DTIC ADA517880: On Maximizing the Lifetime of Delay-Sensitive Wireless Sensor Networks with Anycast

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On Maximizing the Lifetime of Delay-Sensitive
Wireless Sensor Networks with Anycast
Joohwan Kim , Xiaojun Lin , Ness B. Shroff , and Prasun Sinha
∗ † ‡ §
School of Electrical and Computer Engineering, Purdue University
∗†
Departments of ECE and CSE, The Ohio State University
‡§
Email: jhkim, linx @purdue.edu, [e-mail], [e-mail]
∗ † ‡ §
{ }
Abstract—Sleep-wake scheduling is an effective mechanism sensitive applications, such as fire detection or tsunami alarm,
to prolong the lifetime of energy-constrained wireless sensor which require that the event reporting delay be small.
networks. However, it incurs an additional delay for packet Prior work in the literature has proposed the use of anycast
delivery when each node needs to wait for its next-hop relay
to reduce this event reporting delay [6]–[10]. In contrast to
nodetowakeup,whichcouldbeunacceptablefordelay-sensitive
traditional sleep-wake scheduling, where each sending node
applications.Priorworkintheliteraturehasproposedtoreduce
this delay using anycast, where each node opportunistically wakes up a particular next-hop relay node, in anycast each
selects the first neighboring node that wakes up among multiple sending node tries to wake up a group of neighboring nodes
candidatenodes.Inthispaper,westudythejointcontrolproblem in a candidate set, and the sending node then picks the first
ofhowto optimally controlthesleep-wakeschedule, theanycast
nodethatwakesuptorelaypackets.Roughlyspeaking,ifeach
candidate set of next-hop neighbors, and anycast priorities, to
neighboring node wakes up once every T time, by selecting a
maximize the network lifetime subject to a constraint on the
expectedend-to-enddelay.Weprovideanefficientsolutiontothis candidatesetofnnodes,thetimeneededbeforethefirstnode
joint control problem. Our numerical results indicate that the wakes up is on average around T (assuming that the sleep-
n
proposed solution can substantially outperform prior heuristic wake schedules of the n nodes are independent). Thus, the
solutionsintheliterature,especiallyunderthepracticalscenarios
delay to wake up the next-hop neighbors can be significantly
wherethereareobstructionsinthecoverageareaofthewireless
reduced. On the other hand, the end-to-end delay not only
sensor network.
Index Terms—Anycast, Sleep-wake scheduling, Sensor net- depends on the per-hop delay, but also the end-to-end path
work, Energy-efficiency, Delay that packet traverses. Hence, the set of candidate nodes must
be carefully chosen because it will also affect the possible
I. INTRODUCTION routing paths.
The existing anycast schemes in the literature have mainly
Sleep-wakeschedulingisaneffectivemechanismtoprolong
focused on the so-called “MAC-layer anycast” problem, i.e.,
the lifetime of energy-constrained sensor networks. In this
they try to find the candidate set at each node such that some
paper, we are particularly interested in event-driven wireless
local measure of delay is minimized. For the routing path,
sensor networks, where events occur occasionally. Therefore,
theyeitheruseaseparateroutingalgorithm[8],[9],orrelyon
byputtingnodestosleepwhentherearenoevents,theenergy
geographical information [6], [7], [10]. Thus, the interactions
consumptionofthesensornodescanbesignificantlyreduced.
between the choice of the candidate set and the routing path
In the literature, synchronized sleep-wake scheduling pro-
was not systematically studied, and it is then unclear whether
tocols have been proposed in [1]–[3]. In these protocols,
suchapproacheswill minimizetheactualend-to-enddelay.In
sensor nodes periodically or aperiodically exchange synchro-
thispaper,wedirectlyoptimizethesystemwithrespecttothe
nization information with neighboring nodes. However, these
end-to-end delay. In particular, we formulate the joint control
synchronous protocols could incur additional communication
problem of how to optimally control the sleep-wake schedule,
overhead, and consume a considerable amount of energy.
the anycast candidate set of neighboring nodes, and anycast
In this work, we are interested in asynchronous sleep-wake
priorities among neighboring nodes, to maximize the network
scheduling protocols such as those proposed in [4], [5]. In
lifetime subject to a constraint on the end-to-end delay. We
these protocols, the sleep-wake schedule at each node is inde-
provide an efficient solution to this joint control problem, and
pendent of that of other nodes, and thus no synchronization is
as a part of solution, we also show how to optimally choose
required. However, due to the lack of knowledge of the sleep-
the candidate set in order to minimize the end-to-end delay
wake schedule of other nodes, it incurs additional delays for
for all nodes.
