algorithm for equalization of cluster lifetimes in a multi-level beacon enabled 802.15.4 sensor...

Computer Networks 51 (2007) 3252–3264

www.elsevier.com/locate/comnet

Algorithm for equalization of cluster lifetimes in amulti-level Beacon enabled 802.15.4 sensor network

Jelena Misic *

Department of Computer Science, University of Manitoba, Rm E2-406 EITC, Fort Garry Campus, Winnipeg, Manitoba, Canada R3T 2N2

Received 21 November 2006; received in revised form 17 January 2007; accepted 18 January 2007Available online 1 February 2007

Responsible Editor: X.S. Shen

Abstract

We consider the problem of maintaining the prescribed event sensing reliability while maximizing cluster and networklifetime in a multi-cluster 802.15.4 sensor network. Clusters are connected through bridges which also act as cluster coor-dinators; both ordinary nodes and bridges resolve contention using the CSMA-CA algorithm. Cluster lifetime is maxi-mized through the use of redundant sensors which are periodically sent to sleep using a simple distributed activitymanagement algorithm. Network lifetime is maximized by equalizing energy consumption per backoff period in all clustersthrough the adjustment of the number of nodes. We model this problem analytically using the datasheet for tmote_skyultra low power IEEE 802.15.4 compliant wireless sensor module [tmote sky lowpower wireless sensor module, MoteivCorporation, San Francisco, CA, www.moteiv.com, tmote datasheet 802.15.4, 2006] and derive the probability distribu-tion of the network lifetime. We also derive the expression for node count that compensates for the increased load dueto contention caused by the bridge. Numerical results show that this technique easily equalizes cluster lifetimes.� 2007 Elsevier B.V. All rights reserved.

Keywords: Sensor networks; Power management; IEEE 802.15.4

1. Introduction

Achieving and maintaining reliable event sensingin the area under surveillance is the main task ofwireless sensor networks. Furthermore, the powerconsumption should be minimized so as to allowbattery powered sensor nodes to extend their life-time as much as possible. To this end, networks fre-

1389-1286/$ - see front matter � 2007 Elsevier B.V. All rights reserved

doi:10.1016/j.comnet.2007.01.019

* Tel.: +1 204 474 6791; fax: +1 204 474 7609.E-mail address: [email protected]

quently use redundant sensors—more than theminimum number necessary to achieve the desireddata rate at the sink—which, then, randomly alter-nate between active and inactive (sleep) periods;the corresponding reduction of the duty cycle leadsto longer network lifetimes [2].

Energy efficiency of a wireless network is criticallydependent on the design of the MAC layer. Conten-tion-based MACs, such as those based on theCSMA-CA medium access, can handle large numberof devices better than their TDMA-based counter-parts, and are thus better suited for deployment in

.

http://www.moteiv.com

mailto:[email protected]

J. Misic / Computer Networks 51 (2007) 3252–3264 3253

wireless sensor networks. At the same time, conten-tion-based MACs are also susceptible to collisionsthat may waste bandwidth and impair energy effi-ciency [3].

The problem of achieving and maintainingenergy efficiency is further compounded by the lim-ited transmission range of wireless sensor devices,which means that the coverage of larger areasimplies multi-hop transmission of sensor data. Inmulti-hop and multi-level networks, the power con-sumption of the devices at different parts of the net-work may differ—and so will their lifetime. Namely,when the devices at one level exhaust their powersources before those at the other, the network willeffectively cease to function, even though the totalenergy level may still suffice to support operation.Since the desired goal is to maintain the prescribeddata rate for the maximum possible time, it becomesnecessary that all clusters exhaust their powersources at about the same time.

In this paper, we address the problems of activitymanagement and lifetime maximization in the con-text of a sensor network based on the IEEE802.15.4 communication standard for low rate wire-less personal area networks (LR-WPANs) [4]. Thenetwork is formed by three clusters interconnectedin a master–slave regime wherein the coordinatorof a ‘lower’ cluster acts as the bridge to the ‘upper’one, and the coordinator of the topmost cluster actsas the network sink.

All clusters are equipped with redundant sensors,which enables reduction of individual sensor dutycycle through activity management [5]. In otherwords, each node spends most of its time in sleepingmode and wakes up only to transmit its packets.The goal is to maximize the network lifetime whilemaintaining the prescribed data rate R of the trafficsent from each cluster to the network sink.

Individual sensor nodes are battery operated andtheir battery capacity is expressed as a budget of b J.Since the coordinators/bridges have to work with-out ever going to sleep, their power budget isassumed to be infinite; the use of relaying nodeswith larger power resources than ordinary sensingnodes has been shown to increase the useful net-work lifetime [6].

We model this problem under the assumptionsthat (a) each cluster is lightly loaded and (b) eachsensor node autonomously determines its averagevalue of sleep time using the information aboutthe required aggregate reliability and the numberof live nodes obtained from the cluster coordinator.

However, sleeping time needs to be randomized inorder to avoid simultaneous wake up of two ormore nodes and potential packet collisions. Simpleway to randomize the sleeping time is to assume thatit is geometrically distributed with parameter Psleep

where Psleep is equal to the reciprocal value of theaverage sleep time. In our earlier work, we havedeveloped an exact model that provides the proba-bility distribution of node lifetime in a single cluster[3]. This model can be coupled with the bridgingmodel to give an accurate description of networkoperation and performance, but it is rather complexand does not scale well. In this paper, we describe asimplified model that provides sufficient accuracywhilst achieving good scalability.

