I. Introduction

The next generation optical access network (NGOAN) has been considered a solution to the problem of upgrading current congested access networks [1]. However, to realize NGOAN, service providers need to develop more effective operations [2]. The Ethernet passive optical network (EPON), which was standardized in June 2004 as IEEE 802.ah, is currently one of the most common techniques, and it can transmit at 1.25 Gbps bi-directionally [3]. However, the EPON has a problem: the mean waiting time of the arriving packet, i.e., the mean packet delay, is long [4]. This mean packet delay is very high ($\sim 15\mathrm{ms}$) when the traffic load is high [5]. Thus, the mean packet delay must be decreased to achieve more effective NGOAN operation.

The link in an EPON is shared by multiple optical network units (ONUs) at the customer’s premises; an optical line terminal (OLT) at a central office is connected to a single fiber that has an optical splitter/combiner for each ONU. An EPON can enable the development of an optical network at low cost by using a splitter or joint of the optical transmission.

In the downlink of an EPON, which means the communication of an ONU from the OLT, wavelength-division multiplexing (WDM) is used. In contrast, in the uplink, which means the communication to the OLT from the ONU, time division multiple access (TDMA) is used. Thus, dynamic bandwidth allocation (DBA) for time scheduling of appropriate data is required to avoid packet collisions. In the DBA scheme, each ONU sends the transmission request to the OLT, then each ONU can reserve the network resources based on the reply from the OLT.

The exchange of messages for this reservation is defined by the multipoint control protocol (MPCP) of the media access method [6]. In the MPCP, each ONU sends the transmission request amount (the amount of data in the queue) to the OLT as a 64-byte REPORT message. The OLT calculates both the transmission time and the transmission starting time of each ONU. The transmission window (TW) is the data size, which means that an ONU can transmit reserved packets in one cycle time. The OLT sends both the calculated TW and transmission starting time to each ONU as a 64-byte GATE message. Thus, the communication time in the uplink is divided into a reservation interval for packet-transmission control and a data interval for packet transmission.

In an EPON with the MPCP, interleaved polling with adaptive cycle time (IPACT) is a common polling method. With IPACT, the reservation interval is transmitted after the data interval [5], and the MPCP plays an important role in packet transmission. Closed-form mathematical expressions of network performance are also useful for evaluating a communication network during the design process [7]. Thus, theoretical analysis models [7–10] of MPCP in EPON have been proposed. These conventional theoretical analysis models for the MPCP in IPACT can only be applied to the general MPCP.

The mean packet delay is long [11] when the packet arrives after the reservation interval of the MPCP in IPACT. Zhu and Ma proposed a method of decreasing the mean packet delay for IPACT with the MPCP [11]. However, this method can only be applied to light-load traffic situations. The mean packet delay is generally long when the traffic load is heavy. To the best of our knowledge, these analyses cannot solve this problem yet. In the MPCP, the packet transmission time is determined by reservation. Arriving packets are reserved every cycle time, which is the total time of the TW of all ONUs—not every packet arrival. Therefore, depending on the arrival time of the packet, it takes time to reserve the packet and transmit it. This problem occurs more frequently when the traffic load at which the packet arrival rate becomes higher is high.

In this paper, to decrease the mean packet delay time for heavy traffic, we propose a novel reservation interval allocation method called delayed report messages MPCP (DR-MPCP), in which the transmission time of the REPORT message is delayed. Moreover, we model DR-MPCP and derive the mean packet delay time by using the M/G/1 model. Our proposed solution is based on the non-trivial simplification of a previous study [7] on the analytical solution for the general IPACT. We have already proposed DR-MPCP [12,13]. These proposed methods could not minimize the mean packet delay. We also show that mean packet delay can be minimized by considering the distance between an ONU and the OLT. We confirm that there is a close match between the simulation and our theoretical results and show the effectiveness of our method. The main contributions of this paper are as follows.

This paper is organized as follows. Section II describes related studies on DBA for EPON. Section III details the traditional polling model to use as a reference for DR-MPCP. Section IV explains the difference between Bharati and Saengudomlert's [7] derivation and our proposed derivation. Section V explains our assuming system network, and shows our M/G/1 model for DR-MPCP. Section VI presents the numerical and simulation results and concludes the paper.

II. Related Work

The DBA scheme for scheduling uplink transmission in EPONs is composed of a fair method and IPACT by calculating the timing for DBA [14]. With the fair method [15–18], the OLT calculates the transmission scheduling after receiving REPORT messages of all ONUs. Thus, this method can maintain fairness. With IPACT ([5,6–10,19–22]), however, the OLT calculates the scheduling after receiving REPORT messages of an ONU sequentially, and idle time in which data cannot be transmitted does not occur. Thus, network resource efficiency increases.

