Shortest Propagation Delay (SPD) First Scheduling for EPONs with Heterogeneous Propagation Delays

Transcription

1 1 Shortest Propagaton Delay () Frst Schedulng for EPONs wth Heterogeneous Propagaton Delays Mchael P. McGarry, Martn Resslen, Frank Aurzada, and Mchael Scheutzow Abstract Due to the geographc dstrbuton of ts subscrbers, Ethernet Passve Optcal Networks (EPONs) have typcally varyng propagaton delays between the Optcal Network Unts (ONUs) and the Optcal Lne Termnal (OLT). In ths paper, we consder EPONs wth an offlne schedulng framework, whch enables Qualty-of-Servce mechansms by collectng bandwdth requests from all ONUs before the OLT makes dynamc bandwdth allocatons for transmssons on the shared ONUs-to- OLT upstream channel. We propose and evaluate the Shortest Propagaton Delay () frst schedulng polcy whch sequences the ONUs upstream transmssons n ncreasng order of the ONUs propagaton delays,.e., the upstream transmsson of the ONU wth the smallest propagaton delay s scheduled frst. We formally analyze the compettveness of frst schedulng and fnd that t acheves very close to optmal performance. We characterze the stablty lmt for Gated and Lmted grant szng n conjuncton wth grant schedulng. We evaluate the cycle length and packet delay wth schedulng through probablstc analyss and smulatons and fnd sgnfcant reductons n packet delay wth frst schedulng n EPONs wth heterogeneous propagaton delays, especally when Lmted grant szng s employed. Index Terms Ethernet Passve Optcal Network, Grant schedulng, Packet delay, Propagaton delay. I. INTRODUCTION Passve Optcal Networks (PONs) have emerged as an attractve technology for hgh-speed access networks [1] [4]. In partcular, the combnaton of PON technologes wth the ubqutous Ethernet networkng technologes has made Ethernet PON (EPON) a promsng access network choce [5] [14]. In Ethernet Passve Optcal Networks (EPONs), an Optcal Network Unt (ONU) provdes hgh-speed network access to an ndvdual subscrber or a group of subscrbers. Several ONUs connect to a sngle Optcal Lne Termnal (OLT), typcally n the form of a tree topology rooted at the OLT. Due to the geographc dstrbuton of the served subscrbers, the ndvdual ONUs have typcally dfferent dstances, and thus dfferent propagaton delays from the OLT. Wth the emergence of long reach and next-generaton PONs [15] [20] coverng larger geographc areas wth spans of 100 km or hgher (.e., one-way ONU-to-OLT propagaton delays of 0.5 ms or hgher), the dspartes of the propagaton delays are M. McGarry s wth the Dept. of Electrcal and Computer Eng., Unversty of Akron, Akron, OH 44325, Emal: mmcgarry@uakron.edu, Phone: (330) , Fax: (330) M. Resslen s wth the School of Electrcal, Computer, and Energy Engneerng, Arzona State Unversty, Tempe, AZ , Emal: resslen@asu.edu, Phone: (480) , Fax: (480) F. Aurzada and M. Scheutzow are wth the Insttute of Mathematcs, Techncal Unversty Berln, Berln, Germany, Emal: {aurzada, ms}@tuberln.de. lkely becomng more pronounced. Further, trends to consoldate central offces n fewer locatons gve rse to the need to serve ONUs dstrbuted over large geographc areas [21] [23]. EPONs avod collsons on the shared upstream (ONUs-to- OLT) channel through a pollng based medum access control protocol. The ONUs sgnal ther bandwdth demands wth REPORT messages to the OLT, whle the OLT dynamcally allocates bandwdth and schedules the upstream transmssons so as to avod collsons. The OLT sgnals the ONUs wth GATE messages ther upstream transmsson wndows (grants). A key challenge for effcent sharng of the upstream channel s the maskng of the round trp propagaton delay between OLT and ONUs. One of the frst approaches for maskng the propagaton delays has been the Interleaved Pollng wth Adaptve Cycle Tme (IPACT) approach [9], [24] whch nterleaves the REPORT-GATE cycles of the ndvdual ONUs so that they can mask each others propagaton delays. The basc IPACT approach mplements the onlne schedulng framework [25] n that the OLT consders a sngle ONU REPORT when makng bandwdth allocaton and schedulng decsons; the onlne schedulng framework s referred to as nterleaved pollng n [14], [26]. Qualty of Servce (QoS) control generally requres that the OLT consders and trades off requests from several ONUs when makng bandwdth allocaton and schedulng decsons. Wth the offlne schedulng framework [25], whch s referred to as nterleaved pollng wth stop n [14], [26], the OLT collects REPORTs from all ONUs before makng bandwdth allocaton and schedulng decsons. The offlne schedulng framework thus enables the wde varety of QoS mechansms, see for nstance, [27] [35], whch consder jontly all RE- PORTs n ther bandwdth allocaton and schedulng decsons. On the downsde, the offlne schedulng framework mposes an dle perod on the upstream channel between cycles due to the OLT schedule computaton tme, and transmsson tme of the frst GATE message and the round trp propagaton delay to the frst scheduled ONU of a new cycle, as descrbed n more detal n Secton III-B. Further dle perods are possble f an upstream transmsson s not long enough to mask the propagaton delay to the next ONU n the schedule. A few studes have pursued strateges that combne onlne schedulng and offlne schedulng. For nstance, the studes [36] [39] schedule ONUs wth small bandwdth requests mmedately (.e., n onlne fashon), whle ONUs wth large bandwdth requests are only scheduled after REPORTs from all ONUs have been collected (.e., n offlne fashon) and more nformed decsons are possble. When the ONU propagaton delays are farly homogeneous, schedulng small bandwdth

2 2 request (whch have a small mpact on the QoS and farness propertes of the schedule) rght away can ndeed be a good strategy to mask propagaton delays for the larger requests, whch requre more careful nformed decsons. However, when the propagaton delays of the ONUs are vastly dfferent then schedulng a small grant for a far-away ONU can result n large dle tmes. Thus, for EPONs wth heterogeneous propagaton delays, the schedulng decsons need to take the propagaton delays nto consderaton. In ths paper, we examne, to the best of our knowledge, for the frst tme the problem of effcently maskng heterogeneous propagaton delays n EPONs wth offlne schedulng. The problem of accommodatng heterogeneous EPON propagaton delays has prevously only been examned n [40] for an onlne schedulng framework. We propose and evaluate the Shortest Propagaton Delay () frst schedulng polcy. The frst polcy strves to mask the long round trp propagaton delays to far-away ONUs by frst schedulng the upstream transmssons of near-by ONUs. We prove that the frst polcy mnmzes the cycle length to wthn a small tme perod (number of ONUs tmes transmsson delay of GATE message) of an optmal schedulng polcy. We characterze the cycle length and packet delay of frst schedulng for Gated grant szng n low load and hgh load regmes through