The rest of the paper is organized as follows. InSection 2 we briefly discuss bridge operation.Energy consumption for tmote_sky ultra low powerIEEE 802.15.4 compliant wireless sensor module [1]which we will consider in our modeling is consideredin Section 3. In the same Section we present batteryenergy capacity needed to power the tmote_sky

model. Section 4 discusses activity management per-formed by nodes. Section 5 deals with the model ofthe transmission medium from the viewpoint of anode from each of the clusters which independentlycalculates its periods of sleep. In Section 6 we pres-ent the performance of three interconnected clusterswhich have equal number of sensing nodes. In Sec-tion 7, we derive the populations in the top and mid-dle clusters, which result in the same nodeutilization per cluster, and present the performancefigures which show the accuracy of our calculations.Section 8 concludes the paper.

2. Bridging the clusters

Consider the network shown in Fig. 1, operatingin the ISM band at 2.4 GHz (other bands can beused but we do not consider them here). We assumethat all clusters operate in beacon enabled, slottedCSMA-CA mode under the control of their respec-tive cluster (PAN) coordinators. In each cluster, thechannel time is divided into superframes boundedby beacon transmissions from the coordinator [4].All communications in the cluster take place duringthe active portion of the superframe, the duration ofwhich is referred to as the superframe duration SD

as shown in Fig. 2.The basic time unit of the MAC protocol is the

duration of the so-called backoff period. Access tothe channel can occur only at the boundary of the

bottom coordinator+ bridge

middle coordinator+ bridge

top coordinator +network sink

beacon frame

active portion of thesuperframe (CAP only)

coordinator switches to uppercluster as a bridge

bridge returns to its own cluster,resumes coordinator duty

middlecluster

bottomcluster

top (sink)cluster

Fig. 1. Network topology and timing among the clusters.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

guaranteedtime slot(GTS)

GTS

Beacon contention-access period(CAP)

contention-freeperiod (CFP)

superframe duration (SD)

beacon interval (BI)

inactive

Beacon

Fig. 2. The composition of the superframe under IEEE Std. 802.15.4 (adapted from Ref. [4]).

3254 J. Misic / Computer Networks 51 (2007) 3252–3264

backoff period. The actual duration of the backoffperiod depends on the frequency band in whichthe 802.15.4 WPAN is operating. Namely, the stan-dard allows the PAN to use either one of three fre-quency bands: 868–868.6 MHz, 902–928 MHz and2400–2483.5 MHz. In the two lower frequencybands, BPSK modulation is used, giving the datarate of 20 kbps and 40 kbps, respectively. Each databit represents one modulation symbol which is fur-ther spread with the chipping sequence. In the third

Table 1Timing structure of the slotted mode MAC protocol (Note that the valumode)

Type of time period Duration

Modulation symbol 1 data bit in 860 MHz and 915 MHz ba4 data bits in 2.4 GHz band

Unit backoff period 20 symbolsBasic superframe slot

(SO = 0)Three unit backoff periods (60 symbols)

Basic superframe length(SO = 0)

16 basic superframe slots (960 symbols)

(Extended) superframeduration SD

aBaseSuperframeDuration * 2SO

Beacon interval BI aBaseSuperframeDuration * 2BO

band, the O-QPSK modulation is used beforespreading; in this case, four data bits comprise onemodulation symbol which is further spread withthe 32-bit spreading sequence. Table 1 summarizesthe basic timing relationships in the MAC sublayer.Note that the constants and attributes of the MACsublayer, as defined by the standard, are written initalics. Constants have a general prefix of ‘‘a’’, e.g.aUnitBackoffPeriod, while attributes have a generalprefix of ‘‘mac’’, e.g. macMinBE.

es of both BO and SO must be less than 15 in the beacon enabled

MAC constant

nds, N/A

aUnitBackoffPeriod

aBaseSlotDuration

aBaseSuperframeDuration =NumSuperframeSlots Æ aBaseSlotDuration

macSuperframeOrder, SO

macBeaconOrder, BO


As shown in Table 1 the superframe is dividedinto 16 slots of equal size, each of which consistsof 3 · 2SO backoff periods. The variable SO, alsoknown as macSuperframeOrder, determines theduration of the superframe; its default value ofSO = 0 corresponds to the shortest active super-frame duration of 48 backoff periods. In the ISMband, the duration of the backoff period is 0.32 msfor a payload of 10 bytes, which results in the max-imum data rate of 250 kbps. The time intervalbetween successive beacons is BI = aBaseSuper-

frameDuration * 2BO, where aBaseSuperframeDura-tion = 48 backoff periods (SO = 0) and BO

denotes the so-called macBeaconOrder which cantake values between 1 and 14. The duration ofthe inactive period of the superframe can easilybe determined as I = aBaseSuperframeDuration *(2BO � 2SO). The default access mode in beaconenabled operation is slotted CSMA-CA, with someslots optionally reserved for certain nodes.

During the inactive portion of the superframe,any device may enter a low power mode; the clustercoordinator can switch to the upper cluster in orderto perform the bridging function—i.e., deliver thedata to the coordinator of the upper cluster. Alllower cluster coordinators use this facility to per-form the bridging function. As soon as the activepart of the superframe is completed in the lowercluster, the coordinator/bridge switches to the upper

cluster and waits for the beacon so that it can deli-ver the data from the lower cluster to the upper clus-ter coordinator/network sink.

