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On Efficiency and Validity of Previous HomeplugMAC Performance Analysis

Cristina Cano and David Malone

Abstract—The Medium Access Control protocol of Power LineCommunication networks (defined in Homeplug and IEEE 1901standards) has received relatively modest attention from theresearch community. As a consequence, there is only one analyticmodel that complies with the standardised MAC procedures. Weidentify two important limitations of the existing analyti c model:high computational expense and predicted results just prior tothe predicted saturation point do not correspond to long-termnetwork performance. In this work, we present a simplificationof the previously defined analytic model of Homeplug MAC ableto substantially reduce its complexity and demonstrate that theprevious performance results just before predicted saturationcorrespond to a transitory phase. We determine that the causesof previous misprediction are common analytical assumptionsand the potential occurrence of a transitory phase, that we showto be of extremely long duration under certain circumstances.We also provide techniques, both analytical and experimental,to correctly predict long-term behaviour and analyse the effectof specific Homeplug/IEEE 1901 features on the magnitude ofmisprediction errors.

Index Terms—Power Line Communications, Homeplug, IEEE1901, mean field analysis, decoupling approximation.

I. I NTRODUCTION

Data transmission using electrical wires, known as PowerLine Communication (PLC), has the potential to become a use-ful complement and strong competitor to wireless networkingsolutions. The appeal of PLC relies in the high data rates itcan deliver, its low deployment cost (as it is easy to retrofit)and allows communication through obstacles that commonlydegrades wireless signals. Additionally, it also providesa low-cost alternative to complement existing technologies to reachubiquitous coverage. For instance, as a backhaul for wirelesssensor networks or small cells.

Research efforts have been mostly focused on the physicallayer as the characteristics of PLC channels (including fading,impulsive noise and hidden/exposed terminal problems) im-pose several challenges on physical aspects [1]. However, theyalso have an impact on the Medium Access Control (MAC)protocol, which, in contrast, has not received much attention.

PLC standards (we focus on Homeplug [2] and IEEE1901 [3]) define a MAC procedure similar to the DistributedCoordination Function (DCF) defined in the IEEE 802.11standard for Wireless Local Area Networks (WLANs) [4].PLC MAC protocols derive from the vanilla DCF by addinga deferral counter that reduces the attempt rate when highcontention is inferred on the channel (i.e., a certain numberof packets are overheard). Despite being a substantial change

C. Cano and D. Malone are with the Hamilton Institute, Na-tional University of Ireland, Maynooth, Co. Kildare, Ireland. E-mails:{cristina.cano,david.malone}@nuim.ie

over DCF, this feature has not been deeply evaluated. Attemptsto compare DCF and PLC MAC protocols have been made in[5], [6], [7] and [8]. However, there is still much work to doin order to fully demonstrate the conditions under which thedeferral counter improves the performance of the network.

Contributions to the analysis of PLC MAC protocols aimto extend the research on performance evaluation of PLCnetworks, crucial to advance understanding and optimisation.The only complete analytic model of Homeplug MAC ispresented in [9]. This analytic model has been widely used[10], [11], [12], [13], [14], [15], however we identify two mainissues. The first is the high computational complexity, makingit unsuitable for online use or study of complex scenarios.Second, as we will demonstrate, the performance resultsobtained in the regime right before the predicted saturationpoint do not correspond to the long-term network behaviour.In detail, the main contributions of this work are the following:

• We propose a simplified analytic model for HomeplugMAC that provides a 2 orders-of-magnitude improve-ment in runtime compared to the one presented in [9].Specifically, we: i) apply a renewal reward approach[16], [17], ii) allow the most expensive operations to beprecomputed andiii) provide and evaluate the accuracy ofan exponential approximation to the probability to deferin a given backoff stage.

• We demonstrate that the results right before the predictedsaturation point presented in [9] correspond to a transitoryphase of the system instead to the stable (stationary)behaviour. We identify the two causes of misprediction in[9]: i) the decoupling approximation under infinite buffersize considered in the analytical model andii) the pres-ence, under certain circumstances, of an extremely long(of the order of magnitude of hours) transitory phase inexperimental evaluation. This is the first work that showsthe long duration of that transitory phase and, indeed, thecauses of its occurrence suggest it may also be presentin generic random access protocols. Having identifiedthe causes of misprediction, we provide techniques togenerate valid results from analytic models using thedecoupling approximation as well as from experimentalstudies based on simulations.

• We evaluate the impact of the deferral counter on themagnitude of misprediction errors by evaluating thedifferent solutions obtained considering:i) the startingvalues of the deferral counter proposed by the standard,ii) no deferring (as done in IEEE 802.11 DCF) andiii)always deferring after overhearing following the proposalpresented in [18].

2

The remainder of this article is organized as follows. InSection II, we review the Homeplug MAC procedure. Next,in Section III, related work on PLC MAC analytic modelsis discussed. Our simplified analytic model is described inSection IV. Then, in Section V, we demonstrate that theprevious performance results right before predicted saturationdo not correspond to the long-term behaviour of the network.The performance evaluation, including the validation of thepresented analytic model, an evaluation of the effect of thedeferral counter on misprediction errors and a complexitycomparison, are presented in Section VI. Finally, some finalremarks are provided.

