IEEE TRANSACTIONS ON COMMUNICATIONS

Performance Analysis of OCDMA PONsSupporting Multi-Rate Bursty Traffic

John S. Vardakas, Member, IEEE, Ioannis D. Moscholios,Michael D. Logothetis Senior Member, IEEE, and Vassilios G. Stylianakis

Abstract—Optical Code Division Multiple Access (OCDMA)provides increased security communications with large dedicatedbandwidth to end users and simplified network control. Weanalyse the call-level performance of an OCDMA Passive OpticalNetwork (PON) configuration, which accommodates multipleservice-classes with finite traffic source population. The con-sidered user activity is in accordance with the bursty natureof traffic, so that calls may alternate between active (steadytransmission of a burst) and passive states (no transmission atall). Parameters related to multiple access interference, additivenoise, user activity and number of traffic sources are incorporatedto our analysis, which is based on a two-dimensional Markovchain. An approximate recursive formula is derived for efficientcalculation of call blocking probability. Furthermore, we deter-mine the burst blocking probability; burst blocking occurs whena burst delays its returning from passive to active state. Theaccuracy of the model is completely satisfactory and is verifiedthrough simulation. Moreover, we reveal the consistency andnecessity of the proposed model.

Index Terms—passive optical networks, optical code divisionmultiple access, multiple access interference, service differentia-tion, call-burst blocking.

I. INTRODUCTION

THE growing demand for high speed Internet due to theincreased popularity of broadband applications is the pri-

mary driver for the emergence of new access technologies. Thetransition from copper-based to fiber-based access networkinghas offered higher bandwidth capacity, efficiently handling ofmultimedia and interactive services, and Quality of Service(QoS) support [1]. Passive Optical Networks (PONs) is apromising high-performance optical access solution, given thatPONs provide huge bandwidth in a cost-effective manner byincorporating inexpensive passive elements [2].

Over the last years, different PON configurations have beenproposed and have evolved, on the basis of the Time Divi-sion Multiple Access (TDMA) technique. The Ethernet PON(EPON) and the Gigabit PON (GPON) are among the TDMAPONs which have been deployed worldwide [3]. They employonly two wavelengths for the upstream and downstream di-rections. Through Wavelength Division Multiplexing (WDM)we can allocate different wavelengths to each end-user, thusachieving high speed downstream/upstream transmissions. TheOptical Code Division Multiple Access (OCDMA) technique

J. Vardakas, M. Logothetis and V. Stylianakis are with the WCL, Dept.of Electrical and Conputer Engineering, University of Patras, 265 04, Patras,Greece, e-mail: (jvardakas, m-logo, stylian@wcl.ee.upatras.gr

I. Moscholios is with the Dept. of Telecommunications Science andTechnology, University of Peloponnese, 221 00 Tripolis, Greece, email:idm@uop.gr

is another multiple access approach in PONs, which multi-plexes a number of channels on the same wavelength andtime-slot [4]. OCDMA offers many desirable features, suchas dynamic bandwidth assignment, efficiency in bursty traffictransmission, asynchronous and low latency access, as well asbetter security over unauthorized access [5]-[7].

In OCDMA, communication channels are distinguished bya specific optical code. An optical code is a set of (0,1)sequences, known as codewords, with specific length thatsatisfies particular auto-correlation and cross-correlation con-straints [6]. Each 0 or 1 is called a chip, while the numberof 1’ s in a codeword is its weight. The encoding procedureinvolves the multiplication of each data bit by the codeword,either in the time domain [8], or in the wavelength domain[9], [10]. The decoder receives the sum of all encoded signals(from different transmitters) and recovers the data from aspecific encoder, by using the same optical code. All theremaining signals appear as noise to the specific receiver; thisnoise is known as Multiple Access Interference (MAI) and isthe key degrading factor of network performance.

To support multiple service-classes in OCDMA networks,we need to consider multiple rates and/or QoS differen-tiation. For the provision of multi-rate services, a simplesolution is based on the utilization of multi-length codes[11]; however, under multi-length coding, short-length codesintroduce significant interference to long-length codes, whilethere is a considerable Bit Error Rate (BER) for high speedusers. Another solution is the optical fast-frequency technique[12], which, however, requires multi-wavelength transceiverswith high sensitivity on power control. According to [13],a better solution is the parallel mapping technique, wherebyseveral codes are assigned to each service-class. In this case,different data rates are achieved by assigning a number ofoptical codes proportional to the data rate of each service-class. For the provision of differential QoS, one-dimensionaland two-dimensional variable-weight optical codes have beenintroduced in order to control BER at the receiver [14], [15].

There are many research efforts for the performance studyof OCDMA networks, but only very few of them are based onanalytical models [16]-[24]. An analytical framework for thecomputation of blocking probabilities and teletraffic capacityin OCDMA networks is proposed in [16]. Similar studieshave been performed in [17] and [18]. In these references, asingle service-class is considered, while the different sourcesof additive noise that are present in OCDMA systems are nottaken into account. Multiple service-classes in an OCDMAnetwork are considered in [19], where the analysis for the

IEEE TRANSACTIONS ON COMMUNICATIONS

calculation of blocking probabilities includes variable weightmulti-length carrier-hopping prime codes, but neither the useractivity nor the effect of the additive noise are taken intoaccount. Analytical models are presented in [20], [21] forthe call-level performance of a hybrid WDM-OCDMA PON,that are based on a teletraffic model for the performancemodelling of Wireless CDMA networks [22]. In [23], a call-level performance analysis is performed on OCDMA PONs,which accommodates multiple service-classes with infinitetraffic-source population. The shared link is modelled by atwo dimensional Markov chain, whose the solution leads to anapproximate but recursive formula for the calculation of CallBlocking Probability (CBP). Infinite traffic source populationis also assumed in [24], where a code reservation schemeis applied on OCDMA PONs to balance the CBP amongdifferent service-classes.

In this paper we investigate the performance of OCDMAPONs for a more realistic case, where different service-classesof finite traffic-source population are accommodated. Thedifferentiation of service-classes is performed by using theparallel mapping technique, where a different number of fixed-length codewords is allocated to each service-class. Our anal-ysis takes into account a user’ s activity, which is expressedwith different time periods (service-times) regarding active andpassive (silent) states of the user; this user behaviour is inaccordance with the bursty nature of traffic. The maximumnumber of codewords allocated to active users, so that the BERdoes not exceed a minimum value, defines the PON capacity,while passive users are modelled in a fictitious system of afictitious capacity. An arriving call may be blocked and lost,when the resulting number of codewords of all active usersexceeds the PON capacity. This possibility defines the HardBlocking Probability (HBP). A call may also be blocked in anyother system state, because of the presence of additive noise(thermal, fiber-link, beat, or shot noise). This case denotesthe Local Blocking Probability (LBP). The OCDMA PON ismodelled by a two-dimensional Markov chain, which, becauseof local blocking, does not have a Product Form Solution.However, we prove an approximate recursive formula for thedetermination of the shared link occupancy distribution. Basedon it, we efficiently calculate: a) CBP as a function of bothHBP and LBP, and b) the link utilization. Furthermore, weprovide the analysis for the determination of the probabilitythat a passive user cannot return to active state (Burst BlockingProbability (BBP)). The accuracy of the proposed formulasis completely satisfactory as simulative evaluation shows.Moreover, in order to reveal the necessity of the proposedanalysis, we study the influence of various parameters, such asadditive noise interference, user’ s activity and number of traf-fic sources, on both CBP and BBP. We also compare analyticalresults of the proposed analytical model with correspondingresults of two other models in the literature: firstly, we comparethe proposed model with the model of [23], which assumesinfinite number of traffic sources, and show that the model of[23] results in serious CBP/BBP overestimations. Secondly,we compare the proposed model with the model of [19];this comparison reveals that our model achieves lower CBPvalues, while it utilizes codes with fixed length and weight,

Fig. 1. A basic configuration of an OCDMA PON.

in contrast to the variable weight multi-length codes that areused in [19]. Finally, the proposed analysis can be used inorder to determine the minimum number of real and fictitiouscodewords, or the maximum number of traffic sources, forspecific values of CBP and BBP.

