Delay Performance of WDM-EPON for Multi-dimensional Traffic Under the IPACT Fixed Service and the MultiPoint Control Protocol

Transcription

1 Delay Performance of WDM-EPON for Multi-dimensional Traffic Under the IPAT Fixed Service and the MultiPoint ontrol Protocol M. D. Logothetis (1), I. D. Moscholios (2), A.. Boucouvalas (2) and J. S. Vardakas (3) Abstract Passive Optical Networks (PONs) gain ever more ground in the Telecom market, because of their low cost and significant advantages over legacy broadband access systems. We concentrate on the performance evaluation of Ethernet PONs (EPONs) and particularly on Wavelength Division Multiplexing (WDM)-EPONs which support Internet traffic. We consider traffic of multiple service-classes and focus on the calculation of the mean total delay from the so-called Optical Network Unit (ONU) to the Optical Line Terminator (OLT). The PON operates under the fixed service of the IPAT (Interleaved Polling With Adaptive ycle) algorithm, which uses the MultiPoint ontrol Protocol (MPP). The delay analysis is based on a unified approach for EPONs and WDM-EPONs, through the formation of two classical queuing models: an M/D[x]/ for the packets waiting in the local queues of the ONUs and an M/D/ for the frame transmission to the OLT. The number of servers corresponds to the number of wavelengths used in the uplink. Through simulation, we verify our analysis and show the accuracy and the consistency of the results. Fig. 1. A WDM-EPON servicing Internet traffic. I. INTRODUTION In recent years there have been several studies and many results have been presented on Passive Optical Networks (PONs), the most promising and cost-effective fiber based access systems. The latter made ITU-T to issue several related standardizations. The most widely used PON configuration is the Time Division Multiplexing (TDM)-PON, while the Wavelength Division Multiplexing (WDM)-PON is a successor of the TDM-PON capable to convey much more offered traffic load and more Internet users [1]- [3]. A PON consists of a number N of Optical Network Units (ONUs), or Optical Network Terminals (ONTs), which are located in the customers premises and are connected through a Passive Optical ombiner/splitter (PO-S, remote node) to the Optical Line Terminator (OLT). The OLT is located next to an ISP Internet Router and through the router it provides access to the Internet (Fig. 1). ommunication between ONUs is realized through the OLT, only. Figure 1 portrays a WDM- EPON with individual wavelengths per ONU (per traffic-flow direction). However, the number of wavelengths can be much less than the number of ONUs, when the WDM-EPON uses a Dynamic Wavelength Assignment (DWA) algorithm, and it is equipped with a proper device for PO-S [4], [5]; these networking capabilities are considered in this paper. (1) WL, Dept. of Electrical & omputer Engineering, University of Patras, Patras, Greece. mlogo@upatras.gr (2) Dept. of Informatics & Telecommunications, University of Peloponnese, Tripolis , Greece. {idm, acb}@uop.gr (3) Iquadrat, Barcelona, Spain. jvardakas@iquadrat.com In TDM-PONs, only two wavelengths are utilised; one in the uplink and the other in the downlink (usually 1310 nm and 1490 nm, respectively) [6]. The transmission channel (that uses a single wavelength) between the ONUs and the OLT is divided into equal time-slots, according to the TDM principle; a time-slot is assigned to each ONU for the uplink connection. The different time-slots from the different ONUs are multiplexed in the PO-S and transmitted toward the OLT through a single fibre, thanks to the directional properties of the PO-S (that is, they do not reach the other ONUs). The Ethernet protocol fits well to the TDM-PON; it has been standardized as a link layer protocol in the TDM-PON since 2004, and the PON is named EPON (IEEE 802.3ah standard) [7]. Although in the uplink the EPON has a point-to-point architecture, collisions are possible at the PO-S, among packets transmitted simultaneously from different ONUs (see Fig 3). To tackle the problem of collisions, the MultiPoint ontrol Protocol (MPP) has been standardised as a MA sublayer protocol. On the other hand, to enable the transmission of data from all users, and hence from all ONUs, there must be an algorithm which will make a fair partitioning of the provided bandwidth. We make use of the so called Interleaved Polling with Adaptive ycle Time (IPAT) algorithm, which uses the MPP to allocate bandwidth to each ONU in a dynamic manner [8]. According to the IPAT algorithm, the available bandwidth of the wavelength is divided into time-slots. A time-slot is large enough to transmit batches of packets and is an integer multiple of a time-unit c, during which a single packet can

2 be transmitted. The time-slots are allocated to each ONU. There are several methods (called services) of determining the length (duration) of the time-slot, which is assigned to each ONU by the OLT: the fixed, limited, gated, constant credit, linear credit and the elastic service. In this study, we refer to the fixed service, according to which, the time-slot of the maximum length is assigned to each ONU, irrespectively of what length each ONU has asked for. The fixed service has the advantage of simplicity; also, it has been investigated that it performs satisfactorily under a heavy offered traffic-load, as far as the packet-delay performance is concerned [9]. Several other studies on the IPAT services exist in the literature [10]- [13], however, in all these studies, the EPON accommodates a single service-class, only, which is a rather extraordinary case in the traffic environment of contemporary communications networks. In this paper, we first present the delay performance of a TDM-PON (EPON) which accommodates multiple service-classes, and then we proceed to the delay performance of a WDM-EPON. The delay analysis of the TDM-PON with multiple service-classes is not new [14], but it is included in this paper in order to facilitate the presentation of the delay analysis of the WDM-EPON. Moreover, we show that, in the case of the fixed IPAT service, our analysis can cover in a parametric way both EPONs and WDM-EPONs; from the teletraffic point of view, the number of wavelengths used in the uplink (parameter) denotes the EPON or the WDM-EPON. In TDM-PONs, as it has been already mentioned, the bandwidth of each wavelength is shared among the ONUs, according to the TDM principle. In WDM-EPONs multiple wavelengths are accommodated per traffic flow direction, and therefore the transmission bandwidth in the PON is drastically increased (Fig. 1). Specifically, for each connection between the OLT and ONU a different wavelength is allocated statically or dynamically (by a DWA algorithm) to the ONU, depending on the specific device which implements the PO-S. Thanks to a tunable λ router (capable to route different wavelengths), the wavelengths can be technically manageable and used by the users in a cost-effective way. Each wavelength is shared among calls of different service-classes accommodated in each ONU, according to the TDM principle, as in the case of EPONs. Both in EPON and WDM-EPON, when an ONU has to transmit a total number of packets, a frame is formed whose size depends on the fixed-service of the IPAT algorithm. The ONU transmits the frame within the duration of one time-slot. Given that K is the number of different service-classes, the frame is composed by K batches of packets. The greater the size of a batch, the higher the priority of the service-class. In this way, not only multiple service-classes but also priorities among service-classes are supported in a PON. The packet delay analysis in EPONs is based on two queuing models: (a) one model for the packets in the local queues of an ONU, and (b) another model for the complete frame of packets to be transmitted to the OLT. For the first queuing model, we assume the ONU architecture of Fig. 2, where the K service-classes are separated by assigning an individual buffer to each service-class. Batches of packets from each service-class form the frame, while it holds: m 1 >m 2 >... > m k >... > m K, that is, the number of packets from service-class 1 is the highest, which means service-class 1 has the highest priority (contrary to service-class K which has the least priority). The number m k is the same for all ONUs. If we suppose that the portion of the time-slot (frame), which is devoted to service-class k (k =1,...,K) is between a minimum w k and a maximum m k number of packets, then the delay analysis is based on an M/D [w k,m k ] /1 queueing model. By this model we calculate the queueing delay of each service-class in the corresponding queue, or, in other words, the delay in forming the frame. onsequently, since one frame is transmitted within one time-slot, we use another simple queueing model M/D/1 (to calculate the queueing delay by considering the entire frame as a unit). Based on the statistics (queueing delay, queue length) obtained for the entire frame, the corresponding statistics for the individual packets are determined. After having determined the queuing delay, we determine the total delay by adding to it the transmission delay and the propagation delay. Fig. 2. An ONU as a queueing model for the uplink. The basic difference between EPONs and WDM-EPONs is that in WDM-EPONs a multiple of frames are transmitted toward the OLT during a time-slot that correspond to the number of the allocated wavelengths in the uplink ( < N). Therefore, in the case of WDM-EPONs the queueing models comprise servers. Each ONU has transmitters, where each one transmits in a different frequency. In the beginning of each transmission cycle, the transmitter sends a frame to the OLT and frames in total by the end of the transmission cycle. The analytical results, for the uplink, have been compared with simulation results to verify the validity of the analysis and the accuracy of the calculations. As far as the downlink direction is concerned, the transmission procedure is much simpler from a scientific point of view. Signals from OLT to ONUs are split at the PO-S and are received by the ONUs based on the Ethernet addresses. Even in the case of WDM- EPONs, only one wavelength can be used in the downlink. The structure of this paper is as follows. In Section II, we describe the PONs under study, concentrating on the involved protocols. In Section III, we concentrate on the fixed service, and we review the delay performance analysis of EPONs supporting multiple service-classes. In Section IV, we present the delay performance analysis for multiple serviceclasses in WDM-EPONs. In Section V, we present application results which verify the analysis; comparative results between the EPON and WDM-EPON are presented and show the consistency of our analysis. We conclude in Section VI.

3 II. MPP AND IPAT The basic PON architecture, either of an EPON or a WDMEPON, is shown in Fig. 1, as far as the PON topology is concerned, where one OLT supports N ONUs through a PO-S. All data-packets are encapsulated in Ethernet frames (IEEE 802.3ah). As it is shown in Fig. 3, in the downlink where the OLT broadcasts all packets to ONUs (Fig 3a), there is no bottleneck problem, whereas, in the uplink, collisions may appear when packets arrive simultaneously at the PO-S from the ONUs. For collision avoidance the MPP is used. Moreover, the MPP contributes to the Dynamic Bandwidth Allocation (DBA) among the ONUs, which is defined by the IPAT algorithm. Figure 4 shows a communication scenario between the OLT and an ONU according to the MPP, where all five messages are included; both the discovery and the normal mode of operation are shown. Initially the OLT allocates a time-slot, in which unregistered ONUs are allowed to transmit. The start time of the initialization time-slot and its duration are included in the GATE message which is advertised (a multicast address is used) by the OLT (message GAT E Discovery(0) in Fig. 4). An unregistered ONU may respond to this GAT E message by sending the REGIST ER REQ(1) message, when the local timer of the ONU reaches the start time of the initialization slot; the latter is included in the received GAT E message and must not be violated. The OLT answers with the REGIST ER(2) message. Then, the OLT terminates the discovery phase by sending the GAT E grant(3), in which the ONU must answer with REGIST ER AK(4). After a while, but within the time-interval indicated in GAT E grant(3), the ONU asks the OLT for transmission by sending REP ORT (5) and waiting for receiving the GAT E(6). In the REP ORT (5) message, the ONU describes how many packets are stored in its queues; based on this information the OLT assigns the proper bandwidth to the ONU, that is, decides on the duration of a transmission window, which is sent to the ONU in GAT E(6). In this way, the ONU has a first transmission cycle. Since a second cycle is needed, the ONU sends the REP ORT (7), in order for the OLT to issue the GAT E(8) and so on. Fig. 3. (a) Broadcasting in the downlink. (b) In the uplink, since ONUs may transmit packets simultaneously (e.g 2 and 3), the MPP arbitrates the transmission. A. Multi-Point ontrol Protocol The MPP is a protocol of Data Link Layer, and specifically of the MA sublayer, that controls the TDM transmission of the uplink [7]. This is the normal mode of operation whereby transmission opportunities are assigned to ONUs, while there is an additional mode of operation whereby a newly connected ONU in the PON is detected; the OLT is informed about the physical address and the transmision capabilities of the ONU, as well as the round-trip time between the OLT and the ONU. In the normal mode, MPP uses the MA ontrol messages GAT E (from OLT to ONU) and REP ORT (from ONU to OLT). To accomplish the ONU detection, MPP uses three more messages: REGIST ER REQ (from ONU to OLT), REGIST ER (from OLT to ONU) and REGIST ER AK (from ONU to OLT). The discovery mode of the MPP operation is applied by the OLT periodically. A general rule of the MPP is that an ONU is allowed to transmit (either data or a control message) only during the time interval indicated in the GAT E message. Fig. 4. A communication scenario between OLT and ONU. B. The Interleave Polling with Adaptive ycle Time (IPAT) Algorithm The IPAT algorithm is used for dynamic bandwidth allocation by performing inter-onu scheduling. Based on the REP ORT message of the MPP, the OLT knowns the dynamic traffic load in each ONU and can distribute the uplink bandwidth of the PON according to the requirements of the ONUs in a fair way. This is done by the IPAT algorithm; the

4 OLT polls the ONUs and assigns to each ONU transmission timeslots, in round-robin [9]. More precisely, IPAT is a centralized algorithm which runs in a DBA agent of the OLT (at the service layer sitting above the MA protocol layer). IPAT uses the control messages GAT E and REP ORT of the MMP, in order to estimate and allocate bandwidth to each ONU, as well as the Service Level Agreements in order to guaranteeing a minimum degree of service. Needless to say that buffers of a different size exist in ONUs. In case that an ONU of a large buffer was attempting to transmit all the content of the buffer at once (i.e. in one transmission cycle), this would monopolize the transmission and no one else could transmit for a long time. To face this problem in a fair way, the IPAT algorithm applies several methods for bandwidth assignment, which are called services. As it was mentioned in the Introduction, the available bandwidth of the wavelength is divided into time-slots. A timeslot is large enough to transmit a frame of packets and is an integer multiple of a time-unit c, during which a single packet can be transmitted. The length (duration) of the time-slot is determined according to the applied service; the most popular services are the following: Fixed service The maximum possible length of the timeslot is assigned to the ONU, regardless of the requested size. The resultant transmission cycle is fixed and equals to the maximum possible length. Limited service The assigned length of the timeslot equals the required one by the ONU, as long as this length does not exceed the maximum possible (determined by the fixed service). Gated service No certain limit is denoted; each ONU can transmit in one transmission cycle as many packets as they have been stored in its queue. This service is mainly applicable, when the queue buffers of the ONUs have a relatively small length (not exceeding a threshold). onstant credit The assigned length of the timeslot equals the required one by the ONU plus a fixed value. It is supposed that some more packets have been stored in the queue of the ONU, while it waits for the GAT E message in order to start transmission. Linear credit The assigned length of the timeslot equals the required one by the ONU plus a value, not fixed but proportional to the required one. Elastic service The assigned length of the timeslot is not limited. The assigned bandwidth capacity per ONU is determined so that the sum of the last N bandwidth requirements does not exceed the sum of the corresponding lengths of the assigned timeslots. Therefore, if only one ONU needs to transmit, the entire bandwidth of all N ONUs can be assigned to a single ONU. In this paper, we consider the fixed service of the IPAT algorithm and examine the delay performance of the PON (both the EPON and WDM-EPON) when the ONUs support multiple service-classes. The basic principles of the IPAT algorithm for the implementation of the fixed service are the same both for the EPON and the WDM-EPON [15]. III. DELAY ANALYSIS OF EPONS FOR THE IPAT FIXED SERVIE In the uplink, the PON acts as a multipoint-to-point network, where ONUs are able to transmit packets in batches (frames), during different time intervals. The set of packets which have been stored in the local queues of each ONU (Fig. 2) will formulate a frame to be transmitted. The duration of this frame corresponds to the timeslot whose size is assigned by the OLT. Since we consider the fixed service of the IPAT algorithm, the duration of each timeslot (frame) is constant; in other words, the service time of the packets in the frame is constant. As far as the packet arrival process to the local queues is concerned, it is assumed Poisson. Although the Poisson characteristics of independent and identically distributed random arrivals do not perfectly reflect the packet-level traffic characteristics of PONs, the Poisson process is broadly considered as the starting point of a teletraffic analysis. The reason is twofold: (a) The Poisson process is analytically simple, and (b) the teletraffic model in which the Poisson process is incorporated, is usually a well studied model with safe results. Besides, another reason is the fact that, usually, analysis under the Poisson assumption is used as a reference point for comparison against other considerations of the input traffic, because randomness as it is expressed by the Poisson model can easily be understood and compared. Anyway, batch arrivals, event correlations and traffic burstiness are important factors of the packet-level characteristics of PONs [16], which necessitate the use of heavy tailed distributions and of selfsimilarity of Ethernet traffic [17]; therefore, assumptions like a Batch Poisson Arrival Process, or a Markov Modulated Poisson Process are much more realistic than Poisson at packet-level. The consideration of multiple service-classes is a sine qua non in the traffic environment of PONs. The service-classes distinction is done according to the required bandwidth per call, only. In addition, we assign a transmission priority to each service-class, which is the same for all ONUs. This is done as follows: Since the timeslot of each ONU is shared among the K service-classes, we introduce a different transmission priority to each service-class, by assigning a different portion of the timeslot to each service-class. In this way, we can consider as many priorities as the number of service-classes, i.e. K priorities: m n,1 > m n,2 >... > m n,k >... > m n,k, where for the ONU n, (n = 1,..., N), m n,k is the transmission priority of service-class k, and corresponds to the portion of the timeslot (frame), which is devoted to packets of service-class k. More precisely, given that a single packet is transmitted during a time-unit c, the portion of m k means a time-interval of m k c. We can also assume that the portion of the frame, which is devoted to service-class k is between a minimum w n,k and a maximum m n,k number of packets. We shall analyse the total packet delay, when a serviceclass k packet is transmitted from an ONU to the OLT, by considering the queueing delay, the transmission delay and the propagation delay, during the transmission time as it is indicated in Fig. 4. That is, we ignore other type of delay which is introduced by the MPP/IPAT.

5 As it is shown in Fig. 2, the determination of packet delay depends both on the K local queues which correspond to the K service-classes of each ONU, and one more queue for the frame transmission. In what follows, we show that the queuing delay can be determined based on two queuing models: i) the M/D [w k,m k ] /1 for the K local queues and ii) one more model, the M/D/1, for the frame. Let T n denote the entire duration of the frame assigned to ONU n. Then, for each ONU n, it holds: K m n,k = T n (1) k=1 The successful recovery of the arriving frames at the OLT is assisted by the insertion of a safety time interval, with duration f, between two consecutive frames. The elapse time between two consecutive frame transmissions T f,n, of the same ONU n determines the service time of the K local queues of ONU n. That is, T f,n equals to the sum of the duration of the frames from the remaining N 1 ONUs, plus the safety time interval between two consecutive frames: T f,n = N T i T n + N f (2) i=1 From eq. (2) it is clear that T f,n is constant. This is the reason that justifies our assumption that each local queue of service-class k follows the M/D [w k,m k ] /1 queuing model. In this model, given that one frame is transmitted from each ONU, one server is considered as the system s capacity. The calculation of the mean queue length and the mean waiting time of the M/D [w k,m k ] /1 queuing model for serviceclass k is based on the formulation of a second queuing model, which is the M/D/1, because each frame is treated as a single independent unit. Again, a unique server is considered, because within the time-inteval T f,n only one frame of ONU n is serviced. Since we consider Poisson arrivals, if we denote λ n,k to be the arrival rate of individual packets of service-class k for ONU n, then the arrival rate λ n,k of a batch with size m n,k is: λ n,k = λ n,k m n,k (3) while the corresponding equivalent offered traffic load of the batches is: A n,k = λ n,k T f,n c (4) Taking into account the offered traffic load of batches, the mean waiting time of a batch is determined through the M/D/1 queuing system [18]: W n,k = T f,n c A n,k 2 (1 A n,k ) (5) Through Little s law, we can calculate the mean queue length of the M/D/1 queuing system: L n,k = λ n,k W n,k = λ n,k T f,n c A n,k 2 (1 A n,k ) (6) The mean queue length of service-class k packets in the local queue of ONU n is given by the following approximation [19]: L n,k m n,k L n,k+pn,k w mn,k 1 +(1 P w 2 n,k) wn,k 1 (7) 2 where Pn,k w is the probability of delay (waiting) in the corresponding M/D/1 queuing system [18]: P w n,k = P (j 1) = j=1 π n,k j = 1 π n,k 0 = A n,k (8) According to eq. (8), to calculate the probability Pn,k w, the knowledge of the steady-state distribution π n,k i of the M/D/m n,k queueing system is required; it is given by [18]: π n,k i =(1 A n,k) i [( 1) i j (ja e ja n,k )i j n,k ( (i j)! j=1 + (ja n,k )i j 1 )] (i j 1)! (9) From eq. (7) and Little s law, we can calculate the mean waiting time of service-class k packets in the local queue of ONU n, as follows: W n,k = L n,k λ n,k (10) To calculate the average total packet delay, we sum up the mean waiting time in the queue, the transmission delay and the propagation delay. We consider that all ONUs transmit packets with fixed length of l bits, while the bit-rate in the uplink is bits/sec. Then, if the transmission delay of a packet is: T trans = l (11) and the propagation delay of a packet transmitted from ONU n to OLT, in a distance of d n far from ONU n, is: T prop,n = d n (12) c where c is the speed of light in the optical fibre, then, the average total packet delay T n,k of service-class k transmitted from ONU n to OLT is given by: E[T n,k ] = W n,k + T trans + T prop,n (13) IV. DELAY ANALYSIS OF WDM-EPONS FOR THE IPAT FIXED SERVIE The application of the WDM technology in an EPON increases the available bandwidth of each ONU, and consequently of the users, through the utilization of multiple non-overlapping wavelengths. A unique wavelength pair is assigned to each ONU for communication with the OLT: one wavelength for the downlink and another for the uplink. In this way, a separate point-to-point connection is provided between

6 each ONU and the OLT, through the shared point-to-multipoint physical PON architecture [6]. The ONU is characterised colorless, because the wavelength is not fixed but it is assigned to the ONU dynamically, when it is required in the uplink. A key difference between an EPON and a WDM-EPON is the PO-S, which, in the case of WDM-EPON, is a tunable router to any of the wavelengths assigned to the ONU (by a DWA algorithm) in the uplink. By considering again the fixed service of the IPAT algorithm, a time-slot of a certain duration is assigned to each ONU. Within this time-slot, each ONU transmits a batch of packets at a specific wavelength. The set of packets which have been stored in the local queues of each ONU, and are going to be transmitted during a timeslot, formulate a frame. The WDM-EPON supports multiple service-classes with different transmission priorities. That is, each service-class occupies a different portion of the frame according to its transmission priority. The packet arrival process is assumed Poisson, while the service time is deterministic because of the fixed service of the IPAT algorithm. The basic difference with the EPON is the fact that multiple frames are simultaneously transmitted in each time-slot; the number of the transmitted frames equals the number of wavelengths. The total number of simultaneously transmitted frames within a time-slot cannot exceed the total number of wavelengths which are available to ONUs in the uplink. We assume that all ONUs support an equal number of wavelengths. On the other hand, the total number of wavelengths (in the uplink) may be less than N; in that case, which is under consideration in this paper, a DWA scheme is assumed in the WDM-EPON, together with the IPAT algorithm for DBA (alternatively, this combination of DBA and DWA can be considered as an extension of the IPAT algorithm to WDM-EPONs) [15]. We analyse the total packet delay, when a service-class k packet is transmitted from an ONU to the OLT, while ignoring any other delay which is introduced by the DBA/DWA procedures. As in the case of EPON, we shall base our study of queueing delay on two queueing models: i) the M/D [w k,m k ] / for the K local queues of the ONU, and ii) the M/D/ for the transmission of the frames. The number of servers in both queueing systems becomes, since it corresponds to the total number of the supported wavelengths in the uplink and therefore to the number of frames which can be transmitted within a time-slot. Each ONU possesses transmitters which operate in different wavelengths. At the beginning of each transmission cycle, each transmitter is going to send to the OLT one frame. Therefore, in each transmission cycle the maximum number of frames that can be transmitted from each ONU is. Since the fixed service of the IPAT algorithm remains the same as in the EPON, the fixed frame duration is given again by eq. (1). Also, the time interval between two consecutive frames is given by eq. (2). Likewise, since the input process remains Poisson, the frame is formed from batches of packets with a mean arrival rate of λ n,k, given by eq. (3), where m n,k is the batch size. Besides, the corresponding offered traffic load from batches is given by eq. (4). Based on the latter, the mean waiting time of a batch is determined from the M/D/ queueing system [19]: T f,n W n,k = +1 E 2,(A n,k ) A n,k 1 ( A n,k )+1 1 ( A n,k ) (14) where E 2, (A n,k ) is the famous Erlang- formula [18]: E 2, (A n,k ) = 1 r=0 ( A n,k ) A n,k (A n,k )r r! + ( A n,k ) A n,k (15) Based on Little s law, we calculate the mean queue length L n,k of the M/D/ queueing system as follows: L n,k = λ n,k W n,k = = λ n,k +1 E 2,(A n,k ) T f,n 1 ( A n,k A )+1 n,k 1 ( A n,k ) (16) Another way to determine the mean queueing delay of the M/D/ queueing system is through the following approximation, which results from the generalized model M/G/ (and can be compared to eq. (5)) [19]: W n,k = P w n,k Tf,n c A n,k 2 (1 A n,k ) 1 (17) where Pn,k w is the probability of delay (waiting) in the M/D/ queueing system [19]: P w n,k = ( A n,k ) (1 A n,k )! 1 ( A n,k )j + ( A n,k ) j! (1 A n,k )! (18) j=0 This alternative method is preferred, because it provides a unified calculation of the mean queueing delay, for both EPONs and WDM-EPONs. Specifically, when = 1, eq. (17) provides the same results with eq. (5). The mean queue length L n,k of the service-class k packets in the local queue of ONU n is estimated by eq. (7), as in the case of EPON. Likewise, the mean waiting time of serviceclass k packets in the local queue of ONU n is determined by eq. (10). Subsequently, we can use eqs. (11), (12) and (13) to determine the packet transmission delay, the propagation delay and the average total delay (respectively) of service-class k packets transmitted from ONU n to OLT, in a WDM-EPON. V. NUMERIAL RESULTS EVALUATION As an application example, we consider a WDM-EPON consisting of N = 30 ONUs which all are 20 Km far from the OLT. Each ONU supports K = 3 service-classes; the 1 st service-class has the highest transmission priority, while the 3 rd has the lowest transmission priority. In the uplink of this WDM-EPON, = 4 wavelengths are used; when only one wavelength is assumed (instead of four), the system reduces to an EPON. In the case of WDM-EPON, the ONUs are colorless, that is, each ONU employs one tunable transmitter

7 capable of tuning to any of the four uplink wavelengths (channels); tuning times are negligible. We evaluate our analysis for the total packet delay (sum of queueing delay, the transmission delay and the propagation delay) by comparing analytical results with simulation results. Simulation is performed by using the SIMSRIPT III simulation language [20]. The simulation results are obtained as mean values of five runs (replications with a different seed number). For the mean values, confidence intervals of 95% were determined; however, they are not shown in the figures, since they are small enough. As it has been aforementioned, a service-class k packet is transmitted within a frame from an ONU to the OLT; the size of this frame is fixed, 50 time-units (correspond to 50 packets); each time-unit c equals 10 µsec. The safety time interval between consecutive frames is f = 20µsec, i.e. 2 timeunits. The distribution of the frame to service-classes is the same in each ONU, and corresponds to the priorities among the service-classes, as follows: (m 1, m 2, m 3 ) = (25, 15, 10). The length of each data-packet is 1000 bits. All transmission channels (each channel corresponds to a different wavelength) operate at a rate of 1 Gbps. The optical fibres have a reflection index of 1.45, therefore the speed of light in a fibre is determined c = m/s. As far as the local queues of the ONUs are concerned, they are of sufficient length, so that no packet overflow occurs. units, as well as the transmission priorities among the serviceclasses do not alter (the 1 st service-class keeps the highest transmission priority, while the 3 rd service-class has the lowest priority). However, the 50 time-units are distributed to the service-classes, as it is shown in Table I, where the time-units devoted to the 1 st service-classes increase against the timeunits devoted to the other two service-classes. In Fig. 6, we observe that, for the scenario of Table I, as the batch size of the 1 st service-class increases, the total delay of this serviceclass slightly reduces, while the total delay of the other two service-classes (whose the batch size is reduced) increases. The effect of the frame distribution on the total delay of the 3 rd service-class is more important, because the batch size of the 3 rd service-class, as a percentage, has been drastically reduced (from 12 to 7, i.e. it became almost the half). A. Delay results in EPON In Fig. 5, we show the total delay versus the packet arrival rate. For presentation purposes, the three service-classes have the same mean packet arrival rate, which is shown in the x-axis of Fig. 5. The 1 st service-class has the lowest delay, because of its highest transmission priority. In consistency with their transmission priorities, the 3 rd service-class suffers the highest delay. This is clear when the packet arrival rate increases in the x-axis of Fig. 5. The simulation results are very close to the analytical results. In Fig. 6, we present the total delay when different distributions of the frame size to the service-classes is considered. More precisely, the frame size remains constant to 50 time- Fig. 6. Analytical results of the total packet delay vs. different distribution points of the frame size to the service-classes. In Fig. 7, we present the total delay versus the number of ONUs. That is, a different EPON configuration is assumed, as far as the supported ONUs are concerned. The mean packet arrival rate for each service-class is kept constant to 25 packets/sec. Therefore, an increase in the number of ONUs will increase the offered traffic load and the total delay for each service-class, with transmission priorities (m n,1, m n,2, m n,3 ) = (25, 15, 10). This anticipation is verified in Fig. 7. B. Delay results in WDM-EPON Similar to the EPON case, in Fig. 8, we show the total delay versus the packet arrival rate in the WDM-EPON, where the three service-classes have the same mean packet arrival rate, as it shown in the x-axis of Fig. 8. Again, the TABLE I FRAME DISTRIBUTION POINTS SHOWN IN THE X-AXIS OF FIG. 6. Fig. 5. Analytical and simulation results of the total packet delay vs. packet arrival rate in EPON. Point (m n,1, m n,2, m n,3 ) 1 (22, 16, 12) 2 (24, 15, 11) 3 (26, 14, 10) 4 (28, 13, 9) 5 (32, 11, 7)

8 Fig. 7. Total delay versus the number of ONUs. total delay of all service-classes is in consistency with the transmission priorities of the service-classes, with the 1st and the 3rd service-class to have the lowest and the highest delay, respectively. learly, the total delay of the serviceclasses increases together with the mean packet arrival rate. To compare with the EPON case, it is worth-noticing that, in this example, the maximum total delay in WDM-EPON (Fig. 8) is less than the lower total delay in EPON (Fig. 5). As far as the accuracy of the proposed analysis is concerned, it is evaluated absolutely satisfactory, since the simulation results are very close to the analytical results, in Fig. 8. Fig. 9. Analytical results of the total packet delay vs. the number of wavelengths in WDM-EPON. class versus the length of the data-packets. As it is shown in this figure, for the packet arrival rate of 20 packets/sec that achieves the relatively lowest total delay in Fig. 8, the length of the data-packets must be significantly increased (e.g. from 1000 bits to 10,000 bits) in order to cause a remarkable increase in the total delay per service-class. Fig. 10. Analytical results of the total packet delay vs. the length of the data-packets in WDM-EPON. VI. ONLUSION Fig. 8. Analytical and simulation results of the total packet delay vs. packet arrival rate in WDM-EPON. In Fig. 9, we examine how the total delay (analytical results) varies according to the number of wavelengths used in the uplink of a WDM-EPON. The mean packet arrival rate is kept constant to 25 packets/sec. As it was anticipated, because of the increase of the transmission bandwidth when the number of wavelengths increases, the total delay drastically decreases. To facilitate the reader, it is worth-mentioning that the total delay results per service-class of Fig. 9 for one wavelength, coincide with the corresponding results of Fig. 5 when the arrival rate is 25 packets/sec. Also, when the number of wavelengths is 4, the total delay per service-class in Fig. 9 coincide with the corresponding results of Fig. 8 for arrival rate 25 packets/sec. Finally, in Fig. 10, we present the total delay per service- We present an uplink delay performance analysis for both EPONs and WDM-EPONs, which operate under the MPP/IPAT fixed service (and under the cooperation of a DWA in the case of WDM-EPONs). The PON supports multiple service-classes with priorities; the latter can readily be introduced by defining the number of packets per serviceclass (batches) which can be transmitted within a frame (in each transmission period). We determine the mean queueing delay of packet (per service-class), in a unified way for EPONs and WDM-EPONs, in a parametric way, through the formation of two queuing models; an M/D[x]/ model for the packets waiting in the local queues of the ONUs, and another M/D/ model for the frame transmission to the OLT. The parameter corresponds to the number of wavelengths in the uplink. From teletraffic point of view, when = 1, an EPON is

9 assumed, otherwise a WDM-EPON is considered. Having defined the total delay per service-class from an ONU to the OLT, as the sum of the queueing delay, the transmission delay and the propagation delay, we present in figures how it is affected from several factors; this shows the usefulness of the analytical tools. The considered factors which affect the total delay are: the packet arrival rate, the distribution of the batches (number of packets per service-class) in a frame, the number of ONUs, the number of wavelengths in the uplink and the length of the data-packets. The fact that the simulation results of the total delay per service-class are pretty close to the corresponding analytical results verifies our analysis under the assumption of Poisson arrivals; the calculation accuracy is absolutely satisfactory. As far as the anticipated superiority of WDM-EPONs against EPONs is concerned, in respect of the delay performance, it is evident when comparing the results. Moreover, the proposed analysis is proved consistent to parameter, since the delay analysis of WDM-EPONs for = 1 leads to the same results with the delay analysis of EPONs, as it was expected. As a future work, packet arrival processes other than Poisson can be considered in the queueing system of the ONUs. For example, as a first subsequent step, one may consider a Batch Poisson Arrival Process, the packet arrival process from the local queues to the frame, while the packet arrival process to the local queues may remain a Poisson process (Fig. 3). [11] B. Lannoo, L. Verslegers, D. olle, M. Pickavet, M. Gagnaire, and Piet Demeester, Analytical model for the IPAT dynamic bandwidth allocation algorithm for EPONs, Journal of Optical Networking,Vol. 6, No. 6, June 2007, pp [12] S. Bharati, P. Saengudomlert, Analysis and Derivation of Mean Packet Delay for Gated Service in EPONs, ETI Transaction on EE, vol. 8, no. 2, August [13] S. Bharati, P. Saengudomlert, Analysis of Mean Packet Delay for Dynamic Bandwidth Allocation Algorithm in EPONs, IEEE/OSA Lightwave Technology, Journal of, vol. 28, no. 23, pp , December [14] J. Vardakas and M. Logothetis, Packet delay analysis for Prioritybased Passive Optical Networks Proceedings of the IARIA First International onference on Emerging Network Intelligence - IARIA EMERGING 2009, October, 2009, Sliema, Malta. [15] X. Liu, G. Rouskas, MPP-l: Look-Ahead Enhanced MPP for EPON, Proceedings of IEEE I 2013, June 9-13, 2013, Budapest, Hungary. [16] Michela Becchi, From Poisson Processes to Self-Similarity: a Survey of Network Traffic Models, available on-line at: jain/cse567-06/traffic models1.htm [17] W. Leland, M. Taqqu, W. Willinger, and D. Wilson, On the Self- Similar Nature of Ethernet Traffic (Extended Version), IEEE/AM Transactions on Networking, vol. 2, no. 1, pp. 1-15, February [18] V. 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