packetdeliverywheneachnodeneedstowaitforitsnext-hop
The rest of this paper is organized as follows. In Section
node to wake up. This delay could be unacceptable for delay-
II, we describe the system model and introduce the lifetime-
maximization problem that we intend to solve. In Section
ThisworkhasbeenpartiallysupportedbytheNationalScienceFoundation
III, we analyze the end-to-end delay under anycast, and we
throughawardsCNS-0626703,CNS-0721477,CNS-0721434,CCF-0635202,
andAROMURIAwardNo.W911NF-07-10376(SA08-03). develop an optimal distributed anycast algorithm that mini-
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On Maximizing the Lifetime of Delay-Sensitive Wireless Sensor Networks
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mizes the end-to-end delay of all nodes. In Section IV, we assumed to maintain a list of nodes i that use node j as a
solve the lifetime-maximization problem. In Section V, we forwarder (i.e., j ). As shown in Fig. 1, node i starts
i
∈F
provide simulation results that illustrate the performance of sending a beacon signal and an ID signal, successively. All
ourproposedalgorithmcomparedtootherheuristicalgorithms nodesin hearthesesignalsregardlessofwhomthesesignals
i
C
in the literature. are intended for. A node j that wakes up during the beacon
signal or the ID signal will check if it is in the forwarding
II. SYSTEMMODEL set of node i. If it is, node j sends one acknowledgement
We consider a wireless sensor network with N nodes. Let after the ID signal ends. After each ID signal, node i checks
denotethesetofallnodesinthenetwork.Eachsensornode whether there is any acknowledgement from nodes in . If
i
N F
is in charge of both detecting events and relaying packets. If no acknowledgement is detected, node i repeats the beacon-
a node detects an event, the node packs the event information ID-signalingandacknowledgement-detectionprocessesuntilit
into a packet, and delivers the packet to a sink s via multi- hearsone.Ontheotherhand,ifthereisanacknowledgement,
hop relaying. We assume in this paper that there is a single it may take some additional time for node i to identify which
sink,however,theanalysiscanbegeneralizedtothecasewith node acknowledges the beacon-ID signals, especially when
multiple sinks. there are multiple nodes that wake up at the same time. We
We assume that the sensor network employs sleep-wake let t denote the resolution period, during which time node
R
scheduling to improve energy-efficiency and to prolong the i identifies which nodes have sent acknowledgements, and,
network lifetime. With sleep-wake scheduling, nodes sleep if there are multiple awake nodes, chooses one node among
for most of the time and occasionally wake up for a short them that will forward the packet. After the resolution period,
period of time t . When a node i has a packet for node the chosen node receives the packet from node i during the
active
j to relay, it will send a beacon signal followed by an ID packet transmission period t , and then starts the beacon-
P
signal (carrying sender information). Let t and t be the ID-signalingandacknowledgement-detectionprocessestofind
B C
duration of the beacon signal and the ID signal, respectively. thenextforwarder.Sincenodesconsumeenergywhenawake,
When node j wakes up and senses a beacon signal, it keeps t should be as small as possible. However, t has to
active active
awake, waiting for the following ID signal to recognize the belargerthant becauseotherwise aneighboringnodecould
A
sender. When node j wakes up in the middle of an ID signal, wake up after an ID signal and could turn to sleep before the
it keeps awake, waiting for the next ID signal. If node j next beacon signal. In this paper, we set t = t +t
active A C
successfullyrecognizesthesender,anditisthenext-hopnode so that the node that wakes up right before the first beacon
of node i, it then communicates with node i to receive the signalalsohasthesamechanceofdetectingthebeaconsignal
packet. If a node wakes up and does not sense any beacon as nodes that wake up between two beacon signals.
signal or any ID signal, it will then go back to sleep. In
A. Sleep-wake Schedule, Forwarding Set, and Priority
this paper, we assume that the time instants that a node j
wakes up follow a Poisson random process with rate λ . We In this model, there are three control variables that affect
j
also assume that the wake-up processes of different nodes the network lifetime and the end-to-end delay experienced by
are independent. The independence assumption is suitable for a packet.
the scenario where nodes do not synchronize their wake- 1) Sleep-Wake Schedule: The sleep-wake schedule is
up times, which is easier to implement than other schemes determined by the rate λ of the Poisson process with which
j
that require global synchronization [1]–[3]. The advantage of each node j wakes up. If λ increases, the expected one-hop
j
Poisson sleep-wake scheduling is that, due to its memoryless delay will decrease, and so will the end-to-end delay of any
property,sensornodesareabletouseatime-invariantoptimal routing paths that pass though node j. However, it leads to
policy to maximize the network lifetime (see the discussion higher energy consumption at node j so that the network
in Section III-B). While the analysis in this paper focuses on lifetime may decrease. In the rest of the paper, it is more
thecasewhenthewake-uptimesfollowaPoissonprocess,we convenienttoworkwiththenotionofawakeprobabilitywhich
expectthatthemethodologyinthepapercanalsobeextended is a function of λ .
j
to the case with non-Poisson wake-up processes, with more Suppose that node i sends the first beacon signal at time 0,
technically-involved analysis. asinFig.1.Ifnonodesin haveheardthefirstm 1beacon
i
F −
A well-known problem of using sleep-wake scheduling in and ID signals, then node i transmits the m-th beacon and ID
sensor networks is the additional delay incurred when a node
has to wait for its next-hop node to wake up. To reduce this
delay, we use an anycast forwarding scheme described in Fig.
1. Let denote the set of nodes in the transmission range
i
C
of node i. Suppose that node i has a packet, and it needs
to pick up a node in to relay the packet. Each node i
i
C
maintains a list of nodes that node i intends to use as a
forwarder. We call the set of such nodes a forwarding set,
which is denoted by . In addition, each node j is also Fig.1. SystemModel
i
F
signalsinthetime-interval[(t +t +t )(m 1),(t +t + as a forwarder. Note that even though only the nodes in a
B C A B C
−
t )(m 1)+t +t ].Foraneighboringnodej tohearthem- forwarding set need priorities, we here assign priorities to
A B C
−
thsignalsandtorecognizethesender,itshouldwakeupduring all nodes to make priority matrix B an independent control
[(t +t +t )(m 1) t t ,(t +t +t )m t t ]. variable from forwarding matrix A. We let denote the set
B C A A C B C A A C
− − − − − B
Therefore,providedthatnodeiissendingthem-thsignals,the of all possible priority matrices.
probability that node j wakes up and hears these signals
i
∈C B. Performance Metrics
is pj =1 e−λj(tB+tC+tA). We call pj the awake probability
−
of node j. It should be noted that, due to the memoryless We next describe the performance metrics that we are
property of a Poisson random process, p is same for each interested in.
j
beacon-ID signaling iteration, m. 1) End-to-End Delay: We assume that the end-to-end
Note that there is a one-to-one mapping between the awake delay for event delivery is dominated by the cumulative sum
probability p and waking-up frequency λ . Hence, an awake of the delay for each hop to wake up and to relay a packet to
j j
probability is also closely related to both delay and energy its next-hop neighbor. This is a reasonable approximation in
consumption. Let "p=(p ,i ) represent the global awake many event-driven networks. Note that when an event occurs
i
∈N
probability vector. inanevent-drivensensornetwork,thefirstpacketgeneratedby
2) Forwarding Set: The forwarding set i is the set the event usually suffers most of the delay because at every
F
of candidate nodes chosen to forward a packet at node i. hop it has to wait for nodes to wake up so that it can be
In principle, the forwarding set should contain nodes that relayed. Once the first packet goes through, the sensor nodes
can quickly deliver the packet to the sink. However, since can stay awake for a while, hence the delay of the subsequent
the end-to-end delay depends on the forwarding set of all packets are often much smaller. Therefore, in this paper, we
nodes, choosing the correct forwarding set is not easy. We define the end-to-end delay as the delay incurred by the first
use a matrix A to represent the forwarding set of all nodes packet, which is the sum of the delay for each hop to wake
collectively, as follows: up and to relay the packet to its next-hop neighbor.
