Design and Analysis of IPACT-based Bandwidth
Allocation for Delay-Guarantee in OFDMA-PON
Hakjeon Bang, Kyeong-Hwan Doo, Seungil Myong, Giovanni Stea, and Chang-Soo Park
Abstract—To guarantee delay performances for time-sensitive services in an orthogonal frequency-division mul-tiple access passive optical network (OFDMA-PON), we pro-pose a two-dimension (i.e., subcarriers and time) upstream bandwidth allocation method based on interleaved polling with adaptive cycle time (IPACT). We first analyze its delay performance in terms of cycle time, i.e., the length of a polling cycle. Then, by setting the maximum polling cycle so as to guarantee timely transmissions for time-sensitive services, we identify the requirements, i.e., maximum band-width allocation, maximum number of allowed optical net-work units (ONUs), and optimum number of subcarriers, for upstream bandwidth allocation with delay guarantees. The proposed scheme is evaluated both numerically and via simulation.
Index Terms—bandwidth allocation; delay; orthogonal fre-quency division multiple access passive optical network (OFDMA-PON); subcarrier.
I. INTRODUCTION
F
or next-generation optical access, where bandwidth-intensive services must be supported at low operational costs, passive optical networks (PONs) need to increase their transmission rates and distances between core/metro and access networks [1]–[3]. To satisfy these requirements, several multiple access schemes have been proposed, based on time-division multiplexing (TDM), wavelength-division multiplexing (WDM), orthogonal frequency-division multi-plexing (OFDM), and various hybrid options [1]. Time-division multiple access (TDMA)-PONs face limitations in practical applications at high speeds (e.g., 40+ Gb/s), due to the scarcity of available high-speed optical components and burst-mode receivers [2]–[4]. On the other hand, WDM-PONs require modifications in optical distribution networks (ODNs), in order to accommodate expanded WDM devices which provide flexibility for multiple access by allowing a wavelength-based modular expansion function [2], [3]. Sev-eral WDM-PON schemes allowing a hybrid WDM/TDMA operation have been proposed to efficiently use the capacity per available wavelength [5].OFDM-based PONs, which are the subject of this paper, have been studied to increase the network bandwidth by
Manuscript received April 25, 2013.
This research was supported by the IT R&D program of MKE/KEIT (10039170), and it was supported by the MSIP, Korea, under the ITRC support program (NIPA-2013-H0301-13-2005) su-pervised by the NIPA.
Hakjeon Bang and Chang-Soo Park (corresponding author) are with the School of Information and Communications, Gwangju Institute of Science and Technology (GIST), 1 Oryong-dong, Buk-Gu, Gwangju, 560-172, South Korea (e-mail: starm021@gist.ac.kr; csp@gist.ac.kr).
Kyeong-Hwan Doo and Seungil Myong are with the Optical Ac-cess Research Team, Electronics and Telecommunications Research Institute (ETRI), 161 Gajeong-dong, Yuseong-gu, Daejeon, 305-350, South Korea (e-mail: khdoo@etri.re.kr; msi@etri.re.kr).
Giovanni Stea is with Dipartimento di Ingeneria
dell’Informazione, University of Pisa, Largo L. Lazzarino 1, 56122, Pisa, Italy. (e-mail: g.stea@iet.unipi.it).
efficiently using available spectra [4], [6]. Furthermore, they allow for extended network reach due to their high resilience to fiber dispersions [2], [3], [7]. In OFDM-based PONs, the upstream bandwidth is divided into independent subcarri-ers. The Optical Line Terminal (OLT) allocates upstream bandwidth to the Optical Network Units (ONUs) by selecting how many subcarriers are used, and for how much time, by each ONU. We refer to this process as two-dimension (2-D) bandwidth allocation, the two dimensions being subcarriers and time.
Orthogonal frequency-division multiple access (OFDMA)-PONs require a medium access control (MAC) protocol for collision-free multiple access in the upstream. The MAC protocol is also related to the bandwidth allocation method, which should allow for high utilization of the available capacity and satisfy the service requirements expressed by the ONUs. Several MAC protocols and upstream bandwidth allocation methods have been proposed. In [8], two types of MAC protocols, Fixed Burst Transmission (FBT) and Dynamic Circuit Transmission (DCT), are evaluated. FBT has a frame-oriented structure, with frames consisting of possibly many packets. Bandwidth is allocated in a round-robin fashion, with limited quanta. In DCT, an effective bandwidth is computed at the time of connection admission. In [9], a MAC protocol is proposed for an OFDMA-PON which assigns some dedicated subcarriers to ONUs in order to eliminate the need for synchronization. The scheme re-duces the implementation cost by allowing for simpler ONU structure, and it reduces the packet delay by immediately reporting queue information from the ONU to the OLT via dedicated subcarriers. In [6] and [10], MAC-layer issues are introduced with considering physical layer aspects. Further-more, a framework based on a 10 gigabit Ethernet PON (10GE-PON) MAC is presented. A 2-D resource allocation is proposed in [4], which relies on a heuristic search for the optimal rectangle dimension, i.e. the one that minimizes the delay and the unused space. In fact, a wide rectangle, using many subcarriers for a shorter time, wastes a lot of bandwidth due to the Inter-Frame Gap (IFG), i.e. the guard time between adjacent transmissions; on the other hand, a narrow rectangle needs to extend more in time to cover the same area, hence incurs higher delays. The optimal number of subcarriers is determined via simulation.
OFDMA-PONs need to support time-sensitive services, such as VoIP, video conferencing, machine-to-machine appli-cations, etc. These services require firm delay guarantees. Therefore, we need to design MAC-level bandwidth alloca-tion schemes that allow the delay to be predicted based on the traffic load. This can only be done, in turn, if we are able to guarantee that the inter-polling time for an ONU is upper bounded. The contribution of this paper is in fact a 2-D bandwidth allocation method, based on Interleaved Polling with Adaptive Cycle Time (IPACT). The method is designed through mathematical analysis, to cover a wide
(a)
(b)
Fig. 1. Illustrations of (a) an OFDMA-PON architecture, and (b) its MAC protocol for upstream bandwidth allocation.
range of network scenarios. The IPACT approach for 2-D bandwidth allocation is chosen because it can provide high channel utilization with simple processing [11], [12]. In the literature, 2-D bandwidth allocation methods were proposed for a hybrid WDM/TDMA-PON which allows wavelength tuning at each ONU. In [13] and [14], IPACT-based methods with a 2-D polling table were proposed. However, they are not designed based on analyses. In [15], upstream scheduling at the OLT is delayed to receive queue length information from all or some ONUs, in order to achieve higher utilization. However, due to the increased queuing delays, this method is not preferred for the OFDMA-PON, which supports a long-reach option. In [16] and [17], the grant scheduling problem in a multichannel EPON is generalized by using scheduling theory. This can be considered for the OFDMA-PON, but this paper focuses on a simpler 2-D bandwidth allocation with analytically derived delay guarantee. Furthermore, multiple modulation formats are not considered in the above methods for hybrid WDM/TDMA-PON.
The remainder of this paper is organized as follows. In Section II, we describe the OFDMA-PON architecture and MAC protocol. Then, in Section III, we lay down the system model , and in Section IV we analyze its delay performance as a function of the cycle time (i.e., the length of a polling cycle). In Section V, by setting the maximum polling cycle in order to guarantee the delay for time-sensitive services, we identify the constraints, i.e., maximum bandwidth allocation, maximum number of allowed ONUs, and optimum number of subcarriers, for the bandwidth allocation with delay-guarantee. In Section VI, we prove the effectiveness of our scheme through simulations, and improve the IPACT-based 2-D bandwidth allocation. Finally, Section VII concludes the paper.