probablstc analyss and derve stablty lmts for Lmted grant szng [9]. We conduct extensve smulatons to verfy our analyss and to broadly assess the reductons n cycle length and packet delay as well as the ncrease n channel utlzaton acheved wth frst schedulng. Importantly, by ncludng a suffcent number of closeby ONUs wth small propagaton an EPON usng frst schedulng can be engneered to allow for offlne schedulng wth a very small mposed dle tme between schedulng cycles. Further, frst schedulng s very smple n that a gven set of served ONUs needs to be sorted only once n ncreasng propagaton delays. Ths artcle s organzed as follows. In Secton II we provde background on schedulng n EPONs and revew related work. In Secton III we formally model the grant schedulng problem wth heterogeneous propagaton delays and characterze the compettveness of frst schedulng. We also derve approxmatons of the mean cycle length and packet delay. In Secton IV we present numercal results from our analytcal cycle length and delay evaluaton and provde extensve smulaton results for frst schedulng. Fnally, n Secton V we summarze our fndngs. II. BACKGROUND AND RELATED WORK In ths secton we brefly provde background on the dynamc bandwdth allocaton n EPONs and revew related research on schedulng n EPONs. The dynamc bandwdth allocaton (DBA) n EPONs can be dvded nto: 1) the szng of the upstream transmsson wndows (grants), and 2) the schedulng of the grants on the upstream wavelength channel [11]. Wdely used basc grant szng methods are Gated grant szng, where the OLT sets the grant sze equal to the ONU request and Lmted grant szng, where the OLT sets the grant sze equal to the ONU request up to a maxmum grant sze; f request exceeds the maxmum grant sze, then the maxmum grant sze s allocated [9], [24]. QoS mechansms for EPONs, such as [27] [35] typcally control the grant sze to acheve specfc QoS objectves. In ths study, we consder the grant sze as gven and focus on the schedulng of the grants. As noted n the Introducton, the basc onlne schedulng framework consders and schedules one ONU request at a tme, whereas the offlne schedulng framework collects reports from all ONUs before makng schedulng decsons [14], [25], [26]. A few studes have examned a just-ntme schedulng framework, where ONU requests are collected and schedulng decsons are made when the channel s about to become dle [25], [41], [42]. In [43] the ONUs are splt nto two groups whereby each group s scheduled n offlne fashon and the cycles of the two groups are nterleaved to mask the dle tme n between cycles. We also note that a few studes have sought to mprove on the REPORT-GATE traffc sgnalng through traffc predcton, see e.g., [44] [46]. The offlne schedulng framework s mportant as t facltates QoS mechansms, such as [27] [35], by provdng requests from all ONU to be consdered smultaneously n the dynamc bandwdth allocaton. In ths frst study on grant schedulng n EPONs wth heterogeneous propagaton delays we focus on the offlne schedulng framework. We leave the study of grant schedulng for heterogeneous propagaton delays n EPONs wth just-n-tme schedulng, a combnaton of onlne and offlne schedulng, or traffc predcton for future research. We proceed to brefly revew the research on grant schedulng polces n EPONs. The research on grant schedulng has prmarly examned schedulng polces for the offlne schedulng framework, where all ONU requests are consdered n schedulng decsons. Schedulng polces for combnatons of the onlne and offlne schedulng frameworks and for the just-n-tme schedulng framework, where a subset of the ONUs are consdered, have also been studed. A number of studes have examned schedulng polces that provde prescrbed QoS dfferentaton or farness propertes, e.g., [27] [39]. We focus n ths frst study on heterogeneous propagaton delays on mnmzng the average packet delay; consderng heterogeneous propagaton delays n conjuncton wth QoS dfferentaton and farness mechansms are mportant drectons for future research. Exstng schedulng polces for mnmzng the average packet delay nclude: Earlest Arrval Frst (EAF) schedulng [14], [47] whch orders ONUs by the arrval tme of the head of lne packet. Shortest Processng Tme (SPT) frst schedulng [25], [41] whch orders ONUs by ther grant sze. Largest Processng Tme (LPT) frst schedulng [14], [31] whch orders ONUs by ther grant sze (descendng order). Largest Number of Frames () frst schedulng [25] whch orders ONUs by the number of frames queued (descendng order). A recent comparson found that provded slghtly smaller

3 3 or the same average queueng delays [25] than the other polces and we consder therefore as a benchmark n our performance evaluaton n Secton IV. We also note that efforts to mask propagaton and other system delays (such as laser tunng tmes) have been examned for medum access control and schedulng n WDM star networks, e.g., [48], [49]. These WDM star networks provde all-to-all connectvty and are thus fundamentally dfferent from the EPON tree network, where only the OLT can reach all ONUs. III. PERFORMANCE ANALYSIS A. Network Model and Notaton We consder the EPON reportng and grantng cycle wth the offlne schedulng framework, whch s llustrated n Fg. 1 for N = 3 ONUs. We denote t sched for the schedule computaton tme,.e., the tme duraton from the nstant when all REPORT messages have been receved at the OLT to the nstant the transmsson of the frst GATE message commences. We denote t G for the fxed transmsson tme [n seconds] of an MPCP GATE message, t g for the fxed guard tme [n seconds] requred between ONU transmsson wndows, and t R for the fxed transmsson tme of a MPCP REPORT message. We let the constants τ, = 1,..., N, denote the oneway propagaton delays [n seconds] between OLT and ONU (whch we consder to be equal to the ONU to OLT propagaton delay). We let τ (), = 1,..., N, denote the propagaton delays sorted n ascendng order,.e., τ (1) = mn τ and τ (N) = max τ. We note that large propagaton dstance ranges and correspondngly large propagaton delay ranges τ (N) τ (1) can result n large dynamc ranges n the sgnals receved at the OLT. The ntegraton research system desgned n [50] has a dynamc range of 11.6 db (see [50, Fg. 12]) accommodatng a propagaton dstance ranges of over 40 km for 0.25 db/km fber loss and other system parameters (number of spltters and laser power) beng equal. The dynamc range of some commercally avalable OLT recevers s 20 db (e.g., see [51]), allowng for dstance ranges of up to 80 km. Wth the ongong advances n optcal recevers, see for nstance [52] [54], t s reasonable to expect that larger dynamc ranges wll become avalable n the near future. Furthermore, reach extenders for postve power gan [23], [55] [58] and optcal attenuators for negatve power gan can be used to adjust for other loss dfferences and/or for further extendng propagaton delay ranges. For a gven cycle, we let R be a random