Given A, B, and ",pthe stochastic process with which a
A=[a ,i=1,...,N,j =1,...,N]
ij packet traverses the network from the source node to the sink
where a = 1 if j is in node i’s forwarding set, and a = is completely specified, and can be described by a Markov
ij ij
0 otherwise. We call this matrix A the forwarding matrix. process. We define Di("pA, ,B) as the expected end-to-end
Reciprocally, we define (A) as the forwarding set of node delay for a packet from node i to reach sink s, when awake
i
F
iunderforwardingmatrixA,i.e., (A)= j a =1 . probabilityvector",pforwardingmatrixA,andprioritymatrix
i i ij
F { ∈C | }
We let denote the set of all possible forwarding matrices. B are given. Since sink s is the destination of all packets,
A
With anycast, a forwarding matrix determines the paths the delay of packets from sink s is regarded as zero, i.e.,
that packets can potentially traverse. Let g(A) be the di- Ds("pA, ,B) = 0, regardless of ",pA, and B. If node i is
rected graph G(V,E(A)) with vertices V = , and edges disconnected to sink s under forwarding matrix A, packets
N
E(A) = (i,j)j (A) . If there is a path in g(A) that from the node cannot reach sink s. Therefore, the end-to-
i
{ | ∈F }
leads from node i to node j, we say that node i is connected end delay from such a node i is regarded as infinite, i.e.,
to node j. Otherwise, we call it disconnected to node j. For Di("pA, ,B) = . From now on, we call ‘the expected end-
∞
convenience, we call a node that is ‘connected (disconnected) to-end delay from node i to sink s’ simply as ‘the delay from
to sink s’ simply as ‘connected (disconnected).’ An acyclic node i’.
path is the path which does not traverse any node more than 2) Network Lifetime: We assume that node i consumes
once. If g(A) has any cyclic path, we call it a cyclic graph, ui unit of energy each time it wakes up. Let Qi be the energy
otherwise we call it an acyclic graph. available to node i. Then, the expected lifetime of node i is
3) Priority: When multiple nodes send an acknowledge- uQiλii.Byintroducingthepowerconsumptionratioei =ui/Qi,
mentafterthesameIDsignal,thesourcenodeineedstopick we can express the lifetime of node i as
one of them as a forwarder. We assume that node i assigns 1 t +t +t
B C A
hpirgiohreistitepsrtiooriatlyl anmodoensgintheCsie, annoddewsitlhlaptiwckaktheeupn.odCelewariltyh, tthhee Ti(")p= eiλi = eiln(1−1pi) . (1)
priorityassignmentwillalsoaffecttheexpecteddelay.Weuse Here we have used the definition of the awake probability
amatrixBtorepresenttheglobalprioritydecision,asfollows: pi = 1 e−λj(tB+tC+tA) from (1). Note that in this def-
−
inition of lifetime we have chosen not to account for the
B=[b ,i=1,...,N,j =1,...,N]
ij energyconsumptionbydatatransmission.Thisisareasonable
where b 1, , if j , and b = 0 otherwise. approximation for those event-driven sensor networks where
ij i i ij
∈{ ··· |C |} ∈C
This b represents the priority of node j from the viewpoint eventsoccurveryrarely,inwhichcasetheenergyconsumption
ij
of node i. We call this matrix B the priority matrix. The of the sensor nodes is dominated by the energy consumed
priority matrix B further satisfies b = b for all distinct during the sleep-wake scheduling.
ij1 # ij2
nodes j ,j . Among the awake nodes, the node j that We assume that the network lifetime is determined by the
1 2 i
∈C
satisfies b > b for all the other nodes k will be chosen shortest lifetime of all nodes. In other words, the network
ij ik
lifetime for a given awake probability vector "pis given by
T(")p=min T (")p. The methodology of the paper may be
i i
extendedtoh∈aNndleotherdefinitionsoflifetime,e.g.,whenthe Di("pA, ,B)
sensor network is considered operational if less than a certain ∞
= [(t h+t +D ("pA, ,B))P ]
I D j j,h
percentage of nodes are alive. However, we leave this more
general definition of lifetime for future work. h&=1j∈&Fi(A)
t
I
3)ProblemFormulation:Theobjectiveofthispaperisto = tD+
1 (1 p )
choose awake probability vector ",pforwarding matrix A, and − − j
prioritymatrixBtomaximizethenetworklifetime,subjectto j∈F#i(A)
the constraint that the expected delay from each node to sink D ("pA, ,B)p (1 p )
j j k
−
s is below the maximum allowable delay, i.e., +j∈&Fi(A) k∈Fi(A#):bij<bik .(2)
1 (1 p )
j
− −
(P) max T(")p j∈F#i(A)
"pA,,B
We call (2) the local delay relationship, which must hold
subject to Di("pA, ,B) ξ∗, i
≤ ∀ ∈N for all nodes i except the sink s. (Recall D ("pA, ,B) = 0
"p (0,1]N, A , B . s
∈ ∈A ∈B regardless of the delay of neighboring nodes.)
B. The Optimal Forwarding Set and Priority Assignment
where ξ is the maximum allowable delay.
