General QoS-Aware Scheduling Procedure for Passive OpticalNetworks

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Citation for the original published paper (version of record):Hadi, M., Bhar, C., Agrell, E. (2020)General QoS-Aware Scheduling Procedure for Passive Optical NetworksJournal of Optical Communications and Networking, 12(7): 217-226http://dx.doi.org/10.1364/JOCN.390902

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A General QoS-Aware Scheduling Procedure forPassive Optical Networks

Mohammad Hadi, Chayan Bhar, and Erik Agrell, Fellow, IEEE

Abstract—Increasing volume, dynamism, and diversity of theaccess traffic have complicated the challenging problem of dy-namic resource allocation in passive optical networks. We intro-duce a general scheduling procedure for passive optical networks,which optimizes a desired performance metric for an arbitraryset of operational constraints. The proposed scheduling has afast and causal iterative implementation, where each iterationinvolves a local optimization problem followed by a recursiveupdate of some status information. The generality of the platformenables a proper description of the diverse quality of servicerequirements, while its low computational complexity makes agiletracking of the network dynamism possible. To demonstrate itsversatility and generality, the applications of the scheme forservice-differentiated dynamic bandwidth allocation in time- andwavelength-division-multiplexed passive optical networks are dis-cussed. To further reduce the computational complexity, a closed-form solution of the involved optimization in each iteration of thescheduling is derived. We directly incorporate transmission delayin the scheduling and show how the consumed power is tradedfor the tolerable amount of transmission delay. Furthermore, a50% power efficiency improvement is reported by exploiting theinherent service diversity among the subscribers. The impact ofservice prioritization, finite buffer length, and packet drops onthe power efficiency of the scheme is also investigated.

Keywords—dynamic scheduling, passive optical networks, powerefficiency, quality of service

I. INTRODUCTION

PASSIVE optical networks (PONs) play an essential role indeveloping power-efficient and quality of service (QoS)-

aware access networks [1, sec. 2], [2], [3]. Video services and5G applications have been considered as the strongest driversfor changing traffic paradigms in PONs. In fact, video deliveryover PON deals with high capacity demands and diverse QoSrequirements [4]. On the other hand, PON is one of the content-ing technologies for 5G transport, and therefore, the dynamicand diverse behavior of the 5G traffic can have a significant im-pression on traffic paradigms in PON [5], [6]. These changingtraffic patterns have imposed new challenges on the complexproblem of dynamic resource allocation in PONs [1, sec. 5]and consequently, motivated development of new networkcontrol schemes [2], [4]. Unfortunately, network managementin current PONs is not yet mature to address the increasing vol-ume, dynamism, and diversity of the future, and even today’s,traffic streams. In fact, a proper scheduling in PONs needs afine and agile adaptation between the allocated resources and

The authors are with Department of Electrical Engineering, ChalmersUniversity of Technology, Sweden, e-mail: mohadi@chalmers.se. This workwas supported in part by Vinnova under grant no. 2017-05228 and the Knutand Alice Wallenberg Foundation under grant no. 2013.0021.

dynamic network behavior. This adaptation is more importantfor a heterogeneous PON with different service requirementsamong its subscribers. Through the separation of control anddata plane, software-defined networking (SDN) provides adynamically fine-grained traffic control that enhances totalnetwork controllability and manageability [7], [8]. In an SDN-based PON, the data plane can be any type of available PONssuch as gigabit PON (G-PON), time-division-multiplexedPON (TDM-PON), time- and wavelength-division-multiplexedPON (TWDM-PON), and code-division multiple-access PON(CDMA-PON) [1, sec. 2], [7]–[9]. A unified SDN controlplane needs a general scheduling procedure to handle resourcemanagement for different implementations of the data plane.The importance of this general scheduling scheme is doubledby the growing tendency to develop more diverse, tunable,and high-speed devices for next-generation PONs (NG-PONs).Although there are many customized scheduling methods fordifferent PONs [2], [9]–[12], to the best of our knowledge,there is no research work on a general scheduling platform toallocate resources of a PON, regardless of its implementationtechnology.

In [13], we introduced Lyapunov optimization for iterativescheduling in a CDMA-PON system, where transceivers areperiodicity reconfigured such that the transmission rate ismaximized while queue stability and bit error rate constraintsare satisfied. The strategy was generalized and applied toelastic optical networks in [14]. In this work, we use Lya-punov optimization to develop a general scheme to iterativelyschedule a PON, regardless of its implementation technology.With Lyapunov optimization, a differential measure of thePON status and its performance metrics describes a structuredobjective function. Each iteration of the Lyapunov methodinvolves an optimization of the structured objective subjectto a given set of operational constraints [15], [16]. Unlikepure heuristic scheduling algorithms [17]–[20], the proposediterative solution is supported by mathematical analysis. Fur-thermore, its general structure allows to describe not onlythe instantaneous but also the time-averaged behavior of thePON by equality and inequality constraints [18], [21]. More-over, important concepts such as service prioritization, trafficclassification, traffic shaping, and queueing stability can beeffectively described and handled in the proposed schedulingplatform [15], [16].

