IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 4, APRIL 2013 661

PHY-MAC Cross-Layer Approach to Energy-Efficiency and Packet-LossTrade-off in Low-Power, Low-Rate Wireless Communications

Nikola Zogovic, Member, IEEE, Goran Dimic, Member, IEEE, and Dragana Bajic, Member, IEEE

Abstract—In this paper we analyze energy-efficiency andpacket-loss trade-off at physical and medium access controllayers. The trade-off can be tuned by proper setting of packetlength, transmission power and maximal number of allowedtransmissions per packet. Our approach is multi-objective with-out decision making preferences. We show how energy-efficiencyvs. packet-loss Pareto Frontier can be determined. We presentnumerical results for the case when CC1000 transceiver is used.Contrary to intuition, we find that Pareto Frontier is not acontinual locus, meaning that energy-efficiency and packet-losscan not be traded continually in Pareto Optimality sense.

Index Terms—Energy-efficiency, packet-loss, trade-off.

I. INTRODUCTION

AS a contribution to the global goal of energy-efficiency(E2) improvement, communications systems are ex-

pected to operate at lower energy consumption levels. Thiswill influence the quality of service (QoS) parameters [1]–[3].

There are a number of proprietary medium access control(MAC) protocols and supporting physical (PHY) layer tech-nologies for low-power, low-rate communications, predomi-nantly intended for wireless sensor network (WSN) applica-tions [4]. The standardized answer addressing the same issue isIEEE 802.15.4 std. [5]. Since energy-efficiency is the ultimategoal in WSNs, such protocols suffer the serious unreliabilityproblem, [6].

In this paper we focus on energy-efficiency vs. packet-loss trade-off problem, using packet-loss as reliability mea-sure. Our approach is multi-objective with Pareto Optimalityconcept [7], without decision making preferences. We choosethe packet length l, transmission power pt, and maximalnumber of allowed transmissions per packet m, to trade thetwo objectives in a PHY-MAC cross-layer decision variablespace. Note that in Pareto Optimality sense a change of anydecision variable implies concurrent change of the objectives.For example, contention based MAC protocols causing nu-merous packet collisions are found to be the main reason ofunreliability. But, reduction of the packet collision probabilityimproves both E2 and reliability at the same time [8], makingit inadequate for Pareto decision variable. Moreover, thepacket collision probability can be considered as a parameterin the problem and we assume the case when it is zero. Suchassumption is also valid for contention-free MAC protocols.

Manuscript received November 26, 2012. The associate editor coordinatingthe review of this letter and approving it for publication was E. Liu.

N. Zogovic and G. Dimic are with the Institute Mihajlo Pupin, University ofBelgrade, Belgrade, Serbia (e-mail: {nikola.zogovic, goran.dimic}@pupin.rs).

D. Bajic is with the Faculty of Technical Sciences, University of Novi Sad,Novi Sad, Serbia (e-mail: dragana.bajic@gmail.com).

This work was supported in part by grants TR32043 and III43002 of theMinistry of Education and Science of the Republic of Serbia.

Digital Object Identifier 10.1109/LCOMM.2013.021913.122663

Similar reasoning can be applied to interference caused by theother co-located wireless networks.

Energy-efficiency is measured as total energy per transmit-ted bit of useful data after all transmission attempts Eb,data,tot.It is shown in [9] that for any channel attenuation L andcontrol data length lc, within operating range, there exist a(l∗, p∗t ) pair that minimizes Eb,data,tot, when there are no lostpackets, improving E2 of current practice transmission up to86%. We extend the E2 model presented in [9] by introducinga model for packet-loss e, and evaluating the expected numberof transmissions per packet ne, when m is limited.

Packet-loss tolerance is application-dependent. It rangesfrom 0% for file transfer (e.g. WSN node remote reprogram-ming), over 2-5% for audio streaming, and 5-10% for videostreaming. It was shown in [10] that networked control sys-tems can stay stable even for 70% packet-loss. Such toleranceenables reliability to be traded for other QoS parameters orE2. The transmission power is constrained by regulation rules.The constraints of packet length are due to protocol definition.The maximal number of transmission attempts per packet,m, is usually constrained by acceptable delay. Since we donot consider constrained delay, we let m be unbounded. Theenergy per transmitted bit of useful data directly affects theamount of useful data to be transmitted by the communicationsystem powered by the constrained energy resource, such asbattery operated WSN nodes.

II. ENERGY-EFFICIENCY VS. PACKET-LOSS TRADE-OFF

Each packet consists of packet header and payload, l =lc + ldata, ldata ≥ 0, where ldata is a packet payload sizein bits, l ∈ Nl = N \ {1, ..., lc − 1}, and N is the set ofpositive integers. Transmission power takes values from theinterval P= [pt,min , pt,max ], where pt,min and pt,max dependon power amplifier design. m ∈ N.