Mercian et al. [15] and Liu and Rouskas [16] analyzed MPCP behavior for the fair method. Mercian et al. compared the mean packet delay time for an EPON with that for a gigabit PON (GPON) by using the differences in the specifications [15]. An EPON sets the reservation interval after its data interval. However, a GPON sets the data interval after its reservation interval. In general MPCP, a REPORT message is sent to make a transmission reservation of the packets arriving before one cycle. With Liu and Rouskas’s method [16], however, a transmission reservation for the packets that arrived within a few cycles is made by the REPORT message. By using the MPCP, the idle-time interval in which packets cannot be transmitted decreases, and the resource efficiency increases. However, since it is necessary to experience a few cycle times before packet transmission, the delay time increases. The following expanded MPCP methods have been proposed, in which the reserved data are transmitted after a certain number of cycles. Naser and Mouftah [17] focused on the classes of service and proposed the fairness DBA method for every class. Tanaka et al. [18] proposed a packet scheduling method by using the threshold of frame size in 1G/10G mixed EPON. However, these fair methods decrease network resource efficiency. Thus, IPACT is widely used [23].

IPACT is divided into six services: fixed, gated, limited, constant-credit, liner-credit, and elastic services [5]. The fixed service uses bandwidth allocation (BA) that does not change the transmission schedule dynamically. Other services also use DBA. The gated service [7–10] can transmit all data arriving within its data interval. The limited service [19–22] limits the amount of transmission data within its data interval. Constant-credit, liner-credit, and elastic services expand the limited service. These three services, however, do not differ much in performance [5].

Lannoo et al. [19] proposed a theoretical analysis model of mean packet delay in the limited service of IPACT. With this model, however, it is assumed that packets have a fixed size. Thus, when the traffic load is high, analytical results do not closely match the simulation results. Turna et al. [20] proposed a DBA algorithm for the limited service of IPACT. This algorithm decreases the mean packet delay by half the cycle time. However, this algorithm evaluates only simulation results. Andrade et al. [21] proposed a limited service for dynamic bandwidth scheduling to ONUs in an EPON (DDSPON). Moreover, Garfias et al. analyzed DDSPON theoretically in 1G/10G mixed EPON. This analysis modeled the transition of the number of waiting packets by using the closed Jackson network [22]. However, it is difficult to operate as an actual system because this algorithm needs to expand both ONUs and the OLT.

With IPACT, MPCP behavior can be analyzed using the gated service. Various researchers [7–10] have analyzed MPCP behavior. In MPCP, the REPORT message takes into account packets that arrive before the end of the preceding data interval. However, Park et al. assumed that the REPORT message takes into account only packets that arrive before the start of their data interval [8]. This study derived the mean packet delay time. However, the REPORT message assumption differs from basic MPCP behavior. Aurzada et al. showed that MPCP behavior can be modeled using the M/G/1 model [9]. However, the reservation intervals were omitted in this model. Bharati and Saengudomlert [7] modeled the MPCP of IPACT by expanding the traditional study [24]. Mamounakis et al. [10] proposed a prediction method for the future buffer size of each ONU for limited and gated IPACT. This method only analyzes simulation results. Thus, the problem of the long delay in IPACT has not been theoretically solved yet. Zhu and Ma proposed a method of decreasing the mean packet delay for IPACT with the MPCP [11]. However, this method can only be applied to light-load traffic situations. The mean packet delay is generally long when the traffic load is heavy. To the best of our knowledge, these methods cannot solve this problem yet, which means that the mean packet delay increases depending on the packet arrival time when the traffic load is heavy.

III. Queueing Analysis For Polling System

A polling system can transmit the reservation packets in a multiplex manner when the multi-access channel can transmit only one user at a certain moment. In such a system, the communication resources of the channel are divided into a data interval for packet transmission and a reserved interval for message polling or reservation controls. An analysis model in the polling system of $N$ users has already been proposed by Bertsekas and Gallager [24]. We call this polling model the traditional method. The general solution of the mean delay time is derived as a M/G/1 model with an idle interval.

The polling system is classified into three types (gated, exhaustive, and partially gated systems) on the basis of the choice of the packets to be transmitted in its own data interval [24]. In a gated system, a reservation is composed of those packets that arrive before the beginning of the reservation interval. In an exhaustive system, a reservation is made for those packets that arrive before the end of the data interval. In a partially gated system, a reservation is composed of those packets that arrive before the end of the reservation interval. When the reservation interval is assumed to be the same as the interval for the REPORT message in IPACT, IPACT behavior can be modeled by using the gated system [7]. Note that the gated system should not be confused with the gated service for IPACT.