Application examples for TDM- and TWDM-PONs areprovided in this paper. Especially, we demonstrate the ap-plication of the proposed method for power-efficient service-differentiated scheduling in a TDM-PON, and then extendthe results to support a TWDM-PON. As an important con-tribution, we derive a closed-form solution of the involved

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optimization problem in each iteration of the scheduling. Thisclosed-form solution considerably expedites the schedulingand provides much adaptation capability. The average trans-mission delay, as an important QoS parameter, is an explicitinput to the closed-form solution, which can be set to any valuefor facilitating the desired QoS constraint. The scheduling triesto consume the minimum possible power to meet transmissiondelay requirements at every offered load. The smoothness ofthe simulated transmission delay curves confirms the capabilityof the scheme for guaranteeing a desired value of the trans-mission delay. Furthermore, simulation results show that thescheduling method assigns the available resources accordingto the required QoS levels and priorities, and exploits theexisting service diversity among the subscribers to improvethe power efficiency. The scheduling also offers adjustabletradeoffs between power efficiency, transmission delay, anddrop rate to reduce the consumed power at the cost of acontrollable increase in average transmission delay or droprate. To provide practical results, the simulated PON is loadedby a realistic self-similar traffic pattern, and the impact ofphysical limitations, such as finite buffer storage and packetdrops, on the performance metrics is investigated.

The paper evolves by introducing the general schedulingprocedure in Sec. II. The application of the proposed schemeis demonstrated in Sec. III, where a power-efficient service-differentiated scheduling is proposed. Simulation results areprovided in Sec. IV. Finally, concluding remarks are delineatedin Sec. V.

II. GENERAL SCHEDULING

With a few minor terminology changes, the general re-source allocation scheme proposed in [14] can be used forscheduling in PONs, as a novel customized application with itsown operational constraints and computational requirements.However, to make the paper standalone, a brief description isprovided. Consider a general PON, which operates in discretetime intervals n P Z80 , where Zb

a includes the integers betweena and b inclusively. There are I entities (such as ONUs), eachhaving a queue with backlog qirns, which is filled and emptiedby airns arrived and sirns departed bits in every interval n. Theserved bits sirns are determined by reconfiguring the resourcevector αrns (with possible elements such as transmissionwindow) in each interval n such that a targeted performancemetric (such as power consumption) is optimized, while someQoS requirements (such as average transmission delay) areguaranteed and operational constraints (such as guard time)are satisfied.

In each interval n, the resource vector αrns is reconfiguredby solving the optimization problem

minαrns

Γf0rns `ř

iPZI1qirnspairns ´ sirnsq`

ř

lPZL11

pl,1rnsfl,1rns `ř

lPZL21

pl,2rnsfl,2rns s.t. (1a)

fl,3rns ď 0, @l P ZL31 , (1b)

fl,4rns “ 0, @l P ZL41 , (1c)

where f0rns, fl,mrns, and sirns are arbitrary functions of αrnsand the Lyapunov penalty coefficient Γ controls the balance

between f0rns and the summation terms in (1a). The actualqueue length qirns and the so-called virtual queue lengthpl,mrns are updated recursively as

qirn` 1s “ maxtqirns ` airns ´ sirns, 0u, i P ZI1, (2a)

pl,1rn` 1s “ maxtpl,1rns ` fl,1rns, 0u, l P ZL11 , (2b)

pl,2rn` 1s “ pl,2rns ` fl,2rns, l P ZL21 , (2c)

with initialization qir0s “ 0, i P ZI1 and pl,mr0s “ 0,

m P Z21 and l P ZLm

1 . Optimization (1) approaches the mini-mized time-averaged objective f0 with a proximity controlledby Γ, where the overbar denotes time average defined asx “ limnÑ8

1n

ř

n1PZn´10

Etxrn1su with an expectation overrandom arrival events. As proven in [16] using the Lyapunovanalysis, when the arrival process airns is independent andidentically distributed (i.i.d.) over intervals and qirns does notappear in the functions fl,mrns, then the first penalty termin (1a) and update equation (2a) assure the queue stabilityby imposing ai ď si to keep queue lengths finite overtime. Similarly, the second and third penalty terms along withtheir corresponding update equations (2b) and (2c) enforcefl,1 ď 0 and fl,2 “ 0, respectively, which can be used todescribe various QoS and operational constraints using time-averaged inequality or equality expressions. Moreover, (1b)and (1c) define additional instantaneous inequality and equalityconstraints to be satisfied in each interval.

When airns is not i.i.d. or qirns appears in fl,mrns, theperformance of (1) is not rigorously guaranteed. However, asdemonstrated numerically, the proposed general scheme yieldsacceptable results for many practical scenarios with even morerelaxed assumptions [14]. To support this claim and show theapplicability of the scheme for PONs, we provide in the nextsection a few application examples and use simulation resultsto validate their performance.