II. H OMEPLUG MAC BACKGROUND

The Homeplug MAC protocol is similar to the DCF definedin the IEEE 802.11 standard [4]. Each time a node has a newpacket to transmit, the backoff stage (i ∈ [1,m])1 is initialisedto 1 and a random backoff is selected among[0,W1]. Thebackoff countdown is frozen when activity is detected on thechannel and restarted when the medium becomes idle again.The packet is actually transmitted when the backoff countdownexpires. If an acknowledgement is received, the packet isconsidered successfully transmitted. Otherwise, the nodestartsthe retransmission procedure: the backoff stage changes toi = min(i + 1,m) and a new random backoff is selectedamong[0,Wi], beingWi the contention window of stagei.

Additionally, in the Homeplug and IEEE 1901 MAC, a newcounter, called the Deferral Counter (DC), is introduced. Thiscounter is initialised at each backoff stage toMi and decreasedby one after overhearing a data packet or a collision. If a newpacket or a collision are overheard and the value of the DC isequal to zero, the node acts as if a collision had happened: thebackoff stage is increased if it has not yet reached its maximumvalue and a new backoff is selected among[0,Wi]. The goal ofthe DC is to avoid collisions when high contention is inferredby decreasing the aggressiveness of transmission attempts.

To provide channel access differentiation, four access cat-egories (CA) are defined CA0–3. CA3 and CA2 shareWi

and Mi values, as do CA1 and CA0 (see Table I). Twoslots (called PRS0 and PRS1) are allocated to allow nodesto announce the priority of their transmissions. The highestpriority (CA3) is associated to both PRS0 and PRS1, the CA2category is associated to PRS0 only, CA1 to PRS1 and thelowest access category (CA0) does not have any notificationinterval associated. Following this approach, stations know ifthere is a station with a frame that belongs to a higher category.In such a case, they postpone their transmissions until the highpriority frames are released. However, in this work we considerthe case where all stations contending for the channel belongto the same access category, as assumed in [9].

III. R ELATED WORK ON HOMEPLUG MAC M ODELING

The analytic model in [9] takes into account all the featuresof the Homeplug MAC except the channel access prioritisa-tion. It models the node access procedure as a 3-dimensional

1Actually, (i ∈ [0,m−1]) but indexes have been relabelled here for clarityof illustration.

TABLE IPARAMETERS OF THE DIFFERENT ACCESS CATEGORIES INHOMEPLUG

Parameter All CAs Parameter CA3/2 CA1/0M1 0 W1 7 7M2 1 W2 15 15M3 3 W3 15 31M4 15 W4 31 63

Markov Chain in which the backoff and deferral counters, aswell as the different backoff stages are considered. To solve forthe stationary probabilities of the Markov Chain an iterativenumerical method is required. Then, to compute the channelaccess delay, the authors follow a recursive approach, i.e., thedelay at every backoff stage depends on the delay of previousstages. To obtain this performance metric, further iterations areneeded. Moreover, to compute the channel access probabilitiesas well as the delay, computationally expensive operationsarerequired. The reason for this is the geometric nature of thedeferral counter expiration and its dependence on the selectedrandom backoff. The authors also extend the model to considerunsaturated conditions. However, the approach taken is basedon further iterations in which the value of the transmissionattempt rate is reweighted based on the offered load at everyiteration until a reasonable approximation is obtained from thesaturated model.

Two other analytic models of Homeplug exist in the liter-ature ([19] and [18]). However, they model mechanisms thatdiffer from the original standardised Homeplug procedure.In[19], the value of the contention window is fixed for all backoffstages, while in [18] the backoff stage is incremented everytime a new packet is overheard (thus havingMi = 0, ∀i).These new approaches obviously lead to a simplified analysis,as one of the dimensions of the Markov Chain is removed.

Consequently, the analytic model in [9] has been widelyused both to derive performance metrics and as a basis forextension [10], [11], [12], [13], [14], [15] as it strictly followsthe procedure defined in the standard. This widespread usemotivates our improvements in terms ofi) reducing complexitywhile retaining features of the standard [2] andii) under-standing the transitory nature of the results predicted beforesaturation in [9].

IV. SIMPLIFIED HOMEPLUG ANALYTICAL MODEL

In this section, we present the simplified analytic model ofthe Homeplug MAC protocol. To this end we take a renewalreward approach [16], [17] motivated by the fact that theattempt rate of a given node can be viewed as a regenerativeprocess. This approach allows us to compute metrics of interestwithout the need to solve all state probabilities of the MarkovChain embedded in the analysis. Similar approaches havealready been proposed to simplify analytic models of the IEEE802.11 for WLANs, see for instance [20] and [21].