All clusters use the CSMA-CA access, whichmeans that the bridge has to compete for mediumaccess with ordinary nodes in the upper cluster. Assoon as the data is delivered, the bridge can returnto its own cluster. This also means that, shouldthe bridge be unable to transmit its data when the(active portion of the) superframe in the upper clus-ter ends, it will freeze its backoff counter and leavethe upper cluster. The backoff countdown willresume when the bridge returns to the upper clusterfor the next superframe. Upon returning to thelower cluster, the bridge transmits the beacondenoting the beginning of the next superframe,and the lower cluster continues to operate.

Bridge switching is schematically presented inFig. 1. As can be seen, the three clusters have tooperate with the same beacon interval, and the timebetween successive bridge visits to the ‘upper’ clus-ter is therefore the same as the period betweentwo beacons in its own, ‘lower’ cluster. If the top

and bottom cluster are far enough, i.e., beyondthe transmission range of each other, all three clus-ter may use the same RF channel (the 802.15.4 stan-dard uses 16 channels in the ISM band). Note thatobtaining an increased area of coverage is the mainreason for using a multi-cluster configuration. If theclusters are closer to each other, the top and bottomcluster may use different channels, and the middlecluster can use either of these.

3. Model of energy consumption

As we mentioned earlier each node spends mostof the time sleeping i.e. with radio system switchedoff. When node wakes-up and has a packet to trans-mit it turns its receiver on in order to synchronizewith the beacon. If node’s buffer is empty it will startthe new sleep. From the first beacon payload, awak-ened node will obtain the information about thenumber of live nodes and requested event sensingreliability. After receiving the information fromthe beacon, node turns the transmitter on and startsbackoff count in order to transmit the packet. Afterpacket transmission, node turns the receiver inorder to receive acknowledgement. After the posi-tive acknowledgement node starts the new sleep per-iod. If packet was not received correctly, node hasto repeat the transmission.

Since minimal beacon size is 2 backoff periods weassume that additional backoff period (10 bytes) issufficient for transmitting information about thenumber of live nodes and requested event sensingreliability. However, only awakened node willreceive the beacon and go to sleep immediately afterthe transmission. In a lightly loaded network thismeans that node will receive one beacon in a sleepcycle.

Let us denote power consumptions as xs, xr andxt J per one backoff period during sleep, receivingand transmitting, respectively. They can be derivedfrom typical operating conditions reported in docu-mentation for Ultra low power IEEE 802.15.4sensor module operating in ISM band between2400–2483.5 MHz [1] and shown in Table 2.

For example, at 0 dBm transmission range oftmote_sky is 50-meters indoors and 125-meters out-doors [1]. For average supply voltage level of2.85 V, energy consumptions during one backoffperiod (0.32 ms) are shown in Table 3.

According to the specification of tmote_sky mod-ule two AA batteries are needed in order to supplyvoltage between 2.1 and 3.6 V. Battery energy

Table 2Operating conditions for tmote_sky

nom

Supply voltage 2.1–3.6 VCurrent consumption: radio transmitting at 0 dBm 17.4 mACurrent consumption: radio transmitting at �1 dBm 16.5 mACurrent consumption: radio transmitting at �3 dBm 15.2 mACurrent consumption: radio transmitting at �5 dBm 13.9 mACurrent consumption: radio receiving 19.7 mACurrent consumption: idle mode, oscillator off 20 lA

Table 3Energy consumptions per backoff period

Current consumption: radio transmitting at 0 dBm xt 15.8 lJCurrent consumption: radio transmitting at �1 dBm xt 15.0 lJCurrent consumption: radio transmitting at �3 dBm xt 13.8 lJCurrent consumption: radio transmitting at �5 dBm xt 12.6 lJCurrent consumption: radio receiving xr 17.9 lJCurrent consumption: idle mode, oscillator off xs 18.2 nJ

Table 4Energy characteristics of AA batteries

Zinc–carbon 400–900 milli-Amp-hours Non-rechargeableZinc–chloride 1000–1500 milli-Amp-hours Non-rechargeableNickel–cadmium 800 milli-Amp-hours Rechargeable


depends on the battery technology [7] and has theranges shown in Table 4: for cost effectiveness ofour solution we assume that sensing and radio mod-ule is powered with two AA batteries with totalenergy of b = 2.85 * 0.5 * 3600 = 10,260 J.

4. Distributed activity management

Let the clusters contain nbot, nmid, and ntop ordin-ary sensor nodes, respectively, with the packetarrival rate of k per node. (References to specificclusters will use the subscripts bot, mid, and top,respectively.) All nodes have buffers of finite capa-city, L packets for an ordinary sensor node andLbri packets for the two bridge/coordinators; thetop cluster coordinators acts as the network sinkand it does not block packets. Under the activitymanagement algorithm, coordinator broadcastsrequired event sensing reliability [8] and the numberof nodes which are alive. Each ordinary nodes usesthis information to calculate the average time periodbetween the packet transmissions. Since redundantsensors are used, this time period is much largerthan the packet service time. This time period is alsoused as the average value of the sleep period (when

node turns off its radio subsystem, which is the larg-est consumer of energy). In order to avoid synchro-nized wake-up of nodes, sleep time is randomizedusing the geometric probability distribution whichhas the same average value as the average periodbetween packet transmissions. The parameter ofthe geometric distribution will be denoted as Psleep

and average value of sleep time is equal to 1/Psleep

backoff periods. Under known properties of IEEE802.15.4 MAC, Psleep depends on the requestedevent sensing reliability per cluster and number oflive nodes.