varable denotng the reported queue depth (n unts of seconds of upstream transmsson tme) and G [n seconds] be a random varable denotng the duraton of the upstream transmsson wndow (grant) of ONU, = 1,..., N. We suppose that G ncludes all per-ethernet frame overheads, such as Preamble and Inter Packet Gap (IPG). For a gven cycle, we defne the cycle length Γ as the tme perod from the nstant the schedulng commences to the nstant the upstream transmssons of the cycle are completely receved. We defne the upstream channel utlzaton η as the rato of the sum of upstream transmsson wndows to the cycle length,.e., η = N =1 G /Γ B. Problem Overvew Generally, n order to mnmze packet delays and maxmze the utlzaton on the EPON upstream channel, dle perods on the upstream channel should be mnmzed. In turn, wth mnmal dle perods, the cycle length Γ s mnmzed. Wth the offlne schedulng framework, there s an dle perod (stall tme) between the nstant the end of the last upstream transmsson of the precedng cycle arrves at the OLT and the nstant the begnnng of the frst upstream transmsson of the current cycle arrves at the OLT. Clearly, ths frst stall tme s mnmzed by sendng the frst GATE message to the ONU wth the shortest propagaton delay, whch results n t stall 1 = max(t sched + t G + 2τ (1), t g ). (1) For llustraton of the problem suppose that next the GATE messages and upstream transmssons of ONUs 2 and 3 follow, see Fg. 1. If the frst upstream transmsson s too short to mask the round-trp propagaton delay to ONU 2, a stall tme t stall 2 occurs between the end of the frst upstream transmsson and the begnnng of the recepton of the second upstream transmsson. More specfcally, f t stall 1 + G 1 + t R + t g < t sched + 2t G + 2τ 2, (2) then a non-zero stall tme t stall 2 occurs. On the other hand, as llustrated for ONU 3 n Fg. 1, f the round-trp propagaton delay s masked by a precedng upstream transmsson, then there s no stallng. Note that n the llustraton n Fg. 1, the sequence of the GATE message transmssons s equal to the sequence of upstream transmssons reachng the OLT. These two sequences do not necessarly need to be the same. In fact, one can relatvely easly construct examples where frst sendng the GATE message to a far-away ONU, followed by sendng the GATE messages and recevng the upstream transmssons of near-by ONUs, followed by the recepton of the upstream transmsson from the far-away ONU mnmzes dle perods, and thus the cycle length. We also note that [24] brefly mentoned that GATE messages to far-away ONUs may need to be sent before GATE messages to near-by ONUs to acheve close to contnuous utlzaton of the upstream channel, but dd not analyze n any detal the schedulng of the upstream transmsson wndows. For ease of notaton, we nclude the REPORT transmsson tme t R n the duraton of the upstream transmsson grant G n Sectons III-C through III-E. C. Soluton Strategy The schedulng of the upstream transmssons can be vewed as a generalzed verson of the schedulng problem wth release tmes,.e., tmes when a gven job becomes elgble for executon. Even for fxed known release tmes, the problem of mnmzng the total completon tme s strongly NP-hard [59]. Our problem s more general n that the release tmes,.e., the tmes when upstream transmssons can at the very earlest arrve at the OLT depend on the sequencng of the GATE message transmssons.

4 4 t=0 Γ t stall 1 t stall 2 stall t 3 =0 t sched OLT t R G 1 t R t g G 2 t R t g G 3 t R τ (3) t G t G t G τ (1) τ (1) τ (2) τ (3) τ (3) ONU 1 G 1 t R τ (2) ONU 2 τ (3) G 2 t R ONU 3 t R G 3 t R Fg. 1. Illustraton of offlne schedulng wth three ONUs. The horzontal axs represents tme and the nodes OLT, ONU 1, ONU2, and ONU 3 are llustrated on the vertcal axs; the vertcal axs s not scaled for propagaton delay. The upstream channel experences an dle (stall) perod due to the schedulng computaton tme t sched, transmsson tme of frst GATE message t G, and round trp propagaton delay 2τ (1). There are further stall perods f the upstream transmsson of an ONU does not mask the round trp propagaton delay of the next ONU. To the best of our knowledge, the combned problem of schedulng the sequence of GATE message transmssons and upstream transmssons so as to mnmze the cycle tme s mathematcally ntractable. Our soluton strategy s to consder two restrctons: (R1) We suppose that the GATE message transmsson tme t G s equal to the REPORT message transmsson tme t R, Ths s reasonable as both of these MPCP messages are sent n mnmum-length Ethernet frames. Notng that every upstream transmsson must at least contan a REPORT to obtan the current queue depth of the ONU, t G = t R mples that G t G. (R2) We ntally suppose that the sequence of the GATE message transmssons s equal to the sequence of the upstream transmssons arrvng at the OLT. Note that even wth ths restrcton, the problem of schedulng the upstream transmssons s a generalzed verson of the schedulng wth release tmes n that the release tmes of the jobs are not fxed, but rather depend on ther poston n the schedule (.e., the number of t G delays n the release tme vares accordng to the sequence of the upstream transmssons). Gven these two restrctons, we show n Secton III-D that the shortest propagaton delay () frst schedulng polcy mnmzes the cycle tme. Subsequently, n Secton III-E we relax the restrcton R2 on the sequence of the GATE messages and characterze the compettveness of schedulng. D. Shortest Propagaton Delay () Frst Optmalty Theorem 1. If G t G and the GATE message transmsson sequence s equal to the sequence of the upstream transmssons, then Shortest Propagaton Delay () frst schedulng of the upstream transmssons mnmzes the cycle duraton Γ. Proof: Wthout loss of generalty, we neglect the schedulng tme t sched and the guard tmes t g n the followng as they are not affected by the schedulng of the upstream transmssons. a) Comparson of two ONUs: Consder two ONUs 1 and 2. Let Γ 1,2 denote the cycle length when ONU 1 s scheduled before ONU 2. As llustrated n Fg. 2, there are two cases for the evaluaton of Γ 1,2 : (A) the propagaton delay τ 2 governs the cycle length, and (B) the propagaton delay τ 1 governs the cycle length. Clearly, the actual cycle length Γ 1,2 s obtaned as the maxmum of the two cases: Γ 1,2 = max (2t G + 2τ 2, t G + 2τ 1 + G 1 ) + G 2. (3) Analogously, we obtan by symmetry (whch only exchanges the roles of 1 and 2) the cycle length when ONU 2 s scheduled frst, followed by ONU 1: Γ 2,1 = max (2t G + 2τ 1, t G + 2τ 2 + G 2 ) + G 1. (4) We want to show that f G t G then we cannot have τ 1 > τ 2 and Γ 1,2 < Γ 2,1, (5).e., t cannot be that schedulng the ONU wth the longer propagaton delay (no. 1 n ths case) leads to a shorter cycle length. We proceed to show that (5) leads to a contradcton. We dstngush all possble cases accordng to where the maxmum n the defnton of Γ 1,2 and Γ 2,1, respectvely, s attaned.

