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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 17897–17912
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Impact of the MAI and beat noise on the performance of OCDM/WDM Optical Packet Switches using Gold codes

V. Eramo  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 17897-17912 (2010)
http://dx.doi.org/10.1364/OE.18.017897


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Abstract

Recent advances in optical devices greatly enhance the feasibility of Optical Code Division Multiplexing/Wavelength Division Multiplexing (OCDM/WDM) Optical Packet Switch. In this paper, the performance of an OCDM/WDM switch is investigated when impairment due to both Multiple Access Interference and Beat noise are taken into account. Analytical models are proposed to dimension the switch resources as the number of optical codes carried on each wavelength and the number of needed optical converters. The Packet Loss Probability due to output packet contentions is evaluated as a function of the main switch and traffic parameters when Gold coherent optical codes are adopted. When the available bandwidth is fixed for the WDM/OCDM signal, due to a statistical multiplexing effect, we show that the use of more length codes and fewer wavelengths lead to lower packet loss probability, especially for low offered traffic.

© 2010 Optical Society of America

1. Introduction

Implementing the multiplexing in the optical layer, OCDM can further enhance bandwidth granularity of the WDM transport network and gives rise to the OCDM/WDM hybrid network. It is shown that, the spectrum efficiency of the OCDM/WDM could be twice as much of that with WDM only [18

18. K. Kitayama, “Code Division Multiplexing Lightwave Networks Based upon Optical Code Conversion,” IEEE J. Sel. Areas Comm. 16, 1309–1319 (2000). [CrossRef]

]. Bandwidth on one wavelength can be divided into small fractions labeled by optical codes, and assigned to different connections. In this way, provisioning fractional wavelength capacity is achieved by the marriage of OCDM and WDM. Different from the wavelength routing in a conventional WDM network, there are two dimensions of optical code and wavelength used for optical routing. It is called code/wavelength routing network [19–22

19. H. Sotobayashi, W. Chujo, and K. Kitayama, “Transparent Virtual Code/Wavelength Path Network,” IEEE J. Sel. Top. Quantum Electron. 8, 699–704 (2002). [CrossRef]

].

In this paper we evaluate the performance of a bufferless OCDM/WDM Optical Packet Switch equipped with shared Wavelength Converters and in which the output packet contentions are solved in both code and wavelength domain. Two analytical models are proposed to dimension the switch resources. The first one allows us to dimension the number of optical codes supported on each wavelength so that the Packet Loss Probability due to both Multiple Access Interference (MAI) and beat noise is maintained under a threshold value. The second one allows us to dimension the number of Wavelength Converters so that the minimum packet loss probability due to output packet contentions is reached. By means of the proposed analytical models, the optimal parameters, in particular the number of wavelengths and the code length, are evaluated so that the best performance in terms of packet loss probability are reached for a fixed bandwidth. The organization of the paper is as follows. In Section 2 we describe the OCDM/WDM switch in question. In Section 3 we discuss the control algorithm used to schedule the packets and to decide which packets must be code and wavelength translated. The analytical models needed to evaluate the performance of the OCDM/WDM switch and to dimension the switch resources are introduced in Section 4. In Section 5 we discuss the effectiveness of the OCDM/WDM technique in solving output packet contentions. Our main conclusions and further research topics are discussed in Section 6.

2. WDM/OCDM Optical Packet Switch Architecture

We consider the WDM/OCDM optical packet switch with N Input/Output fibers (IF/OF) reported in Fig. 1. Each fiber supports M wavelengths denoted λ 1, …,λM. On each wavelength up to F packets are carried out by using Optical Code Division Multiplexing. Let L be the code length and {OCk,k = 1, …,F} the set of Optical Codes (OC) of a wavelength on which the packets can be carried out. An input (or output) channel is identified by the triple (i,λj,OCk), where i (i = 1, …,N) identifies one of the input (or output) fibers, λj (j = 1, …,M) identifies one of the wavelengths on that IF (OF) and OCk (k = 1, …,F) identifies one of the optical codes on that wavelength.

The operation mode of the architecture is synchronous, meaning that all arriving packets have a fixed size and their arrival on each input channel is synchronized on a time-slot basis [23

23. P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krhenbhl, C. Raffaelli, M. Schilling, A. Talneau, and L. Zucchelli, “Transparent Optical Packet Switching: Network Architecture and Demonstrators in the KEOPS project,” IEEE J. Sel. Areas Comm. 16, 1245–1259 (1998). [CrossRef]

]. The synchronization operation, not shown in Fig. 1, is realized by means of synchronizers located at the ingress of the switch.

Fig. 1. WDM/OCDM Optical Packet Switch.

The WDM/OCDM optical packet switch illustrated in Fig. 1 performs the following operations: (i) the packets are wavelength demultiplexed and code decoded by means of one WDM demultiplexer and M · F OC decoders for each IF; (ii) the unit control, not shown in Fig. 1, processes the packet headers and according to the rules of the scheduling algorithm illustrated in Section 3, decides which packets have to be wavelength converted and which Output Wavelength Channels (OWC) and Output Optical Codes (OOC) are assigned to the packets to transmit; (iii) the Switching Fabric SFc routes the packets towards either the bank of r Wavelength Converters (WC) or the Output Switching Fabrics SFi (i = 1, …,N) according to decisions taken by the control unit; (iv) the converted packets are routed towards the SFi (i = 1, …,N) where in output the packets are code and wavelength multiplexed by means of M · F OC coders and one WDM multiplexer.

One of the advantages of the proposed OPS is to reduce the number of WCs used because the control algorithm, as illustrated in Section 3, first tries to solve the contention in code domain by changing the packet code and only if this operation is unsuccessful, the contention is solved in wavelength domain by using one WC of the shared pool. This strategy, preferring the use of passive devices like OC decoders and coders rather than active elements like WCs, allows for the reduction of the switch cost.

3. Control Algorithm in WDM/OCDM Optical Packet Switches

A flow chart of the Scheduling Algorithm (SA) executed in each time-slot for the WDM/OCDM Optical Packet Switch (OPS) is illustrated in Figs. 2, 3. The SA starts with the Initialization phase in which some sets and variables are initialized:

  • Γ ≡ {1, …,N}: the set of Output Fibers.
  • Λi, j (i = 1, …,N; j = 1, …,M): the set containing the free output channels on i − th OF carried out on the same wavelength λj. These channels are coded on F different OCs. The set is initialized to {(i,λj,OCk) k = 1, …,F} and it is updated during the execution of the SA when packets are scheduled for i − th OF and wavelength λj.
  • Ii, j (i = 1, …,N; j = 1, …,M): the set containing the packets arriving on wavelength λj, which are directed to i − th OF and are yet to be scheduled. The Ii, j (i = 1, …,N; j = 1, …,M) sets initially contain packets arriving on wavelengths λj and directed to i − th OF. They are updated during the execution of the SA when the packets are scheduled to be directed on the output wavelength channels.
  • ra: the available number of wavelength converters during the execution of the SA; it is initialized to r and decremented by 1 each time one of the converters is used.

