Optimal Deployment and Replenishment ofMonitoring Wireless Sensor Networks

David DorseyDept. of Electrical andComputer Engineering

Drexel UniversityPhiladelphia, Pennsylvania 19104

Email: dave@minerva.ece.drexel.edu

Moshe KamDept. of Electrical andComputer Engineering

Drexel UniversityPhiladelphia, Pennsylvania 19104

Email: kam@minerva.ece.drexel.edu

Abstract—Wireless Sensor Networks (WSNs) composed of

inexpensive devices allow the possibility of large deployments

aimed at monitoring remote, or possibly hostile, areas. However,

due to the unique traffic patterns exhibited by monitoring sensor

networks, the lifetime of a large WSN is constrained by the

burden placed on nodes near the sink node to forward additional

traffic as more nodes are deployed. We discuss approaches for

deploying a WSN that will maximize the lifetime of an initial

deployment. We then consider cases where the mission lifetime

is of a duration such that overdeploying an initial network to

meet this mission criteria would become prohibitively expensive.

We then propose a replenishment control framework where

additional nodes are added to an initial deployment in consecutive

batches in order to meet mission lifetimes while reducing cost.

The control framework consists of a failure process model used to

forecast sensor failures due to energy depletion, and a two-stage

limited lookahead controller used to determine the number of

nodes to be added to the network and the approximate locations

of their deployment.

I. INTRODUCTION

A wireless sensor network (WSN) consists of spatially-distributed autonomous sensors that monitor physical or en-vironmental processes. The sensor nodes are equipped withradio transceivers, a processing unit, power supply, and oneor more sensors. They are deployed onto an area of interest inorder to sample the physical environment. After deployment,the nodes communicate with other nearby nodes (within theirtransmission radius) to form a network. The goal of this net-work is to relay data from the sensors to a central processingstation, called a sink node.

Decisions about sensor quantity, placement, and networkorganization are crucial to designing a WSN that can meetmission objectives. These objectives include the quality ofsensing coverage, the quality of estimation, and the lifetimeof the network. The issue of sensing coverage is related to thedensity of sensors over the area, while the issue of estimationis related to both the density of the sensors and the capacityof the network. If there is insufficient bandwidth availableto the nodes in the network, sensors may not be able tocommunicate their estimates of the area. Since sensor nodespossess limited energy supplies, nodes will begin to fail due toenergy depletion; the lifetime of the network is determined bythe rate that nodes fail. Unlike in peer-to-peer ad hoc networks,

data gathered in a monitored area must be delivered from manysources to a single destination (sink node). Because sensornodes have limited communication range, these messages mustbe repeated over multiple hops until they reach the sink node.Since nodes in WSNs must forward traffic for other nodes thatare farther away from the sink node, nodes that are closest tothe sink node will expend their energy at a faster rate. Whenenough nodes have failed, there will be an insufficient numberof sensor nodes to relay traffic (i.e., connectivity loss) and/orprovide sensing coverage.

In research related to WSN deployment there is often anemphasis on random deployment. In a random deployment, asopposed to a deterministic placement where nodes are placedin specific predetermined locations, nodes are placed withcoarse-grain control, either by dropping them from an aircraftor releasing them from a moving vehicle. Recently, there havebeen many efforts to design models and deployment strategiesthat optimize an objective such as maximizing lifetime (e.g.,[1]), minimizing energy consumption (e.g., [2]), maximizingdetection probability (e.g., [3]), and minimizing estimationerror (e.g., [4]), subject to similar constraints. The decisionvariables in many of these works is the density of nodes toplace over the monitored area.

We show that when the expected lifetime of a WSN ismaximized over node densities with respect to the distancefrom the sink node, the lifetime of the network grows approx-imately with the logarithm of the number of nodes initiallydeployed. That is, the marginal increase in the network lifetimediminishes as the number of deployed nodes is increasedimplying that, as a mission duration increases, the cost ofdeploying a sensor network to monitor the area for the missionduration will become prohibitively expensive. We then proposea replenishment control framework where additional nodes areadded to the network in order to meet mission lifetimes whilereducing cost. The control framework consists of a failureprocess model used to forecast sensor failures due to energydepletion, and a two-stage limited lookahead controller usedto determine the number of nodes to be added to the networkand the approximate locations of their deployment.

