Analysis of the Gated IPACT Scheme for EPONs

Transcription

1 Analyss of the Gated IPACT Scheme for EPONs Swapnl Bhata, Dmtr Garbuzov and Radm Bartoš Department of Computer Scence Unv. of New Hampshre Emal: {sbhata, dmtrg, Abstract Interleaved Pollng wth Adaptve Cycle Tme (IPACT s one of the earlest proposed pollng schemes for dynamc bandwdth allocaton n Ethernet Passve Optcal Networks (EPONs and has been extensvely used as a benchmark by many subsequent allocaton schemes. In ths paper, we attempt to construct a mathematcal model of the IPACT scheme under the gated servce dscplne. For N =1ONU, we derve closed-form expresson for the steady state grant sze. For N>1 ONUs, we need to consder separately a small and a large load-dstance rato. For the former case, the N =1ONU model holds even for N>1. For the latter case, we fnd a closed form expresson for the grant sze. Our model shows a reasonable match wth the values obtaned from smulaton for the steady state queue sze and hence the throughput and delay. I. INTRODUCTION An Ethernet Passve Optcal Network (EPON s a pontto-multpont, bdrectonal, hgh rate optcal network for data communcaton. The EPON lnk s shared by multple users. Each user connects to the EPON lnk through a devce known as an Optcal Network Unt (ONU. Snce the lnk s shared, lnk use must be centrally arbtrated. Ths functon s performed by a sngle specal devce called the Optcal Lne Termnator (OLT. The drecton of communcaton from the ONUs to the OLT s known as upstream drecton whereas the drecton from the OLT to the ONUs s known as the downstream drecton. The data rate n each drecton s set to 1 Gbps by the IEEE EPON standard [1]. Overall, the lnk exhbts a tree topology wth the OLT at the root of the tree and the ONUs at the leaves. The EPON lnk s shared by all users n the upstream drecton. The OLT decdes whch ONU s allowed to transmt data and for how many bytes. The OLT uses a specal control message called a Gate to grant transmsson opportuntes to ONUs. Appended to the data traffc, the ONU also transmts a control message contanng a Report of the number of bytes buffered n ts queue, watng for a subsequent transmsson opportunty. An algorthm mplemented n the OLT, whch uses these reports and gate messages to construct a transmsson schedule s known as a dynamc bandwdth allocaton (DBA algorthm. II. THE IPACT PROTOCOL Interleaved Pollng Scheme wth Adaptve Cycle Tme (IPACT s a DBA scheme for EPON proposed by Kramer et al. [2][3]. IPACT s one of the earlest dynamc bandwdth allocaton schemes for EPONs and has been extensvely used as a benchmark by many subsequent allocaton schemes [4][5][6][7][8][9][1]. To our knowledge, ths s the frst OLT ONU-1 ONU-2 d 1 d 2 t 1 g 1 q 1 Delayed Grant g 2 t 2 t 1 +1 q 2 g 1 +1 q 1 +1 g 2 +1 q 2 +1 t 2 +1 g 1 +2 Fg. 1. An example wth two ONUs llustratng notaton used for the recursve model. attempt to provde an analytcal model for the IPACT scheme. The evaluaton n [11] s based on smulatons and focuses on servce dscplnes. IPACT s an algorthm for nterleaved pollng of ONUs desgned to mnmze the walk tmes. For example, f the OLT sends a grant message to an ONU and then wats for the ONU to send data before sendng a grant message to the next ONU, then, the watng wll result n wastage of a sgnfcant amount of data bandwdth thus decreasng lnk utlzaton. IPACT suggests nterleavng the pollng messages to consecutve ONUs n order of decreasng dstance from the OLT such that transmssons arrve at the OLT as tghtly packed as possble. As llustrated n Fg. 1 the OLT has suffcent nformaton to schedule the transmssons for ONU-2 before the data from ONU-1 arrves at the OLT. Ths s because the OLT knows the dstance to each ONU and also knows the sze of the grant t allocated to each ONU. Hence the OLT can calculate the tme of the arrval of the last bt from each ONU and can therefore schedule the transmsson from the next ONU rght after the one from the prevous ONU has termnated. From Fg. 1 for example, at tme t 1, the OLT knows the tme at whch the transmsson from ONU-1 wll end and can therefore send the Gate message to ONU-2 to schedule ts transmsson to arrve at the OLT at tme t 2. In ths way, the nterleavng helps mnmze lnk underutlzaton durng walk tmes. However, ths s not always possble dependng on the dstance of the ONU from the OLT. To address ths ssue, the orgnal IPACT scheme also ncludes other optmzatons as well as procedures for detectng arrvng and departng ONUs. However, we do not nclude these portons n our analyss. We focus on the core IPACT bandwdth allocaton algorthm whch s a smple clent-server protocol wth qute general applcablty. Another parameter of the IPACT algorthm s the allocaton polcy of the server,.e., the OLT. When the OLT receves a report of the queue length from an ONU, t need not /6/$2. (c 26 IEEE Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 26 proceedngs.