5 5 Case (A): Cycle length governed by τ 2 : t G τ 1 τ 1 t G G 1 τ 2 τ 2 Case (B): Cycle length governed by τ 1 : t G t G τ 1 τ 1 τ 2 τ2 G 1 G 2 Fg. 2. Illustraton of cases for evaluaton of cycle length Γ 1,2. Case 1: 2t G + 2τ 2 t G + 2τ 1 + G 1 and 2t G + 2τ 1 t G +2τ 2 +G 2. In ths case, we have (the frst condton comes from Γ 1,2 < Γ 2,1, the other two from the condton for the case). t G + 2τ 1 + G 1 + G 2 < t G + 2τ 2 + G 2 + G 1 (6) 2t G + 2τ 2 t G + 2τ 1 + G 1 (7) 2t G + 2τ 1 t G + 2τ 2 + G 2. (8) Note that equaton (6) s a contradcton to τ 1 > τ 2. Case 2: 2t G + 2τ 2 t G + 2τ 1 + G 1 and 2t G + 2τ 1 t G + 2τ 2 + G 2. Here, we get 2t G + 2τ 2 + G 2 < t G + 2τ 2 + G 2 + G 1 (9) 2t G + 2τ 2 t G + 2τ 1 + G 1 (10) 2t G + 2τ 1 t G + 2τ 2 + G 2. (11) Note that (10) s a contradcton to τ 1 > τ 2 and t G G 1. Case 3: 2t G + 2τ 2 t G + 2τ 1 + G 1 and 2t G + 2τ 1 t G + 2τ 2 + G 2. Here, we get t G + 2τ 1 + G 1 + G 2 < 2t G + 2τ 1 + G 1 (12) 2t G + 2τ 2 t G + 2τ 1 + G 1 (13) 2t G + 2τ 1 t G + 2τ 2 + G 2. (14) Note that (12) s a contradcton to t G G 2. Case 4: 2t G + 2τ 2 t G + 2τ 1 + G 1 and 2t G + 2τ 1 t G + 2τ 2 + G 2. Here, we get 2t G + 2τ 2 + G 2 < 2t G + 2τ 1 + G 1 (15) 2t G + 2τ 2 t G + 2τ 1 + G 1 (16) 2t G + 2τ 1 t G + 2τ 2 + G 2. (17) Note that (16) s a contradcton to τ 1 > τ 2 and t G G 1. b) General case: Now, consder N ONUs wth propagaton delays τ 1,..., τ N, respectvely. We show that t s optmal to schedule them n frst manner. Consder any order of the ONUs 1, 2,..., N and suppose that we had (strctly) G 2 optmal cycle tme over all orders but we dd not have τ 1 τ 2 τ N. If we do not have τ 1 τ 2... τ N, there must be one such that τ > τ +1. However, ndependently Γ 1,2 = 2t G + 2τ 2 + G 2 of the propagaton delays of the precedng ONUs, t would be better (.e., gve lower cycle tme) to have τ τ +1, by the above comparson of two ONUs n part a) of ths proof. (Ths reasonng s not nfluenced by the fact that the transmssons of ONUs and + 1 may be delayed by precedng transmssons of other ONUs. In any case, t cannot be a loss to schedule the ONU wth the shortest propagaton delay frst.) Ths s Γ 1,2 = t G + 2τ 1 + G 1 + G 2 a contradcton to the optmalty of the orderng. Hence, ths contradcton mples the asserton, snce was arbtrary. Note that an mmedate corollary of Theorem 1 s that frst schedulng maxmzes the upstream channel utlzaton. As examned n detal n the next secton, the bound n Theorem 1 shows that frst schedulng s very close to optmal, snce the GATE transmsson tme t G s typcally small compared to traffc. E. Compettveness of Schedulng In ths secton, we examne the compettveness of frst schedulng n comparson to an optmal schedule that mnmzes the cycle length. The optmal schedule may have dfferent sequences of GATE transmssons and upstream transmssons,.e., does not need to meet restrcton R2. We stll requre that the optmal schedule meets restrcton R1 that G t G snce the EPON pollng requres that each upstream transmsson contans at least a REPORT. We frst characterze the absolute dfference of the cycle length wth Γ compared to the mnmal cycle length Γ opt of an optmal schedule. Next, we examne the compettve rato of schedulng,.e., the bound on the rato of the cycle length wth schedulng Γ to the cycle length of an optmal schedule Γ opt. Theorem 2. The cycle length wth frst schedulng exceeds the mnmal cycle length by no more than (N 1)t G,.e., Γ Γ opt + (N 1)t G. Proof: For frst schedulng, let t start, = 1,..., N, denote the nstant when the upstream transmsson of ONU begns to arrve at the OLT. We defne for convenence t start 0 := 0 and G 0 = 0 and note that t start = max(t G + 2τ (), t start 1 + G 1 + t g ). (18) The cycle length s Γ = t start N + G N. (19) Now, consder an magnary EPON where the grants to all ONUs are communcated wth one GATE message requrng only one transmsson tme t G. Ths magnary EPON serves as a comparson for the optmal schedulng n the real EPON. Let Γ opt,c denote the cycle tme n the magnary EPON (the subscrpt c s for comparson strategy ). Clearly, frst

6 6 schedulng of the upstream transmssons s optmal n the magnary EPON. We proceed to show that Γ (N 1)t G Γ opt,c Γ opt Γ, (20) where the last two nequaltes are trvally satsfed. We prove the frst nequalty by nducton. In the magnary EPON let s start denote the nstant when the upstream transmsson of ONU begns to arrve at the OLT. Denote s start 0 := 0 and note that s start = max(t G + 2τ (), s start 1 + G 1 + t g ), for = 1,..., N. (21) We obtan by nducton that t start s start + ( 1)t G. (22) The case = 1 s trval. In general, we get t start = max(t G + 2τ (), t start 1 + G 1 + t g ) (23) max(t G + 2τ (), s start 1 + ( 2)t G + G 1 + t g ) (24) max(( 1)t G + t G + 2τ (), s start 1 + ( 1)t G + G 1 + t g ) (25) = ( 1)t G + max(t G + 2τ (), s start 1 + G 1 + t g ) (26) = ( 1)t G + s start. (27) The asserton follows from (22), snce Γ = t start N + G N (28) s start N + G N + (N 1)t G (29) = Γ opt,c + (N 1)t G. (30) We remark that the bound n Theorem 2 s attaned for an example scenaro wth τ 1 =... = τ N 1 = 0, 2τ N = Nt G, and G 1 =... = G N = t G. For ths example, the cycle length wth optmal schedulng s Γ opt = (N + 2)t G, whereas the cycle length wth schedulng s Γ = (2N + 1)t G. That s, the dfference Γ Γ opt s exactly (N 1)t G n ths example, and thus the bound n Theorem 2 cannot be mproved. Proposton 1. The compettve rato for the cycle length wth Smallest Propagaton Delay () frst schedulng s { Γ tg + 2 max τ + mn G Γ opt t G + 2 mn τ + G, t G + 2 max τ + G }. (31) t G + max (2τ + G ) Proof: Frst, note that the shortest possble cycle length must satsfy Γ opt t G + 2τ (1) + N G (32) =1 snce at least the frst GATE needs to be transmtted and at least the round trp propagaton delay to the nearest ONU s ncurred before all the upstream transmssons (wth aggregate duraton N =1 G ) can arrve at the OLT. Ths bound s attaned when the ONU wth the shortest propagaton delay has a very large upstream transmsson. Second, note that cycle must be long enough to accommodate the GATE message transmsson, round-trp propagaton delay, and upstream transmsson