II. OFDMA-PON ARCHITECTURE ANDMAC PROTOCOL
This section describes aspects of the OFDMA-PON tech-nology that are most relevant to the bandwidth allocation
problem in the upstream. We consider an OFDMA-PON architecture in a point-to-multipoint topology, as shown in Fig. 1(a). An OLT at a central office is connected to multiple ONUs at user premises via optical fiber through a passive optical splitter/combiner. Amplifications with/without multi-ple levels of splitting may be included to cope with a large number of ONUs or to extend the transmission distance in a long-reach option, as presented in [6], [20] or [21].
The downstream/upstream bandwidth is divided into mul-tiple subcarriers. Some of these are reserved for control or link maintenance, e.g., pilot tones to obtain channel state information and guards to avoid interference with the DC component [4], whereas others are used for data transmis-sions. In the downstream direction, broadcast traffic can be conveyed to multiple ONUs by using all the subcarriers on only one wavelength [8]. In the upstream direction, different wavelengths having large enough optical spectral gaps are required to avoid Optical Beat Interference (OBI), especially if direct detection is used at the receivers, this being the simplest and cheapest technique [4], [8], [22]. The OBI occurs when different ONUs transmit data with their own light source and the OLT detects them with a single photodiode during the direct-detection process [23].
For the OFDMA-PON upstream bandwidth allocation, a MAC protocol can be designed by adapting classical TDMA-PON MAC to OFDMA/TDMA operation, in order to ensure evolution from the existing TDMA-PON standards [4]. Based on a frame structure described in [10], which adapts 10GE-PON MAC to a 2-D structure as shown in Fig. 1, the upstream bandwidth is allocated by the OLT to ONUs using gate messages, which indicate which subcarrier(s) can be used by an ONU, and for how long. Furthermore, ONUs send report messages to the OLT to report their queue lengths, so that the OLT can dimension the grant accordingly. The report message is 64-byte long and it is piggybacked in the upstream transmissions to the OLT [10]. The gate message can be sent via separate subcarriers to each ONU. The gate message should be kept as simple as possible, since it should be easily parsable by the ONUs. For this reason, previous works already identified rectangular shapes in the two di-mensions of subcarriers and time as a good compromise between allocation flexibility and compactness. For instance, [4] and [10] adapt a 10GE-PON MAC to OFDMA/TDMA operation: allocation rectangles are represented by a couple of low/high subcarrier indexes and start/stop times, which re-quires adding only two extra fields (i.e., low/high subcarriers) to the conventional message for the 10GE-PON bandwidth grant. Based on physical constraints such as receiver sen-sitivity, each ONU uses an independent modulation format, which affects the transmission rate. For instance, the rate achieved with 4QAM is double as much the one achieved with BPSK for the same amount of resources.
Transmissions of two different ONUs must be separated by an InterFrame Gap (IFG) on all the interested subcarri-ers. In [4] and [8], the IFG is assumed to be equal to 5 µs and 10 µs, respectively. The amount of wasted bandwidth due to the IFG is therefore directly proportional to the number of subcarriers used by an ONU. Just after the IFG, the packets queued at each ONU are transmitted, based on the received gate grant size, to the OLT, trailing the report message. Byte length for each upstream transmission at the ONU is computed from the grant size, which is represented in sub-carriers and time. Multiple packets at the ONU are arranged
TABLE I
NOTATIONS FORANALYSIS
Symbol Description Unit
C Total upstream capacity with BPSK bits/s
S Number of subchannels
-mi Modulation order of i-th ONU
-hi Logarithmic value of mi,
-i.e., hi=log2mi
N Number of ONUs
-Nj Number of ONUs having the same
-modulation order, where index j represents the value of hi
Bpkt Packet size bytes
Brep report message size, i.e. 64 bytes bytes
λAi Arrival rate for time-sensitive services packets/s
ΛAi Incoming traffic rate by λAi and Bpkt bits/s
ΛAi=8·Bpkt·λAi
¯
λA Identical arrival rate for λAi packets/s
¯
ΛA Identical incoming traffic rate for ΛAi bits/s
¯
ΛA=8·Bpkt·¯λA
Vpkt(hi) Service time for a single packet s
Trep(hi) Service time for a report message s ρAi(hi) Time-sensitive traffic load at i-th ONU,
-ρAi(hi)=λAi·Vpkt(hi)
ρAtot Total time-sensitive traffic load of all
-ONUs, ρAtot=
PN
i=1ρAi(hi)
TRT T Round-trip time s
TIF G Time duration of interframe-gap s
and transmitted to the OLT via multiple subcarriers. Packets are processed after the last one arrives at the OLT.
In upstream direction, ONUs’ transmissions can be in-terleaved to increase the total network utilization even at high loads which is inferred from fully occupied upstream channel. At such loads, the inter-polling time increases, so the delay increases as well. In order to keep the inter-polling time low enough to allow time-sensitive services to run smoothly, each ONU’s grant time has to be upper bounded.
III. SYSTEMMODEL ANDPROBLEMSTATEMENT
Consider an OFDMA-PON with an OLT and N ONUs. The ONUs are logically located at the same distance from the OLT. The round-trip time (RTT) between the OLT and the ONUs is denoted as TRT T. The OFDMA-PON provides total upstream capacity of C (bits/s) if BPSK is used as a modulation format. The effective capacity may be different if another modulation is used. For example, with 4QAM the total upstream capacity would be 2·C. As shown in Fig. 1, the total upstream capacity is divided into S sub-channels. Each subchannel is a group of a fixed number of orthogonal subcarriers. We assume that all the subchannels have the same number of subcarriers, in order to simplify implementation and control complexity. In other words, each subchannel has the same upstream capacity when the same modulation format is used. The OLT dynamically allocates each subchannel for a time window to ONUs, which the ONU uses to transmit its own upstream data using its modulation format. Let midenote the order of the i-th ONU’s modulation format, i=1, ..., N , and let hi denote the logarithmic value
of the i-th ONU’s modulation order, i.e., hi=log2mi. For example, hi=2 for mi=4 (i.e., 4QAM), and hi=4 for mi=16 (i.e., 16QAM). To count the number of ONUs having the same modulation order, we used a symbol Nj(N =PalljNj) where index j (j=1, 2, ...) represents the value of hi. For example, N4=8 means 8 ONUs with 16QAM.
We are interested in the performance of time-sensitive services, for which we allocate bandwidth using the IPACT-based method described in the next section. ONUs may also run time-insensitive services, which we assume will be served using the leftover upstream bandwidth. Therefore, from now on, we will assume that only time-sensitive ser-vices are present on the ONUs, without repeating every time. We assume that the packets of the i-th ONU have a fixed size of Bpkt(bytes). We will show later on that our derivations still match simulation results if packets have variable length, provided that Bpkt is their average length. Packets arrive based on a Poisson process with rates λAi (packets/s). For
ease of understanding, we also use the term for an incoming traffic rate, ΛAi (=8·Bpkt·λAi (bits/s)). If each arrival rate is
the same and equal to ¯λA for all ONUs, the identical term ¯
ΛA(=8·Bpkt·¯λA(bits/s)) is used to derive analytic equations. A 64-byte report message, which is defined in [10], is used to convey queue information from each ONU to the OLT, and its size is denoted with Brep (bytes). When the total upstream capacity C and the number of subchannels S are given, the service time Vpkt(hi) for a single packet is obtained as:
Vpkt(hi) =
8·Bpkt·S C·hi
, (1)
which means that the service time depends on the transmis-sion rate, i.e. on the modulation format hi. For example, at 10 Gbits/s with 64 subchannels, a packet of 1518 bytes needs service times of 38.86 µs and 19.43 µs for 4QAM and 16QAM, respectively. We assume that overheads for preamble and link controls are negligible, except the report message for the bandwidth allocation process. Similarly, the service time for a 64-byte report message is Trep(hi)=8·Brep·S/(C·hi). Let ρAi(hi) and ρAtot denote the traffic load at the i-th ONU,
i=1, ..., N , and total traffic load of all ONUs, respectively. ρAi(hi) and ρAtotare obtained as:
ρAi(hi) = λAi·Vpkt(hi), (2) and, ρAtot= N X i=1 ρAi(hi), (3) respectively.