After the Initialization phase, the control unit performs the packet scheduling operations. To allow for a fair assignment of the Wavelength Converters to the various OFs, these ones are randomly selected and for each of them the Code Conversion (CC) andWavelength Conversion (WC) phases are performed. In CC phase, illustrated in Fig. 2, the packets that can be directed without wavelength conversion are scheduled and the contentions are resolved by changing the code. In WC phase, described in Fig. 3, the packet contentions are resolved in the wavelength domain by using the WCs.

In CC phase, for each wavelength λj (j = 1, …,M), the control unit randomly schedules up to F packets, chosen among the packets arriving on wavelength λj, to be transmitted without wavelength conversion. The set Λi, j initially contains all the output channels on wavelength λj and optical codes OCk (k = 1, …,F) which are used to forward without wavelength conversion up to F packets belonging to Ii, j set. The following actions are performed by the control unit for each wavelength λj: (i) one packet a and one output channel (i,λj,OCk) are randomly selected from sets Ii, j and Λi, j respectively; (ii) the packet is scheduled to be forwarded on output channel (i,λj,OCk); (iii) the sets Ii, j and Λi, j are updated by removing the elements a and (i,λj,OCk) respectively. The CC phase complexity is O(F · M) because in the worst case at the least one operation for each output channel is performed.

In WC phase the control unit tries to forward with wavelength conversions the remaining packets on output channels not used in CC phase. The packets not scheduled and the free output channels for i − th OF, are stored in Ii and Λi respectively. The following actions are performed by the control unit until either Ii or Λi (or both) are empty and one WC is available: (i) one packet a and one output channel (i,λj,OCk) are randomly selected from sets Ii and Λi respectively; (ii) the packet is scheduled to be forwarded on output channel (i,λj,OCk); (iii) the sets Ii and Λi are updated by removing the elements a and (i,λj,OCk) respectively; iv) the available number ra of WCs is decremented by one. Before the end of WC phase, the control unit checks if there are packets in Ii which are yet to be scheduled and in this case they are discarded. The WC phase complexity is O(min(r,M · F)) because in the worst case at the least one operation is performed for each output channel until WCs are available.

Because the operations in CC and WC phases are performed for each OF, the computational complexity of the proposed scheduling algorithm equals O(N · F · M).

4. Resource dimensioning in WDM/OCDM Optical Packet Switches

We illustrate the analytical models to dimension the resources in WDM/OCDM Optical Packet Switch. Two analytical models are proposed: the first one allows the dimensioning of the number F of codes so that the Packet Loss Probability Pnoiseloss due to both Multiple Access Interference (MAI) and beat noise on each wavelength is limited to a given threshold value; the second one allows the dimensioning r of the WCs so that the Packet Loss Probability Popcloss due to packet contentions is minimized. We will evaluate Pnoiseloss and Popcloss in Sections 4.1 and 4.2 respectively as a function of the following traffic and switch parameters: (i) N is the number of IF/OF; (ii) M is the number of wavelengths; (iii) F is the number of codes carried out on each wavelength; (iv) p is the offered traffic offered to each input channel; (v) L is the code length; (vi) H is the packet length.

Fig. 2. Scheduling Algorithm in WDM/OCDM Optical Packet Switch; Initialization and Conversion Code phases are described.

4.1. Evaluation of the Packet Loss Probability due to MAI and beat noise

The evaluation of the Packet Loss Probability Pnoiseloss for a target user will be carried out under the assumptions that coherent OCDM techniques are considered and the coding is based upon optical amplitude, where each chip in a code sequence can have a phase 0 or π with a Binary-Phase-Shift-Keying (BPSK) scheme [3

3. G. Manzacca, A. M. Vegni, X. Wang, N. Wada, G. Cincotti, and K. Kitayama, “Performance Analysis of a Multiport Encoder/Decoder in OCDMA Scenario,” IEEE J. Sel. Top. Quantum Electron. 13, 1415–1421 (2007). [CrossRef]

, 9

9. S. Goldberg and P. R. Prucnal, “On the Teletraffic Capacity of Optical CDMA,” IEEE Trans. Commun. 55, 1334–1343 (2007). [CrossRef]

, 10

10. J. A. Salehi, “Code Division Multiple Access Techniques in Optical Fiber Networks I: Fundamental Principles,” IEEE Trans. Commun. 37, 824–833 (1989). [CrossRef]

, 21

21. S. Huang, K. Baba, M. Murata, and K. Kitayama, “Architecture Design and Performance Evaluation of Multi-granularity Optical Networks Based on Optical Code Division Multiplexing,” J. Opt. Netw. 5, 1028–1042 (2006). [CrossRef]

]. Gold Codes are assumed that leads to a maximum number of codes equal to L+2 [24

24. D. L. Sarwate and M. B. Pursley, “Crosscorrelation Properties of Pseudorandom and Related Sequences,” Proc. IEEE 68, 593–619 (1980). [CrossRef]

, 25

25. S. Tamura, S. Nakano, and K. Okazaki, “Optical Code-Multiplex Transmission by Gold Sequences,” J. Lightwave Technol. LT-3, 121–127 (1985). [CrossRef]

]. In evaluating Pnoiseloss we take into account both MAI and beat noise modeled as gaussian processes [26

26. T. Pu, H. Zhang, Y. Guo, and Y. Li, “Evaluation of Beat Noise in OCDMA System with Non-Gaussian Approximated Method,” J. Lightwave Technol. 24, 3574–3582 (2006). [CrossRef]

].

