Performance Analysis of OCDMA PON Configuration

Supporting Multi-Rate Bursty Traffic

with Retrials and QoS Differentiation

John S. Vardakasa, Ioannis D. Moscholiosb,∗, Michael D. Logothetisc,Vassilios G. Stylianakisc

aIquadrat, 08009, Barcelona, SpainbDept. of Informatics & Telecommunications, University of Peloponnese,

221 00, Tripolis, GreececWCL, Dept. of Electrical & Computer Engineering, University of Patras,

265 04 Patras, Greece

Abstract

An Optical Code Division Multiple Access (OCDMA) Passive Optical Net-

work (PON) is a highly promising PON configuration, thanks to the potential

of OCDMA technology in eliminating channel contention and efficiently sup-

porting multi-rate traffic with different QoS requirements. In this paper,

we provide a mathematical framework for a call-level performance analysis

of an OCDMA PON that accommodates multiple service-classes of bursty

traffic with retries. Blocked calls of random arrivals may immediately retry

several times, by requesting less resources (number of codewords) each time,

in order to be admitted for service. The bursty nature of traffic is denoted

by the fact that in-service calls may alternate between transmission and no-

transmission periods. Our analysis takes into account user activity and the

various components of additive noise. An approximate recursive formula is

∗Corresponding author:idm@uop.gr Tel.:+30-2710-372283; Fax:+30-2710-372160

Preprint submitted to Optical Switching and Networking November 11, 2015

derived for the efficient calculation of call-level performance measures. The

proposed analytical model is further extended to assess the performance of

an OCDMA PON that provides QoS differentiation, through the utilization

of multiple-weight optical codes. Both proposed models are verified through

simulation and found to be absolutely satisfactory. Moreover, comparison

of the new models with a model without retries, shows the necessity and

effectiveness of the new models.

Keywords: passive optical networks, optical code division multiple access,

bursty traffic, retrials, quality of service.

1. Introduction

The high volume of diversified data traffic resulted from the emergence of

IP-based multimedia applications, such as IPTV, VoIP and video conferenc-

ing, require access networks that can support in an efficient way multi-rate

applications with different QoS requirements. To this end, the Ethernet PON

(EPON) has been widely spread, mainly due to its ability to provide econom-

ically 1 Gb/s or 10 Gb/s to a number of end users, covering a distance of

about 20 Km [1], [2]. However, future optical access networks will require

to provide “symmetric gigabit-bandwidth”, which is very difficult to be im-

plemented by adapting an EPON solution [3]. To increase the longer-term

PONs’ performance, at 40 Gb/s Wavelength Division Multiplexing (WDM)

PON systems have been proposed [4], but these schemes have wavelength al-

location problems when they co-exist with 1 Gb/s and 10 Gb/s–based PON

systems [5]. A promising access network is the emerged OCDMA PON that

provides high-speed symmetric traffic in a simple and cost-effective manner

2

and advanced characteristics in supporting multiple service-classes [6], [7].

The OCDMA principle is based on the differentiation of each communica-

tion channel by a codeword that is assigned to each user. A codeword is a set

of (0,1) sequences, where the total number of 0 and 1 is the code’s length and

the number of 1’s is the code’s weight [8]. Various OCDMA techniques have

been proposed and implemented that consider different codewords’ length

and weight, in order to support various traffic types with highly variable

performance targets. The OCDMA PON can be considered as a viable net-

work configuration in supporting multiple service-classes with QoS differenti-

ation, and it has received much attention recently [9]. In multi-rate networks,

OCDMA can flexibly support arbitrary data rates and QoS; more precisely,

OCDMA can provide reduced packet latency and packet collisions compared

to EPON approaches, while higher channel utilization is achieved, which is

beneficial especially for bursty traffic [10].

Multi-rate service-classes can be offered in an OCDMA network through

the utilization of multi-length codes [11], the optical fast-frequency technique

[12], the time hopping technique [13], or the parallel mapping (PM) technique

[14]. In multi-length coding methods, different code lengths are assigned to

calls, to achieve different data rates; however, short-length codes introduce

greater interference to long-length codes. In order to increase the code car-

dinality, a two dimensional (2-D) wavelength-time coding technique can be

applied to these methods, where wavelength changes at every pulse of a

time-spreading code sequence [15]. The optical fast-frequency technique uses

variable attenuators to provide data-rate differentiation, but this technique

requires expensive, multi-wavelength transceivers with high sensitivity on

3

power control. Under the time hopping technique, data-rate differentiation

is achieved by utilizing intermittent transmission, in order to accommodate

various data rate signals using only one sequence. However, the changing

time-hopping rate is implemented based on an optical selector device, which

consists of a number of optical hard limiters [16]. In the PM technique, a

number of optical codes proportional to the data rate of each service-class is

assigned to each call. For the provision of QoS differentiation, one- or two-

dimensional multi-weight codes are considered, in order to achieve different

Bit Error Rate (BER) levels [17], [18]. The combination of multi-length and

multi-weight coding may be used for the provision of multi-rate and QoS-

differentiated OCDMA system. Multi-length, variable length, optical orthog-

onal codes are used in [19] for the support of two service-classes. Specifically,

different code lengths are used for the support of different data rates, while

high-weight users transmit at a lower power and low-weight users transmit

at higher power. Different power levels are also considered in [20] and [21],

in order to provide QoS differentiation for two service-classes.