2 grant a sngle transmsson slot n the current cycle for the buffered contents n ther entrety. Instead, the OLT may fx a partcular polcy such as grantng a tme slot: a of fxed length regardless of the reported queue length (statc allocaton, b equal to the reported buffer length but bounded by a maxmum (lmted allocaton, c larger than the reported queue length n antcpaton of future traffc (credt-based allocaton or d equal to the reported queue length (gated allocaton. Other varants are also possble. In ths paper, for smplcty, we focus on the the gated allocaton dscplne. In the followng sectons, we analyze ths scheme usng a recursve model. III. BACKGROUND AND RELATED WORK Pollng systems have been researched wdely for a number of years [12][13][14]. A general analyss of a pollng system seems to be a challengng problem and the results avalable are a product of complex analyses and therefore nvolve many smplfyng assumptons and approxmatons. The analyss s further complcated f one wshes to nclude a realstc traffc model. In ths paper, we attempt to derve a model of the IPACT scheme from ts basc defnton. Such a clean-slate analyss can provde clearer nsght nto the dynamcs of the scheme wth a mnmal set of assumptons and smplfcatons. Our man goal s to provde a clear and smple yet detaled model of the IPACT scheme derved from ts defnton. In the end, we hope to obtan smple closed-form expressons relatng the grant sze (and hence the delay and utlzaton to other parameters such as the load and round-trp tme. We beleve that our work can provde useful gudelnes n the desgn of new bandwdth allocaton schemes currently an area of ntense research [4][5][6][7][8][9][1]. Recently, Park et al. [15] obtaned new results about the performance of the IPACT scheme under the gated allocaton polcy. Ther approach s novel and comprehensve and provdes strong results. However, ther assumptons and results are dfferent and ther method, more sophstcated and complex. IV. A RECURSIVE MODEL FOR IPACT An accurate model for the IPACT scheme must account for the recursve relatonshp between the transmsson tmes, queue lengths and the grant szes n successve schedulng cycles. In ths secton, we develop such a model. We wll use the notaton as defned n Table I and llustrated n Fg. 1. Our goal s to express queue lengths, grant szes and transmsson tmes n terms of parameters such as the nput traffc rate (λ, lnk rate (δ 1, number of ONUs (N, and the dstances of ndvdual ONUs (d j from the OLT. A. Calculaton of grant sze g j The sze of the grant ssued by the OLT to an ONU s based on the queue length of the ONU. The ONU nforms the OLT about the length of ts queue n a Report message. However, the ONU must be granted a transmsson slot to transmt the Report message as well. As per the IPACT protocol, the ONU always appends a Report message to each transmsson. Let r denote the length of a Report message specfed by the IEEE TABLE I DEFINITION OF NOTATION USED Notaton Defnton Unts (value t j Thetmeofarrvalofthe th transmsson by ONU-j at the OLT seconds q j The queue length at ONU-j after completon bts of the th data transmsson g j The sze of the grant allocated by OLT to ONU-j for ts th transmsson bts d j 1 RTT to ONU-j seconds 2 λ The average arrval bt-rate bts/second δ The tme to transmt one bt over the EPON seconds/bt (1 9 r The sze of the Report