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II. WSN DEPLOYMENT

A. Network Model

The monitored area is modeled as a disk of radius R that isdivided into J annular bands of equal width; the sink node islocated at the center. We assume that the nodes are distributedon the region according to a set of homogeneous spatial Pois-son processes (one for each band), with intensity λj for the jth

band. Therefore, the number of nodes in band j is a Poissonrandom variable, Nj , (asymptotically) distributed as P(λ),where P(λ) denotes a Poisson distribution with parameter λ.The area of band j is denoted Aj and the expected number ofnodes in band j is E[Nj ] = λjAj . The widths of the bandsare chosen to be equal to the communication radius of eachnode, r, so that the number of bands, J is �R

r �.1) Hierarchical Network Architecture: We assume that

some fraction of the nodes are designated to collect readingsfrom neighbor nodes and compress these readings into a singlemessage. We refer to these nodes as clusterheads; clusterheadsform a separate sub-network that is used to relay compressedmessages to the sink node. Clusterhead selection is assumed tobe performed in a distributed fashion using an algorithm thatattempts to balance the number of times each node will assumethe clusterhead role e.g., [5] and [6]. In these clusterheadselection strategies, nodes become clusterheads in a givenround with probability p. The decision to become a clusterheadis made at each node by choosing a random number andcomparing it to a threshold calculated as a function of p. If anode becomes a clusterhead, it broadcasts this decision, andthe neighboring nodes who have not become a clusterhead willdecide to align to the clusterhead whose advertisement has thehighest received signal strength (assumed to correspond to thenearest clusterhead node).

The clustered network can be thought of as a randomgeometric graph with vertices distributed according to twodifferent types of Poisson point processes: a process withintensity λjpj for clusterheads and a process of intensity(1−pj)λj for non-clusterheads. The expected number of nodesin band j is λjAj , so the expected number of clusterheadsin band j is therefore λjAjpj . Each non-clusterhead joins thenearest clusterhead to form a Voronoi tessellation, dividing thegraph into cells bounded by lines that are equidistant from twopoints (clusterheads). Thus, each Voronoi cell corresponds to aPoisson process point with intensity pλ. Using the results from[6] and [7], the number of non-clusterheads associated with aclusterhead in band j is a random variable corresponding tothe number of Poisson process points with intensity (1− p)λin each Voronoi cell. Thus, the expected number of nodessending messages to a clusterhead is 1−pj

pj.

B. Energy Consumption and Network Lifetime Model

The expected energy consumed in each band in any periodis the sum of the energy consumed to perform clusterheadselection, the energy consumed to aggregate data at eachclusterhead from its respective non-clusterheads, and the en-ergy consumed to relay packets across the network through

the clusterheads to the sink node. Using the expected valuesdescribed above, an expression, in terms of the node densitiesλ and clusterhead probabilities p for each band, can be writtenfor the expected energy consumption in each band (see [1] forcomplete derivation):

E[Ej ] = (σ + 1)EtxλjpjAj + (σ + 1)EλjAj (1− pj)

+ σEJ�

k=j+1

λkAkpk. (1)

where σ is the rate that nodes send messages to the sink node,Etx is the energy consumed transmit a single packet, Ercv isthe energy consumed to receive a packet, and E = Etx+Ercv .In this model, it is assumed that each band may be deployedwith a unique density and clusterhead probability. Each nodestarts with the same battery level, E0, so the number of roundsuntil a band is expected to run out of energy (the expectedlifetime) is Lj :

E[Lj ] =λjAjE0E[Ej ]

(2)

The lifetime of the network is the lifetime of the first bandto fail, thus:

E[Lnet] = minj

Lj . (3)

The problem is to find a deployment strategy that will maxi-mize E[Lnet] subject to constraints on connectivity, coverage,and cost. A solution to this problem was given in [1]; underthis deployment the increase in network lifetime is shownto exhibit diminishing marginal increases as more nodes aredeployed. Figure 1 shows numerical results for the optimalexpected lifetime for three different deployment strategies:uniform, static p, and dynamic. In the uniform case, nodes aredeployed with the same density in each band and a single valueof p is determined for all bands. The static p strategy refers tothe case where nodes are deployed with varying densities, butdo not allow p to vary. The dynamic deployment allows bothp and λ to vary over all bands. The strategies where varyingnode densities are permitted clearly outperform the uniformdeployment. Furthermore, allowing the clusterhead density tovary further increases the network lifetime. The results showthat, when the dynamic deployment strategies are used, theexpected lifetime trend begins to flatten as the number of nodesdeployed increases.