Two inequalities have to be satisfied in order to guarantee delay performance. In order to guarantee that the delay at the i-th ONU stays finite, a necessary condition is that: ρAi(hi)+ρWi(hi)≤1, where ρWi(hi) is the load that
repre-sents wasted bandwidth, i.e., the guard for interframe-gap (IFG) and bandwidth report. The IFG is denoted as TIF G. Since the amount of ρWi(hi) is normally negligible, we can
substitute the above inequality with the following:
ρAi(hi) < 1. (4)
Second, since all ONUs’ traffic is served in S subchannels simultaneously, the following condition must be satisfied: ρAtot+ρWtot≤S, where ρWtot is the sum of all ρWi(hi),
i=1, ..., N , i.e., ρWtot=
PN
i=1ρWi(hi). Similarly, the condition
can be replaced by the following:
ρAtot < S. (5)
In both (4) and (5), since ρAi(hi) and ρAtot are influenced
by modulation orders of ONUs, it means that a high-order modulation format is required to transmit more traffics in the allocated subchannel to each ONU. The notations for analyses are summarized in Table I.
Time-sensitive services (e.g., VoIP, video conferencing) re-quire bandwidth allocation with delay guarantees. The delay is directly influenced by how much time elapses between two subsequent upstream transmissions of a backlogged ONU [26], which we call the cycle time. In other words, the delays of time-sensitive services can be guaranteed in tolerable ranges by giving an upper bound on the cycle time, either by simulations or analytical techniques (e.g., [26]– [28]). We identify an analytical framework that allows the average cycle time to be computed, allowing for ONUs to have independent modulation formats. Given the above, we can easily compute the following quantities:
• maximum bandwidth allocation with delay guarantees; • maximum number of ONUs that the network can serve
when bandwidth limits are given;
• optimal numbers of subchannels that allow maximum resource utilization with delay guarantees.
The first quantity tells us the admission control limit. More-over, when the bandwidth currently allocated is below the admission control limit, the second item determines how many ONUs the network can serve. The third quantity helps one to increase the total network utilization by controlling the number of subchannels. A higher utilization means ei-ther a higher bandwidth per ONU or more ONUs with the same bandwidth.
IV. IPACT-BASED2-D BANDWIDTHALLOCATION
The MAC protocol for the OFDMA-PON of Fig. 1 is ob-tained by adapting the 10GE-PON MAC to the 2-D struc-ture of Section II. In the 10GE-PON, polling requests from multiple ONUs are overlapped in time and an IPACT-based scheme interleaves bandwidth grants for the ONUs [24], [25]. In the OFDMA-PON, the transmission rate at each ONU is affected by its modulation format. That is, the throughput, delay, and jitter can be varied based on the modulation format as well as the maximum grant, the num-ber of ONUs, and the numnum-ber of subcarriers/subchannels. The IPACT adapts the cycle time (i.e., the length of a polling cycle) to the instantaneous network load (i.e., queue occupancies), so that the total upstream bandwidth is dis-tributed to ONUs based on their loads. At lower loads, the cycle time (and, consequently, the delay) will be shorter, and it will increase proportionally with the load up to a maximum, which should be computed so as to guaran-tee a reasonable delay performance. The IPACT-based 2-D scheme interleaves bandwidth grants in the two dimensions of subchannels and time. Given a subchannel u, call F T (u) the end of the latest grant scheduled in the future for that subchannel. The IPACT-based 2-D algorithm, which is aimed at minimizing the delay of the ONUs, consists in the following: when the OLT receives an ONU’s band-width request at time t, the OLT computes the transmission window T W in time required to satisfy the request, and
(a)
(b)
(c)
Fig. 2. An example of IPACT-based 2-D bandwidth allocation.
locates the subchannel s=argmin(F T (u)). The transmission window size is given by the grant size plus enough time for a report message. It then allocates the transmission window on subchannel s. In doing so, F T (s) is updated as F T (s)=max(t+Tproc+TRT T+T W, F T (s)+TIF G+T W ), where Tproc is the processing time for the upstream bandwidth allocation. An example of IPACT-based 2-D allocation in a three-subchannel case is shown in Fig. 2.
(a) Assume that at time t0 the OLT gives the k-th grant to the i-th ONU, which consists in a subchannel index and time window. In Fig. 2(a), the bandwidth is located in the subchannel-#2, after the additional delay (i.e., round-trip time (RTT), processing time) from t0, and then the F T (2) will be updated.
(b) At time t1, the OLT tries to find the subchannel that minimizes the delay for the (i+1)-th ONU. In Fig. 2(b), since a subchannel-#3 offers the shortest delay, the grant for the (i+1)-th ONU begins an IFG time after the previous grant has terminated.
(c) At time t2, the (k+1)-th grant for ONU i is allocated, again picking the subchannel which warrants the short-est delay. In Fig. 2(c), the cycle time (i.e., the length of the polling cycle) is defined as the time duration between the two subsequent grants k and (k+1) of the same ONU. We analyze the delay performance of the IPACT-based 2-D bandwidth allocation scheme in the following subsection. Later on, in Section V, we optimize the bandwidth allocation for delay guarantees for time-sensitive services.
A. Cycle Time
The cycle time strongly influences bandwidth utilization and delay [26]. Hereafter we derive estimates of the latter in two different regimes, i.e., at high loads and low loads respectively. The distinction makes sense, since in the first case we can assume that the RTT is negligible in one case, not so in the other. The following analyses can be applied – with some modifications – to a hybrid WDM/TDMA-PON. However, in an OFDMA-PON, each ONU has its own modulation format, which affects the time window size for upstream transmissions. This is taken into account in the model by terms, Vpkt(hi) and Trep(hi), which are related to the modulation format.
1) Estimating the cycle time at high loads: In IPACT-based allocation, the cycle time drastically increases when upstream transmissions at high loads are closely inter-leaved. We estimate the cycle time by using average value approaches presented in [27] and [28], using the notation summarized in Table I. Let Ti(hi, k) and Gi(hi, k) denote the k-th cycle time and grant time for the i-th ONU with hi, respectively, where k=0, 1, 2, ..., and i=1, ..., N . We estimate the cycle time under a gated service discipline, i.e. one where the OLT allocates bandwidth to an ONU as much as it has requested [25]. Note that effective values for estimation are started from k=1.
Given Poisson arrivals with rates λAiat the i-th ONU, the
probability of having exactly nAarrivals within a cycle time of Ti(hi, k) follows the Poisson probability density function. Let NAi(k+1) be the random variable that defines the
num-ber of packets which is reported to the OLT at the start of the (k+1)-th polling cycle (i.e., queued during the k-th polling cycle). P r (NAi(k+1)=nA) = e λAi·Ti(hi,k)·(λAi·Ti(hi, k)) nA nA! . (6) The average number of arrivals in a polling cycle is obtained as: E [NAi(k+1)] = ∞ X nA=0 nA·P r (NAi(k+1)=nA) . (7)
By substituting (6) into (7) (See Appendix A),
E [NAi(k+1)] = λAi·E[Ti(hi, k)]. (8)
Then, since – under a gated service discipline – the grant time depends on the number of packet arrivals, it is given by:
E [Gi(hi, k+1)] = Vpkt(hi)·E [NAi(k+1)] . (9)
With (2) and (8), E [Gi(hi, k+1)] can be rewritten as: E [Gi(hi, k+1)] = ρAi(hi)·E [Ti(hi, k)] . (10)
Note that the grant time in this estimation does not include the 64-byte report message, which will be counted in later. In the IPACT-based 2-D bandwidth allocation, the grant time is allocated to a subchannel which can minimize the delay. The i-th ONU’s transmission is interleaved with the interframe-gap TIF G and a report time, after the previous N −1 transmission windows for N −1 ONUs. At high loads, when upstream transmissions are closely interleaved, each
ONU’s transmission window incorporates a grant time, re-port time, and IFG. The rere-port time is given by (1).