III. APPLICATION EXAMPLES

The involved intuition in the general scheme inspires thedevelopment of novel iterative and practical scheduling proce-dures, such as the proposed scheme in [13] for CDMA-PONs.To further affirm this assertion, a power-efficient service-differentiated scheduling scheme for TDM-PONs is estab-lished, and then extended to support TWDM-PONs. The per-formance of the scheme is mathematically guaranteed in somespecial cases [15], [16], as briefly explained in Sec. II, whileits general performance can be evaluated using simulationresults, as will be discussed in Sec. IV. The main notationis summarized in Table I. The time index n of the variables isremoved in coming equations and Table I to emphasize that thescheme only needs to store the status of the current interval.

A. TDM-PONConsider the TDM-PON of Fig. 1(a) with a gross bit rate of

RU in the upstream direction. The time is divided into intervalswith duration TC . The upstream capacity of the PON in aninterval is time-shared by I ONUs. An ONU can operate inactive or sleep mode, with or without transceiving capability,

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TABLE I: Main constants and variables.

Type Notation Definition

Con

stan

tsI “ 32 Number of ONUsVi “ 100 Drop penaltyNW “ 1 Number of wavelengthsΓ “ 10 Lyapunov penaltyRU “ 10 [Gbit/s] Upload bit rateTC “ 2 [ms] Interval durationTD “ 0 [ms] Maximum round trip time differenceTH “ 51.2 [ns] Report timeTG “ 1 [us] Guard timeTO “ 2 [ms] Transition timeTS “ 0 [us] Start timeTP “ 0 [us] Process timeTW “ 50 [us] Wavelength tuning timePS “ 0.75 [W] Sleep powerPA “ 4.2 [W] Active powerTi “ 80 [us] Round trip timeDi “ 6 [ms] Average delayQi “ 8 [Mbit] Delaying buffer capacityEi “ 1 [Mbit] Maximum arrived bits in an intervalAi “ 1.5 [Mbit] Collecting/Shaping buffer capacity

Var

iabl

es

Aw Ă A Active ONU set per wavelengthei Wavelength numberpi Virtual queue backlogqi [bit] Delaying buffer backlogai [bit] Shaping buffer backlogsi [bit] Served bitsbi [bit] Uploaded bitsdi [bit] Dropped bitsci [bit] Number of sleep intervalsti [s] GATE transmission time

Delaying Buffer Shaping Buffer

ONU TX

ONU TX

ONU TX

ONU TX

OLT RX +

ONU Queue

ONU Queue

ONU Queue

ONU Queue

Collecting Buffer

(b)

(a)

Fig. 1: (a) Upstream data flow for the TDM-PON application example. (b) The proposedONU queue structure for the TDM-PON application example.

and different consumed power PA or PS , respectively. EachONU transmitter i is fed by a limited-size queue. As illustratedin Fig. 1(b), the queue is in turn divided into three consecutiveparts, named delaying buffer with storage capacity Qi andbacklog length qi, shaping buffer with storage capacity Ai andbacklog length ai, and collecting buffer with a same storagecapacity Ai as the shaping buffer. The small storage capacityAi can accommodate the maximum number of bits arrived toONU i during an interval Ei. A central scheduler residing inthe OLT commands the ONUs using GATE messages whilethe ONUs narrate their status by sending REPORT messagesto the OLT. The OLT sends a GATE message pbi, di, ciq toONU i at time ti and announces the number of uploaded bitsfrom the delaying buffer bi, the number of dropped bits fromthe shaping buffer di, and the number of sleep intervals ci. TheONU i includes the backlog length of the delaying and shapingbuffers, qi and ai, in its sent REPORT message pai, qiq to theOLT.

The scheduler always remembers the latest REPORT mes-sages pai, qiq and stores the index of the active ONUs of thecurrent interval in a set A Ă ZI

1. As shown in Fig. 2, aninterval begins when the OLT runs its scheduling algorithm.The scheduling, which needs the time TP to be finished,includes three main steps of optimization (3), updating (5),and prediction (6), gets the latest REPORT messages pai, qiqas the input, and determines the corresponding GATE messagespbi, di, ciq and proper transmission times ti as the output.The ONUs require different average transmission delay Di

and the OLT involves these distinct QoS requirements in thescheduling. When the scheduling ends, the OLT sends eachGATE message at its proper time ti. After a propagation delayTi2, where Ti is the round trip time (RTT) of ONU i, theGATE message pbi, di, ciq reaches the ONU i. After the timeTS from receiving the GATE message, ONU i uploads themaximum number of the packets with a sum length no morethan bi bits from the delaying buffer to the OLT, drops theminimum number of packets with a sum length no less thandi bits from the shaping buffer and moves the remaining bits tothe delaying buffer, clears the collecting buffer for upcomingpackets by moving its content to the shaping buffer, announcesthe new volumes of the delaying and shaping buffers to theOLT using the REPORT message pai, qiq in time TH , and thengoes to sleep mode for ci intervals. Since a transition fromsleep to active mode needs the time TO, each ONU starts itsreactivation at a proper time before the end of its sleep interval,when the OLT will send a new GATE message. The OLT alsoconsiders a guard time TG between consecutive uploads of thedifferent ONUs.