This paper is organized as follows: In Section II, we presentthe basic modelling principles for the OCDMA PON understudy. In Section III, we provide the approximate recursiveformula for the determination of the shared link occupancydistribution. Section IV includes the performance metrics ofthe PON and how they are determined. Section V is theevaluation section. We conclude in Section VI. Appendix Acontains the proof of the recursive formula of the shared linkoccupancy distribution.

II. THE OCDMA PON MODEL

We consider the OCDMA PON architecture of Fig. 1 with UONUs. All ONUs are connected to the OLT through a PassiveOptical Splitter/Combiner (PO-SC). The PO-SC is responsiblefor collecting data from all ONUs and transmitting them tothe OLT (upstream direction), as well as for broadcastingdata from the OLT to the ONUs (downstream direction). Weconcentrate on the upstream direction; however our analysiscan be applied to the downstream direction too. Users that areconnected to an ONU, alternate between active and passive(silent) transmission periods. The PON uses (L, W , la, lc)-codewords, which have the same length L and the sameweight W , while the auto-correlation la and cross-correlationlc parameters are defined according to the desired BER atthe receiver. The PON supports K service-classes which aredifferentiated via the parallel mapping technique. Under thistechnique, the OLT assigns bk codewords to a service-class kcall for the entire duration of the call. More precisely, duringthe holding time of a service-class k call, the data bits of thiscall are grouped per bk bits and transmitted in parallel in eachbit period. One codeword is used to encode bit “1”, whiledata bit “0” is not encoded. Thus, the call uses a number of

SUBMITTED PAPER

these bk codewords in each bit period and this number is equalto the number of data bits “1” that are transmitted during abit period. In this way, the complex procedure of assigningcodewords in each data bit period is avoided. Furthermore,since bk bits are transmitted in each data bit period, the datarate of service-class k is bk ·D, where D is the basic data rateof a single codeworded call.

When a single codeword is assigned to an active user (activecall), the received power of this call at the OLT, is denotedby Iunit (Iunit corresponds to the received power per bit, fora specific value of the BER [25]). Since the PON supportsmultiple service-classes of different data rates, a number ofsingle codewords is assigned to each service-class; therefore,the received power Iactk of an active call of service-class kis proportional to Iunit, since bk data bits are simultaneouslytransmitted for service-class k, during a bit period. Therefore:

Iactk = bk · Iunit (1)

The connection establishment between the end-userand the OLT is based on a three-way handshake (Re-quest/ACK/Confirmation). Calls of service-class k arrive atONU u (u=1, . . . , U ) from a finite number of traffic sourcesNu,k; the total number of service-class k traffic sources inthe PON is Nk =

∑Uu=1Nu,k. The mean call arrival rate of

service-class k, is λk =(Nk−n1

k−n2k

)vk, where vk is the

arrival rate per idle source, while n1k and n2

k is the numberof service-class k calls in the PON, in active and passivestate, respectively. Calls that are accepted for service start anactive period and may remain in the active state for their entireduration, or alternate between active and passive periods [26].During an active period, a burst of data is sent to the OLT,while no data transmission occurs throughout a passive period.When a service-class k call is transferred from the active topassive state, the total number of the in-service codewordsis reduced by bk. When a passive call attempts to becomeactive, it re-requests the same number of codewords (but notnecessarily the same codewords) as in the previous active state.If the total number of codewords in use does not exceed amaximum threshold (the PON capacity), the call begins a newactive period; otherwise, burst blocking occurs and the callremains in the passive state for another period. At the end ofan active period, the call is transferred to passive state withprobability αk, or departs from the system (the connection isterminated) with probability 1−αk. The active and passiveperiods of service-class k calls are exponentially distributedwith mean µ−1

i,k (i=1 indicates active state, i=2, passive state).In OCDMA systems, an arriving call should be blocked, if,

after the new call acceptance, the noise of all in-service calls isincreased above a predefined threshold; this noise is MAI. Wedifferentiate the MAI from other forms of noise (thermal, fiber-link, beat and shot noise). The thermal noise and fiber-linknoise are typically modelled as a Gauss distribution (0, σth)and (0, σfb), respectively, while the shot noise is modelledas a Poisson process (p, p) [27]. The beat noise is modelledas a Gauss distribution (0, σb) [28]. According to the centrallimit theorem, we can assume that the total additive noise ismodelled as a Gauss distribution (µN , σN ), considering thatthe number of noise sources in the PON is relatively large.

Therefore, the total interference IN caused by the thermal,the fiber-link, the beat and the shot noise is modelled as aGauss distribution with mean µN = p and variance σN =√σ2th + σ2

fb + p2 + σ2b .

Upon a call arrival at an ONU, a Call Admission Controller(CAC) located at the OLT decides on its acceptance orrejection, according to the total received power at the OLT.More precisely, the CAC estimates the total received power(together with the power of the new call); if it exceeds amaximum threshold Imax, the call is blocked and lost. Themaximum received power is calculated based on the worstcase scenario that all bk data bits transmitted in parallel are“1”, in order to ensure that the BER will never increase abovethe desired value. The value of Imax is also determined by thedesired BER at the receiver [25]. This condition is expressedby the following relation:∑K

k=1(n1kIactk · Pinterf)+I

actk +IN>Imax⇔

INImax

>1−∑Kk=1(n1

kIactk

Imax· Pinterf)− Iact

k

Imax

(2)

The summation in (2) refers to the received power of all in-service active calls of all K service-classes multiplied by theaverage probability of interference Pinterf . This probability isa function of the weight W , the length L and the maximumcross-correlation parameter lc of the codewords, as well as ofthe hit probabilities between two codewords of different users.Specifically, the hit probabilities plc,i of getting i hits duringa bit period out of the maximum cross-correlation value lc aregiven by [29]:

lc∑i=0

i · plc,i =W 2

2L, while

lc∑i=0

plc,i =1 (3)

where the factor 1/2 is due to the fact that data bit “0” is notencoded. In the case of lc = 1, the percentage of the totalpower of a data bit that interferes with a bit of the new callis 1/W, since 1 out of W “1” of the codewords may interfere.In this case Pinterf = (1/W ) plc,1 = W/2L. When lc ≥ 1 theprobability of interference is given by:

Pinterf =

lc∑i=0

i

Wplc,i =

W

2L(4)

The condition expressed by (2) is also examined at thereceiver, when a passive call tries to become active. Basedon (2) , we define the LBP Lk(n1

k) that a service-class k callis blocked due to the presence of the additive noise, when thenumber of in-service active calls is n1

k, as:

Lk(n1k)=P

(INImax

>1−K∑k=1

(n1k

Iactk

Imax·Pinterf

)− Iactk

Imax

)(5)

or,

1−Lk(n1k)=P

(INImax

≤1−K∑k=1

(n1k

Iactk

Imax·Pinterf

)− I

actk

Imax

)(6)

Assuming that the total additive noise IN follows a Gaussdistribution (µN , σN ), the variable IN/Imax follows a Gaussdistribution (µN/Imax, σN/Imax), too [30]. Therefore, theRight Hand Side (RHS) of (6) is the Cumulative Distribution

IEEE TRANSACTIONS ON COMMUNICATIONS

Function (CDF) of the Gauss variable IN/Imax and is denotedas Fn(x):

Fn(x)=P (IN/Imax≤ x)=1

2

(1 + erf

(x− µN

Imax

σN

Imax

√2

))(7)

where erf(•) is the well-known error function. By using (6)and (7) , we can calculate Lk(n1

k) of service-class k calls bysubstituting x = 1−

∑Kk=1(n1

kIactk

Imax· Pinterf) − Iact

k

Imax:

Lk(x) =

1− Fn(x), x ≥ 01, x < 0

(8)

III. SHARED LINK OCCUPANCY DISTRIBUTION

Let C1 be the capacity of the (real) shared link, which isthe PON capacity. This is discrete, because it is expressed bythe total number of codewords, which could be assigned tothe PON users. When a call is transferred to a passive state, itis assumed that a number of fictitious codewords are assignedto it from a total number of fictitious codewords C2. That is,each passive call is accommodated in a fictitious shared link offictitious discrete capacity C2 [24]. The number of codewordsassigned to a passive call is equal to the number of codewordsassigned to the call at the active state.

To show the role of the fictitious system in call admission,let j1 be the number of codewords of all active calls and j2be the number of codewords of all passive calls:

j1 =

K∑k=1

n1kbk and j2 =

K∑k=1

n2kbk (9)

If an arriving call is not blocked because of local blocking,then the CAC works as follows, by taking into account thehard blocking conditions:

j1 + bk ≤ C1 and j1 + j2 + bk ≤ C2 (10)

According to the first condition of (10) , an arriving callof service-class k is accepted in the link, if the number bkof codewords assigned to the call, plus the total number j1of codewords assigned to all in-service active calls, does notexceed C1. The second condition of (10) averts the acceptanceof an arriving call, when a large number of calls are in passivestate: the number of codewords assigned to the arriving calltogether with the total number of codewords assigned to allin-service active and passive calls (j1+j2) should not exceedC2. The value of C2 is chosen so that C1≤C2. If C1=C2,then a passive call can always becomes active, i.e. no burstblocking occurs. If C1<C2, then a passive call becomes activeonly if the first condition of (10) is met.

Let Ω be the set of all permissible states of the whole system(real and fictitious links), then the occupancy distribution of~j = (j1, j2), denoted by qF(~j), is given by the proposed two-dimensional approximate recursive formula:

2∑i=1

K∑k=1

(Nk−n1k−n2

k+1)bi,k,sρik,F(~j)qF(~j−Bik)=jsqF(~j) (11)

where ~j ∈ Ω⇔

(j1 ≤ C1 ∩

(2∑s=1

js ≤ C2

))(12)

The index i refers to the system states (i=1 for active state,i=2 for passive state), while s stands for active system if s=1,and passive system if s=2. The variable bi,k,s is defined as:

bi,k,s =

bk, if s = i0, if s 6= i

(13)

and Bi,k = (bi,k,1, bi,k,2) is the i, k row of the (2K×2) matrixB, with elements bi,k,s; Also, ρik,F(~j) is the utilization of theith system by service-class k:

ρik,F

(~j)

=

vk[1−Lk(j1−bk)]

(1−ak)µ1kfor i = 1

vkak(1−ak)µ2k

for i = 2(14)

Moreover, js is the occupied capacity of the system and isgiven by:

js =

2∑i=1

K∑k=1

nikbi,k,s (15)

For the proof of (11) see Appendix A.The only problem of using (11) is that the number of the

service-class k sources nik is unknown. A calculation methodfor the values of nik is proposed in [33], through the useof an equivalent system, with the same traffic descriptionparameters and exactly the same set of states. The equivalentsystem results in the same CBP values as the original system.However, the state space determination of the equivalentsystem can be complex enough, especially for large capacitysystems that serve calls from many service-classes. Becauseof this, we avoid the previous method and calculate nik as themean number of service-class k calls in state ~j = (j1, j2),when infinite number of traffic sources is considered (Poissonarrivals):

nik(~j) ≈ y ik,INF(~j) =Nkρik,F(~j)qINF(~j −Bi,k)

qINF(~j)(16)

where qINF(~j) is the occupancy distribution for Poisson ar-rivals [23]:

2∑i=1

K∑k=1

bi,k,sNk ρik,F(~j)qINF(~j −Bi,k) = jsqINF(~j) (17)

A similar method for the calculation of nik in the Engset Multi-rate Loss Model has been proposed and investigated in [31].

IV. PERFORMANCE METRICS

The CBP of service-class k, Pbk, is calculated as a functionof both LBP and HBP, as follows:

Pbk =∑

~j∈Ω−ΩH

G−1Lk(~j)qF(~j) +∑~j∈ΩH

G−1qF(~j) (18)

where ΩH =~j|[bi,k,1 + j1 > C1] ∪ [bi,k,2 + j1 + j2 > C2]

.

The first summation of the RHS of (18) signifies theprobability that a connection request will be blocked due tothe presence of the additive noise. The second summationrefers to the HBP, which is derived by summing up theprobabilities of all the blocking states that are definedthrough (10) . Therefore a call can be blocked in any state~j ∈ Ω − ΩH due to the presence of the additive noise,

SUBMITTED PAPER

while, when the system is at any state ~j ∈ ΩH the callis blocked due to the fact that the total number of assignedcodewords exceeds the PON capacity (if the new call wereaccepted).

We now proceed to the derivation of an analytical formulafor the BBP calculation of service-class k. The BBP derivationis based on the fact that burst blocking occurs when a passivecall cannot return to the active state. In particular, this situationtakes place when: a) the number of assigned codewords tothe new (passive-to-) active call together with the numberof assigned codewords to all active calls exceeds the PONcapacity, b) the additive noise causes the total received powerto exceed Imax. The effect of the first reason can be determinedby the number of service-class k calls in passive state, n2

k,when the system is at any burst blocking state:

~j ∈ Ω∗⇔

(C1−bk+1 ≤ j1≤C1∩

(2∑s=1

js≤C2

))(19)

By multiplying n2k by the corresponding q(~j) and the service

rate in the passive state µ2k we calculate the rate that service-

class k calls depart from a burst blocking state, if it waspossible. Then, we sum the rates that a service-class k callwould depart from any burst blocking state:∑

~j∈Ω∗

n2kqF(~j)µ2k (20)

Following the same procedure resulted in (20) , we calculatethe sum of the rates that a service-class k call would departfrom any state, except from the burst blocking states. Dueto local blocking, however, each rate is multiplied by thecorresponding value of Lk(~j):∑

~j∈Ω−Ω∗

n2k Lk(~j) qF(~j)µ2k (21)

This condition refers to the case where the transition from thepassive to the active state is not possible due to the presenceof the additive noise. By normalizing the sum of (20) and (21)(i.e. by taking into account the state-space Ω), we obtain theBBP of service-class k, Bbk :

Bbk=

∑~j∈Ω∗

n2kqF(~j)µ2k +

∑~j∈Ω−Ω∗

n2k Lk(~j) qF(~j)µ2k∑

~j∈Ω

n2kqF(~j)µ2k

(22)

The utilization Rs of the shared link s (s=1 and s=2corresponds to the active and passive link, respectively) isgiven by:

Rs =

Cs∑i=1

iRs(i) (23)

where Rs(i) is the marginal link occupancy distribution of thelink s and is given by:

Rs(i) =∑

~j|js=i

qF(~j) (24)

TABLE IANALYTICAL AND SIMULATION CBP AND BBP RESULTS OF THE 1st

SERVICE-CLASS.