First, let us consider that there is one user in this gated system. As shown in Fig. 1, the $i$th packet needs to wait for a time $W_i$ before all packets are transmitted. We refer to the waiting time of the packet as packet delay. This packet delay $W_i$ consists of three components [7,24].

 

Fig. 1. Calculation of the average waiting time.

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Let $\overline{U}$ be the mean of random variable $U\in \mathbb{R}$, and let the random variable for the delay of packets, the residual time component, the service time component, and the reservation time component be denoted as $W$, $W_F$, $W_Q$, and $W_R$, respectively. Since $W=W_F+W_Q+W_R$, it follows that $\overline{W}=\overline{W_F}+\overline{W_Q}+\overline{W_R}$ [7].

Next, let us consider that there are $N$ users in the gated system. With this, it is assumed that packets from all users arrive in accordance with a Poisson distribution, $\lambda$. Thus, the packet rate from one user is $\frac{\lambda }{N}$. The first two moments of each packet-service time are denoted as $\overline{X}=\frac{1}{\mu }$ and $\overline{X^2}$. The first two moments of each reservation time are denoted as $\overline{V}$ and $\overline{V^2}$. All service times and reservation times are assumed to be independent. Let $N_i$ be the number of packets found waiting in the queue by the $i$th packet upon arrival. The delay expectation $E\{W_i\}$ for the $i$th packet can be derived as follows [24]:

When $i\in \mathbb{N}\to \infty$, $\overline{W_{\text{tr}}}={\lim }_{i\to \infty }E\{W_i\}$ of the traditional method can be calculated using the Pollaczek–Khinchin formula [24]:

IV. Derivation of Mean Packet Delay For IPACT Using The Expectation

A. System Model

As shown in Fig. 2, we assume an EPON with $N$ ONUs that are identical in terms of the statistics of packet arrivals and service time. The distance $d$ between each ONU and OLT is the same, and we assume that $d$ is about 20 km with a short propagation delay. As mentioned in Section I, the packets may collide in the uplink of an EPON. Thus, we analyze the uplink transmission.

 

Fig. 2. Our system model.

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With this system, it is assumed that the OLT controls packet transmission of the packet by using the cyclic polling with the gated system. Let $T_{\text{cycle}}$ be the total one cycle time of the data and reservation intervals for all ONUs. We also assume the gated service type of IPACT. Since the allocated TW size of each ONU is equal to the amount requested in the ONU’s last request, each TW and $T_{\text{cycle}}$ varies in every cycle [5].

Each ONU has a queue for the arriving packets. Each arrival packet waits in the queue until the GATE message is received from the OLT. Waiting packets are transmitted in accordance with the assigned TW in first-in-first-out (FIFO). The buffer of each ONU is larger than the number of arriving packets. The guard time is set between reservation and data intervals to switch to another ONU.

With this system, it is assumed that each packet arrives at each ONU’s queue with rate $\frac{\lambda }{N}$ of the Poisson process. Packet-service times are random in the first two moments $\overline{X}$ and $\overline{X^2}$. Note that each service time of a packet contains an interframe gap, which means that a minimal pause may be required between network packets. Let $\overline{V}$ and $\overline{V^2}$ be the first two moments of the reservation interval and ${\sigma }_v^2$ be the variance. Let $\rho =\lambda \overline{X}$ be the traffic intensity for all ONU’s packets. We assume that all services and reservations are independent.

The mean packet delay in this general IPACT was derived by Bharati and Saengudomlert (Bharati’s method) [7]. In this paper, we propose a novel method (DR-MPCP method) to decrease the mean packet delay in IPACT. However, since Bharati’s method derives the mean packet delay by using the difference between the mean packet delay of the traditional method., i.e., the polling model [24], and that of Bharati’s method, with the traditional method, it is difficult to apply Bharati’s method to the DR-MPCP method. Thus, we simplify Bharati’s method on the non-trivial solution. By using this simplification to decrease the number of cases, we derive the exact mean packet delay for the DR-MPCP method. In the next subsection, we describe the derivation of the mean packet delay [7] for the IPACT method using the difference between IPACT and traditional methods.

B. Derivation Using Difference Between IPACT and Traditional Method

IPACT behavior can be modeled by expanding this cyclic polling system. Bharati and Saengudomlert [7] argued that $\overline{W_F^{ip}}$ and $\overline{W_Q^{ip}}$ in IPACT are the same as $\overline{W_F^{\text{tr}}}$ and $\overline{W_Q^{\text{tr}}}$ of the traditional method in Ref. [24]. Moreover, $\overline{W_R^{ip}}$ was derived using the difference $\mathrm{\Delta }\overline{W_R}$ between the packet delay of the traditional method $\overline{W_R^{\text{tr}}}$ [24] and that of IPACT [7].

The mean reservation packet delays for $\overline{W_R^{ip}}$ are grouped into four classes in accordance with the pattern of packet arrival position: own ONU’s data interval ($=D_{ow}$), own ONU’s reservation interval ($=R_{ow}$), data interval of other ONUs ($=D_{ot}$), and reservation interval of other ONUs ($=R_{ot}$). Let $c\in C_b=\{D_{ow},R_{ow},D_{ot},R_{ot}\}$ be an element of a set of these cases as shown in Fig. 3. Let $\overline{W_{R,c}^{\text{tr}}}$, $\overline{W_{R,c}^{ip}}$, and $\mathrm{\Delta }\overline{W_{R,c}}$ be the reservation packet delay for every case of the traditional method, the reservation packet delay for every case of IPACT, and the difference between $\overline{W_{R,c}^{\text{tr}}}$ and $\overline{W_{R,c}^{ip}}$, respectively. From the mean packet delay of each case, by using Table I, $\mathrm{\Delta }\overline{W_R}$ can be derived as $\mathrm{\Delta }\overline{W_R}=(N-1)\overline{V}$. Thus, by $\mathrm{\Delta }\overline{W_R}$ and Eq. (2), $\overline{W_R^{ip}}$ can be derived as follows:

 

Fig. 3. Cases according to the packet arrival time focusing ONU 1 of Bharati’s method.

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TABLE I. Difference in $\overline{W_R}$ Between Traditional and IPACT Methods [7]

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The mean packet delay can be derived as follows [7]:

C. Mean Packet Delay Without Considering Difference of Mean Packet Delay of Traditional Method

For each case, $W_R^{ip}$ and the probability of occurrence for the case is derived as shown in Table II [12].

TABLE II. Proposed Cases for $\overline{W_R^{ip}}$ and Probability

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We derive $\overline{W_R^{ip}}$ for IPACT without considering $\mathrm{\Delta }\overline{W_R}$. With IPACT, the transmission time of a packet is determined in accordance with the arrival time of the packet because the timing of the REPORT message for the packet is determined in accordance with the arrival time of the packet. Before or after the REPORT message time for the packet, the waiting time of the packet changes greatly. Thus, we classify the waiting time $\overline{W_R^{ip}}$ in accordance with the packet arrival time as shown in Fig. 4. In each allocated TW, the reservation is shaded, while the data interval is labeled by its ONU number $n$.

 

Fig. 4. Cases according to the packet arrival time focusing ONU 1.

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We divide these cases into two categories (before interval and after interval) in accordance with the transmission time of the REPORT message. Moreover, in after-interval cases, it is classified whether the arriving time is a data interval or not. We call these cases before data interval ($D_b$), after data interval ($D_a$), and after reservation interval ($R_a$). Note that the number of our proposed cases for deriving the mean packet delay is less than that of Bharati and Saengudomlert [7] for the mean packet delay. Let $c\in C_m=\{D_b,D_a,R_a\}$ be an element of a set of these cases. Moreover, let $\overline{W_{R,c}^{ip}(n)}$ be the reservation packet delay for every case of our proposed derivation. Note that this reservation packet delay is the function of ONU number $n$.

To derive the mean packet delay by using Bharati’s method [7], the arrival pattern with the traditional method [24] needs to agree with the arrival pattern in IPACT [7] because $\overline{W_R^{ip}}$ is derived using $\mathrm{\Delta }\overline{W_R}$. However, we derive the mean packet delay directly without the difference between our mean packet delay and the traditional mean packet delay. Thus, we can divide the cases more simply than Bharati and Saengudomlert did [7].

The probability when the packet of each ONU arrives in a data interval, such as in $D_b$ and $D_a$, is $\frac{\rho }{N}$. However, the probability when the packet of each ONU arrives in a reservation interval, such as in $R_a$, is $\frac{1-\rho }{N}$. Thus, we can derive $\overline{W_R^{ip}}$ as

Equation (5) is the same as Eq. (3) proposed by Bharati and Saengudomlert [7]. Our new case classification can decrease the number of cases from four to three. Moreover, we can derive the mean packet delay directly without the difference $\mathrm{\Delta }\overline{W_R}$. Thus, our derivation method can be applied to derive the mean packet delay more simply. In the next section, we propose the DR-MPCP method to decrease the mean packet delay by using this IPCT derivation method.