Considering the required average delay Di, the delayingbuffer can postpone packet transmission to put the ONU insleep mode and save power. Although there is no forcedassumption on the delaying buffer storage capacity Qi, intu-itively, it should have a storage capacity proportional to therequired delay Di to achieve the expected power efficiency.The controllable amount of packet drops in the shaping bufferhelps to satisfy very strict delay requirements and enables auseful trade-off between drop rate and power consumption.The storage capacity of the collecting buffer Ai should beenough to accommodate not only the maximum arrived bitsin an interval Ei but also the arriving bits during sleep mode.When there is no free space in the collecting buffer, parts of theincoming packets are dropped. We refer to the packet dropsin the shaping buffer as the controllable drops while packetdrops due to a full collecting buffer are called unwanted drops.Such unwanted drops can be prevented by a suitable choice ofAi and Ei. For instance, when all ONUs have a same arrivalprocess and equally share the upstream capacity, a reasonablechoice is Ai Á Ei, and Ei Á RUTCI . As a result, assuminga suitable value for Ai, we only focus on the controllablepacket drops in the shaping buffer during different steps ofthe scheduling process.

In the optimization step, the OLT calculates bi and di, i P ZI1

by solving

minb,d

ÿ

iPZI1

”

Γpbi ` Vidiq ` pipqi ´Di

TC

`

ai ´ diq˘

ı

s.t. (3a)

4

OLT

ONU2

TXRX

TXRX

TXRX

ONU1

10

11

10

1010

2020

2020 11

11

11

11 11 21 21

21 21

2121 22

22

22 22

22 22

13

13

13 13

13 13

TC TP

T1/2TG

TS

T2/2TH

TO

Fig. 2: Upstream timing diagram for a TDM-PON with two ONUs. REPORT, GATE, and data messages are shown by blue, red, and gray colors, respectively. Green color showsONU active times. The numbers in each message show the ONU and interval indices. Different timing constants are also illustrated.

bi ě 0, di ě 0, i P ZI1, (3b)

ai ` qi ´ bi ´min

Qi,Di

TCai(

ď di, i P A, (3c)ÿ

iPA

bi ď RU

´

TC ´ TD ´|A| pTH ` TGq

¯

, (3d)

bi “ 0, di “ 0, i P ZI1zA, (3e)

where |A| is the cardinality of A. The optimization problem(3) is a simple adoption of (1) obtained by the substitutionsα “ pb,dq, L1 “ I , L2 “ 0, L3 “ 3I ` 1, L4 “ 2I ,

si “ bi ` di, i P ZI1, (4a)

f0 “ÿ

iPZI1

pbi ` Vidiq, (4b)

fi,1 “ qi ´Di

TC

`

ai ´ diq, i P ZI1, (4c)

fi,3 “ ´bi, fI`i,3 “ ´di, i P ZI1, (4d)

f2I`i,3 “ 0, i P ZI1zA, (4e)

f2I`i,3 “ ai ` qi ´ si ´min

Qi,Di

TCai(

, i P A, (4f)

f3I`1,3 “ÿ

iPA

bi ´RU

´

TC ´ TD ´|A| pTH ` TGq

¯

, (4g)

fi,4 “ 0, fI`i,4 “ 0, i P A, (4h)fi,4 “ bi, fI`i,4 “ di, i P ZI

1zA. (4i)

Similar to (1a), the objective function (3a) has a main termmultiplied by the Lyapunov coefficient Γ, and a penalty termto satisfy a time-averaged constraint. The main term bi`Vidiis a weighted sum of the upload bits bi and dropped bitsdi. To shorten the upload time and equivalently, extend thesleep time and improve the power efficiency, the number ofupload bits bi is minimized in (3a). Furthermore, the priorityof the packet delivery over packet drop can be adjusted bythe coefficient Vi. Moreover, distinct Vi for different ONUscreates a service prioritization and allows to drop part of thestored packets of a low-priority ONU at high load conditions.The penalty term of (3a) ensures the time-averaged constraintqi ´ DiTCpai ´ diq ď 0, which is a simple adoption ofLittle’s theorem to guarantee the required average delay Di

[22, sec. 1], [15]. In (3a), pi is the length of the virtualqueue corresponding to the averaged delay Di. Constraint (3c)drops the incoming bits that cannot be accommodated in astorage size of min

Qi, DiaiTC

(

. Intuitively, the number of

bits stored ahead of the new arrivals ai should not exceedDiaiTC to meet the required delay Di. Therefore, we forcean upper bound of DiaiTC to the storage capacity of thedelaying buffer to better control the transmission delay. Notethat since the buffer length is finite in (3c), the queue stability isalready guaranteed and there is no need to add the first penaltyterm of (1a) to (3a). Constraint (3d) specifies that the overallupload in an interval cannot exceed its net capacity. Note thatthe report and guard times are excluded in the calculation ofthe net capacity. Moreover, due to different distances fromthe OLT, an interval is seen with a variable time shift byeach ONU. As can be interpreted from Fig. 2, to ensure thatconsecutive intervals do not overlap in the OLT and ONUs,the maximum RTT difference TD “ maxi,i1 |Ti ´ Ti1 | shouldalso be excluded in constraint (3d). For inactive ONUs i R A,bi “ di “ 0 by constraint (3e). Fortunately, the OLT does notneed to solve (3) numerically since its closed-form solutionis available in (9), as proven in Appendix A. This closed-form solution considerably reduces the process time TP andimproves the scalability of the proposed scheduling, and shouldbe consequently considered as an important novelty comparedto other customized applications of the introduced generalplatform, such as [14].