E [Ti(hi, k+1)] = 1
S· {TIF G+ E [Gi+1(hi+1, k)] + Trep(hi+1) ... +TIF G+ E [GN(hN, k)] + Trep(hN) +TIF G+ E [G1(h1, k+1)] + Trep(h1) ... +TIF G+ E [Gi(hi, k+1)] + Trep(hi)} . (11) In the steady state, the cycle time E [Ti(hi, k+1)] is sta-ble. Since the grant time is determined by the number of arrivals during the cycle time, the index term k can be eliminated. Hence, the transmission windows Gi(hi, k) also have a steady-state average size E[Gi(hi)], which depends on the modulation format. Then, the above equation can be rewritten as: E [Ti(hi)] = 1 S· ( N X i=1 (E [Gi(hi)] + Trep(hi)) + N ·TIF G ) . (12) From (10), E [Gi(hi)] =ρAi(hi)·E [Ti(hi)] at the steady state.
Moreover, it is straightforward to observe that (12) does not depend on the ONU index i. Thus, it can be rewritten as:
E [T ] = 1 S· ( N X i=1 (ρAi(hi)·E [T ] + Trep(hi)) + N ·TIF G ) . (13) Then, by expanding (13) in terms of E [T ], the average cycle time is obtained as:
E [T ] = N ·TIF G+ PN i=1Trep(hi) S −PN i=1ρAi(hi) , (14)
which holds for ρAtot<S, where ρAtot=
PN
i=1ρAi(hi). Now,
E[T ] must be larger than TRT T+Tproc, where TRT T is the round-trip time and Tprocis the processing delay at the OLT. That is, E [T ] = N ·TIF G+ PN i=1Trep(hi) S −PN i=1ρAi(hi) > TRT T+ Tproc. (15) From the above equation, since Trep(hi)=8·Brep·S/(C·hi) and ρAtot=
PN
i=1ρAi(hi), we can find that the minimum
overall load under which (14) makes sense is: ρAtot> S −
N ·TIF G+ (8·Brep·S/C)·PNi=11/hi TRT T+ Tproc
. (16) This confirms that the above derivations only hold at suf-ficiently high loads. For example, assuming TIF G=10µs [8], Tproc=35µs [27], and TRT T=200µs, 256 ONUs using 4QAM, and 10 Gbits/s on 64 subchannels, the minimum ρtotis equal to 51.32. For each ONU, 51.32/256=0.20.
2) Estimating the single cycle time at low loads: Most of the times, (14) is sufficient, since it allows one to locate the knee point, i.e. the load at which the delay starts to increase drastically. Large differences in transmission rates due to different modulations may make ONUs have different knee points. However, (14) can underestimate the cycle time at low loads. In this case, in fact, we can assume that the cycle time is the one corresponding to a single transmission,
with the RTT being the dominant component. Such a low-load cycle time estimate bounds from below the actual cycle time at all regimes, hence can be used to complement (14). When the traffic is not very high, the cycle time is mainly determined by the round trip time, processing delay, and granted transmission window.
E [Ti∗(hi, k)] = TRT T+ Tproc+ E [G ∗
i(hi, k)] + Trep(hi). (17) The above equation at the steady state becomes:
E [Ti∗(hi)] = RT T + Tproc+ E [G ∗
i(hi)] + Trep(hi). (18) Note that the cycle time is influenced by the modulation of the ONU. The cycle time in (14) is determined as the sum of grant times in the steady state while the cycle time in (18) is determined by the grant time affected by the modu-lation format order. Since the grant time depends on packet arrivals and the arrivals are reported based on the previous cycle time of E [T∗
i(hi)], the grant time is represented as: E [G∗i(hi)] = ρAi(hi)·E [T ∗ i(hi)] . (19) By substituting (19) into (20), E [Ti∗(hi)] = TRT T + Tproc+ ρAi(hi)·E [T ∗ i(hi)] + Trep(hi). (20) The cycle time at an independent transmission is therefore:
E [Ti∗(hi)] =
TRT T + Tproc+ Trep(hi) 1 − ρAi
, (21)
which holds for ρAi<1. This can be used to improve the
estimation of the cycle time for each ONU. In addition, taking the average, E [T∗] = 1 N· N X i=1 E [Ti∗(hi)]. (22)
Finally, we can estimate the cycle time by using (14) and (22), respectively. Since (14) is only validated in ranges of (3) and (16), (22) improves the cycle time at a low load and a point which is near to TRT T+Tproc.
B. Delay Performance
By ensuring upstream bandwidth for time-sensitive ser-vices in an OFDMA-PON, the delay performance can be measured with lower and upper bounds. When (2) and (3) are satisfied, the delay experienced by a packet lies in a range of E[T ] + TRT T/2 to 2·E[T ] + TRT T/2. In fact, if a packet arrives at an empty ONU exactly before the report message is transmitted, then its delay will be one cycle time, i.e. E[T ] on average, with a transmission delay of TRT T/2. If instead the packet arrives just after the report has been transmitted, its delay will be two cycle times, i.e., 2·E[T ] on average, plus the transmission delay TRT T/2. Hence the average delay is proportional to the average cycle time in a steady state. This means that the cycle time can be used to evaluate the delay performance of a bandwidth allocation method.
On the other hand, a maximum average delay can be guaranteed by setting a bound on the maximum polling cycle in the network. In the next section, we suggest design guidelines for efficient IPACT-based bandwidth allocation.
V. DESIGN OFIPACT-BASED2-D BANDWIDTH
ALLOCATION WITHDELAY-GUARANTEE
The OFDMA-PON needs to satisfy quality requirements of time-sensitive services. We have shown that, when the load is such that queues are stable, delays lie in a range of one to two times of a polling cycle time. The adaptive polling cycle should be limited by a predefined maximum polling cycle, which affects bandwidth utilization, delay, and jitter [26]. Let Tlim denote the maximum polling cycle, which must be larger than TRT T+Tproc. That quantity needs to be above the average cycle time, at both a high and a low load. Therefore, for high loads, we have:
E [T ] =N ·TIF G+ PN i=1Trep(hi) S −PN i=1ρAi(hi) ≤ Tlim. (23) On the other hand, for low loads, we have:
E [Ti∗(hi)] =
RT T +Tproc+Trep(hi) 1 − ρAi(hi)
≤ Tlim. (24) Equations (23) and (24) allow us to derive constraints on the parameters, such as maximum bandwidth, maximum number of ONUs and minimum number of subchannels, so that a maximum cycle time is not exceeded.
A. Maximum Bandwidth Allocation
At each polling, an ONU conveys its queue information to the OLT, which grants bandwidth based on the queue length. This implies that ONUs with large queues could monopolize the entire bandwidth, thus overly delaying the others. To avoid this, the OLT will limit the maximum transmission window size: every ONU gets a grant for as many bytes as it has requested, up to a maximum transmission window size. The limit can be specified according to several schemes, e.g., inferred from the Service Level Agreement (SLA) for each ONU, or dynamically adjusted based on network conditions. We have already observed that the cycle time needs to be set below a maximum. By expanding in terms of loads, i.e., ρAtot and ρAi(hi), from (23) and (24), we can obtain the maximum transmission size. First, from (23), the following condition is obtained. ρAtot≤ S − N ·TIF G+P N i=1Trep(hi) Tlim . (25)
With the assumption that each arrival rate is the same with ¯
λA in all ONUs, the requirement for an incoming rate ¯ΛA (=8·Bpkt·¯λA(bits/s)) can be obtained as (See Appendix B):
¯ ΛA≤ C·S·Tlim− C·N ·TIF G− 8·Brep·S·P N i=11/hi Tlim·S·PNi=11/hi . (26) Furthermore, at low loads, an additional inequality derived from (24) has to be considered.
