Markov Chain Model of HomePlug CSMA MAC for Determining Optimal Fixed Contention Window Size

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1 Markov Chai Model of HomePlug CSMA MAC for Determiig Optimal Fixed Cotetio Widow Size Eva Krimiger * ad Haiph Latchma Dept. of Electrical ad Computer Egieerig, Uiversity of Florida, Gaiesville, FL, USA evakrimiger@gmail.com * ad latchma@qcslik.com Abstract This paper aalyzes the optimum costat cotetio widow (CW) for the HomePlug 1. ad AV CSMA/CA MAC. A discrete time, homogeous Markov chai, with the state specified by both the backoff couter (BC) ad deferral couter (DC), is used to model a sigle ode cotedig for trasmissio. The structure of the Markov chai admits a geeralized expressio for the statioary state probability mass fuctio (pmf) associated with each state. The recursively defied state pmfs ca be aalytically reduced to a sigle expressio relatig the probability p, of the ode fidig the medium idle, the maximum widow size W, the maximum deferral couter size, ad the umber of odes. Optimizig the MAC efficiecy provides a target value for p, which ca be attaied with the proper selectio of W ad. It is show that a optimal cotetio widow size ca be chose based o a liear relatioship with the umber of odes. Keywords--HomePlug 1.; HomePlug AV; cotetio widow; CMSA-CA; medium access cotrol I. INTRODUCTION The HomePlug 1. ad HomePlug AV MAC protocols both use a carrier sese medium access with collisio avoidace (CSMA-CA) scheme that offers several improvemets over the IEEE CSMA/CA protocol. Whe a ode wishes to sed, but detects the medium busy, it geerates a backoff time, which is a uiform radom variable betwee ad W-1, iclusive, where W is the widow size. This value decremets by oe after every time slot ad whe it reaches zero, the ode trasmits. Aother parameter, the deferral couter, allows the ode to reduce the risk of collisio whe there is high traffic. The deferral couter is set to λ whe the backoff time is established. For every slot i which the medium is detected busy, the DC decremets by oe. If the DC drops below zero, it is reset to a ew, higher value ad the backoff time is recalculated based o a ew W size that correspods to the ew λ. I this way, the ode chages its backoff to accommodate a busier medium [3]. The MAC efficiecy,η, of this system suffers a sigificat drop whe the umber of cotedig odes, icreases [1], as is expected to be the case as more ad more commuicatio ad evirometal moitorig ad cotrol devices are coected to the PLC grid. As the backoff times icrease, the medium is used iefficietly, with efficiecies as low as.2 for 3-5 odes. A modified MAC has bee proposed that sets W ad λ to costat values, which are selected based o to optimize the MAC efficiecy. II. MODEL A discrete time, homogeous Markov chai show i Fig. 1 is used to model a sigle ode i a system of odes. The assumptio is that all odes are cotedig for the medium, which establishes a worst-case-sceario for etwork traffic. The state pmf is a fuctio of the backoff couter, b, ad the deferral couter, d. It will be represeted as Π(d,b). The probability that the ode detects the medium idle is p ad also serves as the state trasitio probability betwee a ode ad the adjacet ode oe less i backoff cout ad of the same deferral cout. The trasitio probability for odes differig by oe i both deferral cout ad backoff cout is 1- p. For odes which have ru the backoff couter to zero, the trasitio probability to ay of the top row states (deferral couter λ) is 1/W. This trasitio probability correspods to the radom selectio of the backoff time as described i the itroductio. Fially, for states with a zero deferral cout ad ozero backoff cout, the trasitio probability to the top row states is (1-p)/W. This is similar to the previously calculated trasitio probability, but icludes the chace of the deferral cout droppig below zero whe the medium is foud to be busy. The probability that the ode trasmits is p, ad correspods to the probability that the backoff couter has ru to zero. It is give by the sum of all states i which b is zero. Π, 1 Likewise, p ca be related to p, sice the medium is detected idle whe o other odes are trasmittig. 1 2 The object of this aalysis is to elimiate the state pmfs ad obtai a relatio betwee p ad the parameters W, λ, ad. III. ANALYSIS The pmf for each state will be determied. This aalysis will begi at the top row. For coveiece, let S λ represet Π(λ,W-1), the pmf of the rightmost top row state. We will show that all other state pmfs ca be expressed as a fuctio of S λ. The (λ,w-1) state arises from ay of the b = states with probability 1/W ad from the d =, b states with probability (1-p)/W. Therefore, S λ ca be represeted