In low-rate systems, wireless channel can be characterizedusing a frequency-nonselective, slow fading channel model[11]. We assume Rayleigh block-fading channel with blockssufficiently long to transmit any copy of a packet. Theprobability of packet received with error is independent andidentically distributed. An average signal-to-noise ratio (SNR)at receiver (RX) front-end, γ, stays stable during all trans-mission trials. Then, the expected packet error probability, letpp := E {pp} for brevity, and the probability that the packetwill be lost after k attempts are

pp (l, pt) = pp ≈ 1− exp

(k2 − k1 ln (l)

γ

), (1)

Pr{lossk} = e (l, pt, k) = (pp)k , (2)

1089-7798/13$31.00 c© 2013 IEEE

662 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 4, APRIL 2013

respectively, cf. [12, approx. (13)] for (1), where k1 and k2are modulation specific parameters. Note that pp = pp (l, pt) .The link budget is pt = Lpr, with L being instantaneousattenuation between transmitter (TX) antenna input and RXfront-end, and pr being signal strength at RX front-end.Keeping transmission power constant, average received SNRis

γ =ptTb

LN0, (3)

where Tb is bit duration, N0/2 is two sided noise spectraldensity and L is the average channel attenuation. From (1)and (3) follows

pp (l, pt) ≈ 1− exp

(LN0

k2 − k1 ln (l)

ptTb

). (4)

The probability that TX stops retransmission after k trials,given m, is equal to

Pr {retrk} =

{(1− pp) · (pp)k−1 , k < m

(pp)k−1

, k = m. (5)

Therefore, the expected number of transmissions per packet is

ne (l, pt,m) = (1− pp)

m−1∑k=1

k · pk−1p +m · pm−1

p

=1− pmp1− pp

=1− e (l, pt,m)

1− pp (l, pt). (6)

Extending the model presented in [9, Sec. II],

Eb,data,tot (l, pt,m) = ne (l, pt,m) · l

ldata· TbpTX (pt) , (7)

where pTX (pt) is the dependence of TX power consumptionpTX on power delivered to TX antenna pt. The dependenceis given by Raised-Fractional-Power TX power consumptionmodel [13]:

pTX (pt) = p0 + ρx (pt)v (8)

where p0 = pTX,min is the constant fraction of TXpower used by transmitter electronic circuitry and poweramplifier bias, ρ = pTX,max − pTX,min, x (pt) =(pt − pt,min) / (pt,max − pt,min), pTX,min and pTX,max arethe minimal and maximal total TX power consumption, re-spectively, and v ∈ [0, 1] is the fitting parameter.

To analyze the E2 vs. packet-loss trade-off, we set a multi-objective optimization (MOO) problem, given by:

min {Z (x)}s.t. x ∈ X (9)

where

Z = {z = (Eb,data,tot (x) , e (x)) | x ∈ X},X = {x = (l , pt ,m) | l ∈ Nl , pt ∈ P, m ∈ N}

are objective and decision spaces, respectively. For chosenX, Z is not a continual R2 subset. The fact becomes moreobvious if we try to find Eb,data,tot(l, pt,m) when e(l, pt,m)is fixed, e(l, pt,m) = e∗. Then, for any combination of land m, pt | pt ∈ (pt,min, pt,max) is uniquely determined andthe resolution of Eb,data,tot(l, pt,m) is determined only by

integer variables l and m. Therefore, Eb,data,tot(l, pt,m) |e(l, pt,m) = e∗ does not take values from a continual intervalof reals.

We perform Pareto optimality concept to MOO problem andlook for all Pareto-optimal (PO) points in X and Z spaces. Theproblem falls within the class of nonlinear, nonconvex MOOproblems. It can be solved by the method of Proper EqualityConstraints (PEC) [14], the variant of ε-Constraint method,where all the other objectives, except one, are convertedinto equality constraints. The parametric-equality-constrainedsingle-objective (PECSO) optimization problem, associated toMOO problem, given by (9), is:

min Eb,data,tot (x)

s.t. x ∈ X, (10)

e (x) = e∗, e∗ ∈ (0, 1) .