Due to the synchronous operation mode of the switch, we can express Pnoiseloss by conditioning to the number of packets arriving on any input wavelength. We can write:

Fig. 3. Scheduling Algorithm in WDM/OCDM Optical Packet Switch; Wavelength Conversion phase is described.
Plossnoise=i=0FPlossnoise,i(Fi)pi(1p)Fi.
(1)

wherein Pnoise,iloss is the Packet Loss Probability due to MAI and beat noise when there are i packets arriving on any input wavelength. If we assume that an error occurring in a single bit of the packet due to MAI and beat noise results in whole packet error, the following expression can be given for Pnoise,iloss:

Plossnoise,i={0i=0,11(1Pbitnoise,i)Hi=2,,F
(2)

where Pnoise,ibit is the probability that a bit is corrupted from MAI and beat noise when i packets are arriving on any input wavelength. If we assume that interfering users transmit bit ‘0’ and ‘1’ with equal probability 12 , the following expression holds for Pnoise,ibit:

Pbitnoise,i={0i=0,1(12)i1h=0i1(i1h)Pbit,hnoise,i,targeti=2,,F
(3)

where Pnoise,i,′target′bit,h is the probability that a bit transmitted by a target user is corrupted for MAI and beat noise when exactly h interfering users transmit bit ‘1’.

Pnoise,i,′target′bit,h is evaluated in Appendix A and its expression is given by:

Pbit,hnoise,i,target=14erfc(thimZ0,h2σZ0,h)+14erfc(mZ1,hthi2σZ1,h).
(4)

wherein:

  • mx and σx denote the average value and standard deviation of the random variable x respectively;
  • Z 0,h and Z 1,h are the random variables denoting the output value from the integrator of the receiver when the ‘target’ user transmits bit ‘0’ and ‘1’ respectively;
  • thi is the receiver threshold when i packets are transmitted on any wavelength.

The following expressions for mZ0,h , mZ1,h , σZ0,h , σZ1,h , thi have been evaluated in Appendix A:

mZ0,h=12hTc(L+11L)MAI noise.
(5)
mZ1,h=12TcL2Signal+12hTc(L+11L)MAI noise.
(6)
σZ0,h=Tc14h(L2+2L2L1L2)MAI noise+14h(h1)(L+11L)2Secondary Beat noise.
(7)
σZ1,h=Tc14h(L2+2L2L1L2)MAI noise+12hL2(L+11L)Primary Beat noise+14h(h1)(L+11L)2Secondary Beat noise.
(8)
thi=σZ0,12(i1)mZ1,12(i1)+σZ1,12(i1)mZ0,12(i1)σZ0,12(i1)+σZ1,12(i1).
(9)

ℜ and Tc being the detector responsitivity and the chip duration respectively. The dependence on signal, MAI noise, Primary Beat noise and Secondary Beat noise [29

29. X. Wang and K. Kitayama, “Analysis of Beat Noise in Coherent and incoherent Time-Spreading OCDMA,” J. Lightwave Technol. 22, 2226–2234 (2004). [CrossRef]

] have been indicated in Eqs. (5)–(8). Finally by inserting Eqs. (5)–(9) in Eq. (4) and by using Eqs. (1)–(3) we can evaluate Pnoiseloss.

4.2. Evaluation of the Packet Loss Probability due to packet contentions

The evaluation of the Packet Loss Probability Popcloss due to packet contentions will be evaluated under the assumption that packet arrivals on the N · M · F input wavelength channels are independent. We also assume uniform and symmetric traffic that is packet arrivals on each input channel occur with probability p and a packet has the same probability 1N to be directed to any given OF. The uniform and symmetric traffic scenario is assumed because it is the one requiring the highest number of WCs and thus the highest conversion cost [27

27. V. Eramo, “An Analytical Model for TOWC Dimensioning in a Multifiber Optical-Packet Switch,” J. Lightwave Technol. 24, 4799–4810 (2006). [CrossRef]

]. The explanation of this is illustrated in the following: (i) the packet loss is either due to the lack of Output Channels or to the lack of WCs and WCs dimensioning must be performed so that the second loss event is negligible with respect to the first one; (ii) the uniform and symmetric traffic assumption involves the lowest packet loss due to lack of Output Channels and then according to the point (i), the highest number of WCs.

Due to the uniform and symmetric traffic assumption, the Packet Loss Probability Popcloss may be evaluated by considering the packet loss occurring at any OF in the following denoted target. In particular by conditioning to the number i (i = 0,1, …, r) of WCs still available when the OF target is selected according to the scheduling algorithm reported in Figs. 2, 3, we can write:

Plossopc=i=0rPlossopc,OF,ipQ(i).
(10)

wherein:

  • Popc,OF,iloss (i = 0,1, …, r) is the Packet Loss Probability of the target OF if i WCs are still available when the OF target is selected;
  • pQ(i) (i = 0,1, …, r) is the probability mass function (p.m.f) of Q, which is the random variable denoting the number of WCs that are still available for the target OF when it is selected.

The probabilities Popc,OF,iloss (i=0,1, …, r) may be computed according to a procedure illustrated in [27

27. V. Eramo, “An Analytical Model for TOWC Dimensioning in a Multifiber Optical-Packet Switch,” J. Lightwave Technol. 24, 4799–4810 (2006). [CrossRef]

] where these probabilities are evaluated for a Multi-Fiber Optical Packet Switch using both wavelength and space domain to solve output packet contentions.

The probabilities pQ(i) (i = 0,1, …, r) can be evaluated by taking into account that: (i) the control algorithm illustrated in Section 3, selects the OFs in a random way and hence the probability that the target OF is the j − th OL (j = 1, …,N) selected is equal to 1N ; (ii) when the OF target is the j − th OF selected, i (i = 1, …, r) WCs are available for the target OF if the first j − 1 OFs selected need (r − i) WCs; conversely WCs are not available for the OF target if the first j − 1 OFs selected need a number of WCs greater than r. According to the following remarks and denoting with Wj and pWj(i) the number of conversions required by the first j selected OFs and the Wj’s p.m.f. respectively, we can write:

pQ(i)={1Nj=1NpWj1(ri)i01Nj=1Nh=rj(M1)pWj1(h)i=0
(11)

The probabilities pWj(i) are evaluated by assuming the number of conversions required in the various OFs statistically independent. Notice that in reality this independence assumption does not hold in fact: (i) the number of conversions required in an OF depends on the number of packets directed to the OF; (ii) the number of packets arriving on a specific wavelength and directed to the various OFs are statistically dependent and negatively correlated [28

28. V. Eramo and M. Listanti, “Packet Loss in a Bufferless WDM Switch Employing Shared Tuneable Wavelength Converters,” J. Lightwave Technol. 18, 1818–1833 (2000). [CrossRef]

] in fact their sum is not larger than N · F and every additional packet destined to a specific OF reduces the likelihood of packets directed to an other OF. Although this dependence, it is shown in [28

28. V. Eramo and M. Listanti, “Packet Loss in a Bufferless WDM Switch Employing Shared Tuneable Wavelength Converters,” J. Lightwave Technol. 18, 1818–1833 (2000). [CrossRef]

] that it is very slight and can be neglected especially when the traffic offered to the switch is low. According to the observations above and denoting with ⊗ the convolution operator, we can express pWj(i) for j = 1, …,N − 1 as follows:

pWj(i)={δ(i)j=0pW(i)……pW(i)(j1)timesj0
(12)

wherein:

δ(i)={1ifi=00ifi0
(13)

W and pW(i) (i = 0, …,F · M) are the number of conversions required by any OF and the W’s p.m.f. respectively. The probabilities pW(i) are evaluated in [27

27. V. Eramo, “An Analytical Model for TOWC Dimensioning in a Multifiber Optical-Packet Switch,” J. Lightwave Technol. 24, 4799–4810 (2006). [CrossRef]

].

In conclusions, by means of Eqs. (10)–(13), we are able to evaluate the Packet Loss Probability Popcloss due to packet contentions.

5. Numerical Results

This section will be devoted to cope with the following issues: (i) sensitivity analysis of the resource dimensioning of an OCDM/WDM switch; (ii) evaluation of optimal parameters for the design of OCDM/WDM switches so that the best performance in terms of Packet Loss Probability Popcloss is obtained.

The Packet Loss Probability Pnoiseloss due to MAI, Primary Beat Noise (PBN) and Secondary Beat Noise (SBN) [29

29. X. Wang and K. Kitayama, “Analysis of Beat Noise in Coherent and incoherent Time-Spreading OCDMA,” J. Lightwave Technol. 22, 2226–2234 (2004). [CrossRef]

] is reported in Figs. 4–6 versus the number F of OCs supported on each wavelength. The offered traffic p is varied from 0.2 to 1 and the packet length H equals 500 bytes. The code length L is chosen to be 511, 1023, 2047 in Figs. 4–6 respectively. To evaluate Pnoiseloss, we have used the model introduced in Section 4.1.

Fig. 4. Packet Loss Probability due to MAI and beat noise as a function of the number F of Optical Codes. The code length is L=511. The traffic parameters are H=500 bytes and p varying from 0.2 to 1.

The dimensioning F of used Optical Codes is reported in Fig. 7 for L=1023 and 2047. In particular the greatest value F is determined so that Pnoiseloss is lower than or equal to the threshold Packet Loss Probability Pnoise,thloss = 10−9. These values have been referred to as F 1023 and F 2047 for the cases L=1023 and L=2047 respectively.

Fig. 5. Packet Loss Probability due to MAI and beat noise as a function of the number F of Optical Codes. The code length is L=1023. The traffic parameters are H=500 bytes and p varying from 0.2 to 1.

In Fig. 7 we also report the ratio F 2047/F 1023 of the number of OCs for L=2047 to the number of OCs for L=1023. Notice as the ratio F 2047/F 1023 is always greater than two, that is F 2047 > 2 · F 1023. The relation follows from the advantages of the statistical multiplexing according to which if a x Hz bandwidth is available, a greater total number of users can be supported when optical codes with length L=2047 are used on the entire band with respect to the case in which the available bandwidth is divided up in two x/2 Hz width parts and shorter optical codes with length L=1023 are used on each sub-band. Starting from the value 1 and by decreasing the offered traffic p, the statistical multiplexing gain is greater and greater and for this reason the ratio F 2047/F 1023 increases.

The Packet Loss Probability Popcloss due to output packet contentions is reported in Fig. 8 as a function of the number r of used WCs for H=500 bytes, N=8 and p=0.6,0.7,0.8. Two case studies are considered with the code length L equal to 1023 and 2047. As earlier mentioned, the increase in code length L from 1023 to 2047 leads to double the bandwidth allocated to each OCDM signal in order to overcome ISI problems. Because we perform the comparison of the OCDM/WDM switch when the same bandwidth is allocated to the entire WDM/OCDM signal, the increase of L from 1023 to 2047 leads to halve the number M of wavelengths from 16 to 8.

Fig. 6. Packet Loss Probability due to MAI and beat noise as a function of the number F of Optical Codes. The code length is L=2047. The traffic parameters are H=500 bytes and p varying from 0.2 to 1.

The number F of used OCs for each wavelength is chosen so that the threshold Packet Loss Probability Pnoise,thloss due to MAI and beat noise is fixed equal to 10−9. For instance that leads for p=0.7 to the use of a number F of OCs equal to 6 and 15 for L=1023, 2047 respectively. From Fig. 8, we can notice that all of the sketched curves have the same trend, decreasing as r increases up to a threshold value rth for which the packet loss probability saturates: this saturation value for the probability denotes the packet loss probability of an OCDM/WDM switch using a full set of WCs, and therefore represents the Packet Loss probability due to lack of output channels. The rth value denotes the lowest number of WCs needed in an OCDM/WDM switch to reach the same Packet Loss Probability of an OCDM/WDM switch using a full set of WCs. The WC dimensioning is less severe as the number M of wavelengths decreases. For instance when p=0.6, rth equals 30 and 46 when M equals 8 and 16 respectively. This is to be expected, as the lower M is, the higher F and hence the probability that a contending packet can be forwarded on the same arriving wavelength by changing the optical code and without the use of a WC.

We report in Table 1 the values of Popcloss for r=8, 78 that is in the case studies in which few and many WCs are used. For each case study we report Popcloss for p=0.6, 0.7, 0.8, and (M,L)={(16,1023),(8,2047)}. From Table 1 we can notice that when few WCs are employed, the lowest Popcloss is reached when fewer wavelengths are used that is in the case (M,F)=(8,2047). For instance in the case p=0.7, Popcloss = 2.75 · 10−3 when (M,F)=(8,2047). The result is not amazing, because when few WCs are used, the packet loss is smaller if fewer wavelength conversions are needed. That occurs when fewer wavelengths and a greater number F of OCs is used. Even when many WCs are used, the lowest Popcloss is reached when fewer wavelengths are used. In this case Popcloss has already reached the saturation value that depends on the total number M · F of available output channels. Popcloss is as much smaller as M · F increases. As previously mentioned, for L increasing, the statistical multiplexing gain leads to higher M · F and consequently smaller values of Popcloss are obtained as shown in Table 1. For example for p=0.7, Popcloss equals 1.28 · 10−6 for (M,F)=(8,2047).

Fig. 7. Dimensioning of the number F of optical codes versus p for H=500 bytes, Pnoise,thloss = 10 −9 and L=1023, 2047. The ratio of the number of optical codes for L=1023 to the number of optical codes for L=2047 is also reported.

Table 1. Packet Loss Probability Popcloss due to output packets contentions versus the number M of wavelengths and the code length L for r=8, 78, N=8, H=500 bytes and Pnoise,thloss=10−9, p=0.6, 0.7, 0.8

table-icon
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Fig. 8. Packet Loss Probability Popcloss due to output packet contentions as a function of the number r of used WCs for p=0.6, 0.7, 0.8, N=8, H=500 bytes and (M,L)={(8,2047),(16,1023)}. The threshold Packet Loss Probability Pnoise,thloss due to MAI and beat noise is 10−9.

6. Conclusion

In this paper, the performance in Packet Loss Probability due to output packet contentions of an OCDM/WDM switch is investigated. The blocking performance arising from the limited number of output channels and the number of converters is analyzed through uniform traffic models. The proposed analytical models allowed us to dimension the switch resources and to suitably choose the number of wavelengths and the code length so that the best performances in terms of packet loss probability are reached for a fixed bandwidth of the WDM/OCDM signal. The numerical results show that due to a statistical multiplexing effect, better performance is obtained when fewer wavelengths and more length code are used. Due to the statistical multiplexing, the effect is especially evident when low traffic is offered to the OCDM/WDM Optical Packet Switch.

Appendix A: Evaluation of Pnoise,i,′target′bit,h (i = 2 …,F;h = 1, …, i − 1)

The receiver model of a target user is reported in Fig. 9 [26

26. T. Pu, H. Zhang, Y. Guo, and Y. Li, “Evaluation of Beat Noise in OCDMA System with Non-Gaussian Approximated Method,” J. Lightwave Technol. 24, 3574–3582 (2006). [CrossRef]

, 29

29. X. Wang and K. Kitayama, “Analysis of Beat Noise in Coherent and incoherent Time-Spreading OCDMA,” J. Lightwave Technol. 22, 2226–2234 (2004). [CrossRef]

]. Three different kinds of noise sources should be taken into account: MAI noise arising from the network, beat noise at the detector and electrical receiver noise (thermal and shot noise). The bandwidth of the receiver is limited to the chip rate and thus is equivalent to an integration over one-chip interval and a thresholder. In this paper, we will focus on beat noise (both Primary and Secondary Beat) and MAI, which are the two most important performance limitations. Other receiver noises such as shot and thermal noise will be discussed in succeeding paper. In this case, if chip-rate squarelow photodetector is used, the output signal Za,h from the integrator when the ‘target’ user transmits bit a (a = 0,1) and exactly h interfering users transmit bit 1 is given by the following expression [26

26. T. Pu, H. Zhang, Y. Guo, and Y. Li, “Evaluation of Beat Noise in OCDMA System with Non-Gaussian Approximated Method,” J. Lightwave Technol. 24, 3574–3582 (2006). [CrossRef]

, 29

29. X. Wang and K. Kitayama, “Analysis of Beat Noise in Coherent and incoherent Time-Spreading OCDMA,” J. Lightwave Technol. 22, 2226–2234 (2004). [CrossRef]

]:

Fig. 9. OCDM Receiver model and noise sources.
Za,h=12aTcSac2signal+12Tcj=1hScc,j2MAI noise+aTcj=1hSacScc,jcosΔφd,jPrimary Beat noise+Tcj=1h1u=j+1hScc,jScc,ucosΔφj,u.Secondary Beat noise
(14)

wherein:

  • ℜ is the responsitivity of the detector;
  • Tc is the chip duration;
  • a is the variable that assumes value 1 if target user transmits bit ‘1’, otherwise assumes value 0;
  • Sac is the autocorrelation of the sequence associated to the target user; if Gold codes are used we have Sac = L [24

    24. D. L. Sarwate and M. B. Pursley, “Crosscorrelation Properties of Pseudorandom and Related Sequences,” Proc. IEEE 68, 593–619 (1980). [CrossRef]

    ]:
  • Scc, j is the cross-correlation between the sequences associated to the target user and the j − th interfering user; as it will be later more clear in order to evaluate Pnoise,i,′target′bit,h, we need to calculate the Scc,j ’s average quadratic value mScc,j2 and the S 2 cc, j ’s variance σScc,j22 ; if Gold codes are used, these terms can be evaluated by taking into account that the random variables Scc, j (j = 1, …,h) are identically distributed with probabilities [24

    24. D. L. Sarwate and M. B. Pursley, “Crosscorrelation Properties of Pseudorandom and Related Sequences,” Proc. IEEE 68, 593–619 (1980). [CrossRef]

    ]:
    Scc={1+twith probabilityαL1with probability1twith probabilityγLβL
    (15)

    with n = log 2(L+1), t=2n+12 , α=2n2+2n32 , β = 2n − 2n-1 -1, γ=2n2+2n32 . According to Eq. (15) after some algebra we obtain the following expressions for mScc2 and σScc22 :
    mScc2=L+11LσScc22=L2+2L2L1L2.
    (16)

  • φd, j = φdφj, φj,u = φjφu with φd and φs (s = 1, …,h) being the carrier phase of the target and s − th interfering user respectively. In coherent systems we can assume the random variables φd,j and φj,u as uniformly distributed during [−π,π] [26

    26. T. Pu, H. Zhang, Y. Guo, and Y. Li, “Evaluation of Beat Noise in OCDMA System with Non-Gaussian Approximated Method,” J. Lightwave Technol. 24, 3574–3582 (2006). [CrossRef]

    ].

The probabilities Pnoise,i,′target′bit,h (i = 2 …, F;h = 1, …, i − 1) are evaluated under the following assumptions:

  • the ‘target’ user transmits bit ‘0’ and ‘1’ with equal probability 12 ;
  • the interfering MAI and beat noises are modeled as Gaussian noise [26

    26. T. Pu, H. Zhang, Y. Guo, and Y. Li, “Evaluation of Beat Noise in OCDMA System with Non-Gaussian Approximated Method,” J. Lightwave Technol. 24, 3574–3582 (2006). [CrossRef]

    , 29

    29. X. Wang and K. Kitayama, “Analysis of Beat Noise in Coherent and incoherent Time-Spreading OCDMA,” J. Lightwave Technol. 22, 2226–2234 (2004). [CrossRef]

    ].

According to the above assumptions we can write:

Pbit,hnoise,i,target=14erfc(thimZ0,h2σZ0,h)+14erfc(mZ1,hthi2σZ1,h).
(17)

The average values Za,h (a=0,1) are obtained from Eq. (14)

mZa,h=12aTcL2+12Tcj=1hmScc,j2+aTcLj=1hmScc,jmcosφd,j+Tcj=1h1u=j+1hmScc,jmScc,umcosφj,u.
(18)

It is possible to prove that the components in Eq. (14) are uncorrelated. For this reason we can write the following expression for σZa,h2 (a=0,1):

σZa,h2=14h2Tc2σScc22+a2Tc2L2j=1hmScc,j2mcosφd,j2+2Tc2j=1h1u=j+1hmScc,j2mScc,u2mcosφj,u2.
(19)

By substituting Eq. (16) and Sac = L in Eqs. (18)–(19) and by taking into account that the random variables φd,j and φj,u are uniformly distributed during [−π,π], we obtain for mZa,h , σZa,h2 (a=0,1):

mZa,h=12aTcL2+12hTc(L+11L).
(20)
σZa,h2=2Tc2(14h(L2+2L2L1L2)+12ahL2(L+11L)+14h(h1)(L+11L)2).
(21)

The variable thi appearing in Eq. (17) is the receiver threshold. In the case of gaussian noise the expression of the optimum threshold has been evaluated [26

26. T. Pu, H. Zhang, Y. Guo, and Y. Li, “Evaluation of Beat Noise in OCDMA System with Non-Gaussian Approximated Method,” J. Lightwave Technol. 24, 3574–3582 (2006). [CrossRef]

] and it needs the knowledge of the number h of active interfering users to the receiver. That is not the case. For this reason by assuming that the number i of scheduled packets is known we set as threshold thi the one obtained when the average number 12(i1) of active interfering users are considered. The following expression holds:

thi=σZ0,12(i1)mZ1,12(i1)+σZ1,12(i1)mZ0,12(i1)σZ0,12(i1)+σZ1,12(i1).
(22)

Finally by inserting Eqs. (20)–(22) in Eq. (17) we can evaluate Pnoise,i,′target′bit,h.

Acknowledgments

The work described in this paper was carried out with the support of the BONE-project (“Building the Future Optical Network in Europe”), a Network of Excellence funded by the European Commission through the 7th ICT-Framework Programme.

References and links

1.

R. Ramaswami and K. N. Sivarjan, Optical Networks (Morgan Kaufmann, New York, 1998).

2.

B. Mukherjee, Optical Communication Networks (Mc Graw-Hill, New York, 1997).

3.

G. Manzacca, A. M. Vegni, X. Wang, N. Wada, G. Cincotti, and K. Kitayama, “Performance Analysis of a Multiport Encoder/Decoder in OCDMA Scenario,” IEEE J. Sel. Top. Quantum Electron. 13, 1415–1421 (2007). [CrossRef]

4.

X. Wang, N. Wada, T. Miyazaki, G. Cincotti, and K. Kitayama, “Asynchronous Multiuser Coherent OCDMA System with Code-Shift-Keying and Balanced Detection,” IEEE J. Sel. Top. Quantum Electron. 13, 1463–1470 (2007). [CrossRef]

5.

C. Tian, Z. Zhang, M. Ibsen, P. Petropoulos, and D. J. Richardson, “A 16-Channel Reconfigurable OCDMA/DWDM System Using Continuous Phase-Shift SSFBGs,” IEEE J. Sel. Top. Quantum Electron. 13, 1480–1486 (2007). [CrossRef]

6.

T. Hamanaka, X. Wang, N. Wada, and K. Kitayama, “Compound Data Rate and Data-Rate-Flexible 622 Mb/s-10Gb/s OCDMA Experiments Using 511-Chip SSFBG and Cascaded SHG-DFG-Based PPLN Waveguide Optical Thresholder,” IEEE J. Sel. Top. Quantum Electron. 13, 1516–1521 (2007). [CrossRef]

7.

W. Amaya, D. Pastor, and J. Capmany, “Modeling of a Time-Spreading OCDMA System Including Nonperfect Time Gating, Optical Thresholding and Fully Asynchronous Signal/Interference Overlapping,” J. Lightwave Technol. 26, 768–776 (2008). [CrossRef]

8.

M. Yoshino, S. Kaneko, T. Taniguchi, N. Miki, K. Kumozaki, T. Imai, N. Yoshimoto, and M. Tsubokawa, “Beat Noise Mitigation of Spectral Amplitude Coding OCDMA Using Heterodyne Detection,” J. Lightwave Technol. 26, 962–970 (2008). [CrossRef]

9.

S. Goldberg and P. R. Prucnal, “On the Teletraffic Capacity of Optical CDMA,” IEEE Trans. Commun. 55, 1334–1343 (2007). [CrossRef]

10.

J. A. Salehi, “Code Division Multiple Access Techniques in Optical Fiber Networks I: Fundamental Principles,” IEEE Trans. Commun. 37, 824–833 (1989). [CrossRef]

11.

G. Manzacca, X. Wang, N. Wada, G. Cincotti, and K. Kitayama, “Comparative Study of Multiencoding Schemes for OCDM Using a Single Multiport Optical Encoder/Decoder,” IEEE Photon. Technol. Lett. 19, 559–561 (2007). [CrossRef]

12.

P. C. Teh, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Phase Encoding and Decoding of Short Pulses at 10Gb/s using Superstructured Fiber Bragg Gratings,” IEEE Photon. Technol. Lett. 13, 154–156 (2001). [CrossRef]

13.

P. C. Teh, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Demonstration of a four-channel WDM/OCDMA system using 255-chip 320 Gchip/s quaternary phase coding gratings,” IEEE Photon. Technol. Lett. 14, 227–229 (2002). [CrossRef]

14.

K. Kitayama, H. Sotobayashi, and N. Wada, “Optical Code Division Multiplexing (OCDM) and its Applications to Photonic Networks,” IEICE Trans. Fundamentals E82-A, 2616–2626 (1999).

15.

K. Kitayama, H. Sotobayashi, and N. Wada, “1,52 Tbit/s OCDM/WDM (4 OCDM×19WDM×20 Gbit/s) Transmission Experiment,” Electron. Lett. 37, 700–701 (2001). [CrossRef]

16.

K. Kitayama, H. Sotobayashi, and N. Wada, “1,6 b/s/Hz 6,4 Tb/s QPSK-OCDM/WDM (4 OCDM×40WDM×40Gb/s) Transmission Experiment using Optical Hard Thresholding,” IEEE Photon. Technol. Lett. 14, 555–557 (2002). [CrossRef]

17.

H. Sotobayashi, W. Chujo, and K. Kitayama, “Highly Spectral-Efficient Optical Code-Division Multiplexing Transmission System,” IEEE J. Sel. Top. Quantum Electron. 10, 250–258 (2004). [CrossRef]

18.

K. Kitayama, “Code Division Multiplexing Lightwave Networks Based upon Optical Code Conversion,” IEEE J. Sel. Areas Comm. 16, 1309–1319 (2000). [CrossRef]

19.

H. Sotobayashi, W. Chujo, and K. Kitayama, “Transparent Virtual Code/Wavelength Path Network,” IEEE J. Sel. Top. Quantum Electron. 8, 699–704 (2002). [CrossRef]

20.

Y. Zhang and L. K. Chen, “Performance Improvement by Code Conversion in a Reconfigurable Optical Code/Wavelength Routing Network,” in Proceedings of Optical Network Design and Management, (2001).

21.

S. Huang, K. Baba, M. Murata, and K. Kitayama, “Architecture Design and Performance Evaluation of Multi-granularity Optical Networks Based on Optical Code Division Multiplexing,” J. Opt. Netw. 5, 1028–1042 (2006). [CrossRef]

22.

Y. G. Wen, Y. Zhang, and L. K. Chen, “On Architecture and Limitation of Optical Multiprotocol Label Switching (MPLS) Networks using Optical-Orthogonal-Code (OCC)/Wavelength Label,” Opt. Fiber Technol. 8, 43–70 (2002). [CrossRef]

23.

P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. B. Hansen, M. Henry, C. Janz, A. Kloch, R. Krhenbhl, C. Raffaelli, M. Schilling, A. Talneau, and L. Zucchelli, “Transparent Optical Packet Switching: Network Architecture and Demonstrators in the KEOPS project,” IEEE J. Sel. Areas Comm. 16, 1245–1259 (1998). [CrossRef]

24.

D. L. Sarwate and M. B. Pursley, “Crosscorrelation Properties of Pseudorandom and Related Sequences,” Proc. IEEE 68, 593–619 (1980). [CrossRef]

25.

S. Tamura, S. Nakano, and K. Okazaki, “Optical Code-Multiplex Transmission by Gold Sequences,” J. Lightwave Technol. LT-3, 121–127 (1985). [CrossRef]

26.

T. Pu, H. Zhang, Y. Guo, and Y. Li, “Evaluation of Beat Noise in OCDMA System with Non-Gaussian Approximated Method,” J. Lightwave Technol. 24, 3574–3582 (2006). [CrossRef]

27.

V. Eramo, “An Analytical Model for TOWC Dimensioning in a Multifiber Optical-Packet Switch,” J. Lightwave Technol. 24, 4799–4810 (2006). [CrossRef]

28.

V. Eramo and M. Listanti, “Packet Loss in a Bufferless WDM Switch Employing Shared Tuneable Wavelength Converters,” J. Lightwave Technol. 18, 1818–1833 (2000). [CrossRef]

29.

X. Wang and K. Kitayama, “Analysis of Beat Noise in Coherent and incoherent Time-Spreading OCDMA,” J. Lightwave Technol. 22, 2226–2234 (2004). [CrossRef]

30.

X. Wang, N. Wada, and K. Kitayama, “Inter-symbol interference and beat noise in flexible data-rate coherent OCDMA and the BER improvement by using optical thresholding,” Opt. Express 13, 10469–10474 (2005). [CrossRef] [PubMed]

31.

S. Yoshima, N. Nakagawa, N. Kataoka, N. Suzuki, M. Noda, M. Nogami, J. Nakagawa, and K.-I. Kitayama, “10Gb/s-Based PON Over OCDMA Uplink Burst Transmission Using SSFBG Encoder/Multi-Port Decoder and Burst Mode Receiver,” J. Lightwave Technol. 28, 365–371 (2010). [CrossRef]

OCIS Codes
(000.2170) General : Equipment and techniques
(230.1150) Optical devices : All-optical devices

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: February 16, 2010
Revised Manuscript: April 28, 2010
Manuscript Accepted: June 17, 2010
Published: August 4, 2010

Citation
V. Eramo, "Impact of the MAI and beat noise on the performance of OCDM/WDM Optical Packet Switches using Gold codes," Opt. Express 18, 17897-17912 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17897