In the updating step, (2b) is adapted to update the virtualqueue length by

pi Ð maxt0, pi ` qi ´Di

TC

`

ai ´ di˘

u, i P A (5a)

pi Ð maxt0, pi ` qi ´Di

TCaiu, i P ZI

1zA, (5b)

where ai and qi are set to their latest values derived fromcoming REPORT messages pai, qiq. Since the REPORT mes-sage automatically provides the latest value of qi, the updateequation (2a) is not needed.

In the prediction step, the number of intervals that ONU ican sleep is predicted by

ci “ tmax

0,min Di

TC,Ei

ai

(

´ 1(

u, i P A, (6)

where t.u is the floor function. In fact, (6) is intuitively obtainedby considering the requirements on transmission delay andbuffer overflow. To protect from overflow, the overall arrivedbits to the collecting buffer during the sleep intervals shouldnot exceed the capacity Ai. In the worst case, Ei bits arriveto ONU i in an interval and as assumed, the capacity Ai is

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enough to accommodate these maximum arrived bits Ei À Ai.Now, if ai ď Ei bits arrive in each interval, on average, ittakes Eiai intervals to have Ei bits in the collecting buffer.Therefore, the ONU can sleep for Eiai´1 intervals, and then,become active to serve the collected bits in the interval afterthe sleep without an overflow in the collecting buffer. On theother hand, it is not reasonable to have a sleep time more thanthe required average delay value, so ciTC ď Di. Note thatthe described intuition is based on the average behavior ofthe traffic and burst arrivals still may create unwanted dropsin the collecting buffer. As stated, such unwanted drops aresuppressed by proper selection of the constants Ai and Ei.Different ONUs can also have distinct Ei or Di in (6) to letexploit the available diversity among the ONUs to improvenetwork performance.

Assume that the upload priority within an interval is with theactive ONU having the largest RTT. Let Yphq, h “ 1, . . . ,|A|return the index of the ONU having the hth largest RTT inthe active set A. As can be perceived from Fig. 2, the firstGATE message is sent at tYp1q “ nTC`TP , where nTC is thebeginning time of interval n. Sequentially for h “ 2, . . . ,|A|,tYphq “ nTC ` TP ` TYp1q ´ TYphq `

řh´1h1“1pbYph1qRU `

TG ` THq. Referring to Fig. 2, ONU i can sleep for pn `ciqTC ` TP ` Ti ` TG ´ TO ´ ti ´ biRU ´ TH , after whichits reactivation takes time TO. Note that we implicitly haveTC ´ TD ´ IpTH ` TGq ě 0 to maintain the feasibility ofthe optimization problem in each interval. The special case ofAlgorithm 1 for NW “ 1 corresponds to the control schemedescribed above.

B. TWDM-PONUpload transmission in a TDWM-PON with the upload

bit rate RU per each of NW wavelengths can be similarlyscheduled using the proposed general scheme. In fact, aTWDM-PON consists of NW individual TDM-PON subsys-tems stacked into a common passive network using severalwavelengths [23], [24]. Therefore, the described TDM-PONscheduling can be used for each wavelength. Let TW denotethe wavelength tuning time in tunable ONUs and add the newelement ei to the GATE message pbi, di, ci, eiq to declare theoperating wavelength of ONU i. Following the same way asin the previous subsection, the describing equations of thescheduling scheme can be derived, as summarized in Algo-rithm 1. Clearly, when NW “ 1, Algorithm 1 corresponds tothe proposed TDM-PON scheduling scheme. After collectingthe REPORT messages pai, qiq, in each iteration of Algorithm1, a few local variables are initialized and the ONUs aresorted decreasingly by xi “ Vi ` piDipTCΓq to computeX phq, h “ 1, . . . ,|A|, which returns the index of the ONUhaving the hth largest coefficient xi in the active set A. Then,an ONU with xi ą 1 and yi “ ai`qi´mintQi, DiaiTCu ą 0is assigned yi bits from the remaining transmission capacityof the first unoccupied wavelength w. To reduce the numberof used wavelengths and consequently the power consump-tion, the highest possible number of ONUs share the chosenwavelength w. If there are not enough resources to servean ONU in wavelength w, the next wavelength w ` 1 is

Algorithm 1: Scheduling Algorithm for TWDM-PON.input: REPORT messages pai, qiq, i P ZI