The detailed queuing model of two bridged clus-ters with power managed nodes is shown in Fig. 3.If the node’s input buffer is empty upon returningfrom sleep, the node will immediately start a newsleep period. If the node buffer is not empty, thenode will transmit a single packet and go to sleepagain. An input buffer should operate in a push-out regime so that it will always contain the mostrecent data from the sensor. The cluster coordinatortransmits the required event sensing reliability percluster R and the estimated number of live nodesnecessary to achieve it. The nodes use this infor-mation to determine the mean of the geometric

middle cluster

serverawake

serversleeps

L

clusternode 1

serverawake

serversleeps

L

clusternode nmid

top cluster

serverawake

serversleeps

L

clusternode 1

serverawake

serversleeps

L

clusternode ntop

bridge +middle

coordinator

Lbri

networksink +

topcoordinator

bottom cluster

serverawake

serversleeps

L

clusternode 1

serverawake

serversleeps

L

clusternode nbot

bridge +bottom

coordinator

Lbri

Fig. 3. Queueing model of the bridging process among three clusters.


distribution and, subsequently, the duration of eachsleep period.

To analyze the performance in this regime, theapproach described in [3] can be used, but it requiressolving the system of equations that describes allclusters and bridges simultaneously. Instead, wepropose a computationally lightweight techniquein which node populations are calculated one clusterat a time, starting from the cluster which is farthestaway from the sink. Namely, when redundant sen-sors are used, it is reasonable to assume that therequired event sensing reliability per cluster R ismuch lower than the capacity of the cluster, whichmeans that at any given time only a small fractionof nodes is active. The mean number of packets sentby the bridge in a single superframe can be approx-imated with

N ¼ R � SD � tboff ; ð1Þ

where R is the desired event sensing reliability (ex-pressed in packets per second), the beacon intervalBI includes both active and inactive portion of thesuperframe, and tboff denotes the backoff period thatlasts for 0.32 ms when the 802.15.4 network operatesin the ISM band [4]. As the bridge alternates betweenclusters, the active and inactive portions of thesuperframe have equal values; in the default case

when SO = 0 and BO = 1, SD = 0.5BI = 48 backoffperiods. Assuming the event sensing reliability ofR = 10 packets per second per cluster, the bridgesends, on the average, N = 10 · 48 · 0.00032 �0.153 packets per superframe. The middle clustercarries twice that amount of traffic, while the top-most cluster carries three times this amount—andeven that is still less than one half of a packet persuperframe.

Under these assumptions, the probability thatthe bridge will block packets from the nodes in itscluster is negligible. Furthermore, the probabilitythat the bridge will succeed in delivering all of itspackets to the upper cluster coordinator during asingle visit to that cluster is close to one; as a result,the bridge buffer will be empty after returning to itscluster. These notions allow for a simplified analy-sis, as follows.

Mean duration of period between node transmis-sions, as determined by each node, as

Bbot ¼ nbot=ðtboff RÞ; ð2ÞBmid ¼ nmid=ðtboff RÞ; ð3ÞBtop ¼ ntop=ðtboff RÞ; ð4Þwhere ni is the number of nodes in a particular clus-ter (bottom, middle or top) and R is the requiredevent sensing reliability.


In order to ‘map’ this value to the actual sleepprobability for a node, we must find the probabilityQc that a new sleep will immediately follow the pre-vious one without a packet transmission. Let pk andqk denote the steady state probabilities that there arek packets in the node buffer immediately uponpacket departure and after returning from sleep,respectively. Then, the conditional probability thatthe Markov point corresponds to a return from thevacation and the queue is empty at that moment is

Qc ¼ q0

XL

i¼0

,qi: ð5Þ

Also, let ak denote the probability that k packetswill arrive to the node buffer during packet servicetime, and let fk stand for the probability thatk packets will arrive to the node buffer during onesleep period. Since network is lightly loaded packetservice time does not exceed a couple of dozens ofbackoff periods and therefore we can assume thata0 � 1 and ak � 0 for k = 1. . .1.

For input buffer capacity of L packets the steadystate equations for state transitions are

q0 ¼ ðq0 þ p0Þf0;

qk ¼ ðq0 þ p0Þfk þXk

j¼1

pjfk�j; for 1 6 k 6 L� 1;

qL ¼ ðq0 þ p0ÞX1k¼L

fk þXL�1

j¼1

pj

X1k¼L�j

fk;

pk ¼Xkþ1

j¼1

qjak�jþ1�l; for 0 6 k 6 L� 2;

pL�1 ¼XL

j¼1

qj

X1k¼L�j

ak�l;

1 ¼XL

k¼0

qk þXL�1

k¼0

pk;

ð6Þwhere subscripts should be used to indicate theactual cluster.