III. WSN REPLENISHMENT

In the remaining sections, we consider a case where a WSNmonitoring network is to be deployed for a long missionduration. In order to reduce the cost of the deployment, wepropose to extend the lifetime of an initial deployment usinga replenishment strategy. A replenishment strategy adds newnodes to the network at subsequent stages in order to meetmission requirements (connectivity, coverage, lifetime, etc.)The objective of the replenishment strategy is to meet missionrequirements while minimizing the total cost of the mission.

The model for the replenishment controller is illustrated inFigure 2. At the beginning of time period t the controller

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Fig. 1: A plot of the expected lifetimes for the uniform,static p, and dynamic deployment strategies with respect tothe number of nodes being deployed (over a fixed area). Thecalculations for the deployments allowing varying densitiessuggest diminishing marginal lifetime increases for additionalnodes

Controller W

XJ

X2

X1

Y

w1

w2

wJ

z1

z2

zJ

{ L

Fig. 2: Overview of the replenishment control system: Thenumber of nodes in each band at any time is given by X Asnodes fail in each band according to the random processes(wj(t)), the controller infers the aggregate process W (t); aforecast of W (t) is used to choose a batch size Y to order,which will be allocated after a leadtime (L) over each of theJ bands.

observes the number of nodes that are still active in each band(xj) (inferred from messages received at the sink node). Thisinformation is used to update a failure model that is used toforecast the number of expected failures in the future. Thecontroller uses the failure model to determine the number ofnodes that should be ordered (y) to replenish failures in eachband in a deployment L periods into the future. If an order wasplaced at period t−L, the controller specifies the allocation (z)for these nodes across J bands. During period t, the numberof nodes in each band is affected by node failures (w); thesefailures are detected by the controller at the beginning of thenext period. The WSN deployment model introduced in Fig.2 bears close resemblance to a classical problem in inventorycontrol called the multi-period multi-location inventory andsupply problem [8]. In this problem, an inventory systemconsists of a central depot which supplies J retailers whererandom demands for a single commodity must be filled. The

inventories are reviewed and decisions are made periodically.The decisions to be made (centrally, at the depot) are 1. theamount of stock to order from the supplier to be delivered tothe depot, and 2. the fraction of the order each retailer willreceive. When demand is assumed to be stationary and nor-mally distributed about a known mean with a constant, knownvariance, the multi-location inventory and supply problem canbe solved using dynamic programming, yielding an optimalpolicy consisting of a simple rule. However, even with thesesimplifying assumptions, the task of computing an optimalpolicy using dynamic programming can be formidable if thenumber of retailers becomes large.

For the WSN replenishment problem, we assume (1) arelatively large fixed cost for deploying a batch of sensors ofany size (2) a relatively long lead time between the placementof an order and the time it can be delivered to the fieldand (3) non-stationarity of node failure rates. Therefore, inaddition to the computational constraints provided by thedimensionality of the problem, our problem suffers from a“curse of uncertainty”, in that we do not possess a failuremodel that can estimate with any meaningful precision thenumber (and location) of failures beyond a short horizon. Wecan simplify problem through the use of a myopic allocation,where an order placed L periods ago is received and dividedup among the bands in order to minimize the expected costs inthe current period, ignoring costs in subsequent periods. Usingresults from [9], an approximation of the myopic allocationproblem in terms of aggregate state variables is described.This allocation approximation is integrated back into the fulldynamic program, to yield a new dynamic program in termsof aggregate state variables.