of each ndvdual ONU,.e., Γ opt t G + 2τ + G (33) for each ONU, = 1,..., N. Snce ths holds for all we can take the maxmum over all ONUs yeldng Γ opt t G + max(2τ + G ). (34) Ths bound s attaned f one ONU wth a large propagaton delay has a large upstream transmsson and all other ONUs have small propagaton delays and upstream transmssons. Thrdly, for the cycle tme wth frst schedulng Γ t G + 2τ (1) + N =1 N 1 G + δ,+1 (35) =1 where we defne δ,+1 as the dfference between the ( + 1)th smallest and the th smallest round trp propagaton delay,.e., δ,+1 = 2τ (+1) 2τ () for = 1,..., N 1. Note that N 1 =1 δ,+1 s the worst case for the stall tmes n between the upstream transmssons arrvng at the OLT. Ths worstcase occurs f each ONU has only a REPORT message to send upstream,.e., G = t G. The δ-sum s telescopng and we get Thus, N 1 =1 δ,+1 = 2τ (N) 2τ (1). (36) N 1 Γ t G + 2τ (N) + G. (37) =1 Combnng (32) and (37) we get Γ t N G + 2τ (N) + =1 G Γ opt t G + 2τ (1) +. (38) N =1 G From (34) and (37) we get Γ t N G + 2τ (N) + =1 G Γ opt t G + max (2τ + G ). (39) The bounds n Proposton 1 show that the frst schedulng polcy has a good compettve rato n most cases. Indeed, the frst bound s a good compettve rato,.e., s close to one, f N =1 G s large,.e., f there s heavy traffc. The second bound s a good compettve rato f τ (N) s large compared to N =1 G,.e., typcally f there s lght traffc. From the dervatons of the precedng bounds we obtan the followng nsghts nto frst schedulng and potental heurstcs to mprove on frst schedulng: a) If there s heavy traffc at ONUs wth short propagaton delays, then t would be favorable to send them ther GATE messages as soon

7 7 as possble followng the frst polcy. Wth ths strategy the traffc from the ONUs wth short propagaton delays can mask the GATE transmssons and round trp propagatons to the far-away ONUs. b) If there s lttle traffc at the ONUs wth short propagaton delays, then GATE messages could frst be sent to the ONUs wth large propagaton delays, even though ther upstream transmsson wndows should, of course, be scheduled after the wndows of the ONUs wth short propagaton delays. Wth ths strategy, the ONUs wth short propagaton delays can fnsh ther transmssons by the tme the upstream transmssons from the ONUs wth long propagaton delays arrve at the OLT. F. Packet Delay Analyss for Gated Grant Szng In ths secton, we analyze the cycle length and packet delay wth frst schedulng for Gated grant szng. We consder approxmatons for lght traffc and heavy traffc. We defne the packet traffc load ρ, = 1,..., N of ONU as the rato of the average traffc bt rate (ncludng Ethernet frames plus per-ethernet frame overhead,.e., preamble and IPG) generated at ONU to the upstream channel bt rate C [bt/s]. Unlke for the analyss n the precedng sectons, we do not consder the REPORT message wth transmsson tme t R = t G as part of a grant G n ths secton, but rather account for the REPORT transmsson tmes separately from the grant duraton. We also explctly consder the guard tme t g n ths secton. We denote ρ t = N =1 ρ for the total load. Notng that wth Gated grant szng the per-grant overhead (REPORT message, guard tme) becomes neglgble as the grants grow very large n heavy traffc, the stablty condton (maxmum throughput) wth gated grant szng s ρ t < 1 (whch holds for arbtrary packet traffc patterns, ncludng self-smlar traffc). We let L and σl [n bt] denote the mean and standard devaton of the packet sze (plus Preamble and IPG). In the delay analyss we consder Posson packet traffc and defne Po[ω] for a random varable wth Posson dstrbuton wth parameter ω. 1) Low Load Scenaro: In a low load stuaton wth schedulng, the cycle tme s determned by the last (longest prop. delay) ONU. Namely, the cycle tme n cycle n + 1 satsfes Γ LL (n + 1) = Nt G + 2τ (N) + L C Po [ ] C ρ (N) L Γ LL(n) + t G, (40) where ρ (N) s the load parameter of the ONU wth the largest propagaton delay τ (N). The cycle conssts of N GATE message transmsson and the data packet transmssons plus REPORT message transmsson of the last ONU. The other transmssons are masked by the long propagaton delay of the last ONU (due to the low load assumpton). Takng expectatons we get Ths yelds EΓ LL = Nt G + 2τ (N) + ρ (N) EΓ LL + t G. (41) EΓ LL = (N + 1)t G + 2τ (N) 1 ρ (N). (42) Snce we need the second moment of the cycle tme for the delay analyss, we obtan n the same way from (40): EΓ 2 LL = VΓ LL + (EΓ LL ) 2 (43) = ρ (N) L C EΓ LL + (EΓ LL ) 2 (44) = EΓ LL (ρ (N) L C + EΓ LL). (45) The delay of a packet generated at ONU conssts of the backward recurrence tme of the cycle length [60, Ch. 5.5] EΓ 2 LL 2EΓ LL,.e., the tme untl the generated packet s ncluded n a REPORT, the tme perod from the nstant the transmsson of the REPORT s complete to the nstant the next upstream transmsson commences, whch s t G + 2τ () due to the low load assumpton, and the tme needed wthn the grant untl the consdered packet commences transmsson ρ EΓ 2 LL 2EΓ LL : D () LL = EΓ2 LL EΓ 2 LL + t G + 2τ () + ρ 2EΓ LL 2EΓ LL +τ () + L C. (46) = (1 + ρ ) ρ (N) The overall delay s then gven by D LL = 1 ρ t L C + EΓ LL 2 + t G +3τ () + L C. (47) N =1 ρ () D () LL, (48) where ρ t := N =1 ρ and ρ () s the load parameter of the ONU wth the -th smallest propagaton delay. 2) Heavy Load Scenaro: In a heavy load scenaro, all ONUs have data to send, whch masks the round-trp delays between OLT and the second to last ONUs n the frst schedule. In ths case, the cycle length satsfes EΓ HL = t G + 2τ (1) + N ρ EΓ HL + Nt G + (N 1)t g, (49) =1 because a cycle conssts of the GATE transmsson plus round trp propagaton delay to the frst ONU (the one wth the shortest propagaton delay) and all the traffc that s sent (whch conssts of the accumulated data traffc and the N REPORT messages). Ths gves EΓ HL = (N + 1)t G + (N 1)t g + 2τ (1) 1 N =1 ρ. (50) Ths scenaro resembles the case of a sngle ONU treated n [61]. Insertng τ HL := N+1 2 t G + N 1 2 t g + τ (1) and ρ t := N =1 ρ n Eqn. (39) n [61], namely ( 2 ρ t ρ t σ 2 L) D HL = 2τ HL + 1 ρ t 2C(1 ρ t ) L L + + L C (51) gves an approxmaton of the packet delay.