Arrival CBP BBPRate Analysis Simulation Analysis Simulation

Points (%) (%) (%) (%)1 0.0093 0.0115±3.22E-03 1.593E-04 2.234E-04±2.21E-052 0.0238 0.0248±6.11E-03 3.564E-04 4.511E-04±5.61E-053 0.0544 0.0612±7.23E-03 7.279E-04 9.438E-04±1.31E-044 0.1133 0.1189±1.52E-02 1.368E-03 1.613E-03±4.91E-045 0.2174 0.2302±1.89E-02 2.388E-03 2.796E-03±6.99E-046 0.3881 0.4011±3.61E-02 3.910E-03 4.331E-03±8.23E-047 0.6497 0.6711±6.45E-02 6.046E-03 6.435E-03±9.20E-048 1.0273 1.0934±6.91E-02 8.889E-03 9.218E-03±3.12E-039 1.5436 1.6005±8.12E-02 1.250E-02 1.365E-02±4.01E-0310 2.2168 2.2981±1.18E-01 1.690E-02 1.754E-02±4.51E-0311 3.0580 3.1149±2.11E-01 2.207E-02 2.401E-02±4.81E-03

V. EVALUATION AND DISCUSSION

We examine the validity of the proposed analysis by com-paring analytical with simulation results. For the simulationof the OCDMA PON under study, we use the Simscript II.5simulation language [32]. The simulation results are presentedas mean values of 8 runs with 95% confidence interval. TheOCDMA PON supports K = 3 service-classes, with U = 20ONUs. The number of traffic sources per service-class andper ONU is Nu,1 = Nu,2 = Nu,3 = 5, therefore the totalnumber of traffic sources in the PON for each service-classis N1 =N2 =N3 = 100. The codewords used in the PON are(F,W, la, lc) = (757, 4, 1, 2); these codewords are analysedin [26]. The PON capacity C1 and the received power fora single codeworded call Iunit are based on the desired BERvalue. Based on the analysis presented in [26] and for a typicalvalue of BER=10−6, the PON capacity results in C1 = 350(expressed in [26] as the “number of simultaneous users”),for Iunit = 0.5 µW. The total number of the fictitiouscodewords is C2 = 430. The traffic description parametersof the three service-classes are: (b1, b2, b3) = (20, 16, 10),(µ−1

11 , µ−112 , µ

−113 ) = (0.8, 1.8, 1.8) sec, (µ−1

21 , µ−122 , µ

−123 ) =

(0.8, 1.9, 0.9) sec, (α1, α2, α3) = (0.85, 0.9, 0.95). As faras the user activity is concerned, it can be expressed by theactivity factor wk, which is the percentage of time that aservice-class k user stays in the active state:

wk=T actk

T actk +T pask

=µ−1

1k

µ−11k +

∞∑l=0

µ−12k Bbk

l=

µ−11k

µ−11k +

µ−12k

1−Bbk

(25)

The total additive noise follows a Gauss distribution (1, 0.1)µW. The maximum power that the receiver of the OLT couldreceive is assumed to be equal to Imax = 13 µW. This valuewas determined by considering the maximum received powerof an active call of service-class k, maxIactk = (maxbk) ·Iunit, and a maximum estimated noise, based on the totaladditive noise plus an additional value for MAI.

In Tables I, II and III, we comparatively present analyticaland simulation CBP and BBP results, for the 1st, 2nd and3rd service-class, respectively. We consider 11 values of thearrival rate per idle source of service-class k, vk. They aredenoted by arrival-rate points (first column of the Tables),as follows: Each point corresponds to a vector of vk, ~v =

IEEE TRANSACTIONS ON COMMUNICATIONS

TABLE IIANALYTICAL AND SIMULATION CBP AND BBP RESULTS OF THE 2nd

SERVICE-CLASS.

Arrival CBP BBPRate Analysis Simulation Analysis Simulation

Points (%) (%) (%) (%)1 0.0065 0.0062±3.51E-03 1.200E-04 1.734E-04±1.25E-052 0.0167 0.0163±3.85E-03 2.738E-04 3.138E-04±8.33E-053 0.0386 0.0373±5.24E-03 5.668E-04 6.425E-04±1.82E-044 0.0812 0.0808±1.35E-02 1.079E-03 1.355E-03±2.14E-045 0.1575 0.1593±1.86E-02 1.909E-03 2.322E-03±4.27E-046 0.2838 0.2925±2.44E-02 3.166E-03 3.467E-03±6.75E-047 0.4792 0.4851±3.51E-02 4.957E-03 5.251E-03±7.81E-048 0.7637 0.7612±3.62E-02 7.374E-03 7.911E-03±8.14E-049 1.1563 1.1625±5.71E-02 1.048E-02 1.291E-02±1.01E-03

10 1.6723 1.6531±5.51E-02 1.432E-02 1.459E-02±2.65E-0311 2.3222 2.3547±5.78E-02 1.888E-02 2.018E-02±2.61E-03

TABLE IIIANALYTICAL AND SIMULATION CBP AND BBP RESULTS OF THE 3rd

SERVICE-CLASS.

Arrival CBP BBPRate Analysis Simulation Analysis Simulation

Points (%) (%) (%) (%)1 0.0033 0.0028±1.53E-03 5.560E-05 8.223E-05±1.29E-052 0.0086 0.0098±3.46E-03 1.298E-04 1.415E-04±6.23E-053 0.0202 0.0191±6.12E-03 2.746E-04 3.288E-04±1.85E-044 0.0432 0.0414±8.12E-03 5.338E-04 6.187E-04±1.44E-045 0.0849 0.0911±7.61E-03 1.909E-03 1.155E-03±2.71E-046 0.1549 0.1576±1.14E-02 1.623E-03 1.881E-03±3.59E-047 0.2646 0.2781±1.36E-02 2.581E-03 3.011E-03±2.81E-048 0.4265 0.4312±1.51E-02 3.894E-03 4.111E-03±4.14E-049 0.6524 0.6681±2.56E-02 5.610E-03 6.161E-03±4.36E-04

10 0.9529 0.9376±3.43E-02 7.755E-03 8.294E-03±6.55E-0411 1.3357 1.3513±4.46E-02 1.034E-02 1.121E-02±7.81E-04

(v1, v2, v3), where Point 1 corresponds to (0.0005, 0.0015,0.0025) calls/sec, while in the successive points the values ofv1, v2, v3 are all increased by 0.0001 calls/sec, so that the lastPoint 11 corresponds to (0.0015, 0.0025, 0.0035) calls/sec.The comparison between the analytical and simulation resultsof Tables I, II and III shows that the accuracy of the proposedmodels is completely satisfactory, for call-level performance.Small declinations between analytical and simulation resultsare owing to the assumptions that were used in our analysisfor providing recursive calculation (formula (11)).

In Fig. 2, we present analytical and simulation results forthe shared link utilization; both the active and passive linksare considered. The arrival-rate points in the x-axis of Fig. 2are the same with those were used in the Tables. As it wasanticipated, the utilization of both active and passive links isincreased with the increase of the arrival rate, since a greaternumber of users’ request access to the network resources.Again, the accuracy of the results is entirely satisfactory.

Subsequently, we investigate the impact of the total additivenoise on the CBP and BBP, by presenting in Fig. 3 and 4,respectively, analytical results together with simulation resultsof the three service-classes, versus different mean values µNof the total additive noise, while σN is kept constant and equalto 0.2. In both Fig. 3 and 4 the arrival rate per idle source of thethree service-classes is (0.001, 0.002, 0.003) calls/sec, whilethe values of all other parameters are the same with thoseused in Fig. 2. As it can be seen, the accuracy of the results iscompletely satisfactory, even for high values of additive noise

Fig. 2. Analytical and simulation results for the utilization of the active andpassive link vs. the arrival rate.

interference. As it was expected from (5) and (6), higher valuesof the mean additive noise generally result in a higher LBP andtherefore both the CBP and BBP are increased. The increaseof the total additive noise results in the increment of boththe CBP and BBP for the first service-class, while the CBPand BBP of the second and third service-classes are slightlydecreased. This is due to the fact that higher values of the meanadditive noise result in a higher LBP, and therefore calls fromthe first service-class (with the highest demand on numberof codewords) will be blocked; in this way more codewordsare available for calls of the second and third service-class(with lower demands on codewords). Further increase of thetotal additive noise will also result in the increase of CBP andBBP of the second and third service-classes. We also noticethat the increase of the additive noise interference has a greaterimpact on the BBP than on the CBP. This is due to the factthat the presence of the additive noise interference has an effecton the active system only, while the capacity of the passivesystem is not affected. Hence, the difference of the capacitiesbetween the active and passive systems increases, more callsare present in the passive system, and therefore BBP increases.Consequently, it is essential to examine the effect of the totalnumber of fictitious codewords C2 to the CBP and BBP, whenthe the total number of codewords of the active system isconstant; these analytical results are presented in Fig. 5 and6, respectively. In both Fig. 5 and 6 the arrival rate per idlesource of the three service-classes is (0.001, 0.002, 0.003)calls/sec, while the values of all other parameters are thesame as the ones were used previously. The increment of C2

results in lower CBP, because more calls can be accommodatedto the passive system than to the active system. However,this increment results in a higher BBP, given that the callcontention for the transition from the passive to active stateis higher. Note that when C1 = C2, no burst blocking occurs(this happens at the expense of higher CBP).

The effect of traffic source population and the user activityon the CBP and BBP is shown in Fig. 7 and 8, respectively,where analytical results of the three service-classes are pre-

SUBMITTED PAPER

Fig. 3. Analytical and simulation CBP results vs. the additive noiseinterference, for the three service-classes.

Fig. 4. Analytical and simulation BBP results vs. the additive noiseinterference, for the three service-classes.

sented. For presentation purposes, the service-classes have thesame population (N1 =N2 =N3) that is presented in the x-axisof Fig. 7 and 8. We consider two cases of the activity factorwk for each service-class k. The first case corresponds to thetraffic parameters that were used in Fig. 2, where, according to(25), (w1, w2, w3) = (0.499, 0.486, 0.666), while in the sec-ond case we consider that the service time µ−1

2k of service-classk in the passive state is reduced by 25% compared to the µ−1

2k

of the first case; hence (w1, w2, w3) = (0.571, 0.558, 0.727).In both cases the arrival rate per idle source is chosen sothat the product (Total Number of Traffic Sources) by (ArrivalRate per Idle Source) is constant for each point of the x-axis(but the last). As an example, we consider N1v1 = 0.1 erl,N2v2 = 0.2 erl, N3v3 = 0.3 erl. The results that correspondto the “infinite” population (last point of the x-axis of Fig. 7and 8) are derived through the analytical model for Poissonarrivals of [23]. We can observe that when µ−1

2k is reduced,then CBP is decreased, while the BBP is increased. We canobserve that the reduction of the service time in the passivestate causes the reduction of the CBP, but also the increase

Fig. 5. Analytical CBP results vs. the total number of fictitious codewords,for the three service-classes.

Fig. 6. Analytical BBP results vs. the total number of fictitious codewords,for the three service-classes.

of the BBP. In this case the activity factor wk is increased.Another way to increase the activity factor is by increasing theservice time in the active period, which results in the incrementof the CBP and the reduction of the BBP. We can also observethat both CBP and BBP increase, when the traffic sourcepopulation increases. This behaviour is explained by the factthat when the PON supports a larger number of traffic sources,the percentage of idle sources is higher; the latter leads to ahigher offered traffic-load and therefore to higher values ofblocking probabilities. On the other hand, according to Fig. 7and 8, important differences do exist between the “infinite”case and the rest cases of the x-axis, especially when thenumber of traffic sources is small. If the “infinite” case modelwas applied to all cases, significant CBP/BBP overestimations(and therefore ineffective allocation of the PON resources toend users) would result. This fact proves the necessity of ourmodel of service-classes with finite traffic-source population.

Finally, we compare the CBP results of our model withthose of the analytical model in [19], by considering anexample of two service-classes. For both models, the total

IEEE TRANSACTIONS ON COMMUNICATIONS

Fig. 7. Analytical CBP results vs. the number of traffic sources, for the threeservice-classes.

number of traffic sources in the OCDMA PON for eachservice-class is N1 = 50, N2 = 50, while the BER is 10−6.For the model of [19], the pair of length F and weight W ofcodewords used by the first and second service-class calls is121, 3 and 363, 3, respectively. Therefore, the data rateof the second service-class is 3 times the data rate of the firstservice-class. Consequently, in order to compare the modelof [19] with our model, we assume that in our model bothservice-classes utilize the same (F,W, 1, 2) codewords, whereF = 121 is the smallest length of the two codewords of[19] and W = 3. Based on the analysis of [25] for the(F,W, 1, 2) codewords, we apply the (121, 3, 1, 2) codewords,with cardinality 165. This value is reduced to C1 = 105 for aBER of 10−6 and Iunit = 0.4 µW. Also, in our model callsthat belong to the first and second service-class require b1 = 1and b2 = 3 codewords, respectively, so that the data rate of thesecond service-class is 3 times the data rate of the first service-class. Furthermore, since in [19] no distinction between activeand passive users exists, we calculate the service time of eachservice-class of [19] as a function of the service times in theactive and the passive period of the same service-class of ourmodel, according to the analysis of [26]:

µ−1k =

ak1− ak

(µ−11k + µ−1

2k ) + µ−11k (26)

Considering that (µ−111 , µ

−112 ) = (0.08, 0.08) sec, (µ−1

21 , µ−122 ) =

(0.122, 0.122) sec and (a1, a2) = (0.82, 0.82) in our model,the service times of the two service-classes of [19] are cal-culated from (26) as (µ−1

1 , µ−12 ) ≈ (1, 1) sec. Furthermore,

we consider that C2 = C1 = 105 where no burst blockingoccurs while the CBP is maximum, (see Fig. 5 and Fig. 6).Moreover, in our model the total additive noise follows a Gaussdistribution (1, 0.1) µW (the model of [19] does not take intoaccount the effect of the additive noise) and Imax = 6 µW.The analytical CBP results of both models versus the arrivalrate per idle traffic source are presented in Fig. 9. The arrivalrate per idle source is considered the same for the two service-classes and is indicated in the x-axis of Fig.9. The comparisonof the two models reveals that the CBP results of our model

Fig. 8. Analytical BBP results vs. the number of traffic sources, for the threeservice-classes.

