: MSEofRADEcomparedtothoseachievedunderADMMandLS
and atdifferentnoisepowerandfordifferentvirtualsensornetworktopologies.
ǫdiual(k)=√gǫabs+ǫrel kρj,iλj,ik.
j∈A variables using the most recent values they received from
X
other sensors.
The tolerances, ǫpri and ǫdual, define the stopping criteria
i i Mean square error and convergence. The asyn-
of sensor i; i.e., sensor i stops updating θ and z when
i i
chronous and randomization design features of RADE do
θ(k+1) z(k+1) <ǫpri(k), (10) not impact the Mean Square Error (MSE) achieved by
k i − i k i RADE whencomparedto ADMM. This is explainedasfol-
and lows. In both ADMM and RADE, the number of necessary
z(k+1) z(k) <ǫdual(k). (11) dual and primal variables updates required until conver-
k i − i k i gence remains unchanged, so that convergence to the same
The stopping criteria of RADE are different from those of
estimate is guaranteed in both algorithms. Figure 2 shows
the conventional ADMM. Unlike the conventional ADMM
the MSE achievable under both RADE and ADMM when
where all sensors shall stop computations all at the same
comparedto LS under each of the three studied sensor net-
time using a common stopping criteria and common pri-
work topologies: complete, star, and cycle. These results
malanddualtolerances,the stopping criteria(10) and(11)
showthe optimalityofRADE thatwe intuitivelydiscussed.
of RADE allow a sensor i to stop its computations asyn-
All approaches have the same accuracy. But of course each
chronouslyandindependentlyfromothersensors. However,
of them does so at a different performance cost, as will be
these criteria are not enough to ensure asynchronous im-
discussed later.
plementation, as synchronization is still required for dual
On the other hand, RADE exhibits a linear convergence
and primal variable updates at iteration k+1 due to their
rate (O(1/k)), similar to what the conventional ADMM
dependencies on k.
does. Figure 3 shows the number of time steps required
To ensure full asynchronous implementation, we use the
forbothRADEandADMMtoconvergeunderdifferentrel-
doubly stochastic transition matrix, T Rg×g, where T
∈ i,j ative tolerance parameters, ǫrel. RADE convergence tends
is the probability that a sensor i contacts another sensor
to be more restricted by the randomization nature of the
j at any iteration, for deciding the communications among
algorithm for smaller values of ǫ , which can be seen by
rel
sensors. We can have the increasing number of steps as g increases if ǫ =10−2.
rel
1 ADMM generally requires a lesser number of steps to con-
if i=j or (i,j) P,
Ti,j = d′i+1 ∈ verge by relaxing the consensus constraint (through reduc-
 ing ǫ ). However, as will be seen in the numerical results
0 otherwise, rel
section later, this increase in the number of convergence
where d′ = j A:(i,j) P is the degree of the virtual stepsisacceptablewhenconsideringtheamountofcommu-
i |{ ∈ ∈ }| nication overheadthat the algorithm saves.
sensor, in Υ, that i virtualizes. At iteration k +1, sensor
i may need to contact only one sensor j, unless both of
i’s stopping criteria, (10) and (11), are already satisfied.
5 Numerical Results
Whereassensorj canbecontactedbymorethanonesensor
if j is not contacting any other sensor, even when both of
Inthissection,weevaluatetheperformanceoftheproposed
j’s stopping criteria are satisfied.
Upon contacting j, sensor i pushes θ(k) to j only if i’s RADV and RADE algorithms through simulations. In our
i simulations, the swarm of sensors, , and the virtual sens-
primalstoppingconditionisnotsatisfiedandpushesz(k) to G
i ing task requests, Υ, are generated using the parameters
j only if i’s dual stopping condition is not satisfied. Also, i
summarized in Table 1. The virtual sensor network topol-
pulls θ(k) fromj only if j’s primalstopping conditionis not
j ogycaneither be complete,cyclic, orstar,with arandomly
satisfied and pulls z(k) only if j’s dual stopping condition chosen central location, c. We consider receiving and ser-
j
is not satisfied. Finally, both i and j update their k +1 vicing only one virtual sensing task request at a time. The
8
180 12
ADMM,10-4
160 ARDAMDME,, 1100--24 11
RADE, 10-2
140
10
120 ost
C
k 100 nefit - 9
80 Be 8
al
60 ot
T 7 Complete
40 Bound - Complete
Cyclic
20 6 Bound - CySctlaicr
Bound - Star
0 5
3 6 9 12 15 18 21 24 0 400 800 1200 1600 2000
g n
Figure 3: Number of time steps (k) needed until convergence of RADE Figure5: VirtualizationcostofRADVwhencomparedtotheupperbound
when comparedto ADMM for completetopology underdifferent relative underdifferenttopologies.
tolerancevaluesǫrel.