1

output: GATE messages pbi, di, ci, eiq, i P ZI1

for n “ 1, 2, ¨ ¨ ¨ dobi Ð 0, di Ð 0, i P ZI

1

Aw Ð ∅, w “ 1, ¨ ¨ ¨ , NW

w Ð 1z Ð RU

`

TC ´ TD ´|A| pTH ` TGq˘

A “ ti P ZI1|ci “ 0u

xi Ð Vi ` piDipTCΓq, i P Ayi Ð ai ` qi ´mintQi, DiaiTCu, i P Asort A decreasingly by xi to compute X p.qfor i “ 1, ¨ ¨ ¨ ,|A| do

bX piq Ð 0if yX piq ą 0 and xX piq ą 1 then

bX piq Ð yX piq

endif bX piq ą z and w ă NW then

w Ð w ` 1z Ð RU

`

TC ´ TD ´ p|A|´ i` 1qpTH ` TGq˘

endbX piq Ð mintbX piq, zueX piq Ð wAw Ð Aw Y tX piquz Ð z ´ bX piq

endfor w “ 1, ¨ ¨ ¨ , NW do

sort Aw decreasingly by RTT to compute Yp.qtÐ 0for i “ 1, ¨ ¨ ¨ ,|Aw| do

tYpiq Ð nTC ` TP ` TYp1q ´ TYpiq ´ ttÐ t` bYpiqRU ` TG ` TH

endenddi Ð maxt0, yi ´ biu, i P Api Ð maxt0, pi ` qi ´Di

`

ai ´ di˘

TCu, i P Api Ð maxt0, pi ` qi ´DiaiTCu, i P ZI

1zAci Ð tmaxt0,min

DiTC , Eiaiu ´ 1(

u, i P Aci Ð maxtci ´ 1, 0u, i P ZI

1

end

selected. When GATE messages pbi, di, ci, eiq are determined,the indices of active ONUs assigned a particular wavelengthw are grouped in corresponding sets Aw. Thereafter, eachAw is sorted by decreasing RTT. Finally, the transmissiontimes of the GATE messages ti are determined, sleep timesci are calculated, and the status variables are updated. Uponreceiving a GATE message, each ONU tunes its transmitterto the declared wavelength during the start time TS “ TW

and follows the same buffering mechanism as the describedTDM-PON buffering policy in Sec. III-A.

IV. NUMERICAL RESULTS

Transmission delay, power efficiency, and drop rate are thekey performance metrics for evaluating a power-efficient PONscheduling scheme. Here, we report these performance metricsin terms of offered load for different values of a constantparameter to analyze the operation of the proposed schemein different scenarios. Transmission delay is the average time

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between a packet arrival to an ONU and its delivery to theOLT, and includes queuing delays in the different buffers aswell as propagation delay. The power efficiency is the amountof saved power compared to a scenario where all the ONUsare permanently active. Drop rate measures the percentage ofthe packets dropped from the overall arrived packets to theONUs. The offered load Ω “

ř

iPZI1ωiRU is the summation

of the average bit-rate ωi injected to the PON by each ONU idivided by the upstream rate RU . Unless otherwise mentioned,the constant parameters are set to their default values in TableI [10], [23], [25], [26]. The input traffic stream to each ONUis modeled from a self-similar traffic pattern [26]. Trafficdemands of random sizes arrive at random times. The numberof packets in each such demand is Pareto-distributed withshape parameter 1.25 and scale parameter 1, whereas thesilence duration between two consecutive demand arrivals lastsfor the time needed to send a virtual demand, whose numberof packets is Pareto-distributed with shape parameter 1.25 andthe scale parameter set to provide the desired value of Ω. Thelength of each packet is randomly and independently selectedfrom a uniform distribution over r512, 12144s bits [27].

For NW “ 1, the introduced method is a G-PON schedulingand comparable with the “Fixed Threshold” and “DynamicThreshold” schemes proposed in [25]. In these benchmarkschemes, the ONU continues the sleep mode until its queuelength reaches a certain threshold. This threshold is fixed at100 frames for the Fixed Threshold scheme while in the Dy-namic Threshold scheme, the threshold is dynamically updatedin each interval with increment step 1 frame and targeted delay6 ms [25]. As explained in [26], our considered traffic modelis more realistic than the one used in [25]. In each figure, themarked points show the average values for a high number ofruns, each having a standard deviation less than 5%.

Fig. 3 shows the performance metrics versus the offeredload Ω for different values of Di, which have been commonlyused in the literature for performance validation. As expected,the power efficiency decreases with the offered load Ω in Fig.3(a), since when the system is highly loaded, there are moreaccumulated data packets in the ONU queues and we needmore power to serve them. The power efficiency of our schemedepends on Di and can be more or less than the benchmarks.Similarly, depending on the values of Ω and Di, the proposedscheme provides lower or more transmission delay than thebenchmark methods, as can be seen in Fig. 3(b). At low loads,the Dynamic Threshold scheme offers a better transmissiondelay at the cost of lower power efficiency. Conversely, at highloads, the proposed scheme delays packets less by consuminga bit more power than the Dynamic Threshold method. Thereis a mismatch of about 4 ms between the reported delays inFig. 3(b) and the set values Di, which is due to the extradelay imposed by the collecting and shaping buffers. Clearly,the proposed scheduling offers smoother average delay curves,especially compared to the Fixed Threshold scheme. Thisis a significant benefit of the proposed method, where thescheduling accepts the transmission delay as an input constant,and then guarantees it at every offered load Ω. In fact, ourscheme seems more capable of supporting dynamic traffic