A number of research efforts appear in the literature for the performance

assessment of OCDMA networks that are based on analytical models. Sig-

nificant work has been carried out for the packet-level analysis of OCDMA

networks (e.g., [22]-[26]), where main performance metrics are the network

throughput and the packet delay. In [22], an analytical model has been

developed for OCDMA PONs that estimates the data throughput of binomi-

ally distributed traffic at the packet-level, while considering the multi-user

interference and code contention in the upstream direction. Furthermore,

in [23]-[26] various access protocols have been proposed for OCDMA net-

4

works; in [23] two different protocols are proposed, which are differentiated

by the “pretransmission” coordination requirement, while this work is ex-

tended in [24], in order to consider multi-packet messages and the connection

establishment effect. The proposed protocol of [24] is called Round Robin

Receiver/Transmitter (R3T ) protocol and has been also studied in [25], in

order to include different noise sources in the protocol analysis. In the con-

tinuance of the above research work, in [26] an OCDMA Medium Access

Control (MAC) protocol is proposed that considers variable size window in

the link layer, in order to support the variable length data transmission. On

the other hand, only a few analytical models have been developed for the call-

level analysis of OCDMA networks, where the main performance metrics are

the blocking probabilities. For Call Blocking Probability (CBP) and teletraf-

fic capacity computation in OCDMA networks, analytical models appear in

[27]-[29], but only a single service-class is considered. Analytical models for

the CBP calculation of multiple-service-classes are proposed in [30], where

an OCDMA network utilizes variable-weight, multi-length optical codes. In

all these references neither the effect of different sources of additive noise,

nor the user activity, are taken into account. In [31], a call-level performance

analysis is presented for OCDMA PONs that accommodate multiple service-

classes, where in-service calls alternate between active (transmission) and

passive (non-transmission) periods. This work is extended in [32] and [33],

to cover a code reservation mechanism (an equivalent mechanism to trunk

reservation of telephone networks). Another significant extension appears in

[34] to include service-classes of finite traffic source population (quasi-random

arrivals), whereas in [30]-[33] the service-classes are of infinite population

5

(random arrivals).

In this paper, we present an analytical framework for the call-level perfor-

mance analysis of OCDMA PONs that keeps the integrity of our study. The

PONs support multi-rate service-classes of bursty traffic with random arrivals

and retries, as well as QoS differentiation, while we take into account many

parameters of the OCDMA-PON configuration. Blocked calls may immedi-

ately retry, one or more times, to enter the system by reducing their demand

for resources. The “retrial” feature largely differentiates the proposed model

from the model of [33] (and [34]). On the other hand, multi-retries make the

performance analysis more complicated, and result in new and more intricate

formulas, which however are computationally efficient, since they are recur-

sive. Moreover, the feature of multi-retries is an important consideration,

because it is realistic in the environment of contemporary networks, where

most of input traffic is elastic and, therefore, necessitates a deep study (i.e.

comparison between models with and without retries). As far as the data

rate / service-class differentiation is concerned, it is performed by adopting

the PM technique, where a different number of fixed-length codewords is

allocated to each service-class.

We adopt the PM technique instead of other multi-rate techniques, be-

cause it has very good features, as stated in [14]; the proposed analysis,

however, is applicable to multi-length codes, too. A PON uses a pool of

codewords; the maximum number of codewords is defined according to the

desired maximum value of BER. A Poisson (random) arriving call in the

PON is blocked if the required number of codewords exceeds the maximum

number of codewords. This blocking case is expressed by the so called Hard

6

Blocking Probability (HBP). Even if the required number of codewords is

available in the PON, a new call may also be blocked due to the presence

of additive noise, composed of thermal, fiber-link, beat and shot noise. This

soft blocking case is expressed by the so called Local Blocking Probability

(LBP). In both cases, a call may also immediately retry one or more times to

access the system, while requesting a smaller number of codewords each time,

with an increased service-time. In-service calls alternate between active and

passive periods, for which different times are considered. Thus, by taking

into account the user activity in this way, we capture the bursty nature of

traffic.

The proposed analysis extends the basic model of [35] to OCDMA systems

in general, and particularly to OCDMA PONs, where our study is concen-

trated on the uplink, i.e. the bottleneck link from the Optical Network Units

(ONUs) to the Optical Line Terminal (OLT) (Fig. 1). The maximum number

of codewords in the PON resembles the link capacity, while the codewords

in use resemble the link occupancy. We provide an approximate recursive

formula for the efficient calculation of the distribution of the codewords in

use. Based on this formula, we calculate the CBP as a function of both

HBP and LBP that occurs at the last attempt of a call to access the sys-

tem. We also determine the probability that a passive call cannot return to

the active state (Burst Blocking Probability-BBP). Moreover, we extend our

analysis to OCDMA PONs which provide the important feature of QoS dif-

ferentiation through the utilization of variable-weight optical codes. With or

without QoS differentiation, the accuracy of the proposed models is evaluated

(through simulation) absolutely satisfactory, while their comparison with a

7

model without retries ([33]) shows their necessity and effectiveness. Specif-

ically, the comparison shows that when retries are permitted, either with

or without QoS differentiation, considerable traffic saving (CBP) is achieved

(at least 10% in our evaluation examples), which is a significant amount in

high-speed networks. On the other hand the increment of the BBP (delay)

that is observed, it can be controlled so as to remain very low, by reducing

the number of calls in the passive state.

The analytical calculation of blocking probabilities is significantly vital

for network designers, since it is performed in relatively very short CPU-

time, in comparison to the time-consuming simulations, which are typically

performed by using complex and expensive simulation tools. Furthermore,

the proposed models are resourceful tools that could be used by network

designers, in answering questions involving trade-offs between the amount

of resources allocated to a specific service-class and the QoS experienced

by subscribers, as well as to predict the network performance under various

traffic conditions. Specifically, they can be used for the derivation of the

maximum number of supported codewords that achieve blocking probabilities

under a predefined upper limit, while considering the end users’ behavior

(arrival rate of connection requests, service times or user activity).

The rest of this paper is organized as follows: In Section 2, we present the

modeling principles for a multi-rate OCDMA PON without QoS differentia-

tion, and we provide the analysis for the calculation of LBP, distribution of

codewords in use, HBP and CBP. In Section 3, we present the analysis for a

multi-rate OCDMA PON that supports QoS differentiation. The evaluation

appears in section 4. We conclude in Section 5.

8

2. System Model of a Multi-Rate OCDMA PON Without QoS

Differentiation

We consider the OCDMA PON of Fig. 1 with U ONUs, which are con-

nected to the OLT through a Passive Optical Splitter/Combiner (PO-SC).

We study the upstream direction of the PON, while our analysis can be ap-

plied to the downstream direction (from the OLT to the ONUs), too. The

PON utilizes (L,W, la, lc)-codewords, which have the same length L, the same

weight W (number of “1’s”), while the auto-correlation and cross-correlation

parameters, la and lc, respectively, are estimated according to the desired

BER at the receiver. The PON supports K service-classes with service dif-

ferentiation (i.e. data-rate differentiation), but without QoS differentiation.

Data-rate differentiation is performed by using the PM technique, where a

different number of codewords is assigned to each service-class. More pre-

cisely, the data bits of a service-class k call are grouped per bk (k=1, . . . ,

K ) and these bk bits are transmitted in parallel during a data bit duration.

On the other hand, the OLT assigns bk codewords to a service-class k call

for the entire duration of the call. One codeword is used to encode data

bit “1”, while data bit “0” is not encoded. Thus, the call uses a number

of these bk codewords that is equal to the number of “1” that are transmit-

ted in each data bit period; therefore at most bk bits are encoded in each

data bit period (see the lower part of Fig. 1 for an example). In this way,

the complex procedure of assigning codewords in each data bit duration is

avoided by assigning the same bk codewords for the entire duration of the

service-class k call. Since bk data bits are simultaneously transmitted in a

data bit duration, the bit rate of service-class k is bkD, where D is the basic

9

bit rate of a single codeworded call and is a function of the length L of the

codeword. The received power per data bit “1” of a single-codeworded call

is denoted by Iunit, and it is a function of the desired BER at the receiver.

A three-way handshake (Request–ACK–Confirmation) is applied for the

connection establishment between an end-user and the OLT. Let λk denote

the total arrival rate of service-class k calls from all ONUs to the OLT; we

assume that calls arrive/originate at an ONU according to a Poisson process.

Accepted calls in the ONU-OLT link, start an active transmission period and

may remain active for their entire duration, or alternate between active and

passive periods. During an active period a burst of data is sent to the OLT

at a fixed rate, whereas during a passive period no data transmission occurs.

When an active service-class k call becomes passive, it releases the assigned

bk codewords, which then become available to new arriving calls. In order for

a passive call to be transferred to the active state again, the same number

of codewords, as in the previous active state, is re-requested by the call.

If this number of codewords is available, then the call is transferred to the

active state; otherwise burst blocking occurs, that is, the call remains in the

passive state for another passive period. There is a probability ak whereby

a service-class k call moves to passive state at the end of an active period,

and a probability 1− ak whereby the call departs from the system. The call

cannot depart from the system via a passive state. The time-periods that

a service-class k call spends in active and passive states are exponentially

distributed with mean µ−11k and µ−12k , respectively.

A new service-class k call that arrives to an ONU requests bk,0 codewords

and if the available codewords are less than bk,0, the call is blocked. In this

10

case, the call may retry several times, while requesting a smaller number of

codewords each time. The number of retries for service-class k calls is denoted

as Rk. If the call is accepted for service in the r-th (r = 0, 1, . . . , Rk) attempt,

then it uses bk,r codewords; note that bk,0 = bk. At the final attempt the call

is blocked and lost, if the available codewords are less than bk,Rk. Based

on the required codewords of a service-class k call, the received power that

corresponds to the r-th attempt is at most Ik,ract = bk,rIunit, since at most bk,r

bits “1” are transmitted in parallel in each bit period.

2.1. Call Admission Control – Soft Blocking

In an OCDMA system a call should be blocked if, after the call admission,

the total received power in the receiver is above a predefined threshold (soft

blocking). The total received power comprises of the noise from all other

calls, which is the Multiple Access Interference (MAI), and of other forms of

additive noise (thermal, fiber-link, beat and shot noise). Thermal, fiber-link

and beat noise are modeled as Gauss distributions with parameters (0, σth),

(0, σfb) and (0, σb), respectively, while the shot noise is modeled as a Poisson

process (p, p) [36], [37]. By using the central limit theorem we can assume

that the total additive noise follows a Gauss distribution with the following

parameters:

µN = p and σN =√σ2th + σ2

fb + σ2b + p2. (1)

A Call Admission Controller (CAC) located at the OLT decides on the

acceptance or the rejection of an arriving call, based on the total maximum

received power at the OLT. The maximum received power is calculated based

on the worst case scenario that all bk,r data bits transmitted in parallel are

11

“1”, in order to ensure that the BER will never increase above a desired value.

If the total maximum received power together with the power of the new call

exceeds a maximum threshold Imax, the call is blocked. The value of Imax

is determined based on the desired BER at the receiver [38]. The following

relation is checked by the CAC, in order to decide on the acceptance or the

rejection of a call in the r -th attempt:

K∑k=1

(n1kI

k,0actPinterf)+Ik,ract+IN>Imax ⇔ IN

Imax>1−

K∑k=1

(n1kbkIunitImax

Pinterf)− Ik,ract

Imax(2)

where n1k is the number of service-class k calls in the active state, IN is the

interference caused by the total additive noise and Pinterf is the probability

of interference. In order to derive eq. (2), we follow the analysis of [34] and

we assume that all in-service calls are accepted for service with their initial

requirements in codewords. This assumption does not affect the accuracy of

eq. (2) at all, because the summation of eq. (2) finally includes the total

number of codewords in use, which would be the same with the total number

of codewords even if all calls were accepted with their initial requirements

in codewords. As far as the probability of interference is concerned Pinterf ,

it is a function of the maximum cross-correlation parameter lc, the weight

W and the length L of the codewords, and the hit probabilities between two

codewords of different users. Let plc,s denote the hit probabilities of getting s

hits during a bit period out of the maximum cross-correlation value lc; then

they are obtained through [39]:

lc∑s=0

s · plc,s =1

2· W

2

L(3)

where∑lc

s=0 plc,s =1, and the factor 1/2 is due to the fact that data-bit “0” is

not encoded. For lc = 1, the percentage of the total power of another’ s user

12

bit that interferes with a bit of the new call is 1/W, since 1 out of W “1” of

the codewords may interfere. In this case, Pinterf = (1/W ) plc,1 = (1/2)W/L.

However, the same expression for Pinterf holds, even in the general case where

the maximum value of the cross-correlation is lc ≥ 1 [34]:

Pinterf =lc∑s=0

s

Wplc,s =

1

2· WL

(4)

Not only when a new call is accepted but also when a passive call tries

to become active, the OLT will examine the condition of eq. (2). Based on

eq. (2), the probability that a call is blocked in its r -th attempt, due to the

presence of the total additive noise, is expressed by:

Lk,r(n1k) = P

[INImax

>1−K∑k=1

(n1kbkIunitImax

Pinterf

)− Ik,ract

Imax

]⇔

1−Lk,r(n1k)=P

[INImax≤1−

K∑k=1

(n1kbkIunitImax

Pinterf

)− Ik,ract

Imax

] (5)

where n1k indicates the number of in-service active calls in the system.

The LBP is a function of the variable IN/Imax, which follows a Gauss dis-

tribution (µN/Imax, σN/Imax), since the total additive noise follows a Gauss

distribution (µN , σN) [40]. Therefore, the right hand side of eq. (5) is the

Cumulative Distribution Function (CDF) of the variable IN/Imax [40]:

Fn(x) =1

2(1 + erf(

x− µN/Imax

(σN/Imax)√

2)) (6)

The LBP can be calculated as:

Lk,r(x) = Fn(x), where x = 1−∑K

k=1(n1kbkIunitImax

Pinterf)−Iactk,r

Imax.

It should be noted that based on eq. (5), higher values of the mean additive

noise result in higher LBP.

13

2.2. Call Admission Control – Hard Blocking

Let C1 be the total number of the supported codewords which can be as-

signed to active calls. This number is a function of the applied type of code-

words and its maximum value is determined by the so called Johnson bound

[41]. For example, by considering the (L,W, λ) optical orthogonal codes

(with la= lc = λ), the maximum number of codewords is upper bounded by

b1/W b(L− 1)/(W − 1) · · · b(L− λ)/(W − λ)c · · ·cc.

When a call is at the passive state, no codewords are utilized by the call,

in the reality. However, in the proposed model, we assume that this call is

assigned a number of fictitious codewords from a total number of fictitious

codewords C2. This fictitious system is used in order to model the transition

of a call from the active to the passive state, and to define the number of calls

which are present in the passive state. The number of codewords that are

assigned to a passive call equals the number of codewords that are assigned

to this call in the active state. The value of C2 is defined so that it is larger

than or equal to C1. If C1 = C2, then a passive call can always become active

(no burst blocking occurs), while if C1 < C2, there is a probability that a

passive call cannot become active; this probability is called BBP [35], [42].

If an arriving call of service-class k is not blocked due to the additive

noise (soft blocking), the CAC checks the following hard blocking conditions

[35]:

j1 ≤ C1 − bk,r ∧ j1 + j2 ≤ C2 − bk,r (7)

where j1 and j2 are the total number of codewords in the active and passive

state, respectively.

If a new arriving call of service-class k is blocked in its first attempt

14

(r = 0) to enter the system, it may immediately retry r = 1, . . . , Rk times to

connect to the system, with traffic parameters (λk, bk,r, µ−1ik,r), where bk,Rk

<

bk,Rk−1 < ...bk,1 < bk and µ−1ik,Rk> µ−1ik,Rk−1 > ...µ−1ik,1 > µ−1ik,0 = µ−1ik [35].

According to the first condition of eq. (7) a new call is accepted for service, if

at least bk,r codewords out of C1 are available for active calls, while, according

to the second condition of eq. (7), the CAC prevents the acceptance of a

new call, when a large number of calls are in the passive state. If eq. (7)

is not satisfied in (the last) attempt r = Rk, then the call is blocked and

lost; otherwise the call is accepted for service and starts alternating between

active and passive states.

2.3. Distribution of Active and Passive Calls

Let Ω be the set of all permissible states of the active and passive states.

The distribution of the codewords in use ~j = (j1, j2), denoted by q(~j), is

given by the proposed two-dimensional approximate recursive formula:

jsq(~j) =2∑i=1

K∑k=1

bi,k,spik,0(~j)q(~j −Bik)+

2∑i=1

K∑k=1

Rk∑r=1

bi,k,r,spik,r(~j)Di,k,r(~j)q(~j −Bretryi,k )

(8)

for ~j = (j1, j2), j1 = 1, ..., C1, j2 = 1, ..., C2−j1, and

Dik,r(~j)=

1, if ~j : [(C1−(bi,k,r−1,s−bi,k,r,s)<j1≤C1) ∧ (j1+j2≤C2)]

∨ [(C2−(bi,k,r−1,s − bi,k,r,s) < j1 + j2 ≤ C2)]

0, otherwise

The variable bi,k,r,s is defined as [35]:

bi,k,r,s =

bk for r = 0 and s = i

bk,r for r > 0 and s = i

0 for r ≥ 0 and s 6= i

(9)

15

Also, Bi,k = (bi,k,1, bi,k,2) is the i,k row of the (2K×2) matrix B with elements

bi,k,s. Similarly, Bretryi,k = (bi,k,r,1, bi,k,r,2) is the i,k row of the (2K×2) matrix

Bretry with elements bi,k,r.s. Finally, pik,r(~j) is the utilization of state i by

service-class k calls in their r -th attempt:

pik,r(~j) =

λk(1−Lk,r(j1−bk,r))

(1−ak)µ1k,rfor i = 1

λkak(1−ak)µ2k,r

for i = 2(10)

Equation (10) gives the utilization pik,0(~j) of state i by service-class k calls

in their initial attempt, for r=0. It should be noted that if Rk = 0 for all

service-classes, eq. (8) coincides with eq. (10) of [33], i.e. for the case where

calls of all service-classes cannot retry.

The CBP that we examine is the probability that a call is blocked in its

last attempt to access the system. The calculation of the CBP of service-class

k, Pbk,Rk, is based on the combination of LBP and HBP:

Pbk,Rk=

∑~j∈Ω−ΩH

G−1Lk,Rk(~j)q(~j) +

∑~j∈ΩH

G−1q(~j) (11)

where G ≡ G(Ω) is the normalization constant of q(~j) and

ΩH =~j : [b1,k,Rk,1 + j1 > C1] ∧ [b1,k,Rk,1 + j1 + j2 > C2]

.

The first summation of eq. (11) signifies the probability that a call cannot

be accepted due to the presence of the additive noise. The second summation

of eq. (11) refers to the HBP that the available codewords are less than the

requirements of the call at its last attempt to access the system.

The derivation of an analytical formula for the BBP is based on the fact

that burst blocking occurs when a passive call cannot return to the active

state. This situation takes place when: a) the codeword requirements of the

16

call together with the number of assigned codewords to all in-service calls

exceed the total number of the supported codewords, b) the additive noise

causes the total received power to exceed Imax. Since the BBP may occur in

the initial attempt or in any one of the Rk retries, the BBP is given by:

Bbk=

Rk∑r=0

∑~j∈Ω∗

r

y2k,r(~j)q(~j)µ2k,r

Rk∑r=0

∑~j∈Ωr

y2k,r(~j)q(~j)µ2k,r

+

Rk∑r=0

∑~j∈Ω−Ω∗

r

y2k,r(~j)Lk,r(~j)q(~j)µ2k,r

Rk∑r=0

∑~j∈Ωr

y2k,r(~j)q(~j)µ2k,r

(12)

where ~j ∈ Ω∗r ⇔

(C1−Bk,r+1≤j1≤C1) ∧(∑2

s=1 js≤C2

)and ~j ∈ Ωr ⇔

(C1−(bk,r−1−bk,r)≤j1≤C1) ∧(∑2

s=1 js≤C2

).

The average number yik,0(~j) of service-class k calls in state i accepted in the

system in their initial attempt is given by:

yik,0(~j) =pik,0q(~j −Bi,k)

q(~j)(13)

while the average number yik,r(~j) of service-class k calls in state i accepted

in the system in the r -th attempt is given by:

yik,r(~j) =pik,r(~j)Dik,r(~j)q(~j −Bretry

ik )

q(~j)(14)

Note that if Rk = 0 for all service-classes, eq.(12) coincides with eq. (30) of

[33].

3. System Model of a Multi-Rate OCDMA PON With QoS Dif-

ferentiation

QoS differentiation can be achieved by assigning codewords with different

weights. We assume that the PON supports K = T ·S service-classes; where

17

S service-classes are differentiated by the data-rate and each one of these

service-classes supports T different QoS levels, which are defined by different

values of the BER. Therefore, there are S groups of service-classes, and

each group comprises of T service-classes. The application of variable-weight

codewords to an OCDMA system that utilizes the PM technique has been

studied in [43]. In the proposed analysis, we consider the (L, F = Wt, la, 1)

codewords of [43], for t = 1, . . . , T , where L is the temporal code length,

while the weight is equal to the number F of the supported wavelengths.

Calls of these T service-classes require the same number bl,t,r (l = 1, . . . , S)

of codewords in the r -th attempt (in order to have the same data rate).

However, calls of these service-classes are differentiated by the weight Wt of

the codewords that they utilize during their service. The traffic parameters of

service-class t, l are denoted as (λt,l, µ−11,t,l,r, µ

−12,t,l,r, σt,l). The received power

per bit “1” of a single-codeworded call of service-class l of the group t is

denoted as I t,lunit, while the received power that corresponds to a service-class

t, l call in the r -th attempt is at most Iactt,l,r = bt,l,rIt,lunit. In order to use one

notation for the service-classes, we denote that the parameters of service-

class k (k = 1, . . . , T · S) are Ikunit = I t,lunit, Ik,ract = I t,l,ract , bk,r = bl,r, λk = λt,l,

µ−1ik,r = µ−1i,t,l,r and σk = σt,l.

The LBP of service-class k can be calculated by eq. (6) and the following

relation which is based on eq. (5):

Lk,r(n1k)=P

[INImax

>1−T ·S∑x=1

(n1xbxIunitImax

P x,kinterf

)− Ik,ract

Imax

]⇔

1−Lk,r(n1k)=P

[INImax≤1−

T ·S∑x=1

(n1kbxIunitImax

P x,kinterf

)− Ik,ract

Imax

] (15)

where the probability of interference P x,kinterf between two codewords with

18

weights Wx and Wk is a function of the hit probability [10]:

px,k =WxWk

2L(16)

Following the same procedure that was used in order to derive eq. (4), the

probability of interference of a codeword of a service-class k new arriving call

and a codeword of service-class x is:

P k,xinterf =

1

Wx

Wx ·Wk

2L=

1

2

Wk

L(17)

The distribution of active and passive calls for the case of the multi-rate

OCDMA PON with QoS differentiation is given by eq. (8), where the upper

bound of the summations that refers to the total number of service-classes

has to be changed (from K) to T · S. The same change has to be applied in

eq. (11) and eq. (12) in order to calculate the CBP and BBP, respectively.

4. Evaluation and Discussion

In this section we examine the accuracy of the proposed analytical mod-

els through simulation, by providing two application examples: (a) for an

OCDMA PON without QoS differentiation support, and (b) for another

OCDMA PON with QoS differentiation. Besides, in both examples, we reveal

the necessity and the effectiveness of the new models, by comparing them

with a model without retries ([33]), based on the resultant performance on

the OCDMA PONs.

4.1. Results of an OCDMA PON without QoS Differentiation

In the first example we simulate a multi-rate OCDMA PON without QoS

differentiation, by using the Simscript III simulation tool [44]. The PON

19

supports three service-classes and utilizes the (211, 4, 1, 2)-codewords. The

number of codewords cannot exceed the Johnson bound, which is 105 for the

specific type of codewords. Based on the analysis presented in [38], while

considering a typical value of BER=10−6, the total number of codewords is

reduced to C1 = 40 for Iunit = 0.4 µW. The traffic description parameters

are (b1, b2, b3)=(8, 5, 2), (µ−111 , µ−112 , µ

−113 ) = (0.6, 1.0, 1.3), (µ−121 , µ

−122 , µ

−123 ) =

(0.5, 1.9, 1.5), (a1, a2, a3) = (0.9, 0.9, 0.95). Calls from the 1st service-class

can retry twice; the retry parameters are (b1,1, b1,2) = (7, 6), (µ−111,1, µ−111,2) =

(0.8, 0.9), (µ−121,1, µ−121,2) = (0.7, 0.9). Calls from the 2nd and 3rd service-class

can retry once; the retry parameters for the 2nd service-class are b2,1 = 3,

µ−112,1 = 1.2 and µ−122,1 = 2.2, while for the 3rd service-class the retry parameters

are b3,1 = 1, µ−113,1 = 1.5 and µ−123,1 = 1.7. The maximum received power

at each receiver is assumed to be equal to 5 µW, while the total number

of fictitious codewords is C2=46. The total additive noise follows a Gauss

distribution (1, 0.1) µW. Note that according to eq. (4), higher values of the

mean additive noise result in higher local blocking probabilities and therefore

to higher call blocking probabilities.

In Figs. 2, 3 and 4 we present analytical and simulation results for the

CBP of the three service-classes, respectively, versus the call arrival rate.

We consider 9 arrival rate points in the x-axis of Figs. 2, 3 and 4. Point 1

corresponds to (λ1, λ2, λ3)=(0.02, 0.06, 0.06) calls/sec, while in the successive

points the arrival rate of all service-classes is increased by 0.005 calls/sec.

Thus, Point 9 corresponds to (λ1, λ2, λ3) = (0.06, 0.1, 0.1) calls/sec. The

simulation considers the retries, while the error bars in all figures denote

reliability ranges of 95% for the mean values of CBP results. In each figure,

20

we present the corresponding analytical results from the model of [33], i.e.

by considering the same parameters, except of the retry parameter (calls are

not allowed to retry in [33]), in order not only to demonstrate that the retry

consideration reduces the CBP, but also to show that this is a substantial

reduction. The latter reveals the necessity of the proposed model. On the

other hand, we show that this substantial CBP reduction is paid by BBP

degradation, which is controllable, however.

The comparison between analytical and simulation results reveals that

the accuracy of the proposed model is quite satisfactory. Small declinations

between analytical and simulation results reveal the fact that the proposed

recursive formula is an approximate one. The comparison of the analytical

results of our proposed model and the corresponding results of [33] shows

that the CBP is significantly decreased. Specifically, by considering the retry

feature, CBP is decreased on average by 28.7%, 32.3% and 10.1% for the three

service-classes, respectively. The average variation V of the CBP results is

calculated by using the expression:

V =|Xretry −X|

X× 100 % (18)

where Xretry and X are the CBP results from the proposed analysis and [33],

respectively.

However, the retry feature has a negative impact on the BBP. In Figs. 5,

6 and 7 we present analytical and simulation results for the BBP of the three

service-classes, respectively, versus the call arrival rate. We also present the

corresponding BBP results of [33]. The comparison of the results reveals that

the BBP is increased when calls are able to retry; this increase is on average

29.3%, 30.2% and 39.8% for the three service-classes, respectively; these

21

values are obtained by (18). Nevertheless, BBP increase can be addressed

by decreasing the number C2 of the fictitious codewords. More precisely, by

decreasing the value of C2 from 46 to 45, we derive BBP results from the

proposed analysis that are approximately equal to the corresponding results

of [33] (the average declination is lower than 10−3%), while the CBP results

of the proposed analysis are still lower than the results of [33] by 23.7%,

28.1% and 8.6% for the three service-classes, respectively. On the other

hand, the proposed analysis gives approximately the same CBP results with

the analysis of [33] if the value of C2 is decreased from 46 to 43 (the average

declination is lower than 0.2%); in this case the BBP results of the proposed

analysis are in average 46.3%, 59.2% and 54.5% for the three service-classes,

respectively, lower than the corresponding results of [33]. The aforementioned

results prove the superiority of the proposed analysis over the analysis of [33],

in terms of lower CBPs.

We also study the effect of the total number of fictitious codewords C2

to the CBP and BBP, when the total number of codewords of the active

system (C1) is constant. To this end, in Figs. 8 and 9 we present analytical

results of the CBP and BBP, respectively, versus the total number of fictitious

codewords C2. In both figures we assume that the arrival rate of the three

service-classes is kept constant and equal to (0.04, 0.08 and 0.08) calls/sec;

the values of all other parameters remain unchanged. From Figs. 8 and 9 we

observe that the increment of C2 results in lower CBP (Fig. 8), since more

calls can be accommodated to the passive system. On the other hand, this

increment results in the increase of BBP (Fig. 9), because a higher number

of passive calls compete for the transition to the active state. Note that when

22

C1 = C2 no burst blocking occurs, but at the expense of higher CBP values.

4.2. Results of an OCDMA PON with QoS Differentiation

In the second example we examine a multi-rate OCDMA PON with QoS

differentiation. The PON supports four service-classes. Service-classes s1

and s3 require the same number of codewords (same data-rate), as well as

service-classes s2 and s4 require the same number of codewords, but different

than that of s1 and s3. Service-classes s1 and s2 utilize the (331, 10, 10,

1, 1)-codewords, while service-classes s3 and s4 utilize the (331, 5, 5, 1, 1)-

codewords. The total number of codewords is assumed to be equal to C1=45,

for a BER value that is equal to 10−7, for I1unit = I2unit = 0.35 µW and

I3unit = I4unit = 0.5 µW. The traffic description parameters of the 4 service-

classes are (b1, b2, b3, b4)=(7, 5, 7, 5), (µ−111 , µ−112 , µ

−113 , µ

−114 ) = (1.0, 1.0, 1.0, 1.1),

(µ−121 , µ−122 , µ

−123 , µ

−124 ) = (1.5, 1.9, 1.5, 1.1), (a1, a2, a3, a4) = (0.9, 0.9, 0.95, 0.95).

Calls from the 1st and the 3rd service-classes can retry, twice, by using the

retry parameters: (b1,1, b1,2) = (6, 5), (µ−111,1, µ−111,2) = (1.2, 1.6), (µ−121,1, µ

−121,2) =

(1.4, 1.8), (b3,1, b3,2) = (5, 3), (µ−113,1, µ−113,2) = (1.2, 1.6), (µ−123,1, µ

−123,2) = (1.4, 1.8).

Calls from the 2nd and 4th service-class can retry one time, by using the

retry parameters b2,1 = 4, µ−112,1 = 1.2, µ−122,1 = 1.4 and b4,1 = 2, µ−114,1 = 1.4,

µ−124,1 = 1.8. The maximum received power at each receiver is assumed to be

equal to 5.25 µW, while the total number of fictitious codewords is 55. The

total additive noise follows a Gauss distribution (1, 0.1) µW.

In Figs. 10–13, we present analytical and simulation results for the CBP

of the 4 service-classes, respectively, versus the call arrival rate. We also

present corresponding analytical results from the model of [33]. We consider 9

arrival rate points: Point 1 corresponds to (λ1,λ2,λ3,λ4)=(0.02, 0.02, 0.02, 0.02)

23

calls/sec, while in the successive points the arrival rate of the service-classes is

increased by 0.0025 calls/sec. That is, Point 9 corresponds to (λ1, λ2, λ3, λ4)=

(0.04, 0.04, 0.04, 0.04) calls/sec. Comparison between analytical and simula-

tion results shows that the accuracy of the proposed model with QoS dif-

ferentiation is quite satisfactory. Furthermore, in Figs. 14–17, we present

analytical and simulation results for the BBP of the four service-classes, re-

spectively, versus the call arrival rate, as well as the corresponding analytical

BBP results from the model of [33]. Note that the obtained CBP and BBP

results are also differentiated because of the different BER, which is expressed

by the different codewords. Moreover, by comparing the CBP and BBP re-

sults of the proposed models with the corresponding results of [33] (assuming

the same traffic parameters but without retries), we derive conclusions simi-

lar to the case of no QoS differentiation. Specifically, the average decrease of

the CBP results under the retry model is 20.5%, 9.6%, 48.5% and 40.6%, for

the four service-classes respectively, while the average increase of the BBP

results are 29.8%, 13.6%, 53% and 29.3% (for each arrival rate point the

relative CBP decrease or the BBP increase is illustrated in each figure). The

increased BBP results may be negated by decreasing the number C2 of ficti-

tious codewords from 55 to 51; by using this value, the CBP results are still

lower than the corresponding results from [33] by 13.2%, 8.5%, 17.4% and

12.1%, for the four service-classes, respectively.

4.3. Discussion

The results presented in the previous subsections reveal that the retry

consideration can significantly reduce the CBP, while results in increased

BBP values. The amount of the CBP reduction and the BBP increase is

24

highly affected by the selection of the retry parameters’ values and the value

of the fictitious capacity C2, whereby we can control (reduce) the number of

calls in the passive state and achieve lower BBP. Therefore, for the applica-

tion of the proposed scheme, the corresponding analytical models should be

used for the derivation of the optimal set of the retry parameters and the

total number of fictitious codewords that guarantees specific CBP and BBP

thresholds. This information can be derived fast and in a cost-effective way,

compared to time-consuming simulation tools.

Another important parameter that affects the performance of OCDMA

PONs is the coding scheme that can be applied to the proposed analysis.

As it is shown in the previous sections, the PM technique is a suitable solu-

tion that fits well in the proposed analysis, since the number of codewords

assigned to service-class k is expressed by the parameter bk. However, the

proposed analysis can be applied to other existing techniques, which have

been implemented in multi-rate OCDMA networks. For the case of multi-

length schemes, the parameter bk can be expressed by the number of chips

of each code; thus, the different code lengths lead to different bk values, as it

is required in our analysis. Furthermore, coding schemes providing only QoS

differentiation can also fit to the proposed analysis, in the following way. If

the coding scheme considers T different QoS levels, then the same number of

service-classes should be considered in the proposed analysis, with parameter

bt equals 1 for all of these T service-classes, since they are not differentiated

based on the data-rate, but only on the QoS. In any case, to apply the pro-

posed analysis for a coding scheme, we have to find the maximum number of

supported codewords, under specific BER thresholds, and thus to define the

25

system capacity (the total number C1 of codewords).

5. Conclusion

We present a mathematical framework for the call-level performance anal-

ysis of OCDMA PONs, while proposing new multi-rate loss models for the

calculation of blocking probabilities. The PON supports multiple service-

classes of bursty traffic with random arrivals. Blocked calls are allowed to

immediately retry several times to access the system, with reduced number

of codewords and increased service time requirements. In-service calls alter-

nate between active and passive states. The proposed models incorporate the

characteristics of the utilized codewords, the user activity and different addi-

tive noise distributions. We provide a recurrent formula for the determination

of the occupancy distribution in the PON (number of utilized codewords).

Based on this formula, we efficiently calculate the probability that the re-

quired number of codewords in the last retry of a call is not available, i.e.

CBP, and the probability that a call cannot return to the active state, i.e.

BBP. Moreover, we assess the call-level performance of a multi-rate OCDMA

PON with differentiated QoS through the use of multi-weight codewords. Fi-

nally, we reveal the necessity and the effectiveness of the proposed models

by comparing their resultant performance on an OCDMA PON against an

existing model without retries.

As a future work, open issue is the study of the effect of different code

sequences, as well as of interference cancellation capability at the receiver,

on the performance of OCDMA PONs. Furthermore, new research activities

can be contacted on comprehensive CAC schemes, by jointly exploiting the

26

call-level and packet-level performance analysis of OCDMA PONs. Having

contacted performance measures for each service-class both at call level and

at packet level, the CAC can decide whether to accept or reject a call based

on the trade-off between the two levels, in order to guarantee specific QoS

requirements of each service-class.

Acknowledgments

Work supported by the Research Program Caratheodory, Research Com-

mittee of the University of Patras, Greece.

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[43] S. Sahuguede, A. Julien-Vergonjanne, and J. P. Cances, “OCDMA code

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[44] Simscript III, http://www.simscript.com

33

Figure 1: An OCDMA PON configuration.

34

Figure 2: CBP of the 1st service-class (1st example)

35

Figure 3: CBP of the 2nd service-class (1st example)

36

Figure 4: CBP of the 3rd service-class (1st example)

37

Figure 5: BBP of the 1st service-class (1st example)

38

Figure 6: BBP of the 2nd service-class (1st example)

39

Figure 7: BBP of the 3rd service-class (1st example)

40

Figure 8: Effect of the fictitious capacity on CBP results of the proposed model (1st

example).

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Figure 9: Effect of the fictitious capacity on BBP results of the proposed model (1st

example).

42

Figure 10: CBP and relative decrease results of the 1st service-class (2nd example).

43

Figure 11: CBP and relative decrease results of the 2nd service-class (2nd example).

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Figure 12: CBP and relative decrease results of the 3rd service-class (2nd example).

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Figure 13: CBP and relative decrease results of the 4th service-class (2nd example).

46

Figure 14: BBP and relative increase results of the 1st service-class (2nd example).

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Figure 15: BBP and relative increase results of the 2nd service-class (2nd example).

48

Figure 16: BBP and relative increase results of the 3rd service-class (2nd example).

49

Figure 17: BBP results of the 4th service-class (2nd example).

50