Solving the system (6) gives Qc as function of fk.The probabilities fk of k packet arrivals during thesleep period can be found as follows. First, we findthe moment generating function for the sleep periodas

V �ðsÞ ¼X1k¼1

ð1� P sleepÞP k�1sleepe�sk ¼ ð1� P sleepÞe�s

1� e�sP sleep

:

ð7Þ

Then, the probability generating function (PGF)for the number of packet arrivals to the buffer dur-ing the sleep period can be found as:

F ðzÞ ¼ V �ðk� zkÞ ð8Þ

and

fk ¼1

k!

dkF ðzÞdzk

��z¼0

: ð9Þ

The probability distribution for the total inactivetime of the node has a geometric distribution withthe parameter Qc, applied at the moments whenthe node returns from sleep. The correspondingmoment generating function is

I�ðsÞ ¼X1k¼1

ð1� QcÞQk�1c V �ðsÞk ¼ ð1� QcÞV �ðsÞ

1� V �ðsÞQc

ð10Þ

and the mean value is I ¼ 1=ðð1� QcÞð1� P sleepÞÞ.Finally, by equating the average period between

the transmissions with the average inactive time,we obtain

Bi ¼ I ; ð11Þ

which can be solved for Psleep using the number ofnodes in the cluster, the required sensing reliability,and packet arrival rate per node as independentvariables.

A node that wakes up has to wait for the beaconfor synchronization. As there may be more than onenode in this mode, increased collisions may resultfor the packets sent immediately after the beacon.To avoid this, we introduce an additional waitingtime, the duration of which is uniformly distributedin the range 0–7 backoff periods [9]. The PGFs forthese two synchronization functions are:

S1ðzÞ ¼ ð1=BIÞXBI

i¼0

zi; and ð12Þ

S2ðzÞ ¼1

8

X7

i¼0

zi; ð13Þ

respectively.

5. Modeling the energy consumption interconnected

clusters

While the activity management achieves theextension of the lifetime separately for each cluster,individual cluster lifetimes may differ. If this is thecase, the network lifetime is determined by the


shortest cluster lifetime; it is maximized if all clus-ters die at approximately the same time. In orderto accomplish that, we have looked into the possi-bility of modifying cluster parameters so as toequalize their respective lifetimes.

The algorithm to calculate node population con-siders one cluster at a time in an iterative fashion,starting with the cluster which is farthest away fromthe sink.

As mentioned above, we assume that all trans-missions are acknowledged; if the acknowledgment(ACK) packet is not received within the time pre-scribed by the standard [4], the transmission willbe repeated. Let the PGF of the time intervalbetween the data and subsequent ACK packet betack(z) = z2; actually its value is between aTurn-

aroundTime and aTurnaroundTime + aUnitBackoff-

Period [4], but we round the exponent to the nexthigher integer for simplicity.

According to the standard [4], transmission hasto be preceded with the backoff procedure andtwo clear channel assessments (CCA) during whichthe radio part is in the receiving mode. Only aftersuccessful CCAs, radio module switches to thetransmitting mode. Standard allows m (defaultvalue is m = 5) backoff attempts during whichbackoff windows take values W0 = 7, W1 = 15,W2 = W3 = W4 = 31 (if the battery saving mode isnot turned on). However, under the sleep manage-ment regime, all transmissions will complete inone or two backoff attempts and battery savingmode is not important. The PGF for the durationof jth backoff time prior to transmission is equal to:

BjðzÞ ¼XW j�1

k¼0

1

W jzk ¼ zW i � 1

W jðz� 1Þ : ð14Þ

In order to find energy consumption during thejth backoff attempt we need to switch to Laplace–Stieltjes Transform (LST) by substitution z ¼ e�sxr

(because PGFs do not allow non-integer exponents)and obtain LST:

E�BjðsÞ ¼ e�sxrW i � 1

W jðe�sxr � 1Þ : ð15Þ

Furthermore, let us denote the probabilities thatthe medium is idle on first and second CCA with aand b, respectively, and the probability that thetransmission is successful with c. Note that the firstCCA may fail because a packet transmission fromanother node is in progress; this particular backoffperiod may be at any position with respect to that

packet. The second CCA, however, will fail only ifsome other node has just started its transmission –i.e., the backoff period in which the second CCAis undertaken must be the first backoff period ofthat packet. Note that the first medium access byany node will happen within the first 16 backoffperiods of the superframe.

Let the PGF of the data packet length beGp(z) = zk, and let Ga(z) = z stand for the PGF ofthe ACK packet duration. Then the PGF for thetotal transmission time of the data packet will bedenoted with Dd(z) = z2Gp(z)tack(z)Ga(z); its meanvalue is Dd ¼ 2þ G0pð1Þ þ t0ackð1Þ þ G0að1Þ. The LSTfor the energy consumption during pure packettransmission time is e�skxt . The LST for energy con-sumption during two CCAs is equal to e�s2xr . TheLST for energy consumption during waiting forand receiving the acknowledgement is e�s3xr . Thesame value has the LST for energy consumptionduring reception of the beacon frame which is threebackoff periods long.

Then, the PGF for the time needed for onecomplete transmission attempt including backoffsbecomes:

AðzÞ¼Pm

i¼0

Qij¼0ðBjðzÞð1�abÞÞz2ðiþ1ÞðabGpðzÞtackðzÞGaðzÞÞPm

i¼0

Qij¼0ð1�abÞab

:

ð16Þ

The LST for energy consumption for one transmis-sion attempt then becomes:

E�AðsÞ¼Pm

i¼0

Qij¼0ðE�Bj

ðzÞð1�abÞÞe�s2xrðiþ1Þðabe�s2xr e�s3xrÞPmi¼0

Qij¼0ð1�abÞab

:

ð17Þ

By taking packet collisions into account, theprobability distribution of the packet service timefollows the geometric distribution, and its PGFbecomes:

T ðzÞ ¼X1k¼0

ðAðzÞð1� cÞÞkAðzÞc

¼ cAðzÞ1�AðzÞ þ cAðzÞ : ð18Þ

In this case, mean packet service time can simplybe written as T ¼ T 0ð1Þ ¼ A0ð1Þ

c .The LST for the energy spent on a packet service

time is then equal to:

E�T ðsÞ ¼cE�AðsÞ

1� E�AðsÞ þ cE�AðsÞ: ð19Þ


5.1. Bottom cluster

In the first step, the access probability for a nodein the bottom cluster is approximated as:

sð1Þbot ¼ 1=Ibot; ð20Þ

where exponent in braces indicates the index ofiteration.