message bts (512 b The sze of the guard band seconds (2 1 6 m The sze of the Gate message bts (512 N Total number of ONUs n the EPON EPON standard [1]. The OLT receves nformaton about the current sze of the queue length of an ONU only at the end of a current transmsson by that ONU. The OLT can use ths nformaton only to decde the sze of the next transmsson slot to be granted to that ONU. Under the gated servce dscplne, the OLT always allocates exhaustve grants,.e., the transmsson slots are always large enough to transmt all the data n the ONU queue reported to the OLT. Then, for gated servce, the sze of a grant s equal to the queue sze reported n the prevous grant, wth an addtonal r bts allocated for the next Report message. Thus, g j = qj 1 + r, > 1. (1 What happens when an ONU s granted ts frst transmsson opportunty? The OLT has no nformaton about the queue sze of the ONU, snce the ONU has not transmtted any Report messages to the OLT at all. In ths case, we assume that the OLT grants the ONU a transmsson slot of a mnmum sze large enough to transmt a Report message. Thus, for any ONU, the sze of ts frst transmsson opportunty wll be g j 1 = r. (2 B. Calculaton of ntal queue length q j 1 Next, we calculate the ntal queue length at any ONU, as requred by (1. Recall that for any ONU-j, the frst transmsson comprses only a Report message and none of the buffered data as per (2. Traffc arrves at the rate of λ bts/second. Suppose an ONU-j s d j seconds away from the OLT. Suppose that the frst bt of the ONU s frst transmsson reaches the OLT at tme t j 1. Snce ONU-j s dj seconds away from the OLT, t must have transmtted the frst bt of ts frst transmsson at tme t j 1 dj. The amount of traffc accumulated at ONU-j durng ths nterval wll therefore be λ(t j 1 dj whch wll be the queue length reported n the frst Report message by ONU-j. Thus, q j 1 = λ (tj 1 dj. (3 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 26 proceedngs.

3 The frst Gate message from the OLT to poll the frst ONU wll reach ONU-1 at tme d 1. Therefore, the frst Report message from ONU-1 wll reach the OLT at tme t 1 1 =2d 1 +δm. Thus, q 1 1 = λ(d 1 + δm. (4 For the frst cycle,.e., all transmssons t j 1, the length of each transmsson wll be equal to the length of the Report message. Thus, an ONU-j, j>1, wll have to wat at least δr after t j 1 1 before transmttng ts own Report message. On the other hand, an ONU-j cannot transmt ts Report message untl t has been polled by the OLT. Ths takes tme at least d j. Hence, for any ONU 1 <j N, the tme at whch ts Report message arrves at the OLT wll be t j 1 = max(tj δr + b, jδm +2d j. (5 C. Calculaton of queue length q j after >1th transmsson The queue length q j at the end of th transmsson by devce j s exactly equal to the new traffc arrved snce the last queue length measurement. To fnd ths queue length, we need to fnd out the tmes at whch queue lengths are measured and then usng the arrval rate λ obtan the queue length. From Fg. 1 we observe that q j 1 s the length of the queue at the tme the Report message s transmtted at the ONU. Snce any transmsson by the ONU takes tme d j to reach the OLT and snce the frst bt of the th transmsson reaches the OLT at tme t j, the frst bt must be transmtted at the ONU at tme t j dj. Further, snce the grant sze s g j, t must take δgj tme for the