## Abstract

Four-wave mixing and in-band crosstalk noise can limit the performance of wavelength division multiplexing optical networks. The statistics of these noises depend on the nature of the traffic. In this paper, the performance of an IP over MPλS-based network is analyzed using the Multi-Canonical Monte Carlo (MCMC) method and it is shown that the Gaussian approximation does not yield accurate results. The importance of the traffic load and the traffic distribution in the optical channels is also investigated. It is shown that a proper traffic distribution in the optical channels, can improve the performance of the network.

©2005 Optical Society of America

## 1. Introduction

Multiprotocol Lambda Switching (MPλS) [1] is a promising architecture for the next generation Internet. In this architecture, IP packets are transported directly over the optical layer without any optoelectronic conversion reducing the traffic bottleneck. However, the performance of the system is expected to be strongly dependent on the bursty nature of IP traffic. This is because in optical networks, there exist many types of signal-dependent noise such as Four Wave Mixing (FWM) and in-band crosstalk noise, whose statistical behavior will depend on the statistics of the signal [2]–[3].

The influence of traffic burstiness in the statistics of the FWM noise has been previously studied in Refs. [4] – [5]. In Ref [4], the impact of IP burstiness was investigated assuming Gaussian statistics for the FWM noise in the optical domain. However, even for continuous traffic, FWM noise is not Gaussian [2] and the Gaussian approximation underestimates the Bit Error Rate (BER). The analysis of Ref. [5], did not consider the randomness of the bits within a packet. On the other hand, in most studies of in-band crosstalk, the interfering bits are either assumed all to be in the mark state [3], which constitutes the worst case scenario or to be in the mark and space states with equal likelihood which corresponds to continuous traffic.

In this work, the impact of traffic burstiness in the performance of both the physical and the higher network layers in systems limited by FWM noise or in-band crosstalk noise is accurately estimated using the Multi-Canonical Monte Carlo (MCMC) method [6] – [8]. The validity of the Gaussian approximation is investigated and it is shown to lead to performance estimation errors in the case of both bursty and continuous traffic. Using the MCMC, the importance of the traffic load and the traffic channel distribution is also investigated. It is shown proper engineering of the traffic in the WDM channels alleviates some of the effects of FWM distortion and improves the BER of the system.

## 2. Modeling of traffic burstiness

The system under consideration is shown in Fig. 1(a). At the edge nodes, the IP packets are aggregated and the wavelength that carries each packet depends on the final destination and its class. At the ingress nodes, the packets are forwarded according to their wavelength. Each wavelength can be modeled as an M/G/1 system that is, a single server (transmitter) and Poisson packet arrival times. The assumption of Poisson arrivals, for the input flows is accurate if the observation interval is not large (short-range dependence). This happens since the aggregate traffic *A* in each wavelength consists of many small and independent arrival processes *A*_{i}
each corresponding to the traffic generated by a single user. As the number of the *A*_{i}
increases, the aggregate traffic *A* asymptotically tends to follow Poisson statistics [9].

The burstiness of each wavelength is characterized by the traffic load *ρ*, defined as the ratio *t*
_{T}/*t*
_{A} of the average time *t*
_{T} taken to transmit a packet (service time) and the average time *t*
_{A} between two consecutive arrivals (inter-arrival time). If *k* is the expected packet length (in bits) and *R*
_{b} is the transmission bit rate, then *t*
_{T}=*k*/*R*
_{b}. Silent periods are modelled as streams of “0”s since the power of the optical source is approximately zero. Within an IP packet, the “1”s and “0”s appear with equal likelihood.

The probability, that at any given time *t* a packet is being transmitted, equals to *ρ*. This result holds regardless of the statistics of the service times [10] (and hence the statistics of the length of the IP packets). The average probability *p*
_{1} therefore to transmit a “1” equals ½*ρ* and the probability of “0” is *p*
_{0}=1-*p*
_{1}=1-½ [Fig. 1(b)]. The dependence of *p*
_{1} and *p*
_{0} on *ρ* is a first indication of the influence of traffic burstiness in the system performance: the decision variable in both FWM limited [2] and in-band crosstalk noise limited [11] systems depends on the bit statistics and hence on *ρ*. One can use the MCMC method in order to evaluate the Probability Density Function (PDF) of the decision variable. The PDF can then be used to estimate the BER (see Refs. [2] and [11] for the details of these computations). Using the BER, the performance of the physical layer and the higher layers can be analyzed. The system parameters used in the simulations are those of Table 1.