IV. DESCRIPTION OF WSN REPLENISHMENT PROBLEM

A. Notation and Definitions

The notation for the replenishment problem will follow theconvention that boldface variables are vectors. We first identifysome variables related to the state of the system, including thenumber of active nodes in a band at the beginning of a period,batches of nodes ordered for deployment, and the allocationof the order over bands.

xjt Number of active nodes in band j at the beginningof period t

xt = (xjt)Jj=1

yt Number of nodes ordered at the beginning of periodt

yt = (ys)t−1s=t−L Vector of orders placed in the last L−1

periods that have not yet been allocatedzjt Number of nodes delivered to band j in period t

(allocation)Using this notation, we can describe the state of the system

at the beginning of period t using the pair of vectors (xt, yt).Next we introduce some notation related to the costs andfailure model.

wjt Number of failures in band j in period t

138

µjt Expected number of failures in band j during periodt

Ct(yt)Cost associated with ordering yt nodes in period tK Fixed cost for deploying a batch of nodescs Cost of an individual sensor nodeh Cost associated with deploying too many nodes in a

bandd Cost associated with deploying too few nodes in a

bandThe costs h and d are incurred in each period whenever thenumber of nodes in a band is above or below a specified targetlevel. Let ρjt denote the target for band j in period t. Sincewe are interested in the deviation of xjt from the target forour cost calculations, we will introduce the variable xjt =xjt−ρjt. Then, the cost associated with the level in band j atthe end of period t is d(xjt)− +h(xjt)+, where the + and −superscripts denote functions that return the absolute value ofthe argument if it is positive or negative, respectively, and zerootherwise. The ordering cost Ct(yt) is I(y)K + csyt, whereI is the indicator function. The single period expected costsrelated to node levels (penalty costs) across all bands in periodt are Qt(zt, xt) =

�Jj=1 qjt (zjt, xjt), where

qjt = hE [xjt + zjt − wjt]+ + dE [xjt + zjt − wjt]

− .

Given the initial state (xt, yt), the state of the system inthe next period will be (xt+1, yt+1). The recursive Bellmanequation whose solution provides the optimal policy for theproblem is

fT+1 = 0 (4)ft(xt, yt) = min

yt,zt

Ct(yt) +Qt(zt, xjt)

+ E [ft+1 ((xt + zt −wt), yt)]

where the domain of yt and zt are constrained by

J�

j=1

zjt = yt−L (5)

yt, zt ≥ 0.

In order to make the problem easier to solve, we willamend the problem so that we may consider separately theproblem of choosing the number of nodes to order yt andthe number of nodes to allocate to each band zt. This willinvolve the introduction of a few aggregate variables, wherethe aggregation takes place over the J bands and L periods.

1) Aggregate variables: We first define two aggregate fail-ure random variables.

Wt =J�

j=1

wjt, and Wt =t+L−1�

s=t

Ws.

These are normal random variables with means

Mt =J�

j=1

µjt, and Mt =t+L−1�

s=t

Mt

and variances

S2t , and S2

t =t+L−1�

s=t

S2t .

Finally, we define variables that store the aggregate numberof active nodes over J bands and L periods, relative to thetarget levels. Let

Xt =J�

j=1

xjt, and X∆t = Xt +

t−1�

s=t−L

ys.

B. Myopic Allocation ProblemFor any period t ≤ T , the myopic allocation problem is

given by

Rt(xt, yt−L) = minzt

Qt(zt, xt)) (6)

subject to: zt ≥ 0,J�

j=1

zjt = yt−L.

Rt is the minimum value of Qt over all possible allocationsof the order placed in period t − L, and the vector zt thatminimizes Qt is the optimal allocation. In [9], the authorsdescribe a method for approximating the minimal cost of anallocation problem, by a simple, closed-form aggregate costfunction. This technique allows us to state Rt as a function ofXt and yt−L instead of the J-vector xt. The approximationinvolves a relaxation of the non-negativity constraint in (6).Under this relaxation, all bands collapse into a single aggregateband. The allowance of negative values for some entries in ztimplies, in effect, that some nodes may be “taken” from oneband and moved to another. This relaxation of the problemis justified by what is termed the “allocation assumption” ininventory control [10]. This assumption states that, in everyperiod t, we can make an allocation such that the probabilityof falling below the target levels in each band will be the samein the next period. The expression for Rt is