We also apply the decoupling approximation to modelthe conditional (given that a packet is transmitted) collisionprobability independently of the backoff stage, as also donein [9] and in a large number of IEEE 802.11 analytic models(see [22] for a complete survey on this topic). Although thisassumption may not be valid [23], in a range of circumstances

3

it allows us to simplify the analysis while still accuratelypredicting the metrics of interest. The decoupling approxima-tion is also used to model the buffer occupancy probabilityright after a transmission as independent of the backoff stageat which the packet was transmitted. This assumption isimplicitly made in [9]. The accuracy of this assumption willbe discussed in detail in Section V as it is contributing factorto misprediction of results right before saturation in [9].

The rest of assumptions and considerations taken intoaccount are as in [9]:i) an infinite, or large enough to beconsidered infinite, buffer size and retry limit,ii) exponentiallydistributed interarrival of packets,iii) ideal channel conditions,iv) contention among a single access category andv) all nodesare in mutual coverage range, that is, all nodes can overheareach other’s transmissions.

A. Renewal Reward Approach

Assuming an infinite buffer size, the mean queue occupancy(ρ) of a node is derived considering the time needed to releasea packet from the queue (X), called service time or MACaccess delay, and the packet arrival rate from the network layer(λ) asρ = min(λX, 1). While λ depends on the application,the service time is computed as the sum of the following threecomponents:i) the total average backoff duration until thesuccessful frame transmission,ii) the total time on averagespent in transmitting packets that result in a collision andiii)the time spent successfully transmitting the packet:

X = E[w]α + (nt − 1)Tc + Ts. (1)

whereE[w] denotes the average number of slots in backoff,α is the average slot duration andnt is the average numberof attempts to successfully transmit a packet. The durationof a successful transmission (Ts) and a collision (Tc) arecomputed as shown in Eq. 2. These durations account for thetransmission notification intervals (PRS0 and PRS1), the timeto transmit the actual frame (Tfra) and the acknowledgement(Tres), as well as the contention and response interframe spaces(CIFS and RIFS)2, see [2] .

Ts = Tc = PRS0+ PRS1+ Tfra + RIFS+ Tres + CIFS. (2)

Under the decoupling assumption with an infinite numberof retries, the average number of attempts to transmit a frame(nt) is computed as shown in Eq. 3.

nt =1

1− p. (3)

where the conditional collision probability (p) is obtainedas the complementary of having at least one of the othern−1nodes transmitting a frame in the same slot (Eq. 4), withτdenoting the attempt rate of a node.

2In contrast to [9], we have consideredTs = Tc to closely model thestandard procedure. Observe that, in case of collision or frame errors, either anNACK is received or nodes wait an EIFS to provide protection from collisionsfor ongoing transmissions [2]. Thus, the duration of a collision can be betterapproximated to that of a successful transmission.

p = 1− (1 − τ)n−1. (4)

We view the attempt rate as a regenerative process, wherethe renewal events are when the MAC begins processing a newframe. Thus, we apply the renewal reward theorem (Eq. 5).

τ =nt

E[w] + nt + I. (5)

The termI in Eq. 5 accounts for the number of slots inidle state (when there is no packet waiting in the queue fortransmission) and is computed as the probability of having anempty queue over the probability of a packet arrival in a slot.Considering an M/M/1 queue, we then computeI as in Eq. 6.

I = max

(

1− ρ

1− e−λα, 0

)

. (6)

The average slot duration while the node is in backoff isderived depending on the type of slot that is overheard (Eq. 7).A slot can be empty if no other node transmits (that occurswith pe probability) and, in such a case, its duration isσ.Otherwise, it can be occupied due to a successful transmission(that happens with probabilityps) or a collision (that occurswith pc probability), with durationsTs andTc, respectively.

α = psTs + pcTc + peσ. (7)

Probabilitiesps, pe andpc are obtained as follows:

ps = (n− 1)τ(1 − τ)n−2.

pe = (1− τ)n−1.

pc = 1− ps − pe. (8)

To computeX , it remains to obtain the value of theaverage number of slots spent in backoff until a packet issuccessfully transmitted (E[w]). With this aim, we follow thesame approach as in [9] but we compute the average numberof slots at every backoff stage. The average number of slotswaiting in backoff at backoff stagei (with 1 ≤ i ≤ m) isshown in Eq. 9. This expression is obtained from [9] but ithas been rearranged to account for the number of backoff slotsinstead of the total delay to transmit a packet. The reader isreferred to [9] for details.

In Eq. 9,pb refers to the probability of overhearing anothertransmission while the node is in backoff:

pb = 1− (1− τ)n−1. (10)

A node moves to the next backoff stage (if it has not reachedits maximum value) whenever it fails to transmit a packet. Wedefine a failure here as the case in which either:i) the backoffexpires leading to the transmission of the packet and thereis a collision or ii) the deferral counter expires so that thenode defers its transmission. The probabilities of each of thesesituations depend on the backoff stage as they are a functionof the value ofWi andMi. Thus, we compute the probabilityof failure at backoff stagei as shown in Eq. 11.

p(i)f = p · p

(i)bo + p

(i)defer. (11)

4

E[wi] =1

Wi + 1

[

M2i +Mi

2+

Wi−Mi∑

k=1

{

k−1∑

l=0

(

Mi + l

l

)

(1− pb)lpMi

b pb(l +Mi + 1)

+

Mi∑

l=0

(

Mi + k

k + l

)

(1− pb)k+lpMi−l

b (k +Mi)

}]

.