2 Figure 1 Markov chai model for backoff procedure with fixed cotetio widow. 1 Π, 1 Π,. 3 The other top row states have pmfs of the form Π, Π, 1, for all 2. We ca remove the recursivity i Π(λ,b) by tracig back to the iitial (λ,w-1) state. The Π(λ,b) is oly a fuctio of S λ ad p: Π,, for all 1. For the secod row state pmfs, we will agai start at the rightmost state of this row, (λ-1,w-2). It ca oly be reached through the state (λ,w-1), ad hece has a probability give by Πλ 1,2 1. The ext state to the left, (λ-1,w-3), is reached either through the previous state of this row, (λ-1,w-2), or from the state (λ,w-2) o the higher row. Πλ 1,3 Πλ 1,2 1 Πλ, Usig the same method, the whole row ca be determied. Πλ 1, Πλ 1, Each state (d,b) depeds o (d,b+1) ad (d+1,b+1). Π, Π, 1 1 Π 1, 1 for λ1. It turs out that this recursive structure yields state pmfs which ca be represeted i terms of polyomials i p, powers of 1-p, ad S λ. Furthermore, the coefficiets of the polyomials i p for each row are diagoal elemets of Pascal s triagle. Pascal s triagle elemets are represeted simply with the biomial coefficiet ad therefore we ca write Π, 1 λ 4 for λ. These equatios are still recursive because they rely o S λ. However, (4) is liear i S λ ad if we ca factor out S λ ad equate it to (3) it ca be elimiated. A o-recursive solutio for S λ is admitted by (4) usig the ormalizatio coditio, 1 Π,. 5 It will be easier to costruct this sum from sums of the idividual rows, because the pmfs of each state i a row share a similar form. This sum, which is a fuctio of the deferral cout of the row i questio, will be represeted by R(d). Π, for d. Substitutig (4) ito (6), yields (7). 6

3 1 λ 1 7 Sice the polyomial compoet of the pmf for all states i a row have the same coefficiets, a term p i will be repeated exactly oe time more tha the ext highest power p i+1 i the sum ad this is captured by the coefficiet iside the summatio of (7). With this ew represetatio of a row sum, (5) ca be expressed 1. 8 We ca factor out S λ from the sum i (8) ad solve for S λ. 1 9 Betwee (3) ad (9), the pmf S λ ca be elimiated, leavig us with a expressio relatig oly the desired parameters. However, (3) must be simplified; it cotais sums of state pmfs. The first sum i (3) ca be elimiated by employig (1), which defies p. 1 1 Π, Usig (2), p ca be expressed i terms of p Π, The secod sum i (3) ca be expressed i terms of R(d). 1 Π, Π, λ 1 λ Substitutig (11) ito (1) ad solvig for S λ, we get λ1. There are ow two idepedet equatios for S λ, which are fuctios of W, λ, p, ad. Equatig (9) ad (12) ad ivertig for clarity, completely removes the state depedecy. 1 λ1 1 Substitutig (7), the right side of (13) becomes 1 λ λ 13. Equatio (13) provides a aalytical relatioship betwee W, λ,, ad p. IV. APPROXIMATION OF NODE TRANSMISSION PROBABILITY I [2], the probability of a ode trasmittig is approximated as 2 2 λ, 14 which was calculated from the ratio of states which have a backoff couter value of zero. Notice that this solutio is costat i. A key cotributio of the preset paper is to obtai the exact aalytical solutio by solvig for p i (13) ad usig (2) to obtai p. This solutio is depedet o, ulike the approximatio. For a give value of W ad λ, the approximatio i (14) approaches the aalytical solutio for large. I Fig. 2, the two p values are plotted agaist for a fixed value of W ad λ. The magitude of this error is quite severe for lower umbers of odes. The geeral shape of Fig. 2 is represetative of the p versus curve for all values of W ad λ. Whe these two parameters are chaged, the error betwee the aalytical ad approximate solutios chages as well. Probability of ode trasmissio, p Compariso of p (W = 5, λ = 3) Aalytical Approximate Number of statios, Figure 2 Compariso of p geerated from the aalytical ad approximate solutios.

4 1 Optimal Idle Probability, p 5 Optimal cotetio widow for λ = p Optimal W Figure 3 Optimal probability of fidig the medium idle [2] Figure 4 Optimal cotetio widow as a fuctio of the umber of statios for λ = 3. V. OPTIMAL CONTENTION WINDOW Our parameters will be optimized i the sese of maximizig the MAC efficiecy, as was performed i [2]. Let P S be the probability of successful trasmissio, P I be the probability of idle passage, ad P C be the collisio probability. I terms of p these are The MAC efficiecy ca be approximated by, 15 where T S, T C, ad T I are the times of successful trasmissio, collisio, ad idle slot passage, respectively. The MAC efficiecy ca be optimized i p for a give. The result for this optimizatio give by [2] is 1 1, 1,. 16 The correspodig optimal value of p ca be foud usig (2). Therefore, for a give ad the value of p opt associated with it through (16), the values of W ad λ which best satisfy (13) are optimal i the sese of MAC efficiecy. that remais costat for all λ, ad a itercept that chages with λ. For example, 5 1 for λ 3, ad 5 35 for λ 15. The aalytical solutio reveals the true relatioship. To fid the optimal parameters, we search through the reasoable values of λ ad W. For a give, each combiatio of these parameters will yield a value of p that solves (13). The combiatio of W ad λ that yields p earest to the optimal value will be optimal. If a set of λ ad W are optimal for a give, the the p that results from (13) must be withi 1% of the optimal value. First, for compariso with [2] this search will be coducted by fidig the optimal W for a predetermied λ. Optimal W Optimal cotetio widow for λ = 15 VI. RESULTS 15 1 I determiig the optimal p from (16), the values of T I ad T C will be set to 2 ad 8 μs, respectively, as was doe i [2]. The optimal parameters were foud i [2] by substitutig (14) ito (16) ad coductig a search for the best W ad λ. I particular, the optimal cotetio widow is obtaied as a fuctio of whe the value of λ was fixed at 3 ad 15. The result was a affie relatioship betwee W ad with a slope Figure 5 Optimal cotetio widow for λ = 15.