∗
Inthissubsection,wefirstconsiderthehypotheticalscenario
where a node i knows the delays D from its neighboring
j
nodes j to sink s, and such delays D are fixed. Under
j
III. MINIMIZATIONOFEND-TO-ENDDELAYSFORGIVEN this hypothesis, we will study how node i should adjust its
AWAKEPROBABILITIES own forwarding set and priority assignment to minimize the
expected delay from node i to sink s. Then, in the next
subsection,wewillusetheinsightfromtheresulttodetermine
In this section, we consider how each node should choose
the optimal forwarding matrix A, and priority matrix B.
itsforwardingsetandassignprioritiestoneighboringnodesto
Consider that a node i has multiple neighboring nodes with
minimizethedelayD ("pA, ,B),whentheawakeprobabilities
i
fixeddelay.Similarto(2),wecancalculatetheexpecteddelay
are given. Then, in Section IV, we relax the fixed-probability
from node i to sink s for a given neighboring delay vector
assumption to solve problem (P).
"π =(D ,j ), forwarding set , and priority assignment
i j i i
"b as ∈C F
i
A. Local Delay Relationship f("πi, i,"bi)
F
t + D p (1 p )
We first derive a recursive relationship for the delay, ! tD+ I j∈F1i j j k∈(1Fi:bipj<)bik − k .(3)
D ("pA, ,B). When node i has a packet, the probability P ’ − j∈(Fi − j
i j,h
that node j in becomes a forwarder right after the h-th We call the function f(, , ) th(e local delay function.
i · · ·
beacon-ID signaCls is equal to the probability that no nodes Wefirstshowthat,inordertominimizef(, , ),theoptimal
in i have woken up for the past h 1 beacon-ID-signaling priority assignment "b∗i can be completely d·et·er·mined by the
iterCations, and node j wakes up at th−e h-th beacon-ID signals neighboring delay vector "πi.
while all nodes with a higher priority than node j remain Proposition 1: Let"b∗i be the priority assignment that gives
sleeping at the h-th iteration, i.e., higher priorities to neighboring nodes with smaller delay, i.e.,
for each pair of nodes j and k satisfying b < b , the
∗ij ∗ik
inequality D D holds. Then, for any given ,
h 1 k ≤ j Fi
−
Pj,h = (1−pk) pj (1−pk). f("πi,Fi,"b∗i)≤f("πi,Fi,"bi) (4)
k∈#Fi(A) k∈Fi(A#):bij<bik for all possible"bi.
 
The detailed proof is provided in Appendix A in our on-line
Conditioned on this event, the expected delay from node i to technicalreport[11].TheintuitionbehindProposition1isthat
sink s is given by (t +t +t )h+t +t +D ("pA, ,B). when multiple nodes sends acknowledgements, selecting the
B C A R P j
For ease of notation, we define the iteration period tI !tB+ node with the smallest delay should minimize the expected
tC +tA and the data transmission period tD ! tR+tP. We delay. Therefore, priorities must be assigned to neighboring
can then calculate the expected delay D ("pA, ,B) of node i nodes according to their (known) delays D , independent of
i j
for given awake probability vector ",pforwarding matrix A, awake probabilities and forwarding sets. In the sequel, we
and priority matrix B as follows: use b ("π ) to denote the optimal priority assignment for given
∗i i
neighboring delay vector "π , i.e., for all nodes j and k in , is a subset of j D = fˆ("π , ) t . This means
if b∗ij("πi)<b∗ik("πi), then Dik ≤Dj. For ease of notation, wCie Gthat if there exist{s a∈nCodi|e jj such thiatFDi∗j−= fDˆ(}"πi,Fi∗)−tD,
define the value of the local delay function with this optimal is not unique. In other words, if such a node j wakes up
priority assignment as fˆ("πi,Fi)!f("πi,Fi,"b∗i("πi)). Ffiri∗st, there is no difference in the overall delay whether node
The following properties characterize the structure of the i transmits a packet to this node or waits for the other nodes
optimal forwarding set. in to wake up.
Fi∗
Proposition 2: For a given "πi, let 1, 2, and 3 be Since the optimal forwarding set consists of nodes whose
J J J
mutually disjoint subsets of Ci satisfying b∗ij2("πi) < b∗ij1("πi) delay is smaller than or equal to some threshold value,
for all nodes j1 k and j2 k+1 (k =1,2). Let the simplest solution to find the optimal forwarding sets is
∈J ∈J
to run an exhaustive search from the highest priority, i.e.,
D p (1 p )
DJk = ’j∈Jk j 1j(−k∈Jj∈kJ:bk∗ij((1"πi−)<pb∗ijk)("πi) − k , mki=nim|Ciiz|e,stofˆt(h"πei,lFowi,kes)twprhieorreityF,ii,.ke.,=k{=j 1∈,Ctoi|fibn∗ijd("πthi)e≥k tkh}at.
denotetheweightedaverage(delayin k fork =1,2,3.Then, If there are multiple optimal forwarding sets, we only need
the following properties related to fˆ(J"π , ) hold to find one of them. In this paper, we chose to use the
i
(a) fDˆ("πi,+Jt1∪<J3fˆ)(<"π ,fˆ("πi,J1)).⇔DJ3·+tD <fˆ("πi,J1)⇔ sfoetrwFair∗di=ng{sjet∈beCcaiu|Dsejit<isfˆt(h"πei,fiFrsi∗t)o−netDth}atawsethceanopotbimtaianl
(b) fˆ(J"π3, D )=ifˆJ("π1∪,J3) D +t =fˆ("π , ) in the exhaustive search. Therefore, we redefine the optimal
(c) fDIˆf(Jf"πˆ3(ii,"π+Ji,t11JD∪1=J∪J33fˆ)3(.")πi<,Jfˆ1(i"∪πiJJ,1J3)1.⇔),theJn3fˆ("πiD,J1∪J2i∪JJ13)⇔< fFfˆo(i∗r"πwia,=rFdi∗ian)rgg−msetiDtn}FF.ii∗⊂NCaoistfeˆt(h"tπheia,tfFowir)witaharndtdhinisgFdi∗seefit=ntihti{aotjns,∈atthCisefiieo|sDptjibmo<tahl
(d) Iffˆ("πJ, ∪J )=fˆ("π , ),thenfˆ("π , ) forwarding set is unique. Then, the following lemma helps us
i 1 3 i 1 i 1 2 3
fˆ("π , J ∪J), and the eqJuality holds onJly∪wJhen∪DJ =≤ to find the optimal forwarding set more quickly.