8 8 G. Stablty of Lmted Grant Szng For Lmted grant szng wth schedulng we evaluate the maxmum throughput (stablty lmt) as follows. Frst, for arbtrary packet traffc, ncludng self-smlar traffc, we calculate the maxmal cycle tme by t start,max 0 := 0 t start,max where G max Then, = max(t G + 2τ (), t start,max 1 + G max 1 + t g ), (52) are the gven maxmal grant szes (G max 0 = 0). Γ max = t start,max N + G max N (53) s the maxmum cycle length (even n the heavest traffc, no cycle wll be longer than ths). The stablty condton for s then ρ Γ max < G max (54) for all. (The stablty lmt for Largest Propagaton Delay () frst schedulng s obtaned analogously by consderng the τ n decreasng order n (52).) For homogeneous ONU loads ρ 1 = ρ 2 = = ρ N and maxmum grant szes G max 1 = = G max N, the total load s ρ t = Nρ, resultng n the stablty condton ρ max t,sp D < NGmax Γ max. (55) IV. EXPERIMENTAL PERFORMANCE ANALYSIS We conducted a set of smulaton experments to: 1) valdate our analytcal models presented n Secton III, and 2) quantfy the mprovements to cycle length and packet delay acheved by grant schedulng under dfferent operatng condtons. We use an EPON smulator that we have developed usng the CSIM dscrete event smulaton lbrary [62]. We smulated an EPON wth 32 ONUs and vared the maxmum propagaton delay to represent dfferent EPON reaches. The followng quad modal packet sze dstrbuton was used for all smulaton experments: 60% 64 bytes, 4% 300 bytes, 11% 580 bytes, and 25% 1518 bytes. In the absence of any dversty n propagaton delay, the mpact of mnmzng cycle length wll be mnor. In our experments, we use a contnuous (unform) dstrbuton for one-way (OLT-to-ONU) propagaton delays wth a mnmum value of 6.68 µs and dfferent maxmum one-way propagaton delay values. We modfy ths dstrbuton slghtly by forcng one ONU to have a mnmum propagaton delay value and another to have a maxmum propagaton delay value. The propagaton delays for the other 30 ONUs are contnuously dstrbuted over the range. We compare schedulng to Largest Number of Frames () frst schedulng whch was prevously demonstrated [25] to provde low average queueng delay compared to other schedulng polces [47]. We also present some results for Largest Propagaton Delay () frst schedulng whch s the opposte of to llustrate the range of possbltes for cycle length and packet delay. The average cycle length and average packet delay values presented n ths secton represent the mean of several ndependent runs that were constructed Fg. 3. szng. 1.8 msec 1.6 msec 1.4 msec 1.2 msec usec usec usec usec Self smlar, smulaton Posson, smulaton Posson, analyss hgh load Posson, analyss low load 1.8 msec 1.6 msec 1.4 msec 1.2 msec usec usec usec a) 50 µs max. prop. delay (.e., up to 10 km) Self smlar, smulaton Posson, smulaton Posson, analyss hgh load Posson, analyss low load usec 1.8 msec 1.6 msec 1.4 msec 1.2 msec usec usec usec b) 250 µs max. prop. delay (.e., up to 50 km) Self smlar, smulaton Posson, smulaton Posson, analyss hgh load Posson, analyss low load usec c) 500 µs max. prop. delay (.e., up to 100 km) Average cycle length for grant schedulng and Gated grant usng the batch means feature n the CSIM dscrete event smulaton lbrary. The resultng statstcal confdence ntervals for Posson traffc are smaller than the pont marks n the plots. We also compare wth onlne IPACT schedulng [9], [24] where the sequence of grants to the ONUs s generally roundrobn n the order n whch the ONUs regstered wth the OLT. Wth the same set of random propagaton delays used n the smulatons, we smulated and averaged many ndependent random permutatons of the ONU grantng sequence. A. Gated Grant Szng In ths secton we present our results of the experments we conducted usng Gated grant szng. 1) Cycle Length: Fgure 3 shows the average cycle length as a functon of the total load for grant schedulng

9 9 2.5 msec 1.5 msec usec 8.0 msec 7.0 msec 6.0 msec 5.0 msec 4.0 msec 3.0 msec msec 5.0 msec 4.0 msec 3.0 msec a) 50 µs max. prop. delay (.e., up to 10 km) 18.0 msec 16.0 msec 14.0 msec msec 6.0 msec 4.0 msec a) 50 µs max. prop. delay (.e., up to 10 km) msec 6.0 msec 4.0 msec b) 250 µs max. prop. delay (.e., up to 50 km) 35.0 msec 30.0 msec 25.0 msec 15.0 msec 5.0 msec b) 250 µs max. prop. delay (.e., up to 50 km) Fg. 4. c) 500 µs max. prop. delay (.e., up to 100 km) Average cycle length wth Gated grant szng for self-smlar traffc c) 500 µs max. prop. delay (.e., up to 100 km) Fg. 5. Average cycle length wth Gated grant szng for self-smlar traffc as the load approaches the channel capacty. by means of analyss usng Eqs. (42) and (50), as well as smulaton experments usng Posson traffc sources and selfsmlar traffc sources. We observe from ths fgure that Eqs. (42) and (50) provde an excellent ft to the average cycle length measured n the smulaton experments. Fgure 4 shows the average cycle length for,, and grant schedulng and varyng propagaton delay confguratons. Fgure 5 shows the same for load values approachng the channel lmt. We observe from these fgures that always provdes a lower average cycle length than or. The dfference ncreases sgnfcantly wth an ncreasng load and ncreasng maxmum propagaton delay. As an example, wth a 500 µsec maxmum propagaton delay and load value of 0.9 the average cycle length was 2.0 mllseconds usng and 6.0 mllseconds usng. 2) Packet Delay: Fgure 6 shows the average packet delay for grant schedulng by means of analyss usng Eqs. (48) and (51), as well as smulaton experments usng Posson traffc sources and self-smlar traffc sources. The measurements for average packet delay wth the self-smlar traffc sources are, as expected, much hgher despte average cycle lengths smlar to those observed wth Posson traffc sources. Ths s a result of a few very long cycles, whose lengths are observed once, that have many packets whose assocated large delay s observed once for each of these packets. Fgure 7 shows the average packet delay for,, and grant schedulng and varyng propagaton delay confguratons for self-smlar traffc. Fgure 8 shows the same for load values approachng the channel lmt. We observe from Fgures 7 and 8 that provdes lower average packet