Since sð1Þbot is very small and the number of nodesis large, we may estimate the per-cluster arrival rateof medium access events as:

kð1Þc;bot ¼ ðnbot � 1Þsð1ÞbotSD=16: ð21Þ

The probability that the medium is not busy atthe first CCA may, then, be approximated with:

að1Þbot ¼1

16

X15

i¼0

e�ikð1Þc;bot : ð22Þ

The probability that the medium is idle on thesecond CCA for a given node is, in fact, equal tothe probability that neither one of the remainingnbot � 1 nodes has started a transmission in thatbackoff period,

bð1Þbot ¼ e�kð1Þc;bot : ð23Þ

By the same token, the overall probability of successof a transmission attempt is:

cð1Þbot ¼ ðbð1ÞbotÞ

Dd : ð24Þ

Using this value, the access probability is thenrevised to:

sð2Þbot ¼ 1=ðIbotcð1ÞbotÞ; ð25Þ

and a new iteration cycle to calculate að2Þbot, bð2Þbot, andcð2Þbot is undertaken. As the success probability underlow load is close to one, only a few iterations sufficeto obtain convergence. (In order to distinguish thelimiting values of medium descriptors from thosefor a particular iteration, we will use the same lettersbut without superscripts.)

The PGF of the time needed to conduct onetransmission attempt is then obtained by substitut-

ing að2Þbot, bð2Þbot and cð2Þbot in Eq. (18). The LST for theenergy spent in packet service is obtained by substi-tuting those values in Eq. (19). Average value ofenergy consumed for packet service is obtained as:ET;bot ¼ � d

ds E�T;botðsÞjs¼0.Finally, the average battery energy consumption

per backoff period can be found as:

ubot ¼S1xr þ S2xr þ 3xr þ ET;bot þ Ibotxs

S1 þ S2 þ 3þ T bot þ Ibot

: ð26Þ

Given the battery budget of b J, the average num-ber of transmission/sleep cycles in bottom clustercan be found as:

nc;bot ¼b

S1xr þ 3xr þ S2xr þ ET;bot þ Ibotxs

� �:

ð27Þ

Given the law of large numbers [10], the PGF fortotal lifetime of the node in bottom cluster becomes:

LbotðzÞ ¼ ðS1ðzÞS2ðzÞT botðzÞIbotðzÞÞnc;bot : ð28Þ

By differentiating the respective PGFs we can obtainthe standard deviation of the node lifetime as well asthe coefficient of skewness, l, which measures thedeviation of a distribution from symmetry [11].

5.2. Middle cluster

The iterative procedure is then applied to themiddle cluster, where we must account for the pres-ence of the bridge, i.e., the coordinator from thebottom cluster. We start by solving (11) for the sleepprobability in the upper cluster, while keeping nmid

as a parameter. The initial node access probabilityin the upper cluster is estimated as a function ofnmid, i.e.,

sð1Þmid ¼ 1=Imid: ð29Þ

Then, the impact of the bridge CSMA access in thetop cluster is modeled as:

sð1Þbri;mid ¼ nbotsbotSD=16: ð30Þ

The success probability for bridge transmissionsdepends on all the nodes in the cluster,

cð1Þbri;mid ¼ ð1� smidÞDd nmid : ð31Þ

The process is then repeated to obtain revisedaccess probability for the bridge as:

sð2Þbri;mid ¼ sð1Þbri;mid=cð1Þbri;mid:

Again, only a few iterations are needed to reachsatisfactory accuracy. The medium access event ratefor a middle cluster node must also account for boththe ordinary nodes and the bridge, hence:

kð1Þc;mid ¼ ðnmid � 1Þsð1ÞmidSD=16þ sbri;mid: ð32Þ


Parameters a, b and c can, then, be calculated in asimilar way as their bottom cluster counterparts,i.e.:

að1Þmid ¼1

16

X15

i¼0

e�ikð1Þc;mid ; ð33Þ

bð1Þmid ¼ e�kð1Þc;mid ; ð34Þ

cð1Þmid ¼ e�kð1Þc;mid

Dd : ð35Þ

The PGFs for a single transmission attempt andfor the overall packet transmission time can be cal-culated as AmidðzÞ and Tmid(z), respectively. BothPGFs depend on the number of nodes nmid as theparameter. Finally, average battery energy con-sumption per backoff period is calculated as:

umid ¼S1xr þ S2xr þ 3xr þ ET;mid þ Imidxs

S1 þ S2 þ 3þ T mid þ Imid

: ð36Þ

Now, if the lifetime of the middle cluster is to bethe same as that of the bottom cluster, the averageenergy which node consumes per backoff period inboth clusters should have equal values:

umid ¼ ubot ð37Þfrom which we can obtain the initial population ofthe middle cluster nmid. This equation is necessaryonly if we want to choose nmid in order to equalizethe lifetimes of bottom and middle cluster. Other-wise, nmid can be chosen using some other policy.