1000
1
RADV-Complete
0.9 RADV-Cyclic
RADV-Star
0.8
0.7 100
e
at 0.6
on R 0.5 k
ejecti 0.4 10
R ADMM-Complete
0.3 RADE-Complete
ADMM-Cyclic
0.2 RADE-Cyclic
ADMM-Star
0.1 RADE-Star
1
0 3 6 9 12 15 18 21 24
0 400 800 1200 1600 2000 g
n
Figure6: NumberoftimestepsuntilconvergenceofRADEwhencompared
Figure4: Rejectionrateencounteredatdifferentn. toADMMunderdifferenttopologies.
absoluteandrelativetolerances,ǫ andǫ ,aresetto10−4
abs rel In Figure 5, we evaluate the virtualization effectiveness
unless specified otherwise.
achieved by RADV under different virtual topologies. As
Table1: SimulationParameters
the swarm gets denser, RADV achieves a Total Benefit
−
Parameter r C(i) R(j) δ h¯ Costthatisveryclosetotheupperbound. Sincethelowest
possible virtualization cost is with star or cyclic topologies,
Value 0.1 U(50,100) U(25,50) 0.2 20 itisdesiredbythecloudtoalwaysarrangeeachvirtualsens-
∼ ∼
ing task in a star or a cyclic topology. On the other hand,
Figure 4 shows the rejection rate encountered with dif- convergenceandcommunicationoverheadofthedistributed
ferent Υ topologies and n values. As we only consider one estimation is also impacted by the cloud agent’s choice of
single request at a time, the results shown in this figure re- the virtual topology. This creates a design trade-off, as we
flect mainly the impact of the virtual sensornetwork topol- will see in the next two paragraphs.
ogy, the number of sensors n, and the simulations parame- Figure 6 shows the impact of the virtual topology choice
ters given in Table 1 on the rejection rate. The denser the on the convergence performance of RADE when compared
swarmofsensorsis,thelowertherejectionrateis,implying to ADMM. If g is small (three to eight), the impact of the
that the cloud is capable of granting higher number of re- virtual topology on convergence of RADE and ADMM is
quests. Asamarginalnote,astartopologyisslightlyeasier minimal. This is because the degree of parallelism(number
to virtualize than a complete or a cyclic topology. ofsensorsactiveatthe sametime)ismorerestrictedbythe
Onewayofassessingtheeffectivenessofthevirtualization small number of virtual sensors g. In such a scenario, it is
algorithm is by measuring the difference between the total convenient for the cloud agent to always arrange the vir-
virtualization benefit given in (2) and the cost associated tualsensorsina star topology. However,asg increases,the
with the sensor virtualization introduced in Section 2.2.2. impact of the virtual topology becomes significant as the
For a given number of virtual sensors, the cost is mainly degree of parallelism is higher in a complete topology, en-
determinedbythe choiceofthetopology(startopologyhas ablingRADEtoconvergemuchfasterasg getslarger. This
the lowestcostandcompletetopologyhasthe highestone). convergencebecomesslowerwithstarandcyclictopologies.
For a given topology, the total benefit is maximized when Thisisbecauseinstarandcyclictopologies,onlyfewsensors
each virtual sensor is assigned to the participatory sensor are active at a time, making RADE and ADMM converge
withthemaximumcapacityandeachvirtuallinkismapped in a number of steps comparable to that of the ADMM’s
to exactly one physical link. We refer to this maximized sequential implementation. In this later scenario, the cloud
benefit as the upper bound. agent shall arrange the virtual sensors as a complete topol-
9
1e+06 motesensing: Aninternetofeverythingenabler. Inter-
LS
ADMM-Complete net of Things Journal, IEEE,1(3):276–288,June2014.
RADE-Complete
ADMM-Cyclic
ges 100000 RAADDMEM-C-ySctlaicr [2] M. T. Beck, A. Fischer, H. de Meer, J. F. Botero, and
essa RADE-Star X. Hesselbach. A distributed, parallel,andgeneric vir-
M
o. of 10000 tual network embedding framework. In Communica-
N tions (ICC), 2013 IEEE International Conference on,
e
g
era pages 3471–3475.IEEE, 2013.
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[3] A. Belbekkouche, M. Hasan, A. Karmouch, et al. Re-
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Figure7: CommunicationoverheadwhencomparingRADEtoADMMand
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ogy unless the SLA permits slower convergence.
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Moreover, RADE converges in a higher number of steps
when compared to the conventional ADMM. This is be- [5] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah. Ran-
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at a time and variables are updated only between pairs of
stein. Distributed optimization and statistical learn-
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ing via the alternatingdirectionmethod ofmultipliers.
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2011.
significant savings in communication overhead.
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sagesexchangedduringestimationwhencomparingRADE, and J. Wang. Virtual network embedding through
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6 Conclusion and Discussion
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