Fig. 9. Analytical CBP results vs. the arrival rate per idle source, for theproposed model and the model of [19].

are lower than the corresponding CBP results of [19], althoughthe proposed model takes into account the presence of additivenoise. This is the advantage of using our model with fixed-length codes and the parallel mapping technique for servicedifferentiation.

VI. CONCLUSION

We propose a new multi-rate loss model for the calculationof blocking probabilities in an OCDMA-PON which supportsbursty traffic of multi service-classes with finite traffic sourcepopulation. Our analysis takes into account crucial parametersof OCDMA PONs, such as different additive noise distribu-tions and the user activity with different service times foractive and passive periods. We prove an approximate butrecurrent formula for the shared link occupancy distributionin the PON. Based on this formula, we calculate CBP (asfunction of LBP and HBP), and BBP, as well as the sharedlink utilization. Comparison between analytical and simulationresults confirms the validity of our analysis and shows highaccuracy of the proposed model. We also show the necessity

SUBMITTED PAPER

and consistency of the model, as well as its advantagescompared to another multirate analytical model.

As a future work, we will include in our study a mecha-nism for guaranteeing QoS requirements among the supportedservice-classes by considering multi-weight codewords, whilewe could consider interference cancellation capability at thereceiver of OCDMA PONs.

ACKNOWLEDGMENT

This work was supported by the Research ProgramCaratheodory of the Research Committee of the Universityof Patras, Greece. The authors would like to thank all theanonymous reviewers for their constructive comments.

APPENDIX A

In order to derive the recursive formula of (11), we definethe system state as ~n = (n1, n2), where n1 and n2 is thenumber of calls in the active and passive system (respectively):ni=(ni1, n

i2, ..., n

iK), for i=1, 2. Subsequently, we introduce

the following notation:

ni =(ni1, ..., nik, ..., n

iK),

nik+ =(ni1, ..., nik + 1, ..., niK),

nik− =(ni1, ..., nik − 1, ..., niK),

~n1k+ =(n1

k+, n2), ~n2

k+ = (n1, n2k+),

~n1k− = (n1

k−, n2), ~n2

k− = (n1, n2k−)

(A-1)

The transition from and to state ~n is depicted in Fig. 10.The horizontal axis of the state transition diagram representsthe arrivals of new connection requests and the termination ofin-service connections. More precisely, when the system is atstate (A) it will be transferred to state (B) upon the arrival ofa new connection request. This transition happens with a rate(Nk−n1

k−n2k+1)vk (the arrival rate of the idle sources of

service-class k) multiplied by the probability 1−Lk(n1k−1)

that this connection request will not be blocked because ofthe presence of the additive noise. The rate from state (C)to state (A) is defined in a similar way. The transition fromstate (B) to state (A) happens µ1,k(n1

k+1)(1−ak) times perunit time, since one of n1

k+1 active users will depart fromthe system with probability 1−ak. Similarly, we define therate from state (A) to state (C). Furthermore, the vertical axisreflects the transitions of a call between the active and passivestate. More specifically, the system will be transferred fromstate (A) to state (D), µ2,kn

2k[1−Lk(n1

k)] times per unit time(a transition from the passive to active state occurs), whichcan only be blocked because of the presence of the additivenoise. The reverse transition, from (D) to (A), occurs whenone of n1

k + 1 active users becomes passive with probabilityak. The transitions between states (A) and (E) are defined ina similar way.

The LBP that appears in the state transition diagram ofFig. 10 destroys the local balance between adjacent states;therefore we cannot obtain a Product Form Solution for thesystem state probability P (~n) (state (A)). To circumvent thisproblem, we assume that local balance does exist betweenadjacent states in order to find an approximate Product FormSolution and subsequently derive a recursive formula for the

Fig. 10. State transition diagram of the OCDMA system with active andpassive users.

efficient calculation of the link occupancy distribution. Underthis assumption, we may write the following local balanceequations:

(A)(E) : P (~n)µikn1kak =

P (~nk−+)µ2k(n2k + 1)[1− Lk(n1

k − 1)](A)(B) : P (~n)[Nk − n1

k − n2k]vk[1− Lk(n1

k)] =P (~nk+)µ1k(n1

k + 1)(1− ak)(A)(D) : P (~n)µ2kn

2k[1− Lk(n1

k)] =P (~nk+−)µ1k(n1

k + 1)ak(A)(C) : P (~n)µ1kn

1k(1− ak) =

P (~nk−)[Nk−n1k−n2

k+1]vk[1−Lk(n1k−1)]

(A-2)

where P (~nk+), P (~nk−), P (~nk+−), P (~nk−+) are the stateprobabilities of (B), (C), (D), (E), respectively.

By solving the above system of (A-2), the following ProductForm Solution stands for P (~n):

P (n)=

1

G

K∏k=1

(Nk

n1k+n2

k

) n1k∏

l=1

l + n2k

l

2∏i=1

ρnik

ik,F(n1k) (A-3)

where G ≡ G(Ω) is the normalization constant and ρik,F(n1k)

is given by:

ρik,F(n1k)=

vk[1−Lk(n1

k−1)](1−ak)µ1k

≈vk[1−Lk(n1k)]

(1−ak)µ1kfor i=1

vkak(1−ak)µ2k

for i=2(A-4)

In order for (A-3) in conjunction with (A-4) to satisfythe system of (A-2), we assume that the acceptance of oneadditional call in the active state does not affect the LBP.

Because of (A-4), we can express the local balance equationbetween states ~n and ~nik−, as:

P (~n)nik = [Nk−nik−ntk+1]ρik,F(nik)fik(~n)P (~nik−) (A-5)

where i = 1, 2, t = 1, 2, with i 6= t and

fik(~n) =

1 if nik ≥ 10 if nik = 0

(A-6)

The system occupancy distribution qF(~j) is given by:

qF(~j) = P (~j = ~n ·B) =∑

~n∈Ω~j

P (~n) (A-7)

IEEE TRANSACTIONS ON COMMUNICATIONS

where Ω~j=~n ∈ Ω:~nB =~j, ni

k ≥0, k=1, ...,K , i=1, 2

isthe set of states where the occupied bandwidth in the activeand passive state is exactly ~j = (j1, j2).

In order to introduce qF(~j) in (A-5), we sum both sides of(A-5) over Ω~j :∑n∈Ω~j

P (~n)nik=∑n∈Ω~j

[Nk−nik−ntk+1]ρik,F(nik)fik(~n)P (~nik−)

(A-8)Assuming that the RHS of (A-8) can be approximately writtenas:∑

n∈Ω~j[Nk−nik−ntk+1]ρik,F(nik)fik(~n)P (~nik−)≈

≈ρik,F(nik)∑n∈Ω~j

[Nk−nik−ntk+1]fik(~n)P (~nik−)(A-9)

then, the RHS of (A-8) becomes:

RHS=ρik,F(nik)Nk∑n∈Ω~j

fik(~n)P (~nik−)−ρik,F(nik)

∑n∈Ω~j

[nik+ntk−1]fik(~n)P (~nik−) =

ρik,F(nik)Nk∑~n∈Ω~j∩n

1k≥1 P (~nik−)−

ρik,F(nik)∑~n∈Ω~j∩n

1k≥1[n

ik + ntk − 1]P (~nik−)

(A-10)

For the set n ∈ Ω~j ∩n1k ≥ 1

=~n : ~n1

k−B = ~j −Bi,k

we introduce the variable n∗im (m = 1, . . . ,K), for therepresentation of nik ≥ 1 (k = 1, . . . ,K):

n∗im =

nik for m 6= knik − 1 for m = k

(A-11)

Therefore, (A-10) is written as follows:

RHS = ρik,F(nik)Nk∑~n∈Ω~j−Bi,k

P (~n∗i)−ρik,F(nik)

∑~n∈Ω~j−Bi,k

[n∗ik + ntk]P (~n∗i)(A-12)

Based on (A-7), (A-12) can be expressed as:

RHS=ρik,F(nik)NkqF(~j−Bi,k)−ρik,F(nik)·∑~n∈Ω~j−Bi,k

[n∗ik + ntk] P (~n∗i)

qF(~j−Bi,k)qF(~j −Bi,k)

= ρik,F(nik)NkqF(~j −Bi,k)−ρik,F(nik)·∑~n∈Ω~j−Bi,k

[n∗ik +ntk]P (~n∗i|~j−Bi,k)qF(~j−Bi,k)

= ρik,F(nik)NkqF(~j −Bi,k)− ρik,F(nik)·E(n∗ik + ntk|~j −Bi,k)qF(~j −Bi,k)

=ρik,F(nik) (Nk−E(n∗ik +ntk|~j−Bi,k))qF(~j−Bi,k)

(A-13)

where E(n∗ik +ntk|~j−Bi,k) is the expected value of n∗ik +ntkgiven ~j.

Likewise, the Left Hand Side (LHS) of (A-8) can beexpressed as:

LHS=∑n∈Ω~j

P (~n)nik=∑n∈Ω~j

P (~n)

qF(~j)qF(~j)nik=∑

n∈Ω~jnikP (~n|~j)qF(~j)=E(~nik|~j)qF(~j)

(A-14)

By combining (A-13) and (A-14) (i.e. LHS=RHS), we have:

E(P (~n|~j)qF(~j)) =

ρik,F(nik) (Nk−E(n∗ik +ntk|~j−Bi,k))q(~j−Bi,k)(A-15)

By multiplying both sides of (A-15) with bi,k,s and summingover k=1,...,K and i=1,2, we get:∑2

i=1

∑Kk=1E(~nikbi,k,s|~j)qF(~j)=

∑2i=1

∑Kk=1bi,k,s·

ρik,F(nik) (Nk−E(n∗ik +ntk|~j−Bi,k))q(~j−Bi,k)(A-16)

Based on (15), the LHS of (A-16) becomes:2∑i=1

K∑k=1

E(~nikbi,k,s|~j)qF(~j) = jsqF(~j) (A-17)

The only parameter that is unknown in (A-16) is the expectedvalue E(n∗ik + ntk|~j − Bi,k). Based on the analysis presentedin [33], this expected value is given by:

E(n∗ik + ntk|~j − Bi,k) = n∗ik − 1 + ntk (A-18)

By substituting (A-16) to (A-18), we have:

jsqF(~j) =∑2i=1

∑Kk=1 bi,k,s·

ρik,F(nik) (Nk − nik − ntk + 1)qF(~j −Bi,k)(A-19)

The utilization of the ith system by service-class k,ρik,F(nik), is actually a function of ~j = (j1, j2), i.e.ρik,F(nik) = ρik,F(~j), given that ρik,F(nik) is a function ofthe LBP (because of (A-4)) and the LBP is expressed as afunction of the variable x (because of (6)):

x=1−∑Kk=1(n1

kIactk

Imax· Pinterf)− Iact

k

Imax=

1−∑Kk=1(n1

kbkIunit

Imax· Pinterf)−bk Iunit

Imax=

1−j1 Iunit

Imax· Pinterf − bk Iunit

Imax

(A-20)

which denotes that x is also a function of ~j. Therefore, bysubstituting ρik,F(nik) with ρik,F(~j) in (A-19), we derive (11).

REFERENCES

[1] M. Conti, S. Chong, S. Fdida, W. Jia, H. Karl, Y.-D. Lin, P. Mahonen,M. Maier, R. Molva, S. Uhlig, M. Zukerman, “Research challengestowards the future Internet”, Computer Communications, vol. 34, no.18, Dec. 2011, pp. 2115-2134.

[2] C. Lam, Passive Optical Networks, Principles and Practice, AcademicPress, 2007.

[3] K. Grobe and J.-P. Elbers, “PON in adolescence: from TDMA toWDM-PON”, IEEE Commun. Mag., vol. 46, no. 1, Jan. 2008, pp.26-34.

[4] P. R. Prucnal, Optical Code Division Multiple Access: Fundamentalsand Applications, Taylor & Francis, 2006.

[5] C. C. Yang, “High speed and secure optical CDMA-based passiveoptical networks”, Comp. Netw., vol. 53, no. 12, Aug. 2009, pp. 2182-2191.

[6] J. A. Salehi, “Code division multiple-access techniques in optical fibernetwork-Part I: Fundamental principles”, IEEE Trans. Commun., vol.37, no. 8, Aug. 1989, pp. 824-833.

[7] K. Fouli and M. Maier, “OCDMA and Optical Coding- Principles,Applications, and Challenges”, IEEE Commun. Mag., vol. 45, no. 8,Aug. 2007, pp. 27-34.

[8] M. Azizoglu, J.A. Salehi, Y. Li, “Optical CDMA via temporal codes”,IEEE Trans. Commun., vol.40, no. 7, Jul. 1992, pp. 1162-1170.

[9] D. Zaccarin, M. Kavehrad, “An optical CDMA system based onspectral encoding of LED”, IEEE Photonics Technol. Let., vol. 5, no.4, Apr. 1993, pp. 479-482.

[10] S, Wan, S. He, Y. Hu, “Time-spreading wavelength-hopping 2D-OCDMA system with optical hard limiter”, Optik, 118 (2007), pp.381-384.

[11] W.C. Kwong and G.-C. Yang, “Multiple-Length extended Carrier-Hopping Prime Codes for Optical CDMA systems supporting MultirateMultimedia services”, IEEE/OSA J. Lightwave Technol., vol. 23, no.11, Nov. 2005, pp. 3653-3662.

SUBMITTED PAPER

[12] E. Intay, H. M. H. Shalaby, P. Fortier, and L. A. Rusch, “Multirateoptical fast frequency-hopping CDMA system using power control”,IEEE/OSA Journal of Lightwave Technology, vol. 20, no. 2, Feb. 2002,pp. 166-177.

[13] A.R. Forouzan, N-K. Masoumeh and N. Rezaee, “Frame Time-Hoppingpatterns in Multirate Optical CDMA Networks using Conventional andMulticode schemes”, IEEE Trans. Commun., vol. 53, no. 5, May 2005,pp. 863-875.