conditions since it provides almost constant transmission delayat every load Ω. Consequently, the scheme can be used as akey enabler for low latency 5G transport over PONs, whichrequires constant and low transmission delays [5], [6], [28]. Asignificant feature is that the proposed scheme can trade powerefficiency for transmission delay. In fact, when the ONUs cantolerate a higher transmission delay, the arrived packets canaccumulate more in the delaying buffer to let the ONUs operatemore in sleep mode and consume less power. When Ω “ 0.5,this tradeoff improves the power efficiency by more than 100%if the ONUs can tolerate a transmission delay of Di “ 18 msrather than Di “ 10 ms. As can be seen in Fig. 3(c), there isno packet drop for Vi “ 100,@i. This strict condition on zerodrop rate creates a sudden increase for the transmission delaycurves near Ω “ 1 in Fig. 3(b).

As illustrated in Fig. 4(c), if the ONUs can tolerate a certainamount of drop rate at high loads, we can decrease Vi toadd a controllable amount of packet drops and improve thesmoothness of the delay curves at high loads without changingthe expected power efficiency, as depicted in Figs. 4(a) and4(b). For more stringent Di constraints, a higher drop rateshould be tolerated at high loads, as can be seen in Fig. 4(c).Incorporating the drop rate in the scheduling offers beneficialtradeoffs and results beyond the capabilities of the benchmarkmethods.

Fig. 5 illustrates how diversity can improve the systemperformance, where for each curve, a certain portion of theONUs have Di “ 6 ms while for the other ONUs Di “ 14ms. Depending on the delay distribution among the ONUs,the power efficiency and delay curves fall between the corre-sponding curves for the two extreme cases with Di “ 6 msand Di “ 14 ms for all ONUs. As the number of ONUs withDi “ 14 ms increases, the power efficiency improves sincethe ONUs with Di “ 14 ms can operate in sleep mode forlonger durations. This improvement results from the inherentcapability of the proposed scheme that can separately take intoaccount the diverse transmission delays of the ONUs.

Infinite storage capacity is an impractical assumption ofmany dynamic bandwidth allocation methods such as thebenchmarks. However, our proposed buffering structure as-sumes a finite storage capacity. Fig. 6 investigates the impactof the limited storage capacity of the delaying buffer Qi

on the performance metrics for a targeted transmission delayDi “ 6 ms,@i. The storage capacity Qi has a negligible effecton the power efficiency and transmission delay. However, whenthe storage capacity is low, the strict transmission delay ofDi “ 6 ms is achieved by dropping a number of packets athigh offered loads Ω. As shown in Fig. 6(c), a storage capacityof Qi “ 80 Mbit,@i can bring the drop-rate to zero.

Service prioritization is an important requirement, whichis well supported by the proposed scheme. Fig. 7 shows theperformance curves for a scenario including four distinct QoSclasses with different offered loads and service priorities. Eachclass includes 8 ONUs. Class 1 and 3 with the lower penaltycoefficient Vi “ 1 have less service priority compared to thetwo other classes with Vi “ 100. Class 3 and 4 have the sametraffic load, twice the traffic loads of class 1 and 2. As shownin Fig. 7(a), class 1 and 2 with lower traffic load provide better

7

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

20%

40%

60%

80%

100%

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

35

40

45

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

0.5%

1%

1.5%

2%

(c)

Fig. 3: Performance evaluation for various values of the transmission delay Di and a high drop penalty Vi “ 100, @i.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

20%

40%

60%

80%

100%

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

35

40

45

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

0.5%

1%

1.5%

2%

(c)

Fig. 4: Performance evaluation for various values of the transmission delay Di and a low drop penalty Vi “ 1, @i.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

20%

40%

60%

80%

100%

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

35

40

45

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

0.5%

1%

1.5%

2%

(c)

Fig. 5: Performance evaluation for a diverse scenario, where some of the ONUs have the transmission delay Di “ 14 ms and the others Di “ 6 ms.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

20%

40%

60%

80%

100%

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

35

40

45

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

0.5%

1%

1.5%

2%

(c)

Fig. 6: Performance evaluation for various values of the delaying buffer size Qi and fixed transmission delay Di “ 6 ms, @i.

power efficiency. Regardless of the class type, all the classesexperience almost the same transmission delay, as can be seenin Fig. 7(b). However, at high loads, the performance of class2 and 4 with higher priority is guaranteed at the cost of a littlebit drop for classes 1 and 3 with lower drop penalty coefficient.Different offered load to classes 1 and 3 or 2 and 4 does notaffect service prioritization.