V. Proposed DR-MPCP

A. Network System Model

With DR-MPCP, the network system model is almost the same as that with IPACT, as discussed in Section IV. To decrease the mean packet delay, we propose a novel allocation method for the reservation interval (DR-MPCP method) that delays the transmission time of $m$ ONUs for REPORT messages, as shown in Fig. 5 [13]. We refer to $m$ as the REPORT shifting amount. Note that the ONU needs to know the value of $m$ to determine when the ONU should transmit its REPORT message. Each GATE message is broadcasted to all ONUs. By writing the REPORT shifting amount to the expansion slot of the GATE message, all ONUs can know $m$. In our system, the data and reservation intervals are not always a successive interval of the same ONU. We need a guard time of about 1 μs every interval. In general, the mean cycle time $\overline{T_{\text{cycle}}}$ is set from 500 to 1500 μs. Thus, the guard time does not affect the system overhead.

 

Fig. 5. Concept of our DR-MPCP method.

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In theoretical analysis, the DR-MPCP method expands the traditional polling model (traditional method) [24], as stated in a previous section. Moreover, in numerical analysis, we compare the mean packet delay of DR-MPCP with that of IPACT (IPACT method) [7]. Thus, we explain the differences between DR-MPCP, traditional, and IPACT methods in the next subsection.

Parameter $m$ of DR-MPCP is limited by propagation delay causing the distance between ONU and OLT as shown in Fig. 6. However, DR-MPCP in Ref. [13] did not consider this point. In this paper, we propose the optimal REPORT shifting amount $m^{*}$ by considering RTT as the following equation:

 

Fig. 6. Relationship RTT and optimal report shift amount $m^{*}$.

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B. Differences Among DR-MPCP, Traditional Method, and IPACT Method

An EPON based on IPACT connecting $N$ ONUs can be treated equally with the traditional method for $N$ users [7]. Figure 7 shows the data and reservation intervals. As shown in this figure, each ONU repeats data and reservation intervals alternately. Without loss of generality, an arriving packet is for the first ONU, i.e., ONU1. In this figure, we assume that the packets of ONU1 arrive within the data interval of ONU2. The upper, middle, and lower levels in this figure show the traditional, IPACT, and DR-MPCP methods, respectively. In this figure, $a$, $r$, and $t$ denote the arrival time, the reported time for the REPORT message, and the transmitted time of the waiting packet, respectively. Thus, the delay time of this packet can be calculated by the difference $t-a$.

 

Fig. 7. Comparison between DR-MPCP and conventional methods ($m=1$).

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With the traditional method, the reservation interval is set before the beginning of the data interval. With the IPACT method, the data interval is set before the beginning of the reservation interval. Let us consider the reason for the reverse between data and reservation intervals. When the round-trip time (RTT) is long, the GATE message arrives with a delay. Thus, when an EPON uses the traditional method, the idle time occurs in which data cannot be transmitted because the waiting packets need to be transmitted immediately after sending the REPORT message. The IPACT method can transmit data without idle time even if the RTT is long by reversing both intervals.

However, there is a problem with the IPACT method. If the packet arrives after the reservation interval has finished as shown in Fig. 7, the request time for the packet (REPORT message) is the next cycle. Thus, the packet transmission time is the cycle after next. In this situation, the packet delay becomes very long.

To solve this problem specific to IPACT, DR-MPCP delays the transmission time for REPORT messages. In Fig. 7, we set $m=1$ (1-ONU delayed timing). By this control, the packet delay in this figure is shorter than the delay of IPACT. The DR-MPCP method can be modeled using the M/G/1 model. In the next section, we derive the mean packet delay of DR-MPCP $\overline{W_{\text{dr}}}$.

VI. Theoretical Analysis For DR-MPCP

A. Derivation of Mean Cycle Time

With the DR-MPCP method, the time to receive the GATE message is delayed. When the GATE message cannot reach the ONU within $T_{\text{cycle}}$ (in other words, when the RTT is larger than $T_{\text{cycle}}$), idle time occurs, which leads to decreased network resource efficiency. As shown in Eq. (6), $T_{\text{cycle}}$ is used for deriving $m^{*}$. Thus, $\overline{T_{\text{cycle}}}$ is an important parameter for an actual network [25]. However, Bharati and Saengudomlert [7] did not derive $\overline{T_{\text{cycle}}}$. Therefore, we first derive $\overline{T_{\text{cycle}}}$.