10 msec msec 6.0 msec 4.0 msec Self smlar, smulaton Posson, smulaton Posson, analyss hgh load Posson, analyss low load 25.0 msec 15.0 msec 5.0 msec msec msec 6.0 msec 4.0 msec a) 50 µs max. prop. delay (.e., up to 10 km) Self smlar, smulaton Posson, smulaton Posson, analyss hgh load Posson, analyss low load 30.0 msec 25.0 msec 15.0 msec a) 50 µs max. prop. delay (.e., up to 10 km) Fg. 6. szng msec msec 6.0 msec 4.0 msec b) 250 µs max. prop. delay (.e., up to 50 km) Self smlar, smulaton Posson, smulaton Posson, analyss hgh load Posson, analyss low load 1 c) 500 µs max. prop. delay (.e., up to 100 km) Average packet delay for grant schedulng and Gated grant delay for the experments wth 250 µs and 500 µs maxmum propagaton delays. The dfference wth becomes more pronounced at hgh load values. As an example, wth a 500 µs maxmum propagaton delay and load value of 0.9 Gbps the average packet delay was 21.3 mllseconds usng and 26.5 mllseconds usng. At 50 µs, the dfference between and s rather nsgnfcant. However, t s worth notng that clearly provdes a lower average cycle length for all load values (see Fgures 4 and 5) whch leads to lower average packet delay. The mean packet delays wth onlne IPACT schedulng (whch are not ncluded n the plots to avod clutter) were only very slghtly lower than the packet delays; namely wthn 3% of the delays. Fg msec 35.0 msec 30.0 msec 25.0 msec 15.0 msec 5.0 msec b) 250 µs max. prop. delay (.e., up to 50 km) c) 500 µs max. prop. delay (.e., up to 100 km) Average packet delay wth Gated grant szng for self-smlar traffc. One-way prop. delay [µs] Γ max SP D Eq. (53) [ms] Γmax SP D sm. [ms] 50 (up to 10 km) (up to 50 km) (up to 100 km) TABLE I MAXIMUM AVERAGE CYCLE LENGTH (IN MILLISECONDS) FOR FOR LIMITED GRANT SIZING WITH G max = 7188 BYTES, FOR SELF-SIMILAR TRAFFIC. B. Lmted Grant Szng In ths secton we present our results of the experments we conducted usng Lmted grant szng wth G max = 7188 bytes,. 1) Cycle Length: Fgure 9 shows the average cycle length for,, and grant schedulng for varyng propagaton delay confguratons. We make two observatons: 1) n

11 msec 65.0 msec 60.0 msec 55.0 msec 50.0 msec 45.0 msec 40.0 msec 35.0 msec 30.0 msec 25.0 msec 15.0 msec msec 80.0 msec 70.0 msec 60.0 msec 50.0 msec 40.0 msec 30.0 msec a) 50 µs max. prop. delay (.e., up to 10 km) msec 90.0 msec 80.0 msec 70.0 msec 60.0 msec 50.0 msec 40.0 msec 30.0 msec b) 250 µs max. prop. delay (.e., up to 50 km) c) 500 µs max. prop. delay (.e., up to 100 km) Fg. 8. Average packet delay wth Gated grant szng for self-smlar traffc as the load approaches the channel capacty. One-way prop. delay [µs] ρ max t,sp D Eq. (55) ρmax t,sp D sm. 50 (up to 10 km) (up to 50 km) (up to 100 km) TABLE II LOAD AT WHICH THE AVERAGE CYCLE LENGTH APPROACHES THE MAXIMUM CYCLE LENGTH, WHICH IS EQUAL TO THE STABILITY LIMIT, FOR WITH LIMITED GRANT SIZING WITH G max = 7188 BYTES, FOR SELF-SIMILAR TRAFFIC. all plots provdes lower average cycle length for all load values, and 2) as the maxmum propagaton delay s ncreased from 50 µs to 500 µs, s able to mantan a maxmum average cycle length around 2 mllseconds. Explorng the second observaton further, we see that the maxmum average cycle length for ncreases from around 2.2 msec 1.8 msec 1.6 msec 1.4 msec 1.2 msec usec usec usec usec msec 2.4 msec 2.2 msec 1.8 msec 1.6 msec 1.4 msec 1.2 msec usec usec a) 50 µs max. prop. delay (.e., up to 10 km) usec msec 3.0 msec 2.8 msec 2.6 msec 2.4 msec 2.2 msec 1.8 msec 1.6 msec 1.4 msec 1.2 msec b) 250 µs max. prop. delay (.e., up to 50 km) c) 500 µs max. prop. delay (.e., up to 100 km) Fg. 9. Average cycle length for dfferent grant schedulng polces wth Lmted grant szng for self-smlar traffc. 2 mllseconds at 50 µs to close to 2.6 mllseconds at 500 µs. Wth a 500 µsec maxmum propagaton delay, the load at whch the cycle length reaches ts maxmum s 0.9 Gbps for and approxmately only 0.62 Gbps for. In Table I we compare the maxmum average cycle length usng Eq. (53) wth the results from our smulaton experments. The data n ths table ndcates that the equaton s wthn 0.7 % of the expermental data. Our expermental data and Eq. (53) ndcate that s able to keep the maxmum cycle length near 2 mllseconds as the maxmum propagaton delay s ncreased. Whereas, s unable to do the same. Eq. (53) llustrates that the maxmum cycle length s a functon of the order of the ONUs as well as ther propagaton delays. constantly changes the order of ONUs wth respect to ther propagaton delays resultng n

12 msec msec msec msec msec msec msec msec msec 50.0 msec msec msec msec msec msec msec msec msec msec 50.0 msec a) 50 µs max. prop. delay (.e., up to 10 km) msec msec msec msec msec msec msec msec msec b) 250 µs max. prop. delay (.e., up to 50 km) 50.0 msec c) 500 µs max. prop. delay (.e., up to 100 km) Fg. 10. Average packet delay for dfferent grant schedulng polces wth Lmted grant szng for self-smlar traffc. the varatons that can be seen n ts maxmum average cycle length. Further, s able to mnmze the cycle length by orderng the ONUs n ncreasng order of propagaton delay. does not order the ONUs accordng to ther propagaton delays and, as a result, does not mnmze the cycle length. 2) Stablty Lmt: As the load ncreases and the grant szes approach the prescrbed maxmum grant sze G max, the cycle length approaches the maxmum cycle length. That s, the average cycle length levels out at the maxmum cycle length plotted n Fg. 9 and tabulated for n Table I. Further ncreases n the load can not ncrease the cycle length, but result n nfnte queue buld-up n the ONUs,.e., nstablty. The load value at whch the average cycle length levels out to the maxmum cycle length n Fg. 9 thus represents the stablty lmt, whch s tabulated for n Table II. We observe from Table II that Eqn. (55) very accurately characterzes the stablty lmt. Turnng to Fg. 9, we observe that the stablty lmt dfference between and ncreases sgnfcantly as the maxmum propagaton delay s ncreased. As an example, wth a 500 µs maxmum propagaton delay, the stablty lmt s approxmately 0.89 Gbps usng (the average cycle length converges to the maxmum cycle length very slowly between a load of 0.85 and 0.89) and approxmately 0.62 Gbps usng. By schedulng the grants to the close-by ONUs frst, masks the long round-trp delays to the ONUs that are further away. Ths more effcent utlzaton of the upstream channel ncreases the stablty lmt substantally as the propagaton delays become more dverse. 