D ifoJr1a∪llJj3 and j . j2 Lemma 1: For all i that satisfies = j i Dj <
j3 2 ∈J2 3 ∈J3 fˆ("π , ) t , F ⊂.C F { ∈C |
Proof:Thispropositioncanbeshownbynotingthateach i F − D} Fi∗ ⊂F
Proof:FromProposition3(a)andthedefinitionof ,all
nodesetJk (k =1,2,3)canberegardedasanodewithdelay nodes k satisfy D <fˆ(π , ) t fˆ(π , )Fi∗t ,
DdtheeJlakpyraofnubdnabcatiwiloitanykecthapanrtobabenaybexinlpiotrydeesPsieJndkJa=ks w1a−ke(s uj∈p.JkT(h1en−, tphje),loi.cea.l, ffˆo(rπia,nFy)∈s−uFbtsDi∗etfoFria⊂llCnokid.esSiknc∈eiFFFi∗i∗.⊂H−CeniDc,ew,≤Fei∗ob⊂itaFiFn.iD−k D<
Lemma 1 implies that when we exhaustively search for the
fˆ("π , K )=t + tI + Kk=1DJkPJk lk=−11(1−PJl) optimal forwarding set from k = |Ci| to k = 1, we can
i ∪k=1Jk D 1 K (1 P ) stop searching if we find the first (largest) k such that for
’ − k=1 (− Jk all nodes j , D <fˆ("π , ) t , and for all nodes
i,k j i i,k D
for K = 1,2,3. Then, by algebrai(c manipulation, we can l / , D∈Ffˆ("π , ) t F. Sinc−e all neighboring nodes
i,k l i i,k D
establish Properties (a)-(d). Details are again available in ∈F ≥ F −
are prioritized by their delays, we do not need to compare
Appendix B in [11].
the delays of all neighboring node with the threshold value.
The interpretation of Proposition 2 is straightforward. For
Hence, the stopping condition can be further simplified as
example,Property(a)impliesthataddinglowerprioritynodes
follows: node i searches the largest k such that for node j
odfelJay3iifnatondthoenlcyurirfetnhtefworewigahrdteidngavseertaFgei d=elJay1idnecreapsleusstthe with b∗ij("πi)=k, Dj <fˆ("πi,Fi,k)−tD, and for node l with
J3 D b ("π )=k 1, D fˆ("π , ) t .
is smaller than the current delay. ∗il i − l ≥ i Fi,k − D
It should be noted that the optimal forwarding set is time-
Using Proposition 2, we can obtain the following main
invariantduetothememorylesspropertyofaPoissonrandom
result.
Prhoapsotshietiofonll3o:wLinegt sFtriu∗ct=uralarpgrompienrtFiei⊂s.Cifˆ("πi,Fi). Then, ptorowceaskse. Suppecisifiaclawllayy,sthteheexspaemcteedastimtIe/pfojrreeagcahrdnleosdseojfinhoCwi
Fi∗ long the source node have waited. Therefore, the strategy to
(a) must contain all nodes j in that satisfy D <
fFˆ(i∗"π , ) t . Ci j minimize the expected delay is also time-invariant.
i Fi∗ − D
(b) cannot contain any nodes j in that satisfy D > C. Globally Optimal Forwarding and Priority Matrices
(c) fFFˆo(i∗"πria,lFl in∗)od−estDj.in Ci that satisfy DCji= fˆ("πi,Fi∗)−jtD, rithWmecnoemxtpuusteintghethiensgilgohbtaollfySoepcttiimonalIIfIo-Brwtaorddienvgelaonpdapnriaolrgioty-
the following relationship holds, matrices for given ".pThis algorithm has the flavor of the
fˆ("π , )=fˆ("π , j )=fˆ("π , j ). distributed Bellman-Ford’s algorithm for finding the shortest
i Fi∗ i Fi∗\{ } i Fi∗∪{ } paths. At each iteration, each node uses the delay estimates
We can prove Proposition 3 by using the result of Proposition fromthepreviousiterationtoupdatetheforwardingsetandthe
2. (See the detail provided in Appendix C in [11].) priorityassignment.Wewillshowthatthealgorithmconverges
FromProportion3,wecancharacterizetheoptimalforward- in N iterations, and the resulting A and B minimize the
ing set as = j D < fˆ("π , ) t , where expected delay D ("pA, ,B).
Fi∗ { ∈C i| j i Fi∗ − D}∪G i
The algorithm is presented next. 2) The OPT-DELAY algorithm converges within N itera-
The OPT-DELAY Algorithm tions.
Step (1) At iteration 0, each node i sets 3) Forgiven",p(A∗(")p,B∗(")p)=argminA,BDi("pA, ,B)
for all nodes i .
Di(0) = 0 if i=s, Proof:Inthispap∈eNr,weshowthebasicideasoftheproof.
otherwise.
) ∞ The detailed version of proof is provided in Appendix E in
and (0) = . Each node arbitrarily assigns priorities to [11].
neighFbioring no∅des. We first show that D(h+1) D(h) for h 1 and all
i ≤ i ≥
Step (2) At iteration h ( 1), each node i sets "b(h) = nodes i. Suppose in contrary that there exists node i such
"bfSSo∗ittre(ew"πppia((hr(4d−3)i)1nI)gfE),Daswcei(hthh)fenor=oerdD"π"πDeii((i(hhih(ih−)−−u1=11)p))df=aˆ≥fan(ot"dπ(erDsi(aahllj(−Flsho1n−i()ho1,u)d)Fp,edbijs(ayhi∈t)e)∈fiCs.nNiDd)i.i(n,hgt)hiatshseaflogoloiplortiiwtmh(s5ma)l tDRitDshh2ienae,i((kpkhktr·e)e−·Das·ktm+,i=i(inihus1kg))sD,.itn·t<(h·Fe1ii·x)oshir,D−spit=1hi(nrh’−onos+tco1de1tderd)es+a.uunirchiTtse1h−mh,/1eiwp∈t,snhsC,eaiDaotfctniri(ahio1akn−)smrnua1yfinc∈h[gint<C1ede1,trh]aiai,akDt.i−etosw.i(1en,h2Deq−)skaui(11nch,e.∈a−dnbnCHce1e)Dcosiawhhi(o<u−khofes1−wev,nDkeno)rttoh,i(dh1hdee<a)inesft.
terminates.Otherwise,eachnodeincreaseshbyoneandgoes ih 1 ih 1 D I s
i −can deli−ver the packet directly to sink s. If sink s is not
back to Step (2). h 1
in− , D(2) D(1) = . This leads to a contradiction.