Our system assumes that each packet arrives at each ONU’s queue with rate $\frac{\lambda }{N}$ of the Poisson process. The assumption of the same arrival rate per ONU may be not realistic. Some studies assumed different arrival rates for each ONU [26,27]. However, in this study, we analyze with a DR-MPCP model assuming that all ONUs have the same arrival rate as a basic study. Packet-service times are random in the first two moments $\overline{X}$ and $\overline{X^2}$. Let $\overline{V}$ and $\overline{V^2}$ be the first two moments of the reservation interval and ${\sigma }_v^2$ be the variance. Let $\rho =\lambda \overline{X}$ be the traffic intensity for all ONU’s packets. We assume that all services and reservations are independent.

The arriving packet waits in the queue of each ONU and is transmitted in the next cycle in accordance with the received GATE message. Thus, the overall data transmission time for all ONUs in the next cycle can be derived from the difference between the average cycle time and the reservation interval of all ONUs $\overline{T_{\text{cycle}}}-N\overline{V}$. Moreover, the overall traffic intensity is $\rho$. Thus, the overall data transmission that all ONUs request to the OLT in a cycle can also be derived by multiplying the average cycle time by overall traffic intensity $\overline{T_{\text{cycle}}}\rho$. These equations are equal; thus, the average cycle time can be derived as follows:

This equation does not depend on the $m$ of DR-MPCP. Thus, we confirmed that $\overline{T_{\text{cycle}}}$ of the DR-MPCP method is the same as $\overline{T_{\text{cycle}}}$ of the IPACT method.

B. Derivation of Mean Packet Delay for DR-MPCP

The mean packet delay $\overline{W_{\text{dr}}}$ of DR-MPCP can be derived from the three types of waiting times [24]: $\overline{W_F^{\text{dr}}}$, $\overline{W_Q^{\text{dr}}}$, and $\overline{W_R^{\text{dr}}}$.

First, we consider the $\overline{W_F^{\text{dr}}}$ of DR-MPCP. The ongoing service time when the packet arrives is composed of a reservation interval and a data interval. The interval component is the same as the component of the IPACT method. Thus, $\overline{W_F^{\text{dr}}}$ is the same as $\overline{W_F^{\text{tr}}}$ and $\overline{W_F^{ip}}$ in previous studies [24] and [7]. Second, we consider the $\overline{W_Q^{\text{dr}}}$ of DR-MPCP. Let $N_i$ be the number of packets found waiting in the queue by the $i$th packet upon arrival. The expectation of time to transmit $N_i$ packets or reservation in progress can be derived from Little’s formula as follows: $\frac{N_i}{\mu }$. Thus, $\overline{W_Q^{\text{dr}}}$ is the same as $\overline{W_Q^{\text{tr}}}$ and $\overline{W_Q^{ip}}$ in previous studies [7,24].

Finally, we consider the $\overline{W_R^{\text{dr}}}$ of DR-MPCP. To derive $\overline{W_R^{\text{dr}}}$, the expectation of the total time of reservation intervals needs to be calculated. This total reservation interval means all reservation intervals in which the packet $i$ must wait before being transmitted. The number of experienced reservation intervals of DR-MPCP differs from that of IPACT, as illustrated in Fig. 7. Thus, $\overline{W_F^{\text{dr}}}$ and $\overline{W_Q^{\text{dr}}}$ are still valid. However, $\overline{W_R^{\text{dr}}}$ needs to be derived for DR-MPCP.

In this study, the packet arrival patterns are divided by using this characteristic for analysis of $\overline{W_R^{\text{dr}}}$. As shown in Fig. 8, the transmission times are divided into two areas in accordance with the packet arrival time before or after its own REPORT message. Moreover, the patterns are divided into four patterns ($D_b$, before data interval; $R_b$, before reservation interval; $D_a$, after data interval; $R_a$, after reservation interval) in accordance with the packet arrival time of the data or reservation interval. By using these patterns, we derive $\overline{W_R^{\text{dr}}}$. Let $c\in C_d=\{D_b,R_b,D_a,R_a\}$ be an element of a set of these patterns. Moreover, let $\overline{W_{R,c}^{\text{dr}}(m,n)}$ be the reservation packet delay for every case of our proposed derivation. Note that this reservation packet delay is the function of ONU number $n$ and REPORT shifting amount $m$. Note that it may be assumed without loss of generality that the ONU number is 1 in Figs. 9–12 [7].

 

Fig. 8. Cases by packet arrival position of DR-MPCP ($m=1$).

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Fig. 9. Case $D_b$: a packet arrives during data intervals before its ONU’s reservation.