[14] V. Baby, W. Kwong, C. Chang, G. Yang, P. R. Prucnal, “Performanceanalysis of variable-weight multilength optical codes for wavelength-time O-CDMA multimedia systems”, IEEE Trans. Commun., vol. 55,no. 6, Jul. 2007, pp. 1325-1333.

[15] G.-C. Yang, “Variable-weight optical orthogonal codes for CDMA net-work with multiple performance requirements”, IEEE Trans. Commun.,vol. 44, no. 1, Jan. 1996, pp. 47-55.

[16] S. Goldberg and P. R. Prucnal, “On the Teletraffic Capacity of OpticalCDMA”, IEEE Trans. Commun., vol. 55, no. 7, Jul. 2007, pp. 1334-1343.

[17] M. Gharaei, C. Lepers, O. Affes, and P. Gallion, “Teletraffic Ca-pacity Performance of WDM/DS-OCDMA Passive Optical Network”,NEW2AN/ruSMART, LNCS 5764, Springer-Verlag Berlin Heidelberg,2009, pp. 132-142.

[18] E. Mutafungwa and S. J. Halme, “Analysis of the Blocking Perfor-mance of Hybrid OCDM-WDM Transport Networks”, Microwave andOptical Technol. Lett., vol. 34, no. 1, Jul. 2002, pp. 61-67.

[19] Y. Deng and P. R. Prucnal, “Performance Analysis of HeterogeneousOptical CDMA Networks with Bursty Traffic and Variable PowerControl”, IEEE/OSA J. Optical Commun. and Netw., vol. 4, no. 4,Jul. 2011, pp. 313-316.

[20] J. S. Vardakas, V. G. Vassilakis and M. D. Logothetis, “Call-LevelAnalysis of Hybrid OCDMA-WDM Passive Optical Networks”, inProc. of the 10th ICTON, Athens, Greece, 22-26 Jul. 2008.

[21] J. S. Vardakas, V. G. Vassilakis and M. D. Logothetis, “BlockingAnalysis for Priority Classes in hybrid WDM-OCDMA Passive OpticalNetworks”, in Proc. of the 5th AICT, Venice, Italy, May 24-28, 2009.

[22] V. G. Vassilakis, G. A. Kallos, I. D. Moscholios, and M. D. Logo-thetis, “The Wireless Engset Multi-Rate Loss Model for the Call-levelAnalysis of W-CDMA Networks”, in Proc. of the 8th IEEE PIMRC,Athens, Greece, Sep. 3-7, 2007.

[23] J. S. Vardakas, I. L. Anagnostopoulos, I. D. Moscholios, M. D.Logothetis and V. G. Stylianakis, “A Multi-Rate Loss Model folOCDMA PONs”, in Proc. of the 13th ICTON, Stockholm, Sweden,Jun. 26-30, 2011.

[24] J. S. Vardakas, I. D. Moscholios, M. D. Logothetis, and V. G.Stylianakis, “On code reservation in multi-rate OCDMA Passive Op-tical Networks”, in Proc. of the 8th IEEE/IET CSNDSP, Poznan,Poland, Jul. 18-20, 2012.

[25] C.-S. Weng, and J. Wu, “Optical Orthogonal Codes With NonidealCross Correlation”, IEEE/OSA J. Lightwave Technol., vol. 19, no. 12,Dec. 2001, pp. 1856-1863.

[26] M. Mehmet Ali, “Call-burst blocking and call admission control in abroadband network with bursty sources”, Performance Evaluation, vol.38, no. 1, Sep. 1999, pp. 1-19.

[27] W. Ma, C. Zuo, and J. Lin, “Performance Analysis on Phase-EncodedOCDMA Communication System”, IEEE/OSA J. Lightwave Technol.,vol. 20, no. 5, May 2002, pp. 798-803.

[28] X. Wang, and K. Kitayama, “Analysis of Beat Noise in Coherentand Incoherent Time-Spreading OCDMA, IEEE/OSA J. LightwaveTechnol., vol. 22, no. 19, Oct 2004, pp. 2226-2235.

[29] H.-W. Chen, G.-C. Yang, C.-Y. Chang, T.-C. Lin and W. C. Kwong,“Spectral Efficiency Study of Two Multirate Schemes for Asyn-chronous Optical CDMA”, IEEE/OSA J. Lightwave Technol., vol. 27,no.14, July 2009, pp.2771-2778.

[30] S. M. Ross, Introduction to Probability Models, Academic Press, 2007.[31] M. Glabowski and M. Stasiak, “An Approximate Model of the Full-

Availability Group with Multi-Rate Traffic and a Finite Source Popu-lation”, in Proc. of 12th MMB&PGTS, Dresden, Germany, Sep. 12-15,2004.

[32] E. Russell, Building Simulation Models with SIMSCRIPT II.5, CACI,Inc., Los Angeles, CA, 1983.

[33] I.Moscholios, M. Logothetis and M. N. Koukias, “An ON-OFF Multi-Rate Loss Model of Finite Sources”, IEICE Trans. Commun., vol. E90-B, no. 7, Jul. 2007, pp. 1608-1619.

John S. Vardakas received the Dipl.-Eng. in Elec-trical & Computer Engineering from the DemocritusUniversity of Thrace, Greece, in 2004 and his Ph.Dfrom the Electrical & Computer Engineering Dept.,University of Patras, Greece. His research interestsinclude teletraffic engineering, performance analysisand simulation of communication networks and es-pecially of optical networks. He is a member of theIEEE, the Optical Society of America (OSA) andthe Technical Chamber of Greece (TEE).

Ioannis D. Moscholios received the Dipl.-Eng. inElectrical & Computer Engineering (ECE), Univer-sity of Patras, Greece, (1999), the M.Sc. degree inSpacecraft Technology & Satellite Communicationsfrom the University College London, UK, (2000),and the Ph.D. in ECE, University of Patras (2005).He is an Assistant Professor in the Dept. of Telecom-munications Science and Technology, University ofPeloponnese, Greece. His research interests includeteletraffic theory, simulation and performance anal-ysis of communication networks. He has published

over 75 papers. He is a member of the Technical Chamber of Greece (TEE).

Michael D. Logothetis received his Dipl.-Eng.(1981) and Ph.D (1990) in Electrical Engineering,both from the University of Patras, Greece. From1991 to 1992 he was Research Associate in NTT’ sTelecommunication Networks Laboratories, Tokyo,Japan. He is a Professor in the Department of Elec-trical & Computer Engineering, University of Patras,Greece. His research interests include teletraffic the-ory and engineering, simulation, and performanceoptimization of communication networks. He haspublished over 150 papers and has over 510 third-

party citations. He has published a teletraffic book in Greek. He is servingon the Technical Program Committee of international conferences, while heorganizes and chairs several technical sessions. He is a member of the SteeringCommittee of CSNDSP. He has become a Guest Editor and participates in theEditorial Board of international journals. He is a member of the IEEE (Senior),IEICE, IARIA (Fellow), FITCE and the Technical Chamber of Greece (TEE).

Vassilios G. Stylianakis received his Dipl.-Eng.(1981) and Ph.D (1990) in Electrical Engineering,both from the University of Patras, Greece. He isan Assistant Professor in the Electrical & ComputerEngineering Department of the University of Patras.He teaches digital communications. His research in-terests include techno-economical design of telecom-munications systems, access networks design andtechnology diffusion models. He is a member of theTechnical Chamber of Greece (TEE).