ρAi≤ 1 −
RT T + Tproc+ Trep(hi) Tlim
. (27)
Similarly, by expanding in terms of ΛAi, an additional
in-equality is obtained from (27) as: ¯
ΛAi ≤
C·hi· (Tlim− (RT T +Tproc)) − 8·Brep·S Tlim·S
. (28) By using the above inequalities (26) and (28), we can compute the maximum transmission size for the delay-guarantee. For example, assume TIF G=10µs [8], Tproc=35µs
[27], and TRT T=200µs, 10 Gbits/s, 256 ONUs using 4QAM, on 64 subchannels. By setting Tlim=2 ms for the delay-guarantee, we obtain two inequalities: ¯ΛA≤76.3065 M bits/s and ¯ΛAi≤275.52525 M bits/s. Then, the maximum
transmis-sion size of 19076 bytes for 2 ms is obtained from the first inequality. By increasing the number of subchannels from 64 to 128 at TRT T=1ms for 100 km, the two inequalities become
¯
ΛA≤77.08775 M bits/s and ¯ΛAi≤75.134625 M bits/s. Then, in
this case, the maximum transmission size of 18783 bytes for 2 ms is obtained from the second inequality.
B. Maximum Number of ONUs
The OFDMA-PON supports multiple modulation formats based on receiver sensitivity, which mainly depends on trans-mission distance. The capacity can be increased by using high-order modulation formats. By using (23) which contains the term N , we obtain the maximum number of supported ONUs at high loads can be derived, in different conditions.
1) Single Modulation Format: Assuming that ONUs are located approximately at the same distance, the network may use a single modulation format. Let hs be the logarith-mic value of that format. Assuming that each arrival rate is the same and equal to ¯λA for all ONUs, we can obtain the following equation from (23) as:
N ·TIF G+ N ·Trep(hs) S − N ·¯λA·8·Bpkt·S/(C·hs)
≤ Tlim. (29) By rewriting for N , we obtain the maximum number of ONUs as: N ≤ Tlim·S·C TIF G·C + S· Tlim· ¯ΛA+ 8·Brep /hs . (30)
For example, under the same assumptions as before, with Tlim=2 ms, 196 ONUs using 4QAM can be served at 100 M bits/s each. By increasing the modulation order from 4QAM to 16QAM, the number of served ONUs increases to 385.
2) Two Modulation Formats: If the OFDMA-PON covers two ONUs groups located at different distances, the network may support two different modulation formats. Let the num-bers of ONUs at the two groups be Na and Nb, respectively, where N =Na+Nb. Call hs the smallest logarithmic modula-tion value and let the other one be hs+ d (d>0), assuming that each arrival rate is the same and equal to ¯λA for all ONUs, we can obtain the following equation from (23) as:
N ·TIF G+ Na·Trep(hs) + (N −Na)·Trep(hs+d) S − Na·¯λA 8BpktS Chs − (N −Na)¯λA 8BpktS C(hs+d) ≤ Tlim. (31)
By rewriting for N , we obtain the maximum number of ONUS from the following equation (See Appendix C):
N ≤ Tlim·S·C − Na·S· Tlim· ¯ΛA+ 8·Brep
1 hs − 1 hs+d TIF G·C + S· Tlim· ¯ΛA+ 8·Brep /(hs+d) . (32) For example, under the same assumptions as before, given 150 ONUs in a group using 4QAM as a lower modulation order, total 241 ONUs (i.e., 150 ONUs using 4QAM plus 91 using 16QAM) can be served at 100 M bits/s.
C. Number of Subchannels for Utilization Maximization The number of subchannels influences the channel uti-lization due to the fact that IFG wastes bandwidth. The more bandwidth you waste, the higher the delay will be. The problem is already mentioned in [4], where it is observed via network simulations. We instead compute the minimum number of subchannels analytically. Assume that each ar-rival rate is the same and equal to ¯λA for all ONUs. When (5) is satisfied, we can obtain the constraint on the number of subchannels by substituting Trep(hi) and ρAi(hi) into (23)
and by rewriting in terms of S (See Appendix D), where Trep(hi)=8·Brep·S/(C·hi) and ρAi(hi)=λAi·8·Bpkt·S/(C·hi).
S ≥ N ·TIF G·C
Tlim·C − Tlim· ¯ΛA+ 8·Brep·PNi=11/hi
, (33) which holds for Tlim>(8·Brep·PNi=11/hi)/(C− ¯ΛA·PNi=11/hi). For example, under the same assumptions as before, 256 ONUs using 4QAM at 60 (respectively, 70) M bits/s each, require a minimum number of subchannels equal to 6 (respectively, 13), in order to reduce the bandwidth waste by increasing bandwidth granularity). As another example, if Tlimchanges from from 2 ms to 1 ms, the minimum number of subchannels required becomes 12 (respectively, 27). Based on results for the minimum number for subchannels, the optimal number for minimizing the delay can be determined by investigating the cycle times.
VI. PERFORMANCEANALYSIS
We now evaluate the accuracy of our derivation by com-paring the analytical results to those obtained via simulation in the same conditions. Simulations are performed using an OMNeT++-based simulator [29].
For cycle time estimation, an OFDMA-PON with N =128 is evaluated. We consider a distance of 20 km, which entails an RTT of 200 µs, and one of 100 km, which entails an RTT of 1 ms. The IFG and processing time are set to 10 µs [8] and 35 µs respectively [27]; the IFG refers to that of a TDM-PON. The total network capacity (assuming BPSK) is 10 Gbits/s. The transmission rate at a subchannel with BPSK is com-puted by C/S (M bits/s), and it is increased proportionally if a higher-order modulation format (e.g., 4QAM, 16QAM) is used. A Poisson traffic model is used to test the effectiveness of the analysis. Packets have a constant size of 1518 bytes, unless specified otherwise.
Figure 3 shows the results obtained through simulation and using our approach in the two above settings. The simulation results exhibit a good match with those drawn by using our equations, (16) and (22). We observe that, at low loads, (16) underestimates the cycle times. As shown in Fig. 3(c), the error increases with the transmission distance. However, (22) makes up for the error by capturing the cycle time at low loads. The largest error occurs when the two lines intersect, and it appears to grow with the distance. In any case, we believe that the proposed analyses offer a satisfactory prediction.
We now show that our derivations can be applied – with a grain of salt – even when some of the hypotheses of the system model are violated. First, we show that the cycle time estimate is still acceptable when ONUs transmit variable-sized packets, provided that their average length is used in the formulas. Fig. 4 compares a fixed- and variable-sized case. For the fixed case, we use a packet size of 1518
30 35 40 45 50 55 60 65 70 75 80 0 1 2 3 4 Cycle time (ms)
Incoming rate (Mbits/s) S=16 (Simulation)
S=64 (Simulation) S=16 (Analysis at high loads) S=64 (Analysis at high loads) S=16 (Analysis at low loads) S=64 (Analysis at low loads)
(a) 30 35 40 45 50 55 60 65 70 75 80 0 1 2 3 4 Cycle time (ms)
Incoming rate (Mbits/s) S=4 (Simulation)
S=32 (Simulation) S=96 (Simulation) S=4 (Analysis at high loads) S=32 (Analysis at high loads) S=96 (Analysis at high loads) S=4 (Analysis at low loads) S=32 (Analysis at low loads) S=96 (Analysis at low loads)
(b) 30 35 40 45 50 55 60 65 70 75 80 0 1 2 3 4 Cycle tim e (ms)
Incoming rate (Mbits/s) S=16 (Simulation)
S=64 (Simulation) S=16 (Analysis at high loads) S=64 (Analysis at high loads) S=16 (Analysis at low loads) S=64 (Analysis at low loads)
(c) 30 35 40 45 50 55 60 65 70 75 80 0 1 2 3 4 Cycle time (ms)
Incoming rate (Mbits/s) S=4 (Simulation)
S=32 (Simulation) S=96 (Simulation) S=4 (Analysis at high loads) S=32 (Analysis at high loads) S=96 (Analysis at high loads) S=4 (Analysis at low loads) S=32 (Analysis at low loads) S=96 (Analysis at low loads)
(d)
Fig. 3. Cycle time comparisons between analyses and network sim-ulations, (a) and (b) for RTT=200 µs, and (c) and (d) for RTT=1 ms, at 10 Gbits/s with BPSK, 128 ONUs.
bytes, whereas for the variable case we let packet size vary uniformly in [64, 1518] bytes, i.e. with an average length of 791 (roughly half the fixed-size case). As the figure shows, both cycle times are very close. Note that the horizontal axis reports the arrival rate in M bits/s, hence the same point
30 35 40 45 50 55 60 65 70 75 80 0 1 2 3 4 Cycle time (ms)
Incoming rate (Mbits/s) S=16 S=64 (Sim.-fixed size at RTT=200us) S=16 S=64 (Sim.-variable size at RTT=200us) S=16 S=64 (Sim.-fixed size at RTT=1ms) S=16 S=64 (Sim.-variable size at RTT=1ms)
Fig. 4. Cycle times for fixed and variable sizes of packets, at 10 Gbits/s with BPSK, 128 ONUs.