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Fig. 10. Case $R_b$: a packet arrives during reservation intervals before its ONU’s reservation.

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Fig. 11. Case $D_a$: a packet arrives during data intervals after its ONU’s reservation.

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Fig. 12. Case $R_a$: a packet arrives during reservation intervals after its ONU’s reservation.

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Figures 9–12 show the arrival and transmitted times under four different packet arrival patterns. In this figure, $a$, $r$, and $t$ denote the arrival time, the reported time of the REPORT message, and the transmitted time of the waiting packet, respectively.

With the DR-MPCP method, the packet is transmitted after the end of processing for $m$-ONUs. The packet in Figs. 9 and 10 can be transmitted in the next cycle because it arrives before its own REPORT message in the current cycle. However, the packet in Figs. 11 and 12 needs to be transmitted in the cycle after next because it arrives after its own REPORT message in the current cycle.

As we mentioned in Section IV, Bharati and Saengudomlert derived $\overline{W_R^{ip}}$ by focusing on the difference between $\overline{W_R^{ip}}$ [7] and $\overline{W_R^{\text{tr}}}$ in the traditional method [24]. However, when we apply the DR-MPCP method to Bharati and Saengudomlert’s method, case analysis must be performed in consideration of the characteristics of both the DR-MPCP and IPACT methods. Thus, the number of cases becomes large. If we apply the old case method of Bharati and Saengudomlert to DR-MPCP analysis when $m=2$, the number of cases is $4+2m=8$ depending on $m$. On the other hand, the number of cases with our analysis is 4, as shown in Table III, because our method does not depend on $m$. Clearly, this reduction effect is even greater when $m$ is large.

TABLE III. $\overline{W_{R,l}^{\text{dr}}(m,n)}$ and Probability of Each Case

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In this study, we derive the expectation of $\overline{W_R^{\text{dr}}}$ directly. Note that we do not use the difference from $\overline{W_R^{\text{tr}}}$ in the traditional method to derive $\overline{W_R^{\text{dr}}}$. This expectation can be derived by calculating $\overline{W_{R,c}^{\text{dr}}(m,n)}$ and the occurrence probability of each case $C_d$. The occurrence probability when the packet arrives in a data interval such as in case $D_b$ and case $D_a$ is $\frac{\rho }{N}$. However, the probability when the packet arrives in a reservation such as in case $R_b$ and $R_a$ is $\frac{1-\rho }{N}$. Each $\overline{W_{R,c}^{\text{dr}}(m,n)}$ is shown in Table III. Note that each case has a difference range of $n$:

Thus, the mean packet delay $\overline{W_{\text{dr}}}$ can be derived as follows:

Equation (9) corresponds to Eq. (2) in the traditional method when $m=N-1$. Equation (9) also corresponds to Eq. (4) for the IPACT method when $m=0$. As a result, the difference between IPACT and DR-MPCP can be derived by using Eqs. (4) and (9) as follows:

VII. Simulation and Numerical Analysis

A. Parameters Setting

In this section, we conduct the numerical simulation to derive the mean packet delay and compare it to the theoretical results. This calculation uses the same values of the reference traffic parameter [7]. The numbers of ONUs were set to 8, 16, and 32. The bandwidth of uplink was set to $C_{\text{up}}=1\mathrm{Gb}/\mathrm{s}$. The guard time $t_g$ was set to 1 μs, and the size of the REPORT message was set to $L_R$, which was set to 64 bytes, in accordance with the MPCP standard. The mean reservation time was $\overline{V}=t_g+\frac{8L_R}{C_{\text{up}}}=1.512\mathrm{\mu s}$ with the variance ${\sigma }_v^2=0$.

We roughly divided all Ethernet packets into five classes according to their service time. Based on the study by Park et al. [8], the mean service time and the occurrence probability of the classes were set to 64 bytes (47%), 300 bytes (5%), 594 bytes (15%), 1300 bytes (5%), and 1518 bytes (28%). Thus, the mean service times are 0.608, 2.496, 4.848, 10.946, and 12.240 μs. We assumed that the interframe gap was 12 bytes.

These mean service times and the corresponding probabilities are used for simulations of real traffic in, for example, [7,28]. Thus we also used these probabilities for our simulation in this study.

Moreover, the first two moments of service time are $\overline{X}=5.090\mathrm{\mu s}$ and $\overline{X^2}=51.468{(\mathrm{\mu s})}^2$. The packet of each ONU arrives with $\frac{\lambda }{N}$. The traffic intensity $\rho$ is varied from 0.05 to 0.95.

Obviously, our simulation is based on the same assumption as our analytical model. Despite the fact that the simulation uses only these assumptions, the analytical solution is derived by logical inference from these assumptions. Therefore, if some inference steps are wrong, the result may not match the simulation results. In this study, we used the simulation to verify the process of derivation of our analytical solution.

B. $\overline{T_{\text{cycle}}}$ Results

Figure 13 shows the theoretical and simulation results of $\overline{T_{\text{cycle}}}$. The horizontal axis represents the traffic intensity $\rho$. As shown this figure, the analytical result for $\overline{T_{\text{cycle}}}$ closely matched the simulation results. Thus, we used this theoretical $\overline{T_{\text{cycle}}}$ to derive $m^{*}$ of Eq. (6).

 

Fig. 13. Comparison of theoretical $T_{\text{cycle}}$ with simulation $T_{\text{cycle}}$.

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C. $\overline{W_{\text{dr}}}$ Results

Figures 14 and 15 show the mean packet delays $\overline{W_{ipc}}$ and $\overline{W_{\text{dr}}}$ when the distance between ONU and OLT $d=5$ or 10 km. We observed a close match between simulation and numerical results for DR-MPCP. These figures also compare the mean packet delays of IPACT and DR-MPCP. The DR-MPCP method decreased the mean packet delay when there was heavy traffic. This is because $\overline{T_{\text{cycle}}}$ is large when traffic load $\rho$ is high as shown in Fig. 13.

 

Fig. 14. Comparison of $\overline{W_{\text{dr}}}$ with $\overline{W_{ipc}}$ when $d=5\mathrm{km}$.

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Fig. 15. Comparison of $\overline{W_{\text{dr}}}$ with $\overline{W_{ipc}}$ when $d=10\mathrm{km}$.

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These figures also show that when the number of ONU $N$ is large and the ONU-OLT distance $d$ is small, the difference between $\overline{W_{\text{dr}}}$ and $\overline{W_{ipc}}$, which is the effectiveness of DR-MPCP, is large. In this situation, $m^{*}$ can be set to large. Thus, we show the optimal report shift amount $m^{*}$ in the next subsection.

D. $m^{*}$ Results

Figures 16 and 17 show $m^{*}$ when $d=5$ or 10 km. The $m^{*}$ was large when $\rho$ was high. Moreover, when $N$ was large and $d$ was small, the optimal mean shift $m^{*}$ was large.

 

Fig. 16. $m^{*}$ when $d=5\mathrm{km}$.

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Fig. 17. $m^{*}$ when $d=10\mathrm{km}$.

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We previously derived Eq. (10) to confirm the effectiveness of DR-MPCP. From this expression, we can find that the effect depends only on $m$ and $\rho$. However, by our numerical solution, we found that $N$ and $d$, which depend on $m$, also affect the performance of DR-MPCP.

We summarize the traffic conditions in which the effectiveness of DR-MPCP is large as follows:

VIII. Conclusion

To minimize the mean packet delay time, we proposed a reservation interval allocation method called DR-MPCP, which delays the transmission time of the REPORT message. We also modeled DR-MPCP and derived the mean packet delay time by using the queuing theory M/G/1 model. We observed that DR-MPCP can decrease the mean packet delay much more than the IPACT when the traffic load is heavy.

There are a number of important directions for future research. First, this model cannot analyze RTT behavior. In the future, long-reached PON (LR-PON), in which the distance between OLT and ONUs is large, may become widely used [29–31]. In this situation, the theoretical analysis with RTT will be required. In future work, we will expand the analysis with RTT behavior and propose an algorithm that can decrease the mean packet delay in a LR-PON environment. Second, we assumed the arriving packets have a Poisson distribution as in a previous study [7]. Other types of arrival processes will be investigated, such as bursty packet arrivals or bulk data [32]. Third, we derived the mean packet delay by using the M/G/1 model. Wei et al. [33] analyzed circuit-level performance, such as circuit blocking probability, in accordance with traffic intensity. This circuit-level performance should also be theoretically analyzed. However, Wei et al. assumed a fixed cycle time. Thus, we will investigate this theoretical performance for IPACT. We will then consider energy consumption in EPON. In general, ONUs need to continually inspect for downstream traffic. Thus, energy is wasted [34]. Bokhari and Saengudomlert derived the exact mean packet delay by using sleep mode [35]. We will propose an algorithm to solve this energy-consumption problem. Finally, we assumed a single channel in an EPON. However, the current effort is WDM EPONs, in which several upstream channels are shared among ONUs [36,37]. Thus, we will theoretically analyze a WDM EPON.