Rt

�Xt, yt−L

�= d

�Mt −

�Xt + yt−L

��

+ (d+ h)

� Xt+yt−L

−∞Φ

�u−Mt

St

�du. (7)

The cost minimizations for orders and their allocations cannow be separated. Thus, we can write an approximation to thefunction ft of (4) using aggregate variables and by replacingminzt {Qt(zt, xt)} with the value of the approximate myopicallocation Rt. This yields a problem with L+ 1 dimensions:

ft�Xt, yt

�= Rt

�Xt, yt−L

�

+ minyt

{Ct(yt)

+ E�ft+1

�(Xt + yt−L −Wt), yt

���(8)

The form of this dynamic program (with fixed plus linearordering costs) is known to result in a solution of type (st, St)

139

[11]. That is, if the number of nodes in the network is belowst in period t, then we order enough nodes to increase Xt upto St. The values s and S are indexed by time because the rateof lead time failures is changing over time. In order to treatthe non-stationarity of the lead time failure rate, we requirea forecasting method that will adapt to trends in the failuredistribution parameters.

V. FAILURE PROCESS MODEL AND SENSOR NODEFAILURE FORECASTING

Since our replenishment opportunities are constrained bythe fixed lead time, L, our primary focus is the lead timefailures (LTF), Wt. We use a trend-corrected exponentialsmoothing method (Holt smoothing) [12] in order to makeinferences about the LTF distribution form the time seriesof observed failures. Since he parameters of the distributionare non-stationary, the smoothing model characterizes thesetrends through a parameter bt, which is updated after eachobservation. The complete smoothing model is

Wt = lt−1 + bt−1 + et (9)lt = lt−1 + bt−1 + αet

bt = bt−1 + αβet

where lt is the level term, bt is the growth rate, and et isthe random variable representing the unpredictable component.The constants α and β are called the smoothing parameters.Essentially this model is a decomposition of the series of leadtime. After each observation of the failures, the prediction erroret is computed and the unknown parameters α and β aredetermined by minimizing the squared prediction error. Themodel is fully specified once we state the distribution of et. Weassumed that these are independent and identically distributed,following a normal distribution with mean 0 and variance σ2.The data set was tested for normality using a Shapiro-Wilktest, resulting in a p-value of 0.98 for a corresponding alphavalue of 0.05.

In order to obtain an estimation of the LTF for period t+kat the end of period t, we simply compute lt+kbt. The meanand variance of the lead time failures are:

Mt = lt + Lbt, and S2t = S2

t

�1 +

L−1�

k=1

(α+ kβ)

�.

Note that the variance depends only on the smoothing param-eters and the length of the lead time, not on the observedfailure rates. In fact, the smoothing parameters themselves aredependent on the lead time, since the values of α and β aredetermined using a least-squares minimization of the forecasterror over the lead time demand from simulation data; differentvalues of L will yield different parameter values.

A. Effect of Replenishments on Failure RatesEach time a replenishment is delivered to the network area,

nodes that have limited energy resources due to assuming therole of clusterhead will be relieved of this role as new nodeswho have not yet been assigned a clusterhead role will be more

0 50 100 150 200Time (periods)

0

5

10

15

20

25

Num

ber o

f Fai

lure

s (s

moo

thed

)

Fig. 3: Plot of the total failures in each period during a typicalsimulation run. The vertical bars represent replenishments.

likely to do so (this is a function of the clusterhead selectionprotocol). Therefore, each replenishment has an effect on thefailure process; without accounting for this effect, the LTFpredictions near the time of replenishment will be subject toerror, and the failure process model parameters will becomeunstable. Figure 3 shows the effect of replenishments on thefailure process. The figure shows the aggregate (summed overall bands) number of failures in each period. Beneath thegraph of the failure process are bars that represent the times abatch of nodes was delivered. The height of the bars indicatethe number of nodes delivered (divided by 100). The failuretrend appears to remain relatively stable, while the base levelabruptly drops by some amount. In order to reduce the effectof the replenishments, we introduce an additional term to thebase level update equation, −ωy0u[t − t0] where y0 is thenumber of nodes scheduled to be delivered at time t0, u isthe unit step function, and ω is a scaling factor whose valueis obtained through simulation.

