1. Introduction

Next generation intelligent transparent and translucent networks enable operators to dynamically configure and provision resources to customers. A key consideration in service level agreements is the probability of outage that an allocated all-optical light path can expect to experience due to physical layer impairments, and on what time scales outages can be expected to occur. Outage probability has been considered in the context of individual impairments such as polarization mode dispersion (PMD) [1] or residual chromatic dispersion (RCD) [2], but not in the presence of multiple impairments. This letter details a method to compute an outage probability from the combined effects of multiple time varying optical impairments, such as PMD, RCD and amplified spontaneous emission (ASE), over multiple unregenerated fiber spans. Combined with suitable impairment monitoring technologies [3], these methods could be applied to improve automated path selection [4] and network design in intelligent optical networks.

2. Optical impairments

Prior work [5] describes a method to combine multiple optical impairments into a single signal quality metric, the well known Q-factor, via the Eye Closure Penalty (ECP) signal degradation metric. Here, the values of the optical impairments such as PMD, RCD and ASE are transformed into values of ECP via relationships that have been derived analytically, numerically and experimentally [6]–[8]. For the NRZ modulation format, the ECP (in dB) due to RCD, PMD and ASE for a link are, respectively:

where L_s=ΔτΔλsin(θ) is the PMD string length, D is the net path residual chromatic dispersion in ps/nm, Δλ is the optical signal bandwidth in nm, Δτ is the differential group delay (DGD) in seconds and θ is the angle of the launch state of polarization relative to the principal states of polarization in Stokes space. OSNR=P_sig/P_ASE, is the signal Optical Signal to Noise Ratio (P_sig is the signal power at the receive node), c is the speed of light and B_o is the optical bandwidth of the signal incident upon the receiver. The constants A_D, A_DGD and B_DGD depend on details of the modulation format and receiver.

Eqns. (1)–(3) describe the impacts of degradations acting independently, however in real networks multiple degradations occur simultaneously. In [5] we presented a simple approximation for assessing the impact of combined degradations on the overall ECP (in dB) that is expected to hold for linear network operation:

The final step is to concatenate the effects of multiple links, to determine the ECP for a total path, P. To determine the total penalty for a path, we require the cumulative values of D, Δτ and OSNR and θ for the total path. These are given by Eqns (5)–(7) for an N link path:

where D_j (ps/nm) is the residual dispersion, $\overline{\Delta {\tau }_j}$ is the mean DGD, P_ASE,j (W/Hz) is the net ASE power and G_j is the net gain/loss of the j^th link. L_Res (dB) is the residual loss, H_Filt the transmission coefficient and B_Filt (Hz) the bandwidth of any optical filter in the last node. θ can be measured at each transmitter or as a worst case set to $\frac{\pi }{4}$ .

The parameters required to calculate ECP_EST are readily available or can be measured [3], thus each link in a network can thus be represented by a vector of the form (D_j, P_ASE,j,G_j, Δτ_j). These vectors are used in (4) to evaluate ECP_EST.

Given the Q-factor at the transmitter Q₀ it is possible to compute the Q-factor from the transmit node 0, to a receive node m from the following [5]:

Here, G=P_m/P₀=net path gain between the transmit node 0 and receive node m and ECP_m is the (8) will give the signal Q-factor at the receive node N.

3. Outage probability

The “outage probability” in optical networks is commonly defined in terms of the probability that a physical impairment, such as Polarization Mode Dispersion (PMD), