I Introduction
The evergrowing Internet traffic generated by emerging services, including video on demand, remote elearning, and online gaming, continuously exacerbates the last mile bottleneck problem in recent decades [1]. Ethernet Passive Optical Network (EPON) has been considered as an attractive solution to this problem due to its low cost, large capacity and ease of upgrade to higher bit rates [2]. It has been deployed widely in many access networks such as FiberToTheHome (FTTH), FiberToTheBuilding (FTTB) and FiberToTheCurb (FTTC) [3, 4, 5].
A typical EPON is plotted in Fig. 1. An EPON is a pointtomultipoint network, where one optical line terminal (OLT) in the central office is connected to multiple optical network units (ONUs) located at the users’ premises via an optical passive splitter. In the downstream direction, the OLT broadcasts the packets to all the ONUs, and each ONU only accepts the packets destined to it. In the upstream direction, the OLT schedules the ONUs to share the bandwidth in a time division multiplexing (TDM) manner. The OLT assigns transmission windows (TWs) to each ONU through sending GATE messages in a roundrobin fashion. Upon receiving the GATE message, the ONU transmits upstream data in the allocated TW. The number of packets that the ONU can send during a TW is called the TW size in this paper. After data transmission, the ONU generates a REPORT message to inform the OLT of its buffer status [2]. The TWs of two successive ONUs are separated by a guard time to avoid data overlapping. The sizes of TWs that the OLT allocates to each ONU depend on the service discipline that the OLT adopts.
The gated service discipline has been widely studied in previous works [6, 7, 8, 9, 10, 11, 12, 13]. In the gated service, each ONU is authorized to transmit the amount of data that it requests in the REPORT [14]. Thus, the gated service may lead to the phenomenon called the capture effect [15], when an ONU with heavy traffic monopolizes the upstream channel for a long time and transmits excessive amounts of data. The capture effect will impose a large delay to other ONUs and thus impair the quality of service (QoS) of other ONUs. With the gatedlimited service, the EPON users have to sign a service level agreement (SLA) with the network operator to specify the upstream traffic rate, and the OLT typically sets a limit of the maximum TW size to guarantee the QoS of each ONU according to their signed SLAs.
The selection of the maximum TW size is a critical choice in the gatedlimited service EPON. On one hand, if the maximum TW size is set too small, the backlog of the ONUs cannot be cleaned up quickly and the upstream bandwidth is wasted by a large number of guard times and REPORT messages. In this case, the ONU will suffer from a large delay. On the other hand, if the maximum TW size is set too large, the capture effect cannot be suppressed effectively. An extreme case is that the gatedlimited service discipline will change to the gated service discipline when the limit goes to infinity.
In existing literature, only a few previous works have studied the selection of the maximum TW size via simulations. The impact of the maximum TW size on delay performance of an ONU is discussed in [16], in which the author points out that the maximum TW size for each ONU can be fixed based on the SLA, but doesn’t provide any concrete scheme for the selection of the maximum TW size. The aim of our paper is to develop a systematic method to select a proper maximum TW size for gatedlimited service EPONs.
The upstream transmission process of each ONU can be described by a vacation queuing system, in which each TW of the ONU is considered as a busy period while the time between two successive TWs of the ONU is treated as a vacation period. In general, the modeling of a vacation queuing system with limited service discipline is quite difficult [17]. In the gatedlimited service EPON, the number of packets that an ONU can transmit in a TW is limited by the maximum TW size. Thus, before the transmission of an arrival packet, it may have to wait multiple vacations, which is typically difficult to analyze [18].
Ia Previous Works Related to EPONs with GatedLimited Service
The exhaustive type limited vacation queuing systems were studied in [19, 20, 21, 22], where the server takes a vacation when either a queue has been emptied or a predefined number of customers have been served during the visit. In [19]
, the distributions of queue length, waiting time and busy period were obtained by using the embedded Markov chain and a combination of the supplementary variables and sample biasing techniques. In
[20, 21], the authors used matrixanalytic techniques to iteratively calculate the queue length distribution. In [22], a polling system with two priority queues and limited service discipline was analyzed, where the high priority queue is served with queue length dependent service time while the low priority queue is served with constant service time. The high priority queue length distribution at departure instants were derived by the embedded Markov chain. However, these models cannot be directly applied to gatedlimited service EPONs, in which the OLT only serves the packets that arrived before the last REPORT message of an ONU up to a predefined number, regardless if the buffer is empty or not.The gated type limited service vacation queuing systems were considered in [23, 24], where the server serves at most customers that present at a queue upon visiting and then begins a vacation. A queuing model based on an embedded Markov chain was developed in [23, 24] to derive the LaplaceStieltjes transforms of waiting time and busy period distributions, but the computation is too complex to give a clear physical insight into the performance of the entire system. To resolve this problem, a simple geometric approach was proposed in [18] to obtain the mean waiting time, but this approach can only solve a special case when the user is allowed to transmit one packet in each busy period.
In exiting literature, a few works were devoted to the modeling of gatedlimited service EPONs. In [6], the authors gave an approximate expression of mean waiting time for a gatedlimited service (which is called limited service in [6]) EPON under the assumption that the maximum TW size in terms of time (instead of the number of packets) is quite large, which is actually similar to analysis of the gated service EPON. In [8], an approximate mean delay of gatedlimited service EPONs is derived by using a discrete Markov chain, which is invalid when traffic load is high.
In summary, none of the previous works have obtained a useful formula of mean waiting time for general gatedlimited service EPONs where the maximum TW size is finite and larger than one, and neither have they discussed how to select a proper maximum TW size for each ONU of gatedlimited service EPONs.
IB Our Approach and Contributions
In this paper, we analyze the polling process of EPONs with gatedlimited service discipline. Our goal is to develop an insightful model to describe the delay performance of gatedlimited service EPONs, and to find a systematic method of selecting the maximum TW size for each ONU based on the SLA.
First, we adopt the geometric approach described in [18]
to derive the mean waiting time of an M/G/1 queue with vacations and gatedlimited service. A key step is to calculate the mean number of whole vacations, excluding the residual vacation, experienced by an arrival before it receives service. The computation of this key parameter is based on an innovative approach that establishes the connection between the mean number of whole vacations and the first and second moments of the number of packets served in a busy period. A distinguished feature of this model is that there are two queues in the buffer of each ONU: one queue is inside the gate and the other one is outside the gate.
Next, we apply the Chernoff bound of queue length to select the optimum TW size. According to the SLA, the delay performance of an ONU shouldn’t be influenced by the TW size limit if its traffic rate does not exceed the subscribed rate. Thus, the criterion of selecting the optimum TW size is to choose the smallest integer that makes the probability of queue length exceeding the TW size limit negligible. That is, when an ONU operates in the subscripted region, its buffer can be emptied with a high probability at the end of every busy period. Otherwise, the ONU will suffer from a large delay when the input traffic rate exceeds the subscribed rate. Our specific contributions are summarized as follows:

We derive a generalized formula of mean waiting time for M/G/1 queue with vacation time and gatedlimited service discipline, which includes the mean waiting times of two queues.

We provide a simple rule to determine a proper optimum TW size for ONUs of the gatedlimited service EPON based on their SLAs, which is proved to be effective by simulations.
The remainder of this paper is organized as follows. In Section II, we demonstrate the capture effect and provide an overview of the polling process between the OLT and ONUs of a gatedlimited service EPON. In Section III, we derive the mean waiting time of the M/G/1 queue with vacation time and gatedlimited service discipline, and apply the result to ONUs of the EPON. Section IV discusses the method of selecting the optimum TW size and the delay performance of gatedlimited service EPONs under the selected TW size. Section V draws the conclusion.
Ii Motivation and Overview
An EPON can be considered as a polling system where a single server visits a set of queues in a cyclic order. A service discipline is one of the three policies, exhaustive service, gated service, and limited service, that specify the criteria of the server when progressing to the next queue [25]. In a polling system with exhaustive service discipline where the server serves a queue until it becomes empty, the capture effect occurs when a heavily loaded user transmits excessive amounts of data and monopolizes the channel for a long time, such that other lightly loaded users suffer from prolonged waiting times.
To alleviate this problem, current EPON systems adopt the gated service discipline where the OLT only transmits the packets that are requested by the ONU in the last REPORT message during a TW. However, the capture effect still persists since the user with heavy traffic can report a large number of packets to the OLT at the end of a TW to secure a large size TW in the next cycle, which will lengthen the delays that other ONUs have to endure. This problem can be completely solved by the gatedlimited service discipline, which is able to cap the TW size assigned to each ONU by a predefined value.
This point can be illustrated by using the example where two ONUs are connected to the OLT, and each of them is equipped with an infinite buffer. The service capacity of EPON is 1000 packets/ms. We assume that these two ONUs have signed the same SLAs with the network operator, meaning that they subscribe to the same upstream traffic rate (300 packets/ms) and have identical TW sizes, 4. Suppose ONU 1 is a disciplined user with a fixed input rate of 300 packets/ms according to the signed SLA, whereas ONU 2 is a malicious user whose traffic input rate is more than 300 packets/ms. Their mean waiting times versus the arrival rate of ONU 2 under the gated and gatedlimited service disciplines are plotted in Fig. 2 and 2, respectively.
With the gated service discipline, despite that the constant loading of ONU 1 is smaller than that of ONU 2, as Fig. 2 shows, the mean waiting time of ONU 1 is not only continuously increasing with the arrival rate of ONU 2, but also is uniformly larger than that of the malicious user ONU 2. Moreover, it is unbounded when the input traffic rate generated by ONU 2 reaches 700 packets/ms. In this case, 70 percent of bandwidth is monopolized by ONU 2.
On the other hand, with the gatedlimited service discipline, each ONU can transmit no more than 4 packets during each busy period in our example. Fig. 2 shows that once the arrival rate of ONU 2 exceeds the subscribed rate 300 packets/ms, it immediately suffers from a larger mean waiting time than ONU 1 and soon becomes unstable when the arrival rate is larger than 400 packets/ms due to the limited TW size. The disciplined user ONU 1 enjoys a small mean waiting time all the time, and it is immune to the malicious behavior of ONU 2. This example clearly shows that the EPON with gatedlimited service discipline can completely avoid the capture effect and provide a fair service to disciplined users while penalizing malicious users.
In an EPON system with ONUs that adopts the gatedlimited service discipline, the packets waiting in the buffer of each ONU are divided into two groups by a fictitious gate. The number of packets inside the gate is bounded by the maximum TW size, denoted by . An arrival packet first waits outside the gate and then enters the gate before it can be transmitted. As Fig. 3 illustrates, the buffer status is represented by a twotuple state , where is the number of packets waiting outside the gate, and is that waiting inside the gate. The number increases by 1 upon a new arrival, and decreases by 1 when a packet inside the gate begins to be transmitted by the ONU.
Fig. 3 plots the polling process of an EPON, where and . A 64byte GATE is employed by the OLT to notify an ONU about the start time and the length of each allocated TW. Upon receiving the GATE message, the ONU transmits all packets inside the gate during the TW. At the end of the TW, the ONU sends a 64byte REPORT to the OLT, which reports the number of packets waiting outside the gate. According to the number stated in the REPORT, the OLT decides the TW size for this ONU in the next cycle, which equals the smaller of and . Thus, the message REPORT offers the admission for packets waiting outside the gate to enter the gate.
After the ONU 1 issued the first REPORT, as Fig. 3 shows, the buffer state changes from to , which means two packets entered the gate. Then the ONU 1 becomes idle while the OLT polls the next ONU. When the OLT finishes the polling of all other ONUs, it sends a GATE message to ONU 1 again to repeat the process. To avoid data overlapping induced by the clock synchronization problem between the OLT and the ONUs [26], two successive TWs are separated by a guard time. As Fig. 3 depicts, the constant interval includes the guard time and the transmission time of a REPORT.
The purpose of limiting the TW size is twofold: to guarantee a bounded delay experienced by disciplined users, and to constrain malicious users from monopolizing the transmission channel. With the gatedlimited service discipline, the TW size limits the maximum number of packets that can be served in a busy period. The gatedlimited service discipline cannot effectively constrain malicious users if the TW size is too large, and it may introduce longer than expected delays for disciplined users if is too small. Therefore, selecting an appropriate maximum TW size is critical to the performance of EPON with gatedlimited service discipline. In the following sections, we devise a queueing model to analyze the mean waiting time of each ONU under the gatedlimited service discipline, and provide a rule to determine the proper TW size of an EPON. The notations used throughout this paper are defined as follows for easy reference.

Number of ONUs

The maximum TW size

The guard time plus the transmission time of a REPORT

Subscribed traffic rate for all the ONUs

Arrival rate of all the ONUs

Offered load to the EPON

Number of packets found waiting outside the gate of the buffer when the th packet arrives

Number of packets found waiting inside the gate of the buffer when the th packet arrives

Number of packets found waiting in the buffer when the th packet arrives,

Residual time seen by the th packet when it arrives during a busy period or a vacation time in progress

Duration of all the whole vacation times experienced by the th packet before it gets service

Service time of the th packet

Waiting time of the th packet
Iii Modeling of EPONs with Gatedlimited Service
In this section, we analyze the mean packet waiting time of an ONU in the EPON with gatedlimited service discipline. As Fig. 3 shows, an ONU is busy with packet transmission during the TW, followed by a vacation period with a duration that equals the sum of and the TWs of other ONUs. Therefore, at the end of each busy period, a predominate number of packets that an ONU reports to the OLT attributes to the number of arrivals during the vacation period before it. Hence, there are dependencies among TW sizes of ONUs. However, in the analysis of multiple access systems, such as Aloha or CSMA, this kind of dependency is weak and can be neglected when is large [27, 28]. Thus, we can treat each ONU independently. We make the following assumptions in the modeling of EPONs with gatedlimited service:

All ONUs in the EPON are statistically identical.

The number of ONUs
is large, such that the TWs of the ONUs can be considered as i.i.d. random variables. In practice, the number of ONUs is usually more than 16 in a typical EPON.

The packet arrival process of the EPON is Poisson, as is the arrival process of each ONU.

The packets are transmitted in a firstinfirstout (FIFO) manner, and the transmission times of the packets are i.i.d. random variables with a general distribution.

The propagation delay between the OLT and an ONU is very small in currently commercial EPONs, and can be ignored in the analysis.
Under these assumptions, each ONU can be considered as an M/G/1 queue with vacations and gatedlimited service. In Section IIIA, we derive the mean waiting time of this queuing system, which provides the key to analyze EPONs with gatedlimited service discipline in this paper.
Iiia M/G/1 Queue with Vacations and GatedLimited Service
We adopt the following notations in the analysis of the M/G/1 queue with vacations and gatedlimited service:

the traffic arrival rate is ,

the service times of the packets , are i.i.d. random variables with the first moment and the second moment , and

the vacation times , are i.i.d. random variables with the first moment and the second moment .
Under the gatedlimited service discipline, up to packets waiting outside the gate will enter the gate at the end of each busy period, and they will be served in the next busy period. After each busy period, the server takes a vacation. When the vacation terminates, the server returns to serve the packets if the buffer inside the gate is not empty; otherwise, the server takes another vacation.
A cycle starts at the end of a busy period, and consists of a vacation period followed by another busy period. As Fig. 4 illustrates, the th packet may arrive at the system during a busy period or a vacation period. The following definitions pertaining to busy periods will be adopted in the derivation of mean waiting time of the packets:

[]

A busy period.

The number of packets served in a busy period, where .

A busy period, during which packets are served, where ( happens when a vacation finishes while the buffer inside the gate is empty).

The probability that a busy period is a .

The probability that a packet is served in a , where .

The number of packets served ahead of the th packet in the same busy period.
We need the following two lemmas to facilitate the derivation of the mean waiting time of packets.
Lemma 1.
The probability that the th packet is served in a busy period is given by
(1) 
Proof:
Suppose there are busy periods during a time interval , where . The probability that a busy period is a is defined by
During the time interval , the number of packets that are served in all busy periods is , and the total number of packets served in the interval is . It follows that the probability that the th packet is served in a busy period can be obtained as follows
∎
Lemma 2.
The mean number of packets served ahead of the th packet in the same busy period is given by
(2) 
Proof:
Conditioning on the event that packet is served in a busy period , we have
∎
As Fig. 4 shows, it may take the server several busy periods to clear all packets waiting in the buffer ahead of packet . Since the server can only transmit up to packets in each busy period, before the starting of service, the waiting time of packet includes the following components:

The time to complete current service or current vacation. When the packet arrives, the residual time, either residual service time or residual vacation time, seen by is denoted by .

The service times of all packets found waiting in the buffer when arrives.

Besides residual vacation time, the duration of the whole vacation times experienced by before the starting of service is denoted by .
It follows from the similar argument given in [18], we have
(3) 
and
(4) 
where is the mean queue length, and is the traffic load. The key to derive the mean waiting time (3) is the third term , which is given in the proof of the following theorem.
Theorem 1.
The mean waiting time of an M/G/1 queue with vacations and gatedlimited service discipline is given by
(5) 
Proof:
Suppose the system is in state when packet arrives, meaning that the number of packets waiting in the buffer are outside and inside the gate. After arrives, all packets inside the gate are sent out during the first busy period, at the end of which the first of the packets enters the gate. The packet enters the gate at the end of the th busy period, and is sent out during the th busy period. That is, the number of whole vacations that has to experience before the starting of service is , where is the largest integer smaller than .
For example, as Fig. 4(a) shows, the state of system is upon the arrival of and is three, thus has to wait for whole vacation times in the buffer before it can be transmitted. It’s the same as that in Fig. 4(b). Thus, we have
(6) 
In the th busy period, the number of packets transmitted ahead of is given by . For example, as Fig. 4(a) shows, packet is the second packet served in the third busy period. Since upon the arrival of and , it follows that . However, in Fig. 4(b), there is no packet transmitted before in the third busy period since the system state is when arrives, and in this case . Therefore, by definition, we have
(7) 
Notice that the mean queue length is the sum of the mean number of packets waiting outside the gate and that waiting inside the gate . It follows that
(8) 
Since the packet is moved into the gate at the end of the th busy period and served in the th busy period, the waiting time of inside the gate, denoted as , includes the vacation time between these two busy periods, and the total service time of packets transmitted ahead of in the th busy period. Thus, the mean waiting time of inside the gate is given by
From Little’s Law and Lemma 2, we obtain the following mean number of packets waiting inside the gate:
(9) 
The theorem is established by combining (3)(4) and (6)(9). ∎
Suppose the distribution of service time is given. The evaluation of the mean waiting time (5) requires the first two moments of the vacation time and the number of packets transmitted in a busy period. Intuitively, they are related to each other because the random variable is dependent on the number of arrivals during the vacation time . Focusing on the application of the above theorem to EPONs, we will discuss the relationship between the first two moments of and in the next Section IIIB.
IiiB Mean Packet Waiting Time of EPONs with GatedLimited Service Discipline
In this subsection, we apply the result of Theorem 1 to calculate the mean packet waiting time of an ONU in the gatedlimited service EPON, where the rate of the traffic input to the network is and to each ONU is . We assume that the distribution of the packet transmission time is given.
1) Moments of Vacation Time V of An ONU: An ONU is busy with probability and idle with probability . The mean busy period of an ONU is given by
(10) 
In an EPON with ONUs, the vacation time of an ONU is equal to the TWs of other ONUs plus . According to our assumption A2, the TWs are i.i.d. random variables. By definition, we have
where is the offered load to the EPON. After some reconfigurations, the first moment of the vacation time for an ONU is given by
(11) 
Similarly, the second moment of the vacation time for an ONU is defined as follows
(12) 
where and
are the variances of
and , respectively. Recall that is a busy period during which packets are transmitted. It follows that , in which are i.i.d. random variables. Let andbe the LaplaceStieltjes transforms of the probability density function (PDF) of the service time
and the busy period , respectively. They are related as follows:where is the generating function of . Therefore, the variance of the busy period can be obtained by
(13) 
Substituting (13) into (12), we obtain the following expression of :
(14) 
From (11) and (14), we know from (5) that the mean packet waiting time of an ONU can now be determined by the first two moments and of the number of packets transmitted in a busy period.
2) Moments of Number of Packets K Transmitted in A Busy Period: The first moment of can be easily derived from (10) and (11), and given as follows:
(15) 
The derivation of the second moment
, however, has to resort to the discrete time Markov chain embedded in the epochs at the end of busy periods. As Fig.
5 shows, the upstream transmission process of an ONU is a sequence of cycles. For example, at the end of cycle and the beginning of cycle , the ONU reports its queue length, denoted as , to the OLT. According to this report, the OLT determines the size of the TW in cycle , denoted as , as follows: if , and , if . That is, the TW size in cycle is determined by the queue length at the start of cycle and given as follows(16) 
where is the number of reported packets that are not transmitted in cycle .
Let . Recall that is the probability that packets are served in a busy period. According to (16), we have
(17) 
Thus, the second moment of can be obtained based on the distribution of the queue length as follows
(18) 
On the other hand, the queue length at the start point of cycle is determined by the number of packets transmitted during the busy period of cycle and the number of arrivals during cycle . Thus, from (16), the queue length at the start point of each cycle satisfies the following Lindley’s equation:
(19) 
Let be the probability that there are arrivals during a cycle time . We immediately derive the following equilibrium equation from (19):
from which we obtain the generating function of queue length:
(20) 
where is the generating function of .
According to our assumption A3 that the packet arrival process of each ONU is a Poisson process, the distribution is completely determined by the cycle time distribution. Let be the PDF of the cycle time . We have
(21) 
A cycle consists of a vacation period and a busy period, which means that the cycle time is the sum of and the duration of the TWs of
ONUs, which are i.i.d. random variables according to our assumption A2. It follows that the distribution of cycle time is approximately a Gaussian distribution according to the central limit theorem
[29]. The LaplaceStieltjes transform of the cycle time distribution is given by(22) 
where the mean cycle time can be obtained from (11), and is given as follows:
(23) 
while the variance of the cycle time is determined by the variance of busy period (13):
(24) 
We know that the second moment in (18) is coupled to the queue length probability , for . For a given , according to Rouche’s theorem and Lagrange’s theorem [23], we can first solve from (20)(24), and then update by substituting into (18). Repeatedly applying this iterative procedure, we can obtain the value of and then obtain the mean waiting time (5) of an ONU by combining (11), (14), (15), and . The procedure that numerically calculates and the mean waiting time is given in APPENDIX A. In the next section, we seek a systematic rule to select the optimum TW size for each ONU of the EPON that satisfies practical operational requirements of EPONs with gatedlimited service.
Iv Optimum Transmission Window Size
As we mentioned in Section I, EPON users usually have to sign a SLA with the network operator to specify the upstream traffic rate. Suppose all ONUs are statistically identical, and each ONU subscribes to a SLA with a maximum traffic rate , where is the total subscribed traffic rate of all the ONUs, which is less than . An ONU is a disciplined user if its input traffic rate is in the admissible region . Otherwise, it is considered a malicious user. An EPON system is regular if all the ONUs are disciplined users. In this section, we first describe the methodology and procedure to select an optimum TW size for a given traffic rate . We then discuss the stability and delay performance of EPON system under this selected TW size .
As we mentioned in Section II, the purpose of limiting the TW size is twofold: to guarantee that the mean delays experienced by disciplined users are bounded, and to penalize malicious users. With gatedlimited service discipline, the TW size limits the maximum number of packets that can be served in a busy period. Ideally, a proper TW size ensures that all packets arrived at a disciplined ONU during a cycle time can be completely served in the next busy period. To achieve this goal, the probability that the queue length at the beginning of a cycle exceeds the limit should be kept very small. Thus, the criterion for the selection of TW size is given by