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References

  1. R. Ramaswami, and K. N. Sivarjan, Optical Networks (Morgan Kaufmann, New York, 1998).
  2. B. Mukherjee, Optical Communication Networks (McGraw-Hill, New York, 1997).
  3. G. Manzacca, A. M. Vegni, X. Wang, N. Wada, G. Cincotti, and K. Kitayama, “Performance Analysis of a Multiport Encoder/Decoder in OCDMA Scenario,” IEEE J. Sel. Top. Quantum Electron. 13, 1415–1421 (2007). [CrossRef]
  4. X. Wang, N. Wada, T. Miyazaki, G. Cincotti, and K. Kitayama, “Asynchronous Multiuser Coherent OCDMA System with Code-Shift-Keying and Balanced Detection,” IEEE J. Sel. Top. Quantum Electron. 13, 1463–1470 (2007). [CrossRef]
  5. C. Tian, Z. Zhang, M. Ibsen, P. Petropoulos, and D. J. Richardson, “A 16-Channel Reconfigurable OCDMA/DWDM System Using Continuous Phase-Shift SSFBGs,” IEEE J. Sel. Top. Quantum Electron. 13, 1480–1486 (2007). [CrossRef]
  6. T. Hamanaka, X. Wang, N. Wada, and K. Kitayama, “Compound Data Rate and Data-Rate-Flexible 622 Mb/s-10Gb/s OCDMA Experiments Using 511-Chip SSFBG and Cascaded SHG-DFG-Based PPLN Waveguide Optical Thresholder,” IEEE J. Sel. Top. Quantum Electron. 13, 1516–1521 (2007). [CrossRef]
  7. W. Amaya, D. Pastor, and J. Capmany, “Modeling of a Time-Spreading OCDMA System Including Nonperfect Time Gating, Optical Thresholding and Fully Asynchronous Signal/Interference Overlapping,” J. Lightwave Technol. 26, 768–776 (2008). [CrossRef]
  8. M. Yoshino, S. Kaneko, T. Taniguchi, N. Miki, K. Kumozaki, T. Imai, N. Yoshimoto, and M. Tsubokawa, “Beat Noise Mitigation of Spectral Amplitude Coding OCDMA Using Heterodyne Detection,” J. Lightwave Technol. 26, 962–970 (2008). [CrossRef]
  9. S. Goldberg, and P. R. Prucnal, “On the Teletraffic Capacity of Optical CDMA,” IEEE Trans. Commun. 55, 1334–1343 (2007). [CrossRef]
  10. J. A. Salehi, “Code Division Multiple Access Techniques in Optical Fiber Networks I: Fundamental Principles,” IEEE Trans. Commun. 37, 824–833 (1989). [CrossRef]
  11. G. Manzacca, X. Wang, N. Wada, G. Cincotti, and K. Kitayama, “Comparative Study of Multiencoding Schemes for OCDM Using a Single Multiport Optical Encoder/Decoder,” IEEE Photon. Technol. Lett. 19, 559–561 (2007). [CrossRef]
  12. P. C. Teh, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Phase Encoding and Decoding of Short Pulses at 10 Gb/s using Superstructured Fiber Bragg Gratings,” IEEE Photon. Technol. Lett. 13, 154–156 (2001). [CrossRef]
  13. P. C. Teh, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Demonstration of a four-channel WDM/OCDMA system using 255-chip 320 Gchip/s quaternary phase coding gratings,” IEEE Photon. Technol. Lett. 14, 227–229 (2002). [CrossRef]
  14. K. Kitayama, H. Sotobayashi, and N. Wada, “Optical Code Division Multiplexing (OCDM) and its Applications to Photonic Networks,” IEICE Trans. Fundamentals 82-A, 2616–2626 (1999).
  15. K. Kitayama, H. Sotobayashi, and N. Wada, “1,52 Tbit/s OCDM/WDM (4 OCDM×19WDM×20 Gbit/s) Transmission Experiment,” Electron. Lett. 37, 700–701 (2001). [CrossRef]
  16. K. Kitayama, H. Sotobayashi, and N. Wada, “1,6 b/s/Hz 6,4 Tb/s QPSK-OCDM/WDM (4 OCDM×40 WDM×40Gb/s) Transmission Experiment using Optical Hard Thresholding,” IEEE Photon. Technol. Lett. 14, 555–557 (2002). [CrossRef]
  17. H. Sotobayashi, W. Chujo, and K. Kitayama, “Highly Spectral-Efficient Optical Code-Division Multiplexing Transmission System,” IEEE J. Sel. Top. Quantum Electron. 10, 250–258 (2004). [CrossRef]
  18. K. Kitayama, “Code Division Multiplexing Lightwave Networks Based upon Optical Code Conversion,” IEEE J. Sel. Areas Comm. 16, 1309–1319 (2000). [CrossRef]
  19. H. Sotobayashi, W. Chujo, and K. Kitayama, “Transparent Virtual Code/Wavelength Path Network,” IEEE J. Sel. Top. Quantum Electron. 8, 699–704 (2002). [CrossRef]
  20. Y. Zhang, and L. K. Chen, “Performance Improvement by Code Conversion in a Reconfigurable Optical Code/Wavelength Routing Network,” in Proceedings of Optical Network Design and Management, (2001).
  21. S. Huang, K. Baba, M. Murata, and K. Kitayama, “Architecture Design and Performance Evaluation of Multigranularity Optical Networks Based on Optical Code Division Multiplexing,” J. Opt. Netw. 5, 1028–1042 (2006). [CrossRef]
  22. Y. G. Wen, Y. Zhang, and L. K. Chen, “On Architecture and Limitation of Optical Multiprotocol Label Switching (MPLS) Networks using Optical-Orthogonal-Code (OCC)/Wavelength Label,” Opt. Fiber Technol. 8, 43–70 (2002). [CrossRef]
  23. P. Gambini, M. Renaud, C. Guillemot, F. Callegati, I. Andonovic, B. Bostica, D. Chiaroni, G. Corazza, S. L. Danielsen, P. B. Hansen, M. Henry, and C. Janz, A.  Kloch, R.  Krähenbühl, C.  Raffaelli, M.  Schilling, A.  Talneau, and L.  Zucchelli, “Transparent Optical Packet Switching: Network Architecture and Demonstrators in the KEOPS project,” IEEE J. Sel. Areas Comm. 16, 1245–1259 (1998). [CrossRef]
  24. D. L. Sarwate, and M. B. Pursley, “Crosscorrelation Properties of Pseudorandom and Related Sequences,” Proc. IEEE 68, 593–619 (1980). [CrossRef]
  25. S. Tamura, S. Nakano, and K. Okazaki, “Optical Code-Multiplex Transmission by Gold Sequences,” J. Lightwave Technol. LT-3, 121–127 (1985). [CrossRef]
  26. T. Pu, H. Zhang, Y. Guo, and Y. Li, “Evaluation of Beat Noise in OCDMA System with Non-Gaussian Approximated Method,” J. Lightwave Technol. 24, 3574–3582 (2006). [CrossRef]
  27. V. Eramo, “An Analytical Model for TOWC Dimensioning in a Multifiber Optical-Packet Switch,” J. Lightwave Technol. 24, 4799–4810 (2006). [CrossRef]
  28. V. Eramo, and M. Listanti, “Packet Loss in a Bufferless WDM Switch Employing Shared Tuneable Wavelength Converters,” J. Lightwave Technol. 18, 1818–1833 (2000). [CrossRef]
  29. X. Wang, and K. Kitayama, “Analysis of Beat Noise in Coherent and incoherent Time-Spreading OCDMA J. Lightwave Technol. 22, 2226–2234 (2004). [CrossRef]
  30. X. Wang, N. Wada, and K. Kitayama, “Inter-symbol interference and beat noise in flexible data-rate coherent OCDMA and the BER improvement by using optical thresholding,” Opt. Express 13, 10469–10474 (2005). [CrossRef] [PubMed]
  31. S. Yoshima, N. Nakagawa, N. Kataoka, N. Suzuki, M. Noda, M. Nogami, J. Nakagawa, and K.-I. Kitayama, “10 Gb/s-Based PON Over OCDMA Uplink Burst Transmission Using SSFBG Encoder/Multi-Port Decoder and Burst Mode Receiver,” J. Lightwave Technol. 28, 365–371 (2010). [CrossRef]

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