transmsson to complete. Thus, the tme at whch the ONU sends ts last bt must be t j dj + δg j. The queue length s measured δr tme before ths tme at the ONU. Thus, the tme at whch the queue length s measured after the th transmsson by devce j s t j dj + δg j δr. Thus, the queue length after the prevous.e., 1 th transmsson would be measured at tme t j tj 1 = δgj 1 +2dj + δm + b (9 t j 1 dj + δg j 1 δr. Substtutng n (6 and smplfyng gves, Therefore, the amount of new traffc accumulated snce the q j last queue measurement upto the current queue measurement + δg j + δm + b, > 1,N =1. wll be Substtutng (1 and smplfyng gves, q j = λ [(t j dj + δg j δr (tj 1 dj + δg j 1 δr] g j = λ [(t +1 = λ (δgj +2dj + δm + b+r. (1 j tj 1 +δ (gj gj 1 ], > 1 (6 Solvng ths recurrence gves the steady state grant sze, D. Calculaton of transmsson tmes t j Next, we calculate the transmsson tmes for any ONU. If the EPON conssts of N =1ONU, then the transmsson g s = λ (2dj + δm + b+r. 1 λδ (11 tme t j depends only on the transmssons n the prevous grants. A new transmsson can only begn after the prevous one has fnshed and the new Gate message sent out by the OLT reaches the ONU. A message from the OLT takes d j tme to reach an ONU-j. The new transmsson from ONU-j wll n turn take tme d j to reach the OLT. Thus, consecutve transmssons from the ONU are separated by 2d j seconds. An ONU s prevous transmsson begnnng at tme t j 1 wll end at tme t j 1 + δgj 1. Thus, a new transmsson, n the sngle ONU case wll begn at t j = tj 1 + δgj 1 +2dj + δm + b, N =1 (7 If N > 1, then consderng ndces modulo N, the transmsson from ONU-j cannot begn unless the one from the prevous ONU-(j 1 has ended, unless, ONU-j s suffcently far away from the OLT. In ths latter case, the tme d j taken by the Gate message to reach ONU-j suffcently delays the transmsson from ONU-j as llustrated by transmsson t 2 2 of ONU-2 n Fg. 1. Suppose ONU-(j 1 transmts at tme t j 1. Its transmsson wll fnsh at t j 1.No other transmsson can begn before ths tme. Now suppose that ONU-j last transmtted n the prevous cycle at tme t j 1. Then, dj must be large enough so that ONU-j s next transmsson begnnng at t j s suffcently delayed past the tme of completon t j 1 of the prevous ONU. Thus, we arrve at the followng condton for the transmsson tme: ( t j = max t j 1 + δ(gj 1 + m+2dj,t j 1 + b (8 where devce ndex j s counted modulo N > 1. Ths completes the specfcaton of the recursve model. V. CLOSED-FORM SOLUTION OF THE RECURSIVE MODEL In the prevous secton, we expressed the queue length reported by the ONU n ts th transmsson n terms of other parameters and varables of our model. In ths secton, we attempt to derve closed form expressons for the throughput and response tme for an ONU. We dvde our dervaton nto separate cases for N = 1(Sec.V-A and N > 1 (Sec.V-B ONU(s. A. Sngle ONU Consder an EPON wth a sngle ONU. Thus, N =1n the recursve model developed n Sec. IV. From, (7 we have The response tme R 1 ONU for a sngle ONU when λδ < 1 wll be R 1 ONU = δm+2d j +δg s +b = 2dj + δ(m + r+b (12 1 λδ Fg. 5 shows the match between the steady state grant sze predcted by (11 and that obtaned by smulaton. We have also verfed that the response tme predcted by (12 matches smulatons but omt the graph due to space lmtatons. Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 26 proceedngs.