To analyze the OPT-DELAY algorithm, we will use the ThuCsih,−D1(h+i1h)−1 ≤D(h)ihf−o1rall∞nodesi anditerationh>1.
following notations. We define the subgraph gi(A) = We noiw pro≤ve tihe first property. S∈uNppose in contrary that
G(V (A),E (A)) as the graph with vertices V (A) =
i i i there is a cyclic path in g(A(h)). Let the sequence of nodes
j V(A) i is connected to j in g(A) andedgesE (A)=
{{is(j∈c,okn)ne∈cteE|d(Ato)|i{tjs,eklf},⊂i.e.,Vii(A∈)}V.iB(Ay})cofonvrenaltlionA, in∈odAe i. isakhlo+onw1g∈tthhFaitsit(ckhhye)cldfioecrlapykasth=alboe1n,gi21,t,h.i.e2.,,cK·y·c.·liT,cihKpean,t,ahnfsrdoatmiiKsf+[y111]=, wi1e, ic.ea.n,
This subgraph g (A) shows all possible paths from node i
i
underforwardingmatrixA.ForanyforwardingmatrixA,the D(h) >D(h) > >D(h) >D(h).
i1 i2 ··· iK i1
numberofdistinctacyclicpathsing(A)isfinitewhenthetotal
Thisisacontradiction.Therefore,g(A(h))isanacyclicgraph.
number of nodes is finite. Let g(A) be the maximum length
| | Let (h) = A g (A) h . (h) denotes the set
of acyclic paths in g(A). Then, the following proposition Ai { ∈ A|| i |≤ } Ai
of forwarding matrices with which the maximum number of
states an important property for analyzing the OPT-DELAY
hops along acyclic paths from node i to sink s is less than h.
algorithm.
We now show that
Proposition 4: For any ",pA , and B such that
ggi((AA))iisscayccylcilci,c,thger(eAex)ist Ag& (∈∈AAA), aanndd B& ∈∈BB such that Di(h) ≤A m(ihn),BDi("pA, ,B). (6)
i & | i & |≤| i | ∈Ai
Di("pA, &,B&)≤Di("pA, ,B). tWe+ptro/vpebiyfisnindkucstion.A.Otittheerarwtioisne,1,F(1i()1)=={asn}daDnd(1D) =i(1) =.
The detailed proof is provided in Appendix D in [11]. Propo- D I s ∈Ci Fi ∅ (1) i ∞
Now consider the right-hand-side of (6). If is not an
sition 4 implies that for any forwarding and priority matrices Ai
empty set, this means that node i has a direct path to sink s,
thatcauseacyclicpathfromanynodeitosinks,therealways
which implies that its expected delay is t +t /p . If (1)
exist other forwarding and priority matrices with which all D I s Ai
is an empty set, this means that there is no path for node i
paths from node i to sink s are acyclic, and the delay from
to reach sink s within 1 hop, which implies that the expected
node i with the new matrices is equal to or smaller than
delay to reach sink s within 1 hop is infinite. Therefore, (6)
the delay with the original matrices. This is intuitively true
holds at iteration 1.
becauseitwillincurhigherdelayifthepacketshavetotraverse
Next, assume that the induction hypothesis (6) holds at
loops.
iteration h, i.e.,
LetA(h) betheforwardingmatrixthatcorrespondsto (h)
ofothrearlwlinsoe.deSsimii∈laNrly,,lie.et.B, a(h(ijh))b=e t1heifpjrio∈rFityi(mh)a,torirxai(injh)wF=hiic0h, Di(h) ≤A∈mA(iihn),BDi("pA, ,B) ∀i∈N . (7)
the transpose of the i-th row is"b(h). Let A (")pand B (")pbe Then, using Proposition 4, we can show that (7) also holds at
i ∗ ∗
iteration h+1. (See the detail in Appendix E in [11].) Hence,
the forwarding and priority matrices when the OPT-DELAY
(6) holds for all nodes.
algorithm converges. (Note that "pis fixed and given.)
We next prove that D ("pA, (h),B(h)) D(h). At iteration
Thefollowingpropositionprovidesthekeypropertiesofthe i ≤ i
algorithm. 1, Fi(1) ={s} if s∈Ci. Otherwise, Fi(1) =∅. Hence,
Proposition 5: The algorithm has the following properties: t +t /p if s ,
1) At iteration h, g(A(h)) is an acyclic graph. Di("pA, (1),B(1))= D I s ∈Ci
otherwise.
) ∞
Thus, Di("pA, (1),B(1)) = Di(1) for all nodes i, and so Proposition 6: If "q∗ is the optimal solution to problem
Di("pA, (h),B(h))≤Di(h) holds at iteration 1. (P2), then so is "qsuch that "q= (qi = maxkqk∗,i ∈N ),
Next assume that the induction hypothesis i.e., we can let every node have the same qi.