## 3. Accuracy of the Gaussian approximation

We first apply the MCMC method in order to study the accuracy of the Gaussian approximation for the receiver statistics. The Gaussian approximation is based on the Central Limit Theorem (CLT) according to which if the decision variable *D* is the sum of a large number of independent random variables, *D* will approximately follow a Gaussian distribution. However, as discussed in Refs. [2] and [11], in the case of a WDM receiver limited by either FWM or in-band crosstalk respectively, *D* can not be written as a sum of independent random variables and hence the Gaussian approximation may not lead to accurate results. Since the traffic load also has an impact on the noise statistics, it is interesting to compare the results of the Gaussian assumption for bursty traffic as well. For a given threshold *a*, the BER of the Gaussian approximation is obtained using the formula *P*
_{e,g}=*P*
_{0}
*Q*((*a*-*m*
_{0})/*σ*
_{0})+*P*
_{1}
*Q*((*m*
_{1}-*a*)/*σ*
_{1}) where *m*
_{1} and *m*
_{0} are the mean values of the decision variable in the mark and space state, while *σ*
_{1} and *σ*
_{0} are corresponding noise variances. The function *Q*=*Q*(*x*) is related to the complementary error function erfc(*x*) by *Q*(*x*)=½erfc(*x*/√2). The receiver threshold a must be chosen so as to minimize the value of *P*
_{e,g}. When*P*
_{1}=*P*
_{0}=½ the minimum BER *P*
_{e,gmin} is approximated by *Q*(*q*) where *q*=(*m*
_{1}-*m*
_{0})/(*σ*
_{1}-*σ*
_{0}).

The BER obtained using either the Gaussian approximation or the MCMC method is illustrated in Fig. 2(a). For the case of the crosstalk noise, the Gaussian model does not provide an accurate estimate of the BER especially for small values of the traffic load. As however the traffic load increases and the BER rises, the error between the Gaussian approximation and the MCMC results is reduced. This is consistent with the fact that for continuous traffic, the difference between the PDF of the receiver and its Gaussian approximation lies primarily in the tails [11], which correspond to low BER values. The error between the Gaussian approximation and the MCMC is much more pronounced (several orders of magnitude) in the case of FWM noise. This is due to the fact that, even in the case of continuous traffic the PDF of the receiver variable differs significantly from a Gaussian shape both in the high values and the tails [2]. To further understand the implications of the error in the Gaussian approximation, the required values SXR of the optical signal to crosstalk ratio, which lead to a BER equal to 10^{-9}, are plotted in Fig. 2(b) for various *ρ*. For light traffic, the difference in the SXR is about 3dB. For continuous traffic (*ρ*=1) the difference is less significant (≅1.5dB). The values of the input power corresponding to the same BER are also plotted for a FWM-limited receiver for both methods. The difference is now more pronounced and, more importantly, the Gaussian approximation overestimates the maximum power that the system can tolerate. For light traffic load, the power that the Gaussian approximation tolerates is about 7dB higher, while for continuous traffic the difference is reduced but still remains high (≅3dB).

## 4. Influence of the traffic load

In Fig. 3(a) the minimum BER obtained by the MCMC is plotted as a function of the input peak power *P*
_{in} for various values of the traffic load in the case of a system limited by FWM. From Fig. 3(a), the influence of the traffic load *ρ* in the performance of the system can be assessed. For *P*
_{in}=9dBm, a BER equal to 6.7×l0^{-10}, 4.2×l0^{-6} and 2.2×l0^{-4} is achieved for *ρ*=0.2, 0.6 and 1 respectively. It is therefore deduced that p strongly influences the performance of the network. Bursty systems (*ρ*<1) can tolerate higher power and can therefore achieve longer reach.