The specified results and statements also hold for theproposed TWDM-PON scheduling scheme with NW ě 1wavelengths. However, with NW wavelengths, the PON almostsustains NW times higher offered load Ω, as shown in Fig.8, where load Ω “ 1 is the offered traffic

ř

iPZI1ωi that

fills the upload transmission capacity of a wavelength RU .The same power efficiency, transmission delay, and drop rate

8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

20%

40%

60%

80%

100%

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

35

40

45

(b)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90%

0.5%

1%

1.5%

2%

(c)

Fig. 7: Performance evaluation for four different QoS classes, where class 1, 2, 3, and 4 own ONUs i P Z81, i P Z16

9 , i P Z2417, i P Z32

25, respectively. Classes 1 and 3 haveVi “ 1, i P Z8

1 Y Z2417, class 2 and 4 have Vi “ 100, i P Z16

9 Y Z3225, and class 3 and 4 have twice the traffic load of class 1 and 2 equal to ωiRU , i P Z16

1 .

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50%

20%

40%

60%

80%

100%

(a)

0.5 1 1.5 2 2.5 3 3.5 4 4.50

5

10

15

20

25

30

35

40

45

(b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50%

0.5%

1%

1.5%

2%

(c)

Fig. 8: Performance evaluation for various numbers of wavelength NW and fixed transmission delay Di “ 12 ms, @i and drop penalty Vi “ 1, @i.

are roughly obtained for Ω ď 1 and every NW . WhenΩ ą 1, all ONUs should be active almost always, possibly withdifferent working wavelengths, to serve the incoming trafficload; therefore, the power efficiency is very low. A TWDM-PON with NW wavelengths can roughly provide a constanttransmission delay up to the load Ω “ NW , where the droprate or transmission delay increases suddenly, depending onthe value of the drop penalty Vi, as depicted in Figs. 8(b) and8(c).

V. CONCLUSIONS

We use stochastic optimization to propose a general schedul-ing framework for PONs, and employ Lyapunov optimizationto establish a structured technique for solving the proposedformulation. The application of the proposed scheme fortransmission scheduling in TDM- and TWDM-PONs is dis-cussed and demonstrated. Particularly, we develop a samplepower-efficient QoS-aware scheduling procedure for TDM-and TWDM-PONs. The example scheduling method is com-pletely described by closed-form mathematical expressionsand accepts the required transmission delays as its inputparameters. According to the simulation results, the proposedscheme consumes the least power to approximately offer aconstant transmission delay at every offered load. The levelof smoothness in the transmission delay curves can be con-trolled by adjusting the buffer capacity or packet drop rate.Moreover, distinct QoS classes with different service prioritiesand features can be defined. A significant capability of thescheme is its inherent inclusion of service diversity, whichenables a typical improvement of 50% in power efficiency atan offered load of 0.5, if half of the subscribers require a

strict transmission delay of 10 ms while the other subscribershave a looser condition of 18 ms. Furthermore, using theembedded tradeoffs of the scheme, the consumed power can bereduced by 20% at an offered load of 0.5 if all subscribers cantolerate an average transmission delay of 18 ms rather than 10ms. The applications of the proposed scheme for transmissionscheduling in other possible types of NG-PONs, which mightbe proposed, demonstrated, or deployed in near future, can bea potential research topic.

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APPENDIX ACLOSED-FORM SOLUTION OF (3)

Since di “ bi “ 0 for i R A, without loss of generality, it isassumed that A “ ZI

1. Substituting xi “ Vi`piDipTCΓq ě 0,yi “ ai ` qi ´mintQi, DiaiTCu, and z “ RU

`

TC ´ TD ´

|A| pTH ` TGq˘

ě 0 in (3), we get

minbi,di,iPA

ÿ

iPA

pbi ` xidiq s.t. (7a)

bi ě 0, di ě 0, i P A (7b)yi ´ bi ´ di ď 0, i P A (7c)ÿ

iPA

bi ´ z ď 0. (7d)

Combining di ě 0 with (7c) yields di ě maxt0, yi ´ biu.To satisfy this bound with equality for all i P A is optimum.Hence, di can be removed from the optimization as follows.

minbi,iPA

ÿ

iPA

pbi ` xi maxt0, yi ´ biuq s.t. (8a)

bi ě 0, i P A (8b)ÿ

iPA

bi ď z. (8c)

For all i P A such that xi ď 1, bi “ 0 is clearly optimal.Similarly, for all i P A such that yi ď 0, bi “ 0 is also optimal.When yi ą 0, then only bi ď yi needs to be considered, sinceany bi ą yi will give a higher objective value and reduce theslack in (8c). Hence, when xi ą 1 and yi ą 0, it is optimal toset bi “ yi for as many i values as (8c) permits, with prioritygiven to those with highest xi.

Formally, assume that X phq, h “ 1, . . . ,|A| returns the indexof the variable having the hth largest coefficient xi in the activeset A. We have

bX p1q “ ItyX p1q ą 0uItxX p1q ą 1umintyX p1q, zu, (9a)

and sequentially, for h “ 2, . . . ,|A|,

bX phq “ ItyX phq ą 0uItxX phq ą 1umintyX phq, z ´h´1ÿ

h1“1

bX ph1qu,

(9b)

where It.u is the indicator function returning 1 for a trueargument, and otherwise 0. Further,

di “ maxt0, yi ´ biu, i P A. (9c)