Proof:Sincebothsolutionshavethesameobjectivevalue,
Di("pA, (l),B(l))≤Di(l) ∀i∈N (8) it is sufficient to show that if "q∗ is in the feasible set, so is ".q
Let "pand "pbe the awakeprobability vectors that correspond
holds for all l h. Then, from [11], we can show that (8)
∗
also holds for l≤= h+1. Hence, D ("pA, (h),B(h)) D(h) to "∗q and ",qrespectively, by (10). Since pi is monotonically
i ≤ i increasingasq increases,and"q ",qwehave "p ".p(The
holds for all i and h. i ∗ - ∗ -
symbol ‘ ’ denotes componentwise inequality, i.e., if "q ",p
From the previous results, we conclude that - -
thenq p foralli,whereq andp arethei-thcomponents
i i i i
Di("pA, (h),B(h))≤Di(h) ≤A∈mA(iihn),BDi("pA, ,B) (9) ofN"qoatned≤th"a,ptrtehsepedcetliavyelyD.)i("pA, ∗(")p,B∗(")p) from each node i
Themaximumlengthofanacyclicpathisequaltoorlessthan is a non-increasing function with respect to each component
N. Therefore, (N) = for all nodes i. Since A(h) , at of ".p(See Appendix F in [11].) Since "p∗ ",pfor all nodes
iteration N, weAoibtain A ∈A i, we have Di("pA, ∗(")p,B∗(")p) Di("p∗,A-∗("p∗),B∗("p∗)).
≤
Hence, if "q is in the feasible set, so is ".q
D ("pA, (N),B(N))=D(N) =minD ("pA, ,B) ∗
i i i Using the above proposition, we can rewrite problem (P2)
A,B
into a problem with one variable q,
from (9). Hence, the algorithm must converge in at most
N iterations, and A(h), B(h), and D(h) converge to the (P3) min q,
i
optimalforwardingmatrix,theoptimalprioritymatrix,andthe subject to maxDi("pA, (")p,B(")p) ξ∗
minimum expected delay from node i to sink s, respectively. i∈N ≤
pi =1 e−q/ei, i
− ∀ ∈N
Proposition 5 shows that there always exists (A, B) that q (0, ).
∈ ∞
can minimize the delay from all nodes at the same time, and
(A(")p,B(")p)correspondstosuchasolution.Furthermore,the If q∗ is the solution to problem (P3), then ("∗p,A("∗p),B("∗p))
graphg(A∗(")p)isacyclic.Thecomplexityofthisalgorithmis (p∗i =1−e−q∗/ei) corresponds to the solution of the original
problem (P).
givenby (N).Moreover,thisalgorithmcanbeimplemented
O Note that max D ("pA, (")p,B(")p) is a non-increasing
in a fully distributed fashion. i i
∈N
function of p . (See the proof of Proposition 6.) Since "pis an
i
IV. SOLUTIONTOTHELIFETIME-MAXIMIZATION increasingvectorofq,thesimplestsolutiontoProblem(P3)is
PROBLEM tolinearlysearchqsuchthatmaxi Di("pA, (")p,B(")p)=ξ∗
In this section, we solve the original lifetime-maximization where pi =1 e−q/ei. ∈N
−
We develop an efficient binary search algorithm for
problem (P), using the results in previous sections. By letting
qi =ln(1 pi)−ei, we can rewrite problem (P) as computing the optimal value of q.
−
The Binary Search Algorithm for Problem (P3)
t
(P1) max min I, Step (1) Initially, sink s sets p(1) =0.5 and k =1.
subjeq"c,At,tBo Di∈iN("qpiA, ,B)≤ξ∗, ∀i∈N SStteepp ((23))SNinokdesssertusnq(tkh)e=OlPnT(-1D−ELpA(kY))−amlgaoxrii∈thNmei.for given
pi =1−e−qi/ei, ∀i∈N (10) "(pk) =(p(ik) =1−e−q(k)/ei,i∈N ).
q (0, ), i Step (4) After N iterations, the optimal forwarding set and
i
∈ ∞ ∀ ∈N the optimal priority assignment under "p(k) are found. Nodes
A , B .
∈A ∈B j that are not in the other node’s forwarding set, i.e., j /
∈
Since for any given ",pA∗(")p and B∗(")p are the optimal Fi∗(A∗("(pk))) for all nodes i, send feedback of their delays
forwarding matrix and the optimal priority matrix, respec- D ("(pk),A ("(pk)),B ("(pk))) to sink s.
j ∗ ∗
tively, that minimize the delay from all nodes, we have Step (5) Let Dmax be the maximum feedback delay arrived
Di("pA, ∗(")p,B∗(")p) Di("pA, ,B) for all A and B. Hence, at sink s.
≤
we can rewrite problem (P1) as follows: IfD >ξ +%,thensinkssetsp(k+1) =p(k)+0.5k+1,
max ∗
•
(P2) max mintI, increases k by one, and goes back to Step (2).
"q i∈N qi • IfDmax <ξ ∗−%,thensinkssetsp(k+1) =p(k)−0.5k+1,
subject to Di("pA, ∗(")p,B∗(")p) ξ∗, i increases k by one, and goes back to Step (2).
pi =1−e−qi/ei, ∀≤i∈N ∀ ∈N • IafndDrmeatuxr∈ns[ξq∗(k−) a%s,ξth∗e+o%p],titmheanl stohleutailognortiothPmrotbelremmin(aPt3es).,
q (0, ), i
i
∈ ∞ ∀ ∈N The reason that we take q(k) with respect to the maximum
Problem (P2) can be further simplified with the following ei in Step (2) is because this makes all p(ik) less than or
proposition. equal to p(k). (Note that we only search p(k) over (0,1].) In
Step (4), only such a node j that does not belong to any well-known Bellman-Ford shortest path algorithm, in which
other forwarding set needs to send the feedback delay to the thelengthofeachlink(i,j)isgivenbyt /p +t .LetD (")p
I j D i
sink s because the node with the maximum delay does not denote the minimum delay from node i under deterministic
belongtoanyotherforwardingsetaccordingtoProperty(a)in routing. Then, D(h) under the modified algorithm converges
i
Proposition 3. Since sink s only needs to know the maximum to D (")p.
i
delay, there is no need for the other nodes to feedback their In this simulation, in order to compare the network lifetime
delays. under the different algorithms, we run the binary search algo-
rithm for Problem (P3), replacing the OPT-DELAY algorithm
V. SIMULATIONRESULTS
in Step (3) with the above mentioned algorithms.