(9)

wherep(i)bo is the probability that the backoff expires in back-off stagei. On the other side,p(i)defer denotes the probabilitythat the deferral counter reaches zero and a new packet isoverheard in the given backoff stage. This probability is alsoobtained from [9] and it is calculated as:

p(i)defer =

1

Wi + 1

Wi−Mi∑

k=1

k−1∑

l=0

(

Mi + l

l

)

(1− pb)lpMi

b pb. (12)

The probability that the backoff expires is just the comple-mentary of the probability to defer (Eq. 13).

p(i)bo = 1− p

(i)defer. (13)

Thus, the total average number of slots in backoff isobtained by summing the number of slots waiting at eachstage weighted by the probability of moving to that backoffstage (see Eq. 14). Note that the probability to move to a givenbackoff stage is the probability to face a failure in the previousones. An extra term is considered for the last backoff stage asthe node returns to it after every failure. Assuming a packetisretransmitted until it is successfully transmitted the last termfollows.

E[w] = E[w1] +

m−1∑

i=2

E[wi]

i−1∏

j=1

p(j)f

+ E[wm]

m−1∏

j=1

p(j)f

1

1− p(m)f

.

(14)

Finally, having the payload length (L), we obtain thethroughput as:

S = ρL

X. (15)

The previous expressions can be solved by using an iterativenumerical method. If compared to the previous analytic modelin [9], the renewal reward approach allow us to reduce thecomplexity by removing two iteration loops: the one involvingthe calculation of the state probabilities of the Markov Chainas well as the one regarding the total average MAC accessdelay computation.

B. Precomputation of p(i)defer and E[wi]

The most computationally expensive operations of oursimplified analytic model are the computation ofp

(i)defer and

E[wi] (Eq. 9 and 12, respectively). However, both are afunction of just three variables:p(i)defer = f(Wi,Mi, pb) andE[wi] = g(Wi,Mi, pb). SinceWi andMi are fixed parameters

andpb ∈ [0, 1], we can easily precompute the values of thesetwo metrics for each possible value ofWi, Mi andpb.

The error of the approximated results can be reducedby applying a linear interpolation between the two closestprecomputed values ofpb. Results presented in this work havebeen obtained using the precomputed values as they have beenfound indistinguishable (using a step size equal to10−4 forpb) from the ones obtained using Eq. 9 and 12.

C. Exponential Approximation to p(i)defer

We consider a further optional step in reducing the complex-ity of the analytic model. We may calculate the probability todefer a packet in a given backoff stage by approximating boththe backoff countdown and the deferral counter with expo-nential distributions with means chosen to match the actualdistributions. We approximate the backoff counter and thedeferral counter exponentially distributed with rateβi = 2/Wi

andγi = pb/(Mi+1), respectively. Then, we can easily derivethe probability to defer as the probability that the deferralcounter expires first, that is:

p(i)defer =

γiβi + γi

. (16)

Although the approximation considerably reduces the com-plexity of the calculation ofp(i)defer in Eq. 12, it only provides arough estimate as shown in Fig. 1(a) (CA3/2) and 1(b) (CA1/0)for an arbitrary value ofpb. The disagreement is more notablefor high values ofMi. The reason for this result is the inabilityof the approximation to model the cases in which the randombackoff selected at stagei is smaller thanMi. In such a case,regardless of the value ofpb, the backoff counter will alwaysexpire first, yet the exponential approximation always givessome probability that either expire first. However, note thatthe estimation is accurate for certain values ofpb, Mi andWi. As will be shown in Section VI, the approximation doesprovide a good estimate of throughput and delay for certainranges ofp and thus, can be safely used in the cases identified.

V. ON THE PERFORMANCERESULTSRIGHT BEFORE

SATURATION

Before the performance evaluation, in this section, wedemonstrate that the results presented in [9] just prior topredicted saturation correspond to a transitory phase of thenetwork. To do so we refer to previous literature regardingthe decoupling approximation and to show that the coupledsystem of queues under infinite buffer size is unstable in thisregime. Thus, with an infinite queue size, higher throughputthan the one found in saturation, as predicted in [9], cannotbe maintained in the long term. Furthermore, we support this

5

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0.1

0.2

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0.4

0.5

0.6

0.7

0.8

0.9

1

p defe

r

pb

Exact CalculationExp. Approx.M=0, W=7M=1, W=15M=3, W=15M=15, W=31

(a) CA3/2

0 0.2 0.4 0.6 0.8 10

0.1

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p defe

r

pb

Exact CalculationExp. Approx.M=0, W=7M=1, W=15M=3, W=31M=15, W=63

(b) CA1/0

Fig. 1. Comparison ofp(i)defer computed following the exact calculation andthe exponential approximation.

demonstration by experimental evaluation in which we show,under certain circumstances, the extremely long time duringwhich the network remains in the transitory period beforemoving to the stable phase.