Let Y = {y = (l , pp ,m) | pp = pp (l, pt) , (l , pt ,m) ∈ X}.Since X → Y is bijection, the problem:

min Eb,data,tot (y)

s.t. y ∈ Y, (11)

e (y) = e∗, e∗ ∈ (0, 1)

has the same solutions in objective space as (10).We solve (11) as follows. For a given e∗ and m∗ we

find p∗p as p∗p = m∗√e∗, according to (2). We calculate n∗

e

corresponding to e∗ by substituting pp and m with p∗p and m∗

into (6), respectively. Let m∗ be a vector of m∗ values andl a vector of l values. Since e

(l, p∗p,m∗) = e∗ for any l, we

evaluate Eb,data,tot

(l, p∗p,m

∗) numerically to find y∗opt:

y∗opt =

(l∗opt, p

∗p,opt,m

∗opt

)= argmin Eb,data,tot

(l, p∗p,m

∗) ,by setting appropriate m∗ and l. Then we calculate optimaltransmission power p∗t,opt by substituting pp and l in (4) withp∗p,opt and l∗opt, respectively. Having that m∗

opt and l∗opt must befinite, since lim e (x) = 0

m→∝, and lim Eb,data,tot (x) =∝

l→∝, respec-

tively, x∗opt =

(l∗opt, p

∗t,opt,m

∗opt

) ∈ X can be found exactlyby setting finite m∗ and l. Since, x∗

opt ∈ X, Eb,data,tot

(x∗opt

)is infimum of Eb,data,tot (x) | x ∈ X, e (x) = e∗, e∗ ∈ (0, 1)and Eb,data,tot

(x∗opt

)is finite, according to [14], x∗

opt isPECSO-optimal solution of (10). Solving (10) for e∗ inthe interval (0, 1), we obtain the set of all PECSO-optimalsolutions that contains the entire set of PO points of MOOproblem (9), see [14]. We obtain the entire Pareto Frontier(PF), the set of all PO points, by selecting all those PECSO-optimal solutions that satisfy the conditions 1 and 2 of theTheorem 1 from [14].

III. THE TRADE-OFF FOR CC1000 BY PEC METHOD

We solve (9) for the communication system based onCC10001 low-power transceiver. Relevant parameters aregiven in Table I. We assume that the output power canbe set to any value between pt,min and pt,max. CC1000supports non-coherent BFSK modulation where k1 = 2 andk2 ≈ 2 · (0.577 + ln(0.5)), cf. [12].

1CC1000 datasheet is available on www.ti.com

ZOGOVIC et al.: PHY-MAC CROSS-LAYER APPROACH TO ENERGY-EFFICIENCY AND PACKET-LOSS TRADE-OFF IN LOW-POWER, LOW-RATE . . . 663

TABLE ICC1000 PARAMETERS

parameter value

pt,min, pt,max 0.01, 3.16 mW

pTX,min, pTX,max 25.8, 76.2 mW

v 0.64

Tb 19200−1 s

N0 1.054 · 10−18 J

L 94 dB

lc 15 Byte

We evaluate solutions of the associated PECSO prob-lem given by (10). Let m∗ = (1, 2, ..., 50), l =(1000, 1001, ..., 10 000) bits, and e∗ ∈ {

10−6, ..., 0.9}

with20 points per decimal resolution. PESCO-optimal and POsolutions of MOO problem given by (9), visualized in theobjective space, are presented in Fig. 1. The problem solutionsvisualized in the decision space are presented in Fig. 2. Sincethe optimal packet length is more than 2500 bits, long packets- sufficient condition for the approximation (1) - is satisfied,cf. [12].

Fig. 1 shows that E2 and packet-loss can be traded for eachother. But, contrary to intuition, PECSO-optimal curve is notmonotone, yielding PO trade-off curve not being continuallocus. It means that transmission with lower packet-loss prob-ability can be more energy-efficient than with higher, e.g. pointG has lower packet-loss probability and it is more energy-efficient than point F. From point A to B, C to D, E to F etc.,PECSO-optimal solutions and PO are achieved by maximallyone, two, three, etc., transmission attempts, respectively. Thereare 1 to 2, 2 to 3, etc. maximal number of transmissionattempts transitions at B to C, D to E, etc. points, respectively.There are PECSO-optimal but Pareto-suboptimal segments ofpoints around the transitions, starting from 3 to 4 transitionwith more evident segments at higher transitions.

Since Z is not a continual R2 subset, PF discontinuityexist wherever the difference of compared PECSO pointsfalls below the resolution of Z space. It can be seen inFig. 1 around the transitions, starting from 3 to 4 transi-tion, when e falls below 1% and Eb,data,tot varies less than10−6 mJ. We also found 68 Pareto suboptimal points ine ∈ (20.787711279004, 20.787711279005)% interval wherewe checked 100 points with resolution 10−16. In that in-terval m changes from 1 to 2. Another example is e ∈(6.17335941636, 6.17335941637)% interval where we found3 Pareto suboptimal points when we checked 100 pointswith resolution 10−15. In that interval m changes from 2to 3. Unlike the case where e is small, resolution of 10−16

around e ≈ 21% or 10−15 around e ≈ 6.2% does nothave practical importance. Moreover, better resolution of PFdetermination will not eliminate its discontinuity, since, oncedetermined Pareto-suboptimal point will stay suboptimal evenif we increase the number of checked points for e∗.