3) Packet Delay: Fgure 10 shows the average packet delay for,, and grant schedulng and varyng propagaton delay confguratons. We observe that provdes lower average packet delay for all load values and reconfrm the hgher stablty lmt for. From addtonal smulatons we found that onlne IPACT schedulng gves mean packet delays that are generally around 15 20% lower than for offlne schedulng. For nstance, for 250 µs maxmum propagaton delay and a load of 0.8, gves a mean packet delay of s compared to s wth onlne IPACT. Wthn the offlne schedulng framework, the schedulng polcy vastly reduces the packet delay compared to the schedulng polcy. Thus, the schedulng polcy makes t possble to reap the benefts of the offlne schedulng framework wth only relatvely modest delay penaltes compared to onlne schedulng. It s nstructve to compare the packet delay reductons wth compared to grant schedulng for Gated grant szng n Fgs. 7 and 8 to the correspondng delay reductons for Lmted grant szng n Fg. 10. Clearly, for Lmted grant szng we observe substantally larger delay reductons. Wth Gated grant szng, any reported queue sze s served n one upstream transmsson. In contrast, wth Lmted grant szng, a large reported queue requres several maxmum szed grants, and thus several cycles to transmt the traffc to the OLT. The cycle length mnmzng schedulng polcy leads hence to sgnfcantly more pronounced delay reductons for Lmted grant szng than for Gated grant szng. of sze G max C. Engneerng EPONs for Better Channel Utlzaton In ths fnal set of smulaton experments we wsh to llustrate the utlty of grant schedulng n allowng close-by ONUs to be added to an EPON wthout takng bandwdth from exstng ONUs. Essentally, the maxmum channel utlzaton s sgnfcantly mproved when addng close-by ONUs and utlzng schedulng. We consder an EPON wth Lmted grant szng wth 4 ONUs wth ONU-to-OLT propagaton delays contnuously dstrbuted between 250 µs and 500 µs (.e., ONU-to-OLT dstances of 50 km to 100 km). We then added a varyng number of close-by ONUs wth ONU-to-OLT propagaton delays contnuously dstrbuted between 2.5 µs and 25 µs (.e., ONU-to-OLT dstances of 0.5 km to 5 km).

13 sec msec msec msec msec msec msec msec msec msec, 0, 0, 4, 4, 8, 8, 16, 16 Fg. 11. Average packet delay for EPON wth 4 ONUs between 50 km and 100 km from OLT wth varyng number of close-by ONUs (.e., 0.5 km to 5 km from OLT) added. Fgure 11 shows the average packet delay for dfferent numbers of added close-by ONUs for both and. We observe that addng close-by ONUs ncreases the maxmum achevable channel utlzaton (stablty lmt). The shorter propagaton delays of the close-by ONUs allow these ONUs to be servced whle the GATE messages are propagatng to the ONUs wth the larger propagaton delays and ther upstream transmssons are propagatng up to the OLT. Thus, the added close-by ONUs ncrease the channel utlzaton whle almost not ncreasng the packet delay experenced by the far-away ONUs. Wth schedulng the ONUs are ordered by ther grant sze, rrespectve of ther propagaton delays leadng to poor explotaton of ths ablty., on the other hand, always servces the ONUs wth shorter propagaton delays frst allowng them to mask the round-trp tme to the ONUs wth the larger propagaton delays, and thus achevng substantally hgher channel utlzaton than. V. CONCLUSION In concluson, we ntroduced a new EPON grant schedulng technque called Shortest Propagaton Delay () frst grant schedulng to explot heterogenous propagaton delays. We proved that mnmzes grantng cycle length to wthn a small tme perod (number of ONUs tmes GATE message transmsson tme) and maxmzes channel utlzaton. We analytcally characterzed the stablty lmt (maxmum packet throughput or equvalently maxmum channel utlzaton) for both Gated and Lmted grant szng for arbtrary traffc and characterzed the cycle length and packet delay for Gated grant szng for Posson traffc. We have llustrated the utlty of through a set of smulaton experments. Specfcally, we found that can mprove performance measures when usng Gated grant szng as well as Lmted grant szng. The most sgnfcant mprovements came from ts use wth Lmted grant szng and long reach EPONs. In those crcumstances, packet delay and channel utlzaton were sgnfcantly mproved. Another sgnfcant fndng s the potental channel utlzaton mprovement that s possble when usng grant schedulng n conjuncton wth certan EPON desgn prncples. Specfcally, suppose there are numerous subscrber nodes that must be connected to a central offce and a network engneer has some choces n how to layout several EPONs to connect all of the subscrber nodes to a central offce. Our fndngs ndcate that channel utlzaton can be sgnfcantly ncreased f network engneers construct EPONs such that each EPON contans some ONUs close to the OLT as well as ONUs that are further away from the OLT. ONUs that are wthn a short range of the OLT can fll the dle tmes n whch the OLT wats for data from the ONUs that are further away. An nterestng avenue for future research arses from the convergence of fber-based and wreless access networks, see e.g., [63], [64], whch wll potentally cover large geographc areas and thus have hghly heterogeneous propagaton delays. Further, the ntegraton of medum access control on the fber and wreless meda may lump the propagaton delay on the fber and the wreless medum access delay together to lead to addtonal dversfcaton of the round-trp delays experenced by the OLT. REFERENCES [1] F. Effenberger, An ntroducton to PON technologes, IEEE Communcatons Magazne, vol. 45, no. 3, pp. S17 S25, Mar [2] P. Green, Fber to the Home: The New Empowerment. Wley- Interscence, [3] G. Keser, FTTX Concepts and Applcatons. Wley-IEEE Press, [4] C. Lam, Passve Optcal Networks: Prncples and Practce. Academc Press, [5] M. De Andrade, L. Guterrez, and S. 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