Inthissection,weprovidesimulationresultstoillustratethe
performance advantage of our optimal anycast algorithm. We B. Performance Comparison
simulate a wireless sensor network with 400 nodes deployed In Fig. 2, we compare the network lifetime under the
randomly over a 10-by-10 area with uniform distribution, and different algorithms, where x-axis represents different max-
thesinksislocatedat(0,0).Weassumethatthetransmission imum allowable delays ξ∗ in our original Problem (P), and y-
range from each node i is a disc with radius 1.5, i.e., j ∈Ci, axis represents the maximum lifetime for each ξ∗. The curve
if the distance between node j and node i is less than 1.5. labeled ‘Anycast (optimal)’ represents the lifetime under the
The parameters tI and tD are set to 1 and 5, respectively. We optimal anycast algorithm, i.e., the OPT-DELAY algorithm.
also assume that power consumption ratio ei is identical for The curves labeled ‘Anycast (norm)’ and ‘Anycast (naive)’
all nodes i. represent the lifetime under the normalized-latency anycast
algorithm,andunderthenaiveanycastalgorithm,respectively.
A. Existing Algorithms Proposed in the Literature
The curve labeled ‘Deterministic routing’ represents the life-
In this subsection, we review some existing algorithms that time under the deterministic routing algorithm.
we will compare with our optimal algorithm.
Normalized-latency Anycast Algorithm:Thenormalized- 20
latencyalgorithmproposedin[10]isananycast-basedheuris-
18
tic that exploits geographic information to reduce the delay
16
from each node. Let d be the distance from node i to sink
i
s, and let r be the progress from node i to node j toward 14
ij e
m
sthinekosn,e-ih.eo.,prdijel=aydfiro−mdnj.odIfeaintoodae nheaxst-ahoppacnkoetd,el,eatnDd lbeet k Lifeti1102
R be the progress between two nodes. Since node i selects etwor 8
thenext-hopnodeprobabilistically,bothD andR arerandom N
6
variables. The objective of the normalized latency algorithm
4 Anycast (optimal)
is to find the forwarding set that minimizes the expectation
Anycast (norm)
of normalized one-hop delay, i.e., E[D]. The idea behind this 2 Anycast (naive)
R Deterministic Routing
algorithm is to minimize the expected delay per unit distance, 0
0 50 100 150 200 250
which might help to reduce the actual end-to-end delay. Maximum Allowable Delay ξ*
Naive Anycast Algorithm: The naive algorithm proposed
Fig.2. Thenetworklifetimeaccordingtodifferentallowabledelayξ∗when
in [10] is also an anycast-based heuristic algorithm that nodesareuniformlydeployed.
exploits geographic information. Under this algorithm, each
node includes all neighboring nodes with positive progress in From Fig. 2, we observe that all anycast algorithms sig-
the forwarding set. nificantly extend the lifetime compared to the deterministic
Deterministic Routing Algorithm: By deterministic rout- routing algorithm. We also observe that the performance of
ing, we mean that each node has only one designated next- theoptimalandthenormalized-latencyalgorithmisveryclose.
hop forwarding node. Therefore, deterministic routing can be Notethatthenormalized-latencyalgorithmgivespreferenceto
viewed as a special case of anycast, in which the size of nodes with larger progress, while our optimal algorithm gives
the forwarding set at each node is restricted to one. There- preference to nodes with smaller delays. The results in Fig. 2
fore, instead of finding the optimal forwarding set (h) = seem to suggest that there is a correlation between progress
Fi
argmin fˆ("π(h−1), ) in Step (3) of the OPT-DELAY and delay when nodes are deployed uniformly. Finally, the
algorithmF⊂,Cwie updiate F(h) according to reason for the performance gap between the optimal and the
Fi naive algorithms is that transmitting a packet to a neighbor
(h) = argmin fˆ("π(h−1), ),. (11) with small progress is often not a good decision if a node
Fi i F
F⊂Ci:|F|=1 with higher progress is expected to wake up soon.
After the above modification, the OPT-DELAY algorithm be- We next simulate a topology where there is a hole in the
comes one that finds the optimal next hop under deterministic sensor field as shown in Fig. 3. This is motivated by practical
routing. Note that this modified algorithm is equivalent to the scenarios,wherethereareobstructionsinthesensorfield,e.g.,
a lake or a mountain where sensor nodes cannot be deployed. normalized-latency algorithm, all packet are forwarded only
The simulation result based on this topology is provided in to nodes with positive progress, and hence they take longer
Fig. 4. detours. Therefore, the result of Fig 3 shows that when the
node distribution is not uniform, there may not be a strong
10 correlation between progress and delay. Thus, the anycast-
based heuristic algorithms depending only on geographical
9
information could perform poorly.
8 Maximum
Delay
Node VI. CONCLUSION
7
In this paper, we study how to use anycast to reduce the
6
end-to-enddelayandtoprolongthelifetimeofwirelesssensor
5
networks employing asynchronous sleep-wake scheduling. In
4 Lake particular, we study the joint control problem of how to opti-
3 mally control the sleep-wake schedule, the anycast candidate
set of next-hop neighbors, and the anycast priorities, in order
2
to maximize the network lifetime subject to a upper limit on
1
theexpectedend-to-enddelay.Weprovideanefficientsolution
0 to this joint control problem, and as a part of the solution, we
0 2 4 6 8 10
also show how to optimally choose the anycast candidate set
Fig. 3. Node deployment and routing paths under different forwarding
to minimize the end-to-end delay from all sensor nodes. Our
algorithmswhenpi=0.5:Thedottedlinesillustrateallroutingpathsunder
theoptimalanycastalgorithm,thethicksolidlinesillustratetheuniquerouting numerical results suggest that the proposed solution can sub-
path under the deterministic routing path, and thin solid lines illustrate all stantially outperform prior heuristic solutions in the literature
routingpathsunderthenormalized-latencyanycastalgorithm
under practical scenarios where there are obstructions in the
coverage area of the wireless sensor network.
Thealgorithmsthatwehavedevelopedcanbeeasilyapplied
20
to energy-constrained event-driven wireless sensor networks.
18 In future work, we plan to extend the result to the case with
16 non-Poisson sleep-wake patterns, and to handle more general
14 notions of network lifetime.
e
m
eti12 REFERENCES
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