4 OLT g 2 g ONU-{1, 2} g 1 g Fg. 2. The low load-dstance rato 5 25 B. Multple ONUs at an dentcal dstance For N>1, we also assume that all N ONUs are located at an dentcal dstance d = d j. (13 We consder two cases based on the rato of the load to the RTT to any ONU. 1 Low -Dstance rato: Frst, consder the case where for N > 1 ONUs, the l.h.s. term of the two expressons compared n (8 s the maxmum,.e., t j 1 <t j 1 + δ(gj 1 + m+2d. (14 Then by (8 and (14: t j tj 1 = δ(gj 1 + m+2d + b, (15 and, t j > t j 1 > t j 2 + b, + δg j 2 +2b, and so on. Contnung, we can wrte: N t j > t j 1 + j 1 δg 1 k + δg k + Nb. (16 In other words, usng (14, (15 and (16, we can wrte: N j 1 δg 1+ k δg k +Nb t = δ(g j 1 +m+2d+b. (17 where t = t j tj 1 s the cycle tme. Ths corresponds to the example shown n Fg. 2 wth two ONUs. Clearly, f the grant sze to each ONU s smaller than the RTT to the ONU, then nether ONU wll ever have to wat for the other ONU. The dervaton would proceed after (15 exactly as for the sngle ONU case n the prevous secton, resultng n the same steady state grant sze as arrved at n (11,.e., g j λ (2d + δm + b+r = g s =. (18 1 λδ Snce the steady state grant sze g j s the same for all N ONUs, usng (17 and (18, we can wrte: N (δg j + b δ(g j + m+2d + b, or (19 (N 1 (δg j + b 2d + δm. (2 Substtutng Eqn. (18 shows that the steady state grant sze g j from (18 holds when: λ 1 Nδ ( 1 (N 1(δr + b, (21 2d + δm 1 2 No. ONUs RTT (mcro sec. Fg. 3. Steady state grant sze for non-negatve load ponts beneath the surface s gven by (18 and for those above s gven by (26..e., our assumpton (14 translates nto the above condton. In other words, the sngle-onu model apples to the mult-onu case for the load-dstance relatonshp descrbed by (21. The steady state grant sze of any non-negatve load pont beneath the surface shown n Fg. 3 wll be gven by (21. 2 Hgh -Dstance rato: Alternatvely, consder the r.h.s. term of the two expressons compared n (8 to be larger,.e., t j 1 + δ(gj 1 + m+2d<tj 1 Then by (8 and (22: N t j tj 1 = j 1 δg 1 k + and, by smlar reasonng as before, N δ(g j 1 +m+2d+b t =. (22 δg k + Nb, (23 j 1 δg 1+ k δg k +Nb. (24 If all ONUs are located at the same dstance, the cycle tme t s solely comprsed of transmsson grants of other ONUs,.e., those that transmtted after ONU-j n the prevous cycle ( 1 and those that wll transmt before ONU-j n the current cycle (. (See the example wth two ONUs n Fg. (4. Substtutng (23 nto (6 and smplfyng gves ( N q j = λδ j g 1 k + + λnb. (25 +1 Assume that a steady state exsts (see appendx for detals and that the steady state grant sze for each ONU s dentcal. Then, n steady state, g j r = λδng j + λnb or, g j = λnb + r 1 Nλδ. (26 Agan, from (24 we know that (26 holds when δ(g j + m+2d + b < N(δg j + b. (27 g k Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 26 proceedngs.