Since MOO related parameters can not affect X nor theycan change the fact that Z is not a continual R2 subset, PFdiscontinuity exist whenever X and Z are defined as in (9),regardless of MOO related parameters.

10−4

10−3

10−2

10−1

100

101

102

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

−3 Objective space, average L = 94 dB

packet−loss [%]

Min

imal

ene

rgy

per

tran

smitt

ed b

it of

dat

a [m

J]

PECSO optimalPO

I

H

FC

B

A

ED

JKL

G

Fig. 1. Objective space - minimal energy per transmitted bit of data vs.packet-loss.

00.5

11.5

22.5

3

2500

3000

3500

4000

4500

50000

2

4

6

8

10

12

optimal transmission power [mW]

Decision space, average L = 94 dB

optimal packet length [bit]

optim

al m

axim

al n

umbe

rof

allo

wed

tran

smis

sion

s

PECSO optimalPO

CE

D

HF

J

G

KL

I

A

B

Fig. 2. Decision space - optimal packet length, maximal number of allowedtransmissions and transmission power settings.

IV. CONCLUSION

We set the E2 vs. packet-loss MOO problem at PHY andMAC layers and find PF - the set of all points with a propertythat there are no other points improving any objective withoutdegrading another one. We show that E2 vs. packet-losstrade-off can be tuned by choosing proper decision variables:packet length, transmission power and maximal number ofallowed transmission attempts. We use CC1000 transmitter toshow an example of the formal PF determination procedure.Contrary to intuition, we find that PF is not continual locus,meaning that not every packet-loss has corresponding POenergy-efficiency and vice versa. Moreover, PF discontinuityexist regardless of related MOO parameters whenever E2vs. packet-loss trade-off is controlled by the three decisionvariables.

REFERENCES

[1] C. Han et al., “Green radio: radio techniques to enable energy-efficientwireless networks,” IEEE Commun. Mag., vol. 49, pp. 46–54, June 2011.

[2] Y. Chen, S. Zhang, S. Xu, and G. Li, “Fundamental trade-offs on greenwireless networks,” IEEE Commun. Mag., vol. 49, pp. 30–37, June 2011.

664 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 4, APRIL 2013

[3] H. Bogucka and A. Conti, “Degrees of freedom for energy savings inpractical adaptive wireless systems,” IEEE Commun. Mag., vol. 49, pp.38–45, June 2011.

[4] K. Langendoen and A. Meier, “Analyzing MAC protocols for low datarateapplications,” ACM Trans. Sen. Netw., vol. 7, pp. 10:1–10:34, Aug. 2010.

[5] “IEEE Standard for Information Technology, Part 15.4; Wireless MediumAccess Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (WPANs),” 2006.

[6] G. Anastasi, M. Conti, and M. DiFrancesco, “A comprihensive analysisof the MAC unreliability problem in IEEE 802.15.4 wireless sensornetworks,” IEEE Trans. Ind. Inf., vol. 7, pp. 52–65, Feb. 2011.

[7] R. Marler and J. Arora, “Survey of multi objective optimization methodsfor engineering,” Structural and Multidisciplinary Optimization, vol. 26,pp. 369–395, Apr. 2004.

[8] G. Anastasi et al., “Reliability and energy-efficiency in IEEE802.15.4/ZigBee sensor networks: an adaptive and cross-layer approach,”IEEE J. Sel. Areas Commun., vol. 29, pp. 1508–1524, Sept. 2011.

[9] N. Zogovic, G. Dimic, and D. Bajic, “PHY-MAC cross-layer approach toenergy-efficiency improvement in low-power communications,” in Proc.2011 Int. Symp. on Wireless Communication Systems, pp. 402–406.

[10] W. Zhang, M. Branicky, and S. Phillips, “Stability of networked controlsystems,” IEEE Control. Syst. Mag., vol. 21, pp. 84–99, Feb. 2001.

[11] J. Proakis and M. Salehi, Digital Communications, 5th edition. McGraw-Hill, 2008.

[12] S. Liu, X. Wu, Y. Xi, and J. Wei, “On the throughput and optimal packetlength of an uncoded ARQ system over slow Rayleigh fading channels,”IEEE Commun. Lett., vol. 16, pp. 1173–1175, Aug. 2012.

[13] G. Dimic, N. Zogovic, and D. Bajic, “Energy-efficiency of supportiverelay with novel wireless transmitter power consumption model,” in Proc.2012 Int. Conf. on Selected Topics in Mobile and Wireless Networking,pp. 1–6.

[14] J. Lin, “Multiple-objective problems: Pareto-optimal solutions bymethod of proper equality constraints,” IEEE Trans, Autom. Control, vol.ac-21, pp. 641–650, Oct. 1976.