3 30 35 40 45 50 55 60 65 70 75 80 0 1 2 3 4 Cycle time (ms)
Incoming rate (Mbits/s) S=16 (Sim.:equidistant) S=64 (Sim.:equidistant) S=16 (Sim.:non-equidistant) S=64 (Sim.:non-equidistant) (a) 30 35 40 45 50 55 60 65 70 75 80 0 1 2 3 4 Cycle time (ms)
Incoming rate (Mbits/s) S=4 (Sim.:equidistant) S=32 (Sim.:equidistant) S=96 (Sim.:equidistant) S=4 (Sim.:non-equidistant) S=32 (Sim.:non-equidistant) S=96 (Sim.:non-equidistant) (b)
Fig. 5. Cycle times comparisons between equidistant and non-equidistant ONU scenarios, obtained by simulating 128 ONUs with (a) S=16, 64 and (b) S=4, 32, 96. In the equidistant case, all ONUs have RTT=128 µs. In the non-equidistant case, four groups of 32 ONUs have RTT=50, 100, 150, 200 µs respectively.
corresponds to different packet arrival rates in the two cases, packets being shorter on average for the variable-size case.
In Fig. 5, cycle times are plotted with both equidistant and non-equidistant ONUs. In the equidistant-ONU case, 128 ONUs are located at the same distance of 20 km. In the non-equidistant-ONU case, four groups of 32 ONUs are located at 5, 10, 15, and 20 km each. As shown in Fig. 5, cycle times match at high loads, since the effect of the RTT is negligible and upstream transmissions are interleaved. At low loads, where the cycle times are mainly affected by the RTT and grant size, some difference can be observed.
Figure 6 shows the cycle time as a function of the arrival rate for various values of S. As the figure shows, an optimal value of S can be identified, which depends on the network parameters, especially the distance, as shown in Fig. 6(b).
(a)
(b)
Fig. 6. Cycle times obtained by simulation, for (a) RTT=200 µs and (b) RTT=1 ms, at 10 Gbits/s with BPSK, 128 ONUs.
Fig. 7. Delay obtained by simulation, RT T = 200 µs, at 10 Gbits/s with BPSK, 128 ONUs.
Cycle times for RTTs of 200 µs and 1 ms are relatively low at the subchannels of 64 and 32, respectively, over all the range of arrivals. We will show that the optimal numbers can be easily determined using results that are presented in Fig. 7.
Figure 7 shows the relationship between the cycle time and the delay, mentioned in [26]. By comparing Fig. 6(a) and Fig. 7, we can observe that the cycle time directly influence the delay in IPACT-based bandwidth allocation.
Figure 8 shows cycle times obtained analytically as a function of the number of subchannels, with average arrival rates of 45 to 75 M bits/s for an ONU. The results of Fig. 8 can be used to determine the optimal number of subchannels without resorting to simulation. In Fig. ?? and Fig. ??, the minimum points at the highest rate of 75 M bits/s occur with 80 (i.e., near 64) and 32 subchannels, respectively. The numbers obtained are the same as those of Fig. 6(a) and
(a)
(b)
Fig. 8. Cycle time vs. number of subchannels, for (a) RTT=200 µs and (b) RTT=1 ms, at 10 Gbits/s with BPSK, 128 ONUs.
Fig. 9. Cycle time vs. number of subchannels for RTT=200 µs with two ONU groups having different modulation orders, at 10 Gbits/s with ratios of BPSK:4QAM are 3:1, 1:1, and 1:3 128 ONUs.
Fig. 6(b). Obviously, the analysis can be extended to other network designs quite easily. The high cycle time observed with a small number of subchannels (e.g., S=4) is due to the wasted bandwidth incurred by assigning guard bandwidths to surplus subcarriers. For large number of subchannels, the increase is steeper at 100 km than at 20 km because the size of the bandwidth request increases with the trans-mission distance. Furthermore, although the dependence on the number of subchannels is similar in both cases, the increase at longer distances occurs for a higher number of subchannels. This result can also be utilized to determine the number of subcarriers for a subchannel, in order to minimize the cycle time.
Next, we investigate the dependence of the cycle time on the modulation format for the 20 km case. In Fig. 9, the cycle time of 128 ONUs with BPSK are compared with those of the
4 8 16 32 48 64 80 96 112 40 50 60 70 80 90
Maximum grant size (Mbits/s)
Number of subchannels
RTT=200us, IFG=10us, BPSK RTT=200us, IFG=5us, BPSK RTT=1ms, IFG=10us, BPSK RTT=200us, IFG=10us, BPSK:4QAM=3:1
Fig. 10. Maximum grant size.
4 8 16 32 48 64 80 96 112 100 110 120 130 140 150 160
Maximum number of allowed ONUs
Number of subchannels
IFG=10us, A=70Mbits/s, BPSK IFG=10us, A=80Mbits/s, BPSK IFG=10us, A=80Mbits/s, BPSK:4QAM=3:1 IFG=10us, A=80Mbits/s, BPSK:4QAM=7:1
Fig. 11. Maximum number of allowed ONUs.
60 65 70 75 80 85 90 95 100 0 5 10 15 20 25 30 35 40
Minimum number of subchannels
Maximum grant size (Mbits/s)
IFG=10us, Tlim=2ms, N=128 IFG=10us, Tlim=2ms, N=96 IFG=10us, Tlim=1ms, N=128 IFG=5us, Tlim=2ms, N=128
IFG=10us, Tlim=2ms, N=128, BPSK:4QAM=3:1 IFG=10us, Tlim=2ms, N=128, BPSK:4QAM=1:1
Fig. 12. Minimum number of subchannels.
same number of ONUs divided into two groups with BPSK and 4QAM. Since a high-order modulation format increases the total network capacity in time, the cycle time decreases for the same arrival rate. The proposed analyses can be used to determine the optimal number of subchannels even in an OFDMA-PON supporting multiple modulation formats.
The maximum polling cycle has been set in (23) and (24). In Fig. 10, by using (26) and (28), we present the maximum grant size for delay guarantees, to avoid channel monopolization by ONUs with long queues. The results can be used to verify whether or not satisfying SLA requirements in an OFDMA-PON. For the 20 km cases, the maximum grant size increases with the number of subchannels, since the allocation is more fine-grained, hence the bandwidth wasted because of the IFGs decreases as well. For the 100 km cases, however, the maximum grant size is limited by the accumulated upstream traffic volume during a longer cycle time due to increased RTT. We are interested in how many ONUs can be serviced with delay guarantees. In Fig. 11, the maximum number of allowed ONUs at high traffic
rates (i.e., at 70 M bits/s and 80 M bits/s) is investigated by using (32). The maximum number decreases with the maximum upstream service rate (i.e., for 80 M bits/s) which occupies the total channel capacity in time. Furthermore, the maximum number increases if a higher-order modu-lation format is used. In Fig. 12, the minimum number of subchannels for delay guarantees is investigated as a function of the maximum bandwidth allocation limit. Since the wasted bandwidth by IFGs decreases with the number of subchannels, the required minimum number increases to satisfy the maximum polling cycle time which is set to ensure time-sensitive services. The lines are drawn in the effective range for satisfying Tlim in (33).
VII. CONCLUSIONS
We presented the IPACT-based 2-D bandwidth allocation method for the OFDMA-PON. Cycle times in the 2-D band-width allocation are estimated by the proposed analyses, and verified through simulation. Furthermore, by setting the maximum polling cycle in order to guarantee delay performances for time-sensitive services, we identified the requirements, i.e., the maximum bandwidth allocation, the maximum number of allowed ONUs, and the optimum num-ber of subcarriers, for the bandwidth allocation with delay-guarantee. The obtained results can be used to design an OFDMA-PON. Furthermore, the analyses can be extended to similar network designs and performance estimations. As part of the future work, we plan to generalize our model to incorporate an unequal number of subcarriers per subchan-nel.
APPENDIXA
The average number of packet arrivals during a polling cycle is represented with (4) as:
E [NAi(k+1)] = ∞ X nA=0 nA·P r (NAi(k + 1)=nA) = ∞ X nA=1 nA·P r (NAi(k + 1)=nA) = ∞ X nA=1 nA· (λAi·Ti(hi, k)) nA nA! ·eλAi·Ti(hi,k).
At a steady state, since Ti(hi, k) ' E [Ti(hi, k)], E [NAi(k+1)]
is rewritten as: E [NAi(k + 1)] ' ∞ X nA=1 (λAi·E [Ti(hi, k)]) nA (nA− 1)! ·eλAi·E[Ti(hi,k)].
Since the sum of probabilities of all states is equal to 1, i.e., ∞ X nA=1 (λAi·E [Ti(hi, k)]) nA (nA− 1)! ·eλAi·E[Ti(hi,k)] = ∞ X nA=1 (λAi·E [Ti(hi, k)]) nA−1 (nA− 1)! ·eλAi·E[Ti(hi,k)]·λ Ai·E [Ti(hi, k)] =1·(λAi·E [Ti(hi, k)]),
the E [NAi(k + 1)] is obtained as:
APPENDIXB Since ρAtot= PN i=1λAi·Vpkt(hi)= PN i=1λAi·8·Bpkt·S/(C·hi),
when each arrival rate is the same with ¯λAin all ONUs, (25) is expanded as: ¯ λA· 8·Bpkt·S C · N X i=1 1 hi ≤ S −N ·TIF G+ PN i=1Trep(hi) Tlim . With the denotation of ΛAi (=8·Bpkt·λAi (bits/s)), by letting
¯
ΛA be the incoming traffic rate at the same packet arrival rate of ¯λA, the above equation is represented as:
¯ ΛA≤
C·S·Tlim− C·N ·TIF G− 8·Brep·S·PNi=1h1
i
Tlim·S·PNi=1h1
i
.
APPENDIXC
By expanding in terms of N from (31), we can get (32). N ·TIF G+ Na· 8BrepS Chs + (N −Na)· 8BrepS C(hs+d) ≤ Tlim S − Na·¯λA 8BpktS Chs − (N −Na)·¯λA 8BpktS C(hs+d) . N ·TIF G+ N · 8BrepS C(hs+d) + Tlim·N ·¯λA 8BpktS C(hs+d) ≤ Tlim·S + Tlim·Na·¯λA 8BpktS C(hs+d) −8BpktS Chs + Na 8BrepS C(hs+d) −8BrepS Chs . N TIF G+ 8BrepS C(hs+d) + Tlim·¯λA· 8BpktS C(hs+d) ≤ Tlim·S + Na· S C· Tlim·¯λA·8·Bpkt+8·Brep 1 hs+d − 1 hs .
N ≤ Tlim·S·C + Na·S· Tlim·¯λA·8·Bpkt+8·Brep
1 hs+d− 1 hs TIF G·C + Tlim·¯λA· 8BpktS hs+d + 8BrepS hs+d .
N ≤ Tlim·S·C + Na·S· Tlim· ¯ΛA+ 8·Brep
1 hs+d− 1 hs TIF G·C + S· Tlim· ¯ΛA+ 8·Brep /(hs+d) . (34) APPENDIXD
By substituting Trep(hi) (=8·Brep·S/(C·hi)) and ρAi(hi)
(=λAi·8·Bpkt·S/(C·hi)) into (23), the following equation is obtained as: N ·TIF G+ N X i=1 8·Brep·S C·hi ≤ Tlim·S − Tlim N X i=1 λi 8·Bpkt·S C·hi . Then, by expanding in terms of S,
S ≥ N ·TIF G·C Tlim·C − TlimP N i=1λi 8Bpkt hi − PN i=1 8Brep hi . With the assumption that the each arrival rate is the same in all ONUs, the above equation can be written as:
S ≥ N ·TIF GC
TlimC − Tlim· ¯ΛAPNi=1h1
i− 8·Brep PN i=1 1 hi . (35) S ≥ N ·TIF GC
TlimC − Tlim· ¯ΛA+ 8·Brep ·PN i=1 1 hi . (36) REFERENCES
[1] J.-i. Kani, F. Bourgart, A. Cui, A. Rafel, M. Campbell, R. Davey, and S. Rodrigues, “Next-generation PON-part I: Technology roadmap and general requirements,” IEEE Commun. Mag., vol. 47, no. 11, pp. 43-49, Nov. 2009.
[2] N. Cvijetic, D. Qian, and J. Hu, “100 Gb/s optical access based on optical orthogonal frequency-division multiplexing,” IEEE Commun. Mag., vol. 48, no. 7, pp. 70-77, July 2010.
[3] N. Cvijetic, “OFDM for Next-Generation Optical Access Net-works,” IEEE/OSA J. Lightw. Technol., vol. 30, no. 4, pp. 384-398, Feb. 2012.
[4] K. Kanonakis, E. Giacoumidis, and I. Tomkos, “Physical-layer-aware MAC schemes for dynamic subcarrier assignment in OFDMA-PON networks,” IEEE/OSA J. Lightw. Technol., vol. 30, no. 12, pp. 1915-1923, June 2012.
[5] A. Banerjee, Y. Park, F. Clarke, H. Song, S. Yang, G. Kramer, K. Kim, and B. Mukherjee, “Wavelength-division-multiplexed passive optical network (WDM-PON) technologies for broad-band access: a review,” J. of Opt. Networking, vol. 4, no. 11, pp. 737–785, Nov. 2005.
[6] K. Kanonakis, I. Tomkos, T. Pfeiffer, J. Prat, and P. Kourtessis, “ACCORDANCE: a novel OFDMA-PON paradigm for ultra-high capacity converged wireline-wireless access networks,” Int. Conf. Transparent Opt. Netw. (ICTON), Munich, Germany, June 27–July 1, 2010.
[7] N. Cvijetic, D. Qian, J. Hu, and T. Wang, “Orthogonal fre-quency division multiple access PON (OFDMA-PON) for col-orless upstream transmission beyond 10 Gb/s,” IEEE J. Sel. Area. Comm., vol. 28, no. 6, pp. 781-790, Aug. 2010.
[8] W. Wei, T. Wang, D. Qian, and J. Hu, “MAC Protocols for Optical Orthogonal Frequency Division Multiple Access (OFDMA)-based Passive Optical Networks,” Optical Fiber Communication/National Fiber Optic Engineers Conference (OFC/NFOEC), San Diego, CA, Feb. 24–28, 2008.
[9] J. Zhang, T. Wang, and N. Ansari, “An efficient MAC proto-col for asynchronous ONUs in OFDMA PONs,” Optical Fiber Communication/National Fiber Optic Engineers Conference (OFC/NFOEC), pp. 1–3, March 6-10, 2011.
[10] K. Kanonakis and I. Tomkos, “An overview of MAC issues in OFDMA-PON networks,” Int. Conf. Transparent Opt. Netw. (ICTON), Stockholm, Sweden, June 26–30, 2011.