Given the battery budget of b backoff periods,the average number of transmission/sleep cycles inbottom cluster can be found as:

nc;mid ¼b

S1xr þ 3xr þ S2xr þ ET;mid þ Imidxs

� �:

ð38ÞThe PGF for total lifetime of the node in bottomcluster becomes:

LmidðzÞ ¼ ðS1ðzÞS2ðzÞT midðzÞImidðzÞÞnc;mid : ð39Þ

Table 5Initial network parameters for R = 10

Parameter Top Middle Bottom

Number of nodes 100 100 100Inactive period (s) 10.00 10.00 10.00Utilization 0.00236 0.00226 0.00218c 0.7872 0.8649 0.9420Lifetime (days) 314.05 325.35 336.78Std. deviation 0.06% 3.09% 2.46%Skewness l 1.33E�9 0.912E�14 1.68E�14

5.3. Top (sink) cluster

The procedure is then repeated for the top clus-ter, starting from

sð1Þbri;top ¼ ðnbotsbot þ nmidsmidÞSD=16: ð40Þ

We note that this algorithm can easily be scaled tonetworks with several clusters and/or several levels,provided that the clusters are not operating in thesaturation condition.

6. Performance of activity management under

uniform node populations in all clusters

In order to verify the algorithms for distributedcalculation of the sleep interval, for finding thetop population, and for determining the lifetime ofthe network, we have obtained numerical resultsusing Maple 10 from Maplesoft. As mentionedabove, the clusters are interconnected by the mas-ter–slave bridges which also perform the duties ofthe lower cluster coordinators.

The network operates in the ISM band at2.4 GHz, with the maximum data rate of 250 kbps.Packet arrivals to each node follow the Poisson dis-tribution with the arrival rate of k = 1 packet persecond. The packet size is fixed at 30 bytes, includ-ing all PHY and MAC layer headers. Ordinarynodes have buffers that can hold a two packets,L = 2, while the capacity of the bridge buffer isLbri = 6 packets. Event sensing reliability in eachcluster were set to R = 10 and R = 12 packets persecond respectively; note that the middle and topclusters effectively support the traffic load of 2R

and 3R packets per second, respectively. The batterybudget for ordinary nodes is set to b = 10,260 J,while the coordinators/bridges were assumed tohave infinite power supply, as explained above.Each node maintains the counter of the remainingbackoff periods which counts down for each backoffperiod when the radio subsystem was turned on. Allother parameters are set to default values prescribedin the 802.15.4 standard [4].

The number of nodes in each of the clusters wasset to 100, and initial network parameters were cal-culated; the corresponding values that describe thestate of the network for R = 10 when all nodes arealive is given in Table 5. As can be seen, the stan-dard deviation of the node lifetime is small, andthe skewness l is close to zero. The obvious conclu-sion is, then, that all nodes in a given cluster will die

Lifetimes in source, middle and sink cluster

150

200

250

300

350

400

450

40 60 80 100 120 140 160 180population in source cluster


150

200

250

300

350

400


Relative deviation of lifetime in source, middle and sink cluster

0.00054

0.00056

0.00058

0.0006

0.00062

0.00064

0.00066

40 60 80 100 120 140 160 180

population in source cluster


0.00052

0.00054

0.00056

0.00058

0.0006

0.00062

0.00064

40 60 80 100 120 140 160 180


Fig. 4. Cluster performance with identical initial population in all three clusters. Solid line represents source cluster, dashed line representsmiddle cluster and dotted line represents sink cluster. (a) Average lifetime in days for R = 10. (b) Average lifetime in days for R = 12.(c) Ratio of standard deviation and average lifetime for R = 10. (d) Ratio of standard deviation and average lifetime for R = 12.

Table 6Initial network parameters revised for R = 10

Parameter Top Middle Bottom

Number of nodes 109 104 100Inactive period (s) 10.799 10.399 10.00Utilization 0.00218 0.00217 0.00218c 0.7872 0.8649 0.9420Lifetime (days) 333.12 333.86 336.78Std. deviation 0.061% 0.059% 2.465%Skewness l 1.38E�9 1.3E�9 1.68E�14


within a short interval centered around the meanlifetime value for that cluster.

The numerical results for three clusters withidentical initial population of nodes are shown inFig. 4. We can notice that lifetime decreases withincreased remote traffic load in the cluster and dem-onstrates the need for population balancing. Thedifferences in the lifetime are much more expressedas the event sensing reliability grows. Also, rela-tively short sleep times are the result of the highpacket arrival rate per node which was set tok = 1 and scheduling policy which will transmitthe packet if node’s buffer is not empty uponwake-up. Sleep time would be much larger for smal-ler packet arrival rates. Ratio of standard deviationand average value of the lifetime is of order of a per-cent or less which shows that despite of randomiza-

tion of the sleep time, all the nodes will die within avery short time period. This ratio decreases whenrequired event sensing reliability increases, whichis expected since the total number of transmissioncycles decreases.