(25) 
for some positive , when . In the following, we show that an optimum TW size that satisfies the criterion (25) can be selected by using the Chernoff bound of queue length.
Iva Chernoff Bound of Queue Length
The Chernoff bound of the tail distribution of queue length at the beginning of a cycle is given as follows [30]:
(26) 
for any , where is the generating function of the queue length distribution and is the mean queue length.
Suppose all the ONUs are disciplined users with input traffic rate , and the TW size satisfies the criterion (25), then each time the queue length reported by an ONU should be typically smaller than with a high probability . It follows that equations (16) and (19) will respectively degenerate to the following approximate equations:
(27)  
(28) 
which implies that the following generating functions of , and are approximately equal:
(29) 
Thus, according to (21)(24), we have
(30) 
In this equation, the second moment of the number of packets served in each busy period can be obtained by
(31) 
Substituting (30) into (31), we have
which yields
(32) 
Substituting (32) into (30), we obtain in the regular case as follows:
(33) 
The mean and variance of queue length are respectively given as follows:
(34) 
and
(35) 
It follows from the first equation of generating function of queue length in (30), the Chernoff bound (26) is given by
for any . Substituting into the above Chernoff bound, the criterion (25) can be fulfilled if is the smallest integer that satisfies the following inequality:
(36) 
where . We discuss the procedure to find the optimum TW size that satisfies (36) in the next subsection.
IvB Optimum Transmission Window Size
Solving the optimum TW size from (36) involves a complicated transcendental equation, therefore it can only be solved numerically. To initialize the computation procedure, we provide a lower bound and an upper bound of in the following theorem.
Theorem 2.
The proof of the above theorem is given in APPENDIX B. An accurate approximation of the optimum TW size can be derived from the upper deviation inequality of normal random variables. We know that the cycle time approaches a normal random variable when is large. The relation (28) indicates that the queue length at the beginning of each cycle is approximately equal to the number of arrivals during a cycle time , or . As expected, the mean queue length given by (34) is the product of the arrival rate and the mean cycle time . It is also interesting to note that the variance of the queue length given by (35) is the sum of and the variance of a Poisson random variable with parameter . Thus, the queue length can be approximated by a normal random variable that is discretized by a Poisson process with rate . That is, we adopt the following approximation of queue length distribution:
(38) 
As Fig. 6 shows, the bigger the gap between these two distributions, the smaller the probability , where is obtained through the inverse transform of in (33). We use the same set of parameters, including the number of ONUs , a guard time and a REPORT message transmission time , the first and second moments of service time , in all figures of this subsection.
It is wellknown that any normal random variable satisfies the following upper deviation inequality [31]:
for . Since the distribution of queue length is close to that of the normal random variable , from the above inequality, the optimum TW size
can be estimated by the smallest integer
that satisfies the following relation:and it is explicitly given by
(39) 
The following inequality immediately follows from (35) and (39):
That is, the approximation of the optimum TW window size also lies between the two bounds and .
As we mentioned before, the optimum TW size that satisfies the inequality (36) can only be solved numerically from the following equation:
where is obtained from the proof of theorem 2 in APPENDIX B and given as follows:
(40) 
The following procedure is used to solve the optimum TW size that satisfies the inequality (36).

;

, calculate by (40);

If , ; else ;
/* If is too large, we update the lower bound of searching region to decrease , otherwise we update the upper bound. */ 
If , , go to Step 2;

, output .
In the practical operation of EPON, the parameter can be selected from the region , which implies that the buffered packets of an ONU can be emptied with a probability between 0.9 to 0.999 at the end of every busy period. A too large causes a too small that impairs the delay performance of disciplined users. On the other hand, a too small causes a too large that cannot effectively suppress the capture effect.
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