The values of the BER were also obtained using MCMC simulations in the case of the in-band crosstalk noise assuming ${c}_{0}^{2}$ =100 photoelectrons in the mark state [Fig. 3(b)]. It is deduced that an increase in the traffic load *ρ*, causes degradation in the performance of the system. For example the BER obtained for SXR=12dB is 4.3×10^{-11}, 1.1×10^{-6} and 4.7×10^{-5} in the cases where *ρ*=0.2, 0.6 and 1 respectively. This implies that as the traffic becomes lighter the required component crosstalk levels are relaxed and more nodes can be concatenated in the network.

## 5. Performance of higher network layers

The performance of the higher layers can be quantified in terms of the packet error rate (PER). Figures 4(a)–4(b) illustrate the effects of FWM and in-band crosstalk noise on the PER, for various values of the traffic load *ρ*. Both the Gaussian and the MCMC results are presented. The PER for a packet of *k* bits is calculated using PER=1-(1-BER)^{k}, while the average length of IP packets is considered *k*=256bytes=2048bits (short packets) and *k*=1500bytes=12000bits (long packets). As it can be seen from Figs. 4(a)–(b), the PER has almost the same behaviour as the BER. This is not surprising since using the fact that (1-*x*)^{n}=1-*nx*+O(*x*
^{2}) one can deduce that for small BER values, PER≅*k*BER. This expression also justifies the fact that, as seen by Figs. 4(a)–4(b), the PER is higher for longer packets. It is obvious that longer packets should be segmented in order to reduce the probability of erroneous reception. Depending on the protocols used in the higher layers, erroneous receptions could cause packet retransmissions and/or loss of quality of service.

The figures once again illustrate the inaccuracy of the Gaussian model, especially in the case of FWM noise. For the in-band crosstalk noise [Fig. 4(a)] the Gaussian approximation can lead to an error of about 5 orders of magnitude in the case of low traffic load that tends to diminish as the traffic load increases. In the case of FWM [Fig. 4(b)] the error is much larger, more than 6 orders of magnitude even in the high load area.

## 6. Reducing the FWM degradation using proper traffic management

In Ref. [2] it was shown that the FWM degradation was more severe for the central channels than the edge channels of a WDM system. It is then obvious that one way to reduce the FWM noise and increase the transmission power is to redistribute the traffic so that the edge channels carry heavier traffic than the central ones.

To illustrate the above remarks, two types of channel traffic distribution are considered. In the first type, all the channels are loaded with the same traffic, *ρ*=0.6 (balanced distribution). In the second, the channels are unequally loaded (unbalanced distribution) so that the average traffic is equal to 0.6 as shown in Fig. 5(a). The obtained values of PER for the central channel, are plotted in Fig. 5(b). It is deduced that for *P*_{in}
=8dBm and short packets, the PER is reduced from 5.2×10^{-4} for the balanced distribution to 2×10^{-5} for the unbalanced one. Also, a PER=10^{-4} is achieved for *P*_{in}
=8.6dBm and P_{in}=7.5dBm for the balanced and the unbalanced distribution respectively (corresponding to a power gain of ≅1dB). It is interesting to note that the unbalanced load distribution lowers the PER and BER values for all channels and not only for the central one. This is illustrated in Fig 5(c) where the BER for all 16 channels are plotted for the two load distributions for *P*_{in}
=7.6dBm. It is deduced that the BER is lower for all channels and the improvement is higher near the central channel (where the BER for uniform traffic is higher). Hence, careful traffic engineering can play an important role in improving the performance of an IP over WDM network. A similar behavior is expected for a network limited by in-band crosstalk noise as well, provided of course that the crosstalk levels for the different wavelength channels are different. If this is so, a larger amount of traffic should be directed to the low crosstalk channels.

## 7. Conclusion

In this paper, the MCMC method was applied to study the impact of IP traffic burstiness on the performance of an IP over MPλS-based DWDM network limited by FWM or in-band crosstalk noise. It was deduced that the Gaussian approximation does not lead to accurate results, especially in the case of light traffic. It was also shown that careful traffic engineering can improve the system performance in terms of the BER by at least one order of magnitude.

## Acknowledgments

This work was partly funded by “PYTHAGORAS I” grant from the Hellenic Ministry of Education. The authors would like to thank Mr. G. Kakaletris and Dr. P. Rizomiliotis for their helpful discussions.

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