7. Performance of cluster lifetime equalization

Our second experiment includes the adjustment ofthe number of nodes in the top cluster. For eventsensing reliability per cluster R = 10, and using

populations in source, middle and sink cluster

40

60

80

100

120

140

160

180

40 60 80 100 120 140 160 180


1

1

1

1

1

2


200

250

300

350

400

450

40 60 80 100 120 140 160 180


1

2

2

3

3

4


0.00054

0.00056

0.00058

0.0006

0.00062

0.00064

0.00066

0.00068


R

0

0

0

0

0

0

0

a b

c

e

d

Fig. 5. Cluster performance with initial population in all three clusters tcluster, dashed line represents middle cluster and dotted line represents spopulations for R = 12. (c) Average lifetime in days for R = 10. (d) Avand average lifetime for R = 10. (f) Ratio of standard deviation and av

nbot = 100 nodes in the bottom cluster, we havesolved (37) to obtain the nmid = 104 nodes in the mid-dle cluster, and ntop = 109 nodes in the top cluster.Initial network parameters for all three clusters forR = 10 are shown in Table 6, with node count

populations in source, middle and sink cluster

40

60

80

00

20

40

60

80

00

40 60 80 100 120 140 160 180



50

00

50

00

50

00

40 60 80 100 120 140 160 180


elative deviation of lifetime in source, middle and sink cluster

.00052

.00054

.00056

.00058

0.0006

.00062

.00064

.00066


f

uned to achieve equal average lifetime. Solid line represents sourceink cluster. (a) Initial node populations for R = 10. (b) Initial nodeerage lifetime in days for R = 12. (e) Ratio of standard deviationerage lifetime for R = 12.


rounded to the next highest integer. The numbersshow that we have extended the lifetime of thethree-cluster network by 7% by increasing the num-ber of nodes by only 4.3% (313 vs. the original 300).Interestingly enough, the values for node utilizationand individual node lifetime do not differ much acrossclusters, due to the longer sleep times made possibleby the increase in the number of nodes in the upwarddirection. While the skewness values are different, itshould be noted that all three values are well below0.1%, and it is the vicinity to zero that counts.

The numerical results for three clusters with bal-anced initial population of nodes are shown inFig. 5. We can notice that all clusters now havealmost the same average lifetime. We also note thatnetwork lifetime decreases when requested eventsensing reliability increases. The ratio of standarddeviation with cluster lifetime is now even smallerwhich shows that network will operate with fullnode population almost up to its death.

8. Conclusion

We have developed simplified analytical expres-sions to model the lifetime performance of multi-level 802.15.4 clusters. The model can be used forestimating the initial node population which leadsto the equalization of cluster lifetime and, thus,maximizes the lifetime of the entire multi-level net-work. The calculation is computationally efficient,and might even be suitable for topology controland restructuring of existing networks. Validity ofthe model is confirmed through numerical resultsconducted for tmote_sky ultra low power IEEE802.15.4 compliant wireless sensor module [1].Numerical results show that simple adjustments ofnode population can give cost-effective increase innetwork lifetime.

Acknowledgement

This research is partly supported by the NSERCDiscovery Grant.

References

[1] tmote sky lowpower wireless sensor module, Moteiv Corpo-ration, San Francisco, CA, www.moteiv.com, tmote data-sheet 802.15.4, 2006.

[2] J. Frolik, QoS control for random access wireless sensornetworks, in: Proceedings of WCNC 2004, Atlanta, GA,March 2004.

[3] J. Misic, S. Shafi, V.B. Misic, Maintaining reliability throughactivity management in an 802.15.4 sensor cluster, IEEETransactions on Vehicular Technology 55 (3) (2006) 779–788.

[4] Standard for part 15.4: Wireless MAC and PHY specifica-tions for low rate WPAN, IEEE, New York, NY, IEEE Std802.15.4, October 2003.

[5] J. Misic, S. Shafi, V.B. Misic, Cross-layer activity manage-ment in a 802.15.4 sensor network, IEEE CommunicationsMagazine 44 (1) (2006) 131–136.

[6] M. Yarvis, N. Kushalnagar, H. Singh, A. Rangarajan,Y. Liu, S. Singh, Exploiting heterogeneity in sensornetworks, in: Proceedings of INFOCOM05, vol. 2, Miami,FL, March 2005, pp. 878–890.

[7] http://en.wikipedia.org:80/wiki/AA_battery, ‘‘Aa battery’’.[8] Y. Sankarasubramaniam, O. B. Akan, I.F. Akyildiz, ESRT:

event-to-sink reliable transport in wireless sensor networks,in: Proceedings of the 4th ACM MobiHoc, Annapolis, MD,June 2003, pp. 177–188.

[9] J. Misic, S. Shafi, V.B. Misic, Avoiding the bottlenecks in theMAC layer in 802.15.4 low rate WPAN, in: ProceedingsHWISE2005, vol. 2, Fukuoka, Japan, July 2005, pp. 363–367.

[10] G.R. Grimmett, D.R. Stirzaker, Probability and RandomProcesses, second ed., Oxford University Press, Oxford, UK,1992.

[11] P.Z. Pebbles Jr., Probability, Random Variables, andRandom Signal Principles, McGraw-Hill Inc., New York,NY, 1993.

Jelena Misic received her Ph.D. degree inComputer Engineering from the Uni-versity of Belgrade, Yugoslavia, in 1993.She is currently Associate Professorof Computer Science at the Universityof Manitoba in Winnipeg, Manitoba,Canada. Her current research interestsinclude wireless networks and security inwireless networks. She is a member ofIEEE and ACM.

http://www.moteiv.com

http://en.wikipedia.org:80/wiki/AA_battery

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