A. The Decoupling Approximation

The decoupling approximation allows to make the analy-sis tractable by assuming that each queue can be modelledindependently of the dynamics of the rest of queues in thenetwork. However, one has to be careful when interpretingthe results obtained. As already observed in [22], unsaturatedanalytic models that decouple the queue dynamics can providetwo different solutions in certain regimes. In particular,whenthe packet arrival rate is slightly higher than the maximumload that the system could serve in saturation. These analyticmodels do not consider the number of instantaneous contend-ing stations, i.e., the number of queues that have at least apacket buffered at the same time. Neglecting this fact makesit impossible to identify the actual regime at which the queuesoperate in this region. This is caused by the possibility offacing two extreme cases: the queues being mostly empty

or saturated conditions. On one hand, if a reduced numberof nodes are contending for the channel, the channel accessdelay is small, thus, making unlikely that a high numberof packets accumulate in the buffer for transmission. Thisimplies higher throughput than found in saturation, as theconditional collision probability is reduced. On the otherhand,if a higher number of nodes are contending, the channelaccess delay increases and so does the probability of havingmore packets accumulating for transmission. The effect ofthe latter case, under infinite buffer size, is the saturationof the buffer. Analytical models that do not consider thenumber of instantaneous contending stations are not alwaysable to differentiate among the two different solutions andthus, iterative algorithms may converge to either, dependingon the starting parameters.

1) Which is the stable solution of the network?: As alreadyconjectured in [22], when considering infinite buffer size,thelowest throughput solution is the one that corresponds to thestable behaviour of the network, while the other solution isjust transitory. This has been demonstrated in [24], where acoupled system of parallel queues with infinite buffer sizeand state-dependent service rates is analysed. The systemof queues is found to be unstable when the packet arrivalrate (λ) is higher than the service rate that can be achievedwhen all nodes are simultaneously contending for the channel.This service rate corresponds to the one found in saturation.Observe that, the instability of the queues translates hereto thestability of the network (all queues have at least a packet totransmit). Thus, from a networking point of view, the solutionthat gives the saturation throughput is the only stable solutionof the network. Therefore, the region just before the saturationpoint identified in [9], is actually justafter the stability limitof the queues.

10 15 20 25 30 35 40 45 500

10

20

30

40

50

60

Number of stations (n)

Sat

urat

ed S

ervi

ce R

ate

[pac

kets

/s]

CA3/2CA1/0

Fig. 2. Service rate in saturated conditions.

In order to identify the stability limit regime, Fig. 2 shows,for different number of nodes contending for the channeland access categories, the service rate obtained in saturationconditions. The parameters used are shown in Table II andcorrespond to those in [9], which are defined in Homeplug1.0 [2]. Note that the service rate in saturated conditions isjust µsat = 1/Xsat, where Xsat denotes here the service

6

time considering that all nodes always have a packet readyfor transmission. From [24] (for a homogeneous system), weidentify the stability condition of the system of coupled queuesasλ < µsat. As will be shown in Section VI, when the stabilitycondition of the network is not satisfied, there is a potential forobtaining two solutions from the analytic model. The stabilitylimit is surpassed ifλ is higher than the service rate depicted.It is important to emphasise that the potential for obtainingtwo solutions decreases whenλ >> µsat as the probability ofhaving a large number of the nodes simultaneously contendingfor the channel increases.

When using iterative numerical solvers, setting certain start-ing parameters can provide a particular solution among the twopossible ones. Setting the metrics assuming a highly-loadednetwork starts the numerics closer to the saturated solution.On the contrary, assuming lightly-loaded conditions, starts theiterative loop closer to the transitory phase. When there isonly one solution, the solver converges to it independentlyof starting under any of these initial conditions. We will usethis technique in Section VI, to obtain both solutions fromthe analysis. This technique, although useful to distinguishthe two solutions, is not able to differentiate them when bothare very close. However, we know that after surpassing thestability limit (as described above) the throughput that shouldbe obtained is the one corresponding to saturation.

B. The Long Transitory Phase in Experimental Evaluation

When λ >> µsat, the system rapidly moves to the stablesolution. However, performing an experimental evaluationright after the stability limit can provide wrong results since, aswe have found in this work, the length of the transitory phasecan be extremely long. We have found that if the experimentsare started with the queues empty, it can take a long timeto reach the stable solution since the system has to reach apoint at which a large number of nodes are simultaneouslycontending for the channel and start to have their queues filledwith an increasing number of packets. The possibility of a longtransitory phase for random MAC protocols in this regime waspostulated in [25], however no experimental findings or proofwas provided. Here we take the step of showing experimentallythat for Homeplug MAC this is exactly the case.

For that purpose we have run long simulations with loadjust above the stability limit with queues long enough to beconsidered infinite (1000 packets). From the simulations wehave recorded the instantaneous throughput and the queuesize (maximum, average and minimum) of all the nodesin the network in1s-long time intervals. Simulation resultsare obtained using a custom simulator based on the SENSEframework [26]. We have considered the case in which50nodes with packets belonging to CA3/2 access categories areaccessing the channel, as larger numbers of nodes help ushighlight the discrepancy between the previously presentedresults (see [9] Fig. 6 and 7) and the saturated solution. Webegin the simulations with the queues empty.

Observe in Fig. 3(a,c,e) that, for different values of thepacket arrival rate, there is a sharp change from the transitorybehaviour to the stable solution. In Fig. 3(b,d,f), we zoom in

TABLE IIPARAMETERSHOMEPLUG1.0

Parameter Value in Homeplug 1.0Data rate (R) 14 Mbps

Frame transmission time (Tfra)/L 1153.5 µs/1500 bytesACK transmission time (Tres) 72 µs

Slot time (σ) 35.84 µsData-ACK interframe space (RIFS) 26 µsContention interframe space (CIFS) 35.84 µsTx. indication slots (PRS0= PRS1) 35.84 µs

on the queue behaviour around this sharp change. We see thatthe queues start to fill with packets until the maximum sizeis reached. The conjecture that the transitory phase is of longduration is also supported, as we observe a duration of theorder of hours in the runs shown.

It is important to emphasise that the results presented inFig. 3 are each from a single simulation run. The time at whichthe change in behaviour occurs varies in different runs due toits stochastic nature. To characterise the length of the transitoryperiod one must perform analysis considering the coupleddynamics of the system of queues. Due to its complexity,especially considering the Homeplug MAC access procedure,we leave this analysis as future work.

Because of this long transitory period, care is required indesigning the simulations. One way to obtain the stable resultsis to start with the queues empty, run the simulation for a longtime until the system changes to the stable solution and thenstart taking the statistics of the performance metrics of interest.However, we suggest starting the simulations with a numberof packets preloaded in the queues as a more practical way toforce the system to enter into the stable operation. If the queuesare stable, nodes will be able to release these preloaded packetsin a reasonable amount of time. If the queues are unstable, wehave started the simulation closer to the stationary regime, andwill see the long-term throughput more quickly. We use thistechnique to avoid the transitory period in the next sectioninorder to suppress the effect of the transitory on performanceresults.

VI. PERFORMANCEEVALUATION

In this section, we present the validation of our analyticmodel. The validation is divided according to saturated andunsaturated conditions. The later also show the two differentsolutions the analytic model may provide as well as thestable solution to which the system converges to, obtainedfrom experimental evaluation. We then evaluate the effectof the deferral counter on misprediction errors and, in thelast subsection, we show the complexity reduction of oursimplified analysis.

Simulations are performed, as in the last section, using theSENSE framework and parameters shown in Table II. Resultsshow averages from simulation runs of10000 s.

A. Saturated Conditions

Throughput and channel access delay metrics in saturationconditions for different number of nodes are shown in Fig. 4(a)and 4(b), respectively. Results fromi) the exact computation

7

0 0.5 1 1.5 2 2.5 3

2

2.5

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3.5

4

4.5

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5.5

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x 106

Time [h]

Inst

anta

neou

s T

hrou

ghpu

t [bi

ts/s

]

(a) λ = 8.5 packets/s

0.5 0.52 0.54 0.56 0.58 0.60

100

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ue S

ize

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kets

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t [bi

ts/s

]

(c) λ = 8 packets/s

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anta

neou

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hrou

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]

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15.45 15.5 15.55 15.6 15.650

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ue S

ize

[pac

kets

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←minimum

←average

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Fig. 3. Evolution of instantaneous throughput and queue size (CA3/2,n = 50 nodes).

(obtained from precomputing the values ofp(i)defer andE[wi]),

ii) the exponential approximation top(i)defer andiii) simulationsare depicted. Observe that the results obtained from the exactcalculation are in agreement with the performance predictionsand simulations presented in [9] (slight differences appear dueto ourTs = Tc consideration). Moreover, we also see how the

approximation is able to accurately predict both throughputand channel access delay in all the cases considered foraccess categories CA3/2. For access categories CA1/0, theexponential approximation is accurate for a small number ofcontending nodes (less than15). For a large number of nodes,the approximation provides a rough estimate.

8

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(a) Throughput

0 10 20 30 40 500

0.05

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Number of stations (n)

MA

C A

ccce

ss D

elay

[s]

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CA3/2

Exact CalculationExp. Approx.Simulations

(b) MAC access delay

Fig. 4. Performance results in saturation conditions. Comparison among the exact calculation, the exponential approximation of p(i)defer and simulations.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

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p

MA

C A

ccce

ss D

elay

[s]

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CA3/2

Exact CalculationExp. Approx.

Fig. 5. Channel access delay vs.p.

To study when the exponential approximation is useful,Fig. 5 shows the channel access delay vs. the conditionalcollision probability, obtained using the exact calculation andthe exponential approximation. This plot allows us to identifywhen the exponential approximation gives a good estimate, in-dependent of the exact configuration, since the channel accessdelay as a function of the conditional collision probability isindependent of the number of nodes contending [27]. Thus,we observe that the exponential approximation is accurate foraccess categories CA3/2 for all configurations, while for accesscategories CA1/0 it provides a good estimate whenp < 0.4.

B. Unsaturated Conditions

Results in unsaturated conditions forn = 10, 30 and 50and varying the packet arrival rateλ are shown in Fig. 6.Values of throughput and channel access delay for the differentaccess categories obtained from the exact computation andsimulations are depicted.

For the analytic results, we show two solutions, correspond-ing to beginning the numerics for solving the model with

I = 0 slots (no idle periods between a departure and the nextpacket arrival, labelledAnalysis 1) or I = 1000 slots (long idleperiods between departures and next packet arrivals, labelledAnalysis 2). Observe in Fig. 6 that the model may converge todifferent solutions. As expected, just after the stabilitylimit,the analytic model provides two different solutions dependingon the starting values we provide to the iterative loop. We seethatAnalysis 1 provides the lowest throughput, whileAnalysis2 shows higher throughput.

We use here the technique of preloading the queues as al-ready discussed in the last section. Thus, we have averaged thevalues of10000 s-long simulations with the queues preloadedwith 50 packets. We have compared results fromAnalysis 1and find agreement with the stable solution obtained after thetransitory phase.

Finally, we compare the exponential approximation and theexact calculation (both starting withI = 0 slots) in unsaturatedconditions in Fig. 7. When the system is saturated, we obtainthe accuracy presented in the last subsection. Here it can beobserved how the exponential approximation is able to predictthe saturation point with a difference, for all cases evaluated,smaller than∆λ = 2 packets/s.

C. Effect of the Deferral Counter

We now evaluate the effect of the the deferral counter onthe difference between the two different solutions obtainedfrom the analytic model right after the stability limit. Theseresults allow us to quantify the potential misprediction errors.We consideri) the results with no deferral (i.e., correspondingto a DCF-like access procedure [4]) andii) increasing thebackoff stage every time a packet from a neighbouring nodeis overheard (i.e., the proposal presented in [18] for improvingthe performance of Homeplug MAC).

1) No deferral: To obtain the results without the deferralcounter (i.e.,Mi = ∞, ∀i), we run the analytic modeldescribed in Section IV withp(i)defer = 0, ∀i and E[wi] =(Wi+1)/2. Note that the performance obtained will match thatof DCF with the same parameters. However, for comparison

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(b) MAC access delay CA3/2

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(c) Throughput CA1/0

0 10 20 30 40 50 60 70 800

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cces

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elay

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Analysis 1Analysis 2Simulations

(d) MAC access delay CA1/0

Fig. 6. Performance results in unsaturated conditions. Comparison among the two solutions derived from the exact analysis and simulations.

purposes, we use the MAC parameters of Homeplug MACshown in Table II. We start the numerics for solving themodel as in the last subsection:Analysis 1 corresponds toan initial value ofI = 0 slots andAnalysis 2 correspondsto the results starting withI = 1000 slots. The two differentsolutions for throughput obtained from the analytic model areshown in Fig. 8. Note that compared to the results depictedin Fig. 6, the difference between the two solutions providedby the analytical model is higher in Fig. 8. The saturationthroughput obtained with no deferral is smaller compared tothe one obtained using the deferral counter. Thus, with nodeferral (Fig. 8), the saturation throughput (Analysis 1) andthe throughput obtained under a lower contention assumption(Analysis 2) show a bigger difference than the one found withdeferral (Fig. 6). Thus, under the same configuration, protocolssuch as the DCF have the potential to be affected by highermisprediction errors than Homeplug MAC, especially whensmall values ofW are used as in the case of CA3/2 categories.

2) Deferring always after overhearing : In accordancewith the conclusions drawn for no deferral, we expect thatdeferring always after overhearing will show a smaller dif-ference among the two solutions as this setting increases the

saturation throughput with higher contending nodes, especiallywhen considering an increased number of backoff stages andlarger contention windows. The improvement in the saturationthroughput by settingMi = 0, ∀i has already been studied fordifferent configurations ofi, W and payload lengths (see Fig. 3in [18]). However, in order to perform a direct comparison withthe outcomes presented in previous sections, we show in Fig.9 the results using the same parameters considered throughoutthis work (i.e., those shown in Table I and II) instead ofreferring to results in [18]. Moreover, this configuration allowsus to evaluate the benefit of increasing the backoff stage afterevery packet overheard when the number of backoff stages areset to those recommended by the standard.

Observe that, under these particular settings, the saturationthroughput compared to Fig. 6 is substantially improved butthe potential misprediction errors are still considerable.

D. Reduction of Complexity

Here, we evaluate the reduction in complexity obtainedusing our simplified analysis (with precomputedp(i)defer andE[wi]) and also our proposed exponential approximation in

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elay

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Exact CalculationExp. Approx.

(d) MAC access delay CA1/0

Fig. 7. Performance results in unsaturated conditions. Comparison among the exact analysis and the exponential approximation top(i)defer.

comparison to the analysis presented in [9]. Moreover, wealso compare the time needed to run the analytic models and10000 s-long simulations, as a crucial value of analysis is asa fast way to predict performance metrics. Since the analyticmodels are solved by means of iterative numerical methods, aregular complexity analysis based on the number of operationsis not useful as the number of iterations needed may varyand cannot be predicted in advance. For this reason, we haveobtained the elapsed times to run the analytic models (usingticandtoc commands in Matlab R2011b) and simulations (usingthe time command in Ubuntu). Tests have been performed inan Intel Core i5 at 3 GHz with 6 GB of RAM running a 12.0432-bit Ubuntu operating system. Results of a single run con-sidering10 and50 nodes using access categories CA1/0 and insaturated conditions are depicted in Table III. Obviously,theseresults may be affected by implementation issues. However,ascan be observed, the improvement of our simplified analysis(both the exact analysis and the exponential approximation)is of 2 orders of magnitude. Furthermore, the analysis in [9]has been shown to be of a similar computational expense tosimulations, placing reasonable doubts on its efficiency.

TABLE IIIELAPSEDRUNTIME COMPARISON FOR10/50NODES

n Analysis in [9] Exact Analysis Exp. Approx. Simulations10 584.5 s 3.7 s 1.7 s 165.5 s50 420.0 s 4.2 s 3.5 s 866.2 s

VII. F INAL REMARKS

In this work we have simplified the analytic model ofHomeplug MAC presented in [9]. To that end, we have applieda renewal reward approach to model the channel attempt rateinstead of solving for the state probabilities associated with theMarkov chain. Furthermore, following our approach, the mostcomputationally expensive operations can be precomputed.Wehave considerably reduced the complexity of the analysis,resulting in an orders-of-magnitude improvement in runtime.An exponential approximation has also been proposed andshown to be generally accurate for access categories CA3/2and accurate under a small-conditional-collision-probabilitycondition for access categories CA1/0.

Building upon previous literature on mean field analysis ofrandom access MAC protocols and on stability analysis of

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Analysis 1Analysis 2

(b) Throughput CA1/0

Fig. 8. Throughput in unsaturated conditions with no deferral (Mi = ∞, ∀i). Comparison among the two solutions derived from the exactanalysis.

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Fig. 9. Throughput in unsaturated conditions deferring always after overhearing (Mi = 0, ∀i). Comparison among the two solutions derived from the exactanalysis.

systems of parallel queues with state-dependent service rates,we have demonstrated that the results right before the predictedsaturation point in [9] correspond to a higher-throughputtransitory phase of the network. We have also experimentallysupported this finding and shown the extremely long durationof the transitory period.

The simplified analytic model has been validated in sat-urated and unsaturated conditions. For the latter, we haveshown the two solutions that the analytic model provides rightafter the stability limit as well as the stable solution thatthenetwork converges to by means of experimental evaluation. Asexpected, the stable solution is in agreement with one of thesolutions provided by the analytic model.

In this article we have presented the first long-term per-formance evaluation of Homeplug MAC under infinite (orlarge enough to be considered infinite) buffer size. We havealso highlighted the complexity of analysing a network rightafter the stability limit as analytic models that consider the

decoupling approximation can provide two different solutionsand simulations have to be run for a long time to allow thesystem to move to the stable phase. Moreover, we have alsoprovided techniques to obtain the stable (stationary) results,both analytical and experimentally. Computing the maximumservice rate the network can support allows us to identifythe range of packet arrival rates for which a potential fortwo solutions exist. We have also suggested how to set thestarting parameters of iterative numerical solvers to obtain bothsolutions. Experimentally, we have shown that the transitoryphase can be drastically ameliorated if stations begin withanumber of packets preloaded in their buffers. Indeed, this isa useful technique to identify MAC protocols which may besubject to a transitory phase via simulation.

The potential occurrence of the extremely long transitoryphase in generic random access protocols, as well as thecharacterisation of its duration, are important aspects tobeanalysed. They can provide further insights on the behaviour

12

of networks based on random access protocols and serveas practical information to perform experimental evaluations.However, this study has been left as future work due to thecomplexity of analysing a system of coupled queues with state-dependent service rates.

We have also evaluated the effect of the deferral counteron the magnitude of misprediction errors and shown that,under the same conditions, protocols such as the DCF havethe potential to be affected by higher misprediction errorsthan the Homeplug MAC. On the other side, improvements ofHomeplug MAC that increase the saturation throughput as thenumber of nodes increases show smaller misprediction errors.

ACKNOWLEDGMENT

This work has been partially supported by the ScienceFoundation Ireland grant 08/SRC/I1403 and 07/SK/I1216a.

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Cristina Cano obtained theTelecommunications EngineeringDegree at the Universitat Politecnicade Catalunya (UPC) in February2006. Then, she received herM.Sc. (2007) and Ph.D. (2011) onInformation, Communication andAudiovisual Media Technologiesfrom the Universitat Pompeu Fabra(UPF). Since July 2012, she has

been working as a research fellow at the Hamilton Institute(NUIM) in wireless networks, sensor networks, power linecommunications and MAC layer design.

David Malone received B.A.(mod), M.Sc. and Ph.D. degrees inmathematics from Trinity CollegeDublin. During his time as apostgraduate, he became a member ofthe FreeBSD development team. He iscurrently a SFI Stokes Lecturer at theHamilton Institute, NUI Maynooth.His interests include mathematics ofnetworks, network measurement, IPv6

and systems administration. He is a coauthor ofO’Reilly’sIPv6 Network Administration.