[11] M.P. McGarry, M. Reisslein, and M. Maier, “WDM Ethernet passive optical networks,” IEEE Commun. Mag., vol. 44, no. 2, pp. 15–22, Feb. 2006.
[12] M.P. McGarry, M. Reisslein, and M. Maier, “Ethernet passive optical network architectures and dynamic bandwidth alloca-tion algorithms,” IEEE Commun. Surv. Tutor., vol. 10, no. 3, pp. 46-60, Third Quarter 2008.
[13] K.H. Kwong, D. Harle, and I. Andonovic, “Dynamic bandwidth allocation algorithm for differentiated sservices over WDM EPONs,” Int. Conf. on Communications Systems (ICCS), Sin-gapore, China, Sep. 7, 2004, pp. 116-120.
[14] F. Clarke, S. Sarkar, and B. Mukherjee, “Simultaneous and interleaved polling: an upstream protocol for WDM-PON,” Op-tical Fiber Communication Conf. and National Fiber Optic Engineers Conf. (OFC/NFOEC), March 5-10, 2006.
[15] A.R. Dhaini, C.M. Assi, M. Maier, and A. Shami, “Dynamic wavelength and bandwidth allocation in hybrid TDM/WDM EPON networks,” J. Lightwave Technol., vol. 25, no. 1, pp. 277-286, Jan. 2007.
[16] M.P. McGarry, M. Reisslein, M. Maier, and A. Keha, “Band-width management for WDM EPONs,” J. Opt. Netw., vol. 5, no. 9, pp. 637-654, Sep. 2006.
[17] M.P. McGarry, M. Reisslein, C.J. Colbourn, M. Maier, F. Au-rzada, and M. Scheutzow, “Just-in-time scheduling for multi-channel EPONs,” J. Lightwave Technol., vol. 26, no. 10, pp. 1204-1216, May 15, 2008.
[18] C.H. Yeh, C.W. Chow, H.Y. Chen, and B.W. Chen, “Using adap-tive four-band OFDM modulation with 40 Gb/s downstream and 10 Gb/s upstream signals for next generation long-reach PON,” Opt. Express, vol. 19, no. 27, pp. 26150-26160, Dec. 7, 2011.
[19] D.-Z. Hsu, C.-C. Wei, H.-Y. Chen, W.-Y. Li, and J. Chen, “Cost-effective 33-Gbps intensity modulation direct detection multi-band OFDM LR-PON system employing a 10-GHz-based
transceiver,” Opt. Express, vol. 19, no. 18, pp. 17546-17556, Aug. 22, 2011.
[20] F. Saliou, P. Chanclou, F. Laurent, N. Genay, J.A. Lazaro, F. Bonada, and J. Prat, “Reach extension strategies for passive optical networks,” IEEE/OSA J. Opt. Commun. Netw., vol. 1, no. 4, pp. C51-C60, Sep., 2009.
[21] J. Kim, H. Bang, and C.-S. Park, “Design and performance anal-ysis of passively extended XG-PON with CWDM upstream,” IEEE/OSA J. Lightw. Technol., vol. 30, no. 11, pp. 1677–1684, June, 2012.
[22] S. L. Jansen, B. Spinnler, I. Morita, S. Randel, and H. Tanaka, “100 GbE: QPSK versus OFDM,” Opt. Fiber Technol., vol. 15, no. 5-6, pp. 407-413, Oct., 2009.
[23] I. Cano, M. C. Santos, V. Polo, F. X. Escayola, and J. Prat, “Di-mensioning of OFDMA PON with non-preselected independent ONUs sources and wavelength-control,” Opt. Express, vol. 20, no. 1, pp. 607-613, Jan. 2, 2012.
[24] G. Kramer, B. Mukherjee, and G. Pesavento, “IPACT a dynamic protocol for an Ethernet PON (EPON),” IEEE Commun. Mag., vol. 40, no. 2, pp. 74-80, Feb. 2012.
[25] G. Kramer, B. Mukherjee, and G. Pesavento, “Interleaved polling with adaptive cycle time (IPACT): a dynamic bandwidth distribution scheme in an optical access network,” Photonic Netw. Commun., vol. 4, no. 1, pp. 89–107, Jan. 2012.
[26] B. Skubic, J. Chen, J. Ahmed, L. Wosinska, and B. Mukherjee, “A comparison of dynamic bandwidth allocation for EPON, GPON, and next-generation TDM PON,” IEEE Commun. Mag., vol. 47, no. 3, pp. S40-S48, Mar. 2009.
[27] B. Lannoo, L. Verslegers, D. Colle, M. Pickavet, M. Gagnaire, and P. Demeester, “Analytical model for the IPACT dynamic bandwidth allocation algorithm for EPONs,” J. Opt. Netw., vol. 6, no. 6, pp. 677-688, June 2007.
[28] E. M. M. Winands, I. J. B. F. Adan, and G. J. van Houtum, “Mean value analysis for polling systems,” Queueing Syst., vol. 54, no. 1, pp. 35-44, Sep. 2006.
[29] OMNeT++ Community website: http://www.omnetpp.org/.
Hakjeon Bang received the B.S. degree in computer science engineering from Hanyang University, Ansan, Korea, in February 2006, and the M.S. degree in information and com-munications from the Gwangju Institute of Science and Technology (GIST), Gwangju, Ko-rea, in February 2009. Since 2009, he has been working toward the Ph.D. degree at GIST. His research interests include integra-tion of optical and wireless networks, perform-ance analysis, and resource management.
Kyeong-Hwan Doo received his BS and MS in electronic engineering from Chonbuk Na-tional University, Jeonju, Rep. of Korea, in 1996 and 1998, respectively, and his PhD in electronic engineering from Chungnam Uni-versity, Daejeon, Rep. of Korea, in 2013. He joined ETRI in 2000, Daejeon, Rep. of Korea, where he has worked on several projects, in-cluding PON MAC device design, flow proces-sor design, and QoS router development . His current research interests include scheduling in high speed networks, passive optical networks, and OFDM sys-tem.
Seungil Myong received the B.S. and M.S. in Electronics Engineering from Myong-ji Uni-versity, Korea, in 1997 and 1999, respectively. He received the Ph.D. degree in Electrical Engineering from Myong-ji University, Korea, in 2010. Since 2000, he has been a senior member of the research staff at ETRI, Ko-rea. His current research interests include OFDMA-PON, OTDR System, RFID System, RTLS and modem designs for communication systems.
Giovanni Stea has been a researcher at the Department of Information Engineering of the University of Pisa, Italy, since 2004. His current research focuses include Quality of Service and resource allocation in wireline and wireless networks, performance evalua-tion through simulaevalua-tion and analytical mod-els, traffic engineering. In these fields he has coauthored more than 50 peer-reviewed pa-pers. He is also co-author of several patents. He has been involved in national and Euro-pean research projects, including IST-EuQoS. He has served as a member of the technical and/or organization committees for several international conferences, including SIGCOMM, WoWMoM, and VALUETOOLS, and has served as a Guest Editor for the Elsevier Performance Evaluation journal. He is a member of IEEE, ACM and ICST.
Chang-Soo Park received the B.S degree from Hanyang University in 1979, the M.S. degree from the Seoul National University, Seoul, Korea, in 1981, and the Ph.D. degree from Texas A&M University, College Station, in 1990, respectively. From 1982 to 1987, he was Principal Member of Technical Staff at the Electronics and Tele-communications Research Institute (ETRI). He joined the Gwangju Institute of Science and Technol-ogy (GIST), Gwangju, Korea, as an Associate Professor in the Department of Information and Communications, where he is currently a Professor. He is also Director of the Photonics Research Center (PRC), which is sponsored by the Ministry of Commerce, Industry, and Energy, Republic of Korea. His current research areas are in wired and wireless optical communications, optical access networks, and microwave photonics. He has over 90 publications in these fields, including 1 book. Prof. Park is a Member of the Institute of Electronics, Information and Communication Engineers (IEICE), Japan, and of the IEEE Lasers & Electro-Optics Society (LEOS).