5 ONU-{1, 2} OLT g 1 g 1 +1 Fg. 4. g 2 g 2 +1 Hgh load-dstance rato. Substtutng (26 and smplfyng shows that (26 holds for the remanng loads λ> 1 ( (N 1(δr + b 1. (28 δn 2d + δm Ths completes the soluton of our recursve model for IPACT under the gated servce scheme. VI. SIMULATIONS AND A COMPARISON Smulatons were conducted n order to valdate the above model of the IPACT scheme under a gated servce polcy. The smulatons were conducted for N {1, 2, 4, 2} ONUs d {25, 5, 1} µs away from the OLT for loads λ N 1. Each run smulated 1 seconds of real tme. For smulatons wth self-smlar traffc, the duraton of the smulaton must be very large (of the order of hundreds to a few thousands of seconds. Dependng on the load to be generated and the number of ONUs, such smulatons can be computatonally ntensve n both tme and space. More rgorous smulatons of much longer duraton are currently n progress. Fg. 5 shows the results of the smulaton. The self-smlar traffc curve conssts of 3 ponts, one obtaned from each run for a specfc load. Ffteen load values were used. Traffc njected nto the ONUs was generated by an aggregated source wth a measured Hurst parameter of.8 [16]. The source generated bursts whose ON and OFF perod lengths followed the Pareto dstrbuton. The nteger lengths of the packets n each burst were drawn unformly at random from the nterval [64, 1518] bytes. Fg. 5 shows the match between the steady state queue sze predcted by our model and that measured from smulaton wth both Posson as well as self-smlar/lrd [16] traffc models. The match aganst the Posson model s better than that aganst the self-smlar model snce our model s based smply on the average load and hence cannot account for the nfnte varance exhbted by the bursty self-smlar traffc. Nonetheless, the model shows a reasonably and suffcently good match wth both models as far as verfyng the correctness of the model s concerned. VII. CONCLUSIONS AND FUTURE WORK In ths paper, we attempted to analyze the IPACT protocol for allocaton of EPON bandwdth usng a recursve formulaton. IPACT s one of the earlest and most wdely used benchmarks for bandwdth allocaton n EPONs. However, there was no analytcal model descrbng the IPACT scheme. In ths paper, we developed such a smple model. We derved closed-form expressons for the grant sze allocated by IPACT under the gated servce scheme as a functon of the nput load, ONU RTT and other protocol parameters. The man assumpton of our analyss s that the flud traffc arrvng n an nterval t s λt where λ s the average traffc arrval rate, s known and fxed. However, just as wth other approaches, ths s a drawback of our analyss as well. For a well-behaved traffc source such as Posson traffc, a small sample from a small nterval t wll converge to the actual mean λ wth hgh probablty. However, for self-smlar, heavy taled traffc, ths does not hold. Snce the tal decays only polynomally, a small sample can contan a large devaton from the mean. In such cases, the predctons of the average value from our analyss wll exhbt error. Thus, the assumpton that the queue reported at the end of a transmsson of sze t wll be λt wll ntroduce consderable error for a heavy taled traffc source. ACKNOWLEDGMENTS We thank Csco for supportng our work under the Csco Unversty Research Program. We thank Dr. G. Kramer for makng the self-smlar traffc generaton tool publcly avalable. We thank Dr. E. Vark for useful comments durng the ntal porton of ths project. APPENDIX Consder (25. We can wrte a smlar equaton for ONUj 1 (usng modulo N ndexng: ( N j 1 q j 1 = λδ g 1 k + g k + λnb. (29 Subtractng, (29 from (25 gves: q j = qj 1 + λδ(q j 1 qj 2, (3 whch can be rewrtten as: a n = a n 1 + λδ(a n N a n 2N. (31 Equaton (31 can be nterpreted as the recurrence descrbng the IPACT scheme under gated servce and hgh load-dstance rato (as defned by (28 snce t captures the essental behavor of the bandwdth allocaton process of the scheme. The characterstc equaton of (31 s p(x =x 2N x 2N 1 cx N + c =, (32 where c = λδ. By showng that all the roots of (32 wll be no greater than unty for c<n 1, we can argue that a steady state exsts n the hgh load-dstance rato regme as well. REFERENCES [1] IEEE Draft P82.3ah (tm, IEEE Standard, 24. [2] G. Kramer, B. Mukherjee, and G. Pesavento, Interleaved Pollng wth Adaptve Cycle tme (IPACT: A Dynamc Bandwdth Dstrbuton Scheme n an Optcal Access Network, Photonc Network Communcatons, vol. 4, no. 1, pp , Jan. 22. [3], IPACT: A Dynamc Protocol for an Ethernet PON (EPON, IEEE Communcatons, vol. 4, no. 2, pp. 74 8, February 22. [4] M. Ma, Y. Zhu, and T. H. Cheng, A Bandwdth Guaranteed Pollng MAC Protocol for Ethernet Passve Optcal Networks, n IEEE INFO- COM, 23. Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 26 proceedngs.

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