VI. ALLOCATION STRATEGY

Once a batch order is received at the controller, the totalbatch size is to be divided among the individual bands ac-cording to an allocation strategy. The solution to the initialdeployment problem prescribes over-deploying nodes near thesink node. It is neither necessary nor desirable to maintainthese high densities as the network approaches the missionlifetime. Therefore, we introduced a moving target level ρjtfor each band; these target levels decrease linearly from theinitial deployment levels in each band down to the criticallevels required for connectivity and coverage over the length ofthe mission time. The intent is to have the minimum number ofnodes remaining on the field as the mission ends. We considertwo variables in our allocation decision. The first is the fractionof failures that have occurred in a band j over the last Lperiods; this is denoted v1,j . The second is the fraction ofdeviation from the moving target level ρjt for band j, denotedv2,i = xjt/

�Jj=1 Xt. The allocation is then given as a linear

combination of v1,j and v2,j :

zj = yt−L [av1,j + (1− a)v2,j ] , (10)

140

where a is a constant expressing the weight given to the twovariables.

VII. SIMULATION AND RESULTS

To test the replenishment strategy, we implemented the re-plenishment controller in simulation ( implemented in Matlab).The values for the smoothing parameters were computed usinga separate series of simulation runs where we allowed thenetwork to run to failure. We also fixed the value of a, theallocation parameter in the previous section to a constant valueof 0.25. With the exception of a, α, and β, and the initialvalue for ω, the remaining parameters were computed in “realtime” as the simulation progressed. We used 6 bands and aninitial deployment of 1700 nodes. The replenisher computedforecasts of failures for 20 periods into the future at eachstage, hence the dynamic program was solved for a 20-periodplanning horizon. In order to see the effects of varying leadtimes, we ran the simulation for lead time values of 1, 5, 10,15, and 20. In each simulation, the mission lifetime was setto 250, more than 3 times the maximum lifetime of the initialdeployment. We set the penalty costs to d = 3 and h = 1.5,

0 50 100 150 200 250

1000

1200

1400

1600

1800

Activ

e N

odes

Fig. 4: Simulation results: node level of the entire network over time,with replenishments for lead time L=15.

0 50 100 150 200 250Time (periods)

0

0.05

0.1

0.15

0.2

0.25

0.3

Nod

e D

ensi

ty

Band 1Band 2Band 3Band 4Band 5Band 6

Fig. 5: Node densities in each band over time with allocations.

0 5 10 15 20Leadtime (periods)

10

20

30

40

50

Cos

t (x1

000)

Fig. 6: The effect of the lead time on the total cost of the mission.

and the fixed cost K was set to 500. Figure 4 shows the totalnumber of nodes active in the network with respect to timefor a 15 period lead time and a 20-period planning horizon.The straight lines represent the sum of the target levels forall bands over time,

�6j=1 ρjt. Figure 5 shows the number

of active nodes in each band scaled by the area of each bandover time. The allocation levels given to each band are evidentfrom the “jumps” in the density. Note that at the end of themission lifetime, the bands densities are near equal (i.e., thenetwork ends in a uniform deployment). These figures showdata for a typical run. The simulations were repeated for 100different random topologies for each lead-time; the averagevalues of the total costs are shown in Figure 6.

In cases where the lead time was equal to the forecastinginterval, or planning horizon, the replenisher consistentlyunder-ordered, resulting in node levels well below the targets.A trend between the lead time and the amounts that nodelevels fall below the targets becomes evident from the totalcosts. Since a penalty is incurred for node levels below thetarget in each period, the effects of the lead time can be easilyseen by comparing the total cost (penalties and actual costs)of a replenishment versus the lead time as shown in Figure 6.Our results show that, for moderate leadtimes, the allocationassumption is valid for this problem, and the full dynamicprogram required for solving the problem can be approximatedby a single-dimensional problem without leadtimes.

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