5 6 Slope of affie segmet 1 MAC efficiecy versus for λ = Slope MAC efficiecy, η W W = 251 W = 59 Stadard HP λ Figure 6 Slope of the best-fit lie as a fuctio of λ. For sufficietly large, the relatioship betwee W ad is affie as i [2], but it is oliear for low. The umber of odes at which liearity ca be assumed icreases proportioally to λ. Sice our work revolves aroud the MAC efficiecy at high, liearity ca be assumed for reasoable values of λ. For compariso, the aalytical result yielded the followig relatioships for the affie segmet, for λ 3, ad for λ 15. I Fig. 4, we have a example of a lower λ, where the optimal W relates to liearly. I Fig. 5 is a example of a higher λ, where the liear relatio fails for small. For λ = 15, we ca assume liearity for > Itercept of affie segmet Figure 8 MAC efficiecy as a fuctio of for three choices of W, compared with the stadard HomePlug 1. MAC efficiecy. The parameters of the best-fit lie for the liear segmet of the -W curve ca be easily visualized by plottig them agaist a chagig λ. From Fig. 6, the slope remais approximately costat for all λ. I Fig. 7 it is apparet that the itercept icreases early liearly with λ. The choice of λ is ot sigificat. We simply select a reasoable value, ad the have the optimal W as a affie fuctio of. I Fig. 8 we show the beefits of adaptively selectig the optimal widow size for a give. For the purposes of compariso, we also show the MAC efficiecy for the stadard HomePlug 1. CSMA/CA protocol. I this plot, λ was set to 3. Three differet selectio schemes for W are show. We first chage W to the optimal value for each, which is approximately for λ = 3. The W is fixed at 251 ad 59, which are its optimal values at = 5 ad = 1, respectively. The MAC efficiecy is determied from (15), usig T S = 11 μs ad T Data = 1 μs, which are reasoable values i accordace to the protocol [3]. By adaptig W with, we maitai a high MAC efficiecy. Ay fixed value of W will suffer serious degradatios i efficiecy for some or all values of. Itercept λ Figure 7 Itercept of best-fit lie as a fuctio of λ. VII. CONCLUSIONS The Markov chai model for HomePlug 1. ad AV CSMA/CA backoff has a solutio free of recursio. Therefore, approximatios ca be avoided i the optimum cotetio widow calculatio. The optimum cotetio widow calculatio provided by [2] uses a simplified probability of ode trasmissio, which approaches the aalytical value oly for large. This approximate solutio geerates a optimum cotetio widow for a give λ that ca be calculated as a affie fuctio of with a costat slope. This leds itself to easy adaptatio usig ode estimatio strategies. The aalytical solutio determies that for a give λ, the cotetio widow is a affie fuctio of, as log as is past a threshold value. The threshold however, is small for

6 values of λ cosistet with the curret HomePlug protocol. The umber of odes at which MAC efficiecy drops should exceed this value, so liearity is a good assumptio for the cases of iterest. The drop i MAC efficiecy with a large umber of cotedig statios ca ow be avoided by estimatig, ad the updatig the backoff widow accordigly. Oly a sigle slope ad itercept eed be stored, ad the optimum cotetio widow ca be geerated for all. REFERENCES [1] K. Tripathi, J. D. Lee, H. A. Latchma, ad J. McNair, Cotetio widow based parameter selectio to improve powerlie MAC efficiecy for large umber of users, accepted for publicatio i IEEE It. Sym. o Power Lie Commuicatios ad its Applicatios, March 26. [2] J. D. Lee, K. Tripathi, H. A. Latchma, ad J. McNair, Populatio based adaptive tuig of costat cotetio widow for HomePlug 1., It. J. of Power ad Eergy Systems, Vol , pp , 28. [3] M. K. Lee, R. E. Newma, H. A. Latchma, S. Katar, ad L. Yoge, HomePlug 1. powerlie commuicatio LANs protocol descriptio ad performace results, It. J. of Commuicatio Systems, 23. [4] A. Leo-Garcia, Probability, Statistics, ad Radom Processes for Electrical Egieerig, Pretice Hall, 3 rd ed., 28. [5] F. Cali, M. Coti, E. Gregori, Dyamic tuig of the IEEE protocol to achieve a theoretical throughput limit, IEEE/ACM Tras. O Networkig 8 (6) (2) [6] G. Biachi, Performig aalysis of the IEEE distributed coordiatio fuctio, IEEE J. o Selected Areas i Commuicatio, 18(3), March 2, [7] F. Cali, M. Coti, ad E. Gregori, IEEE protocol: desig ad performace evaluatio of a adaptive backoff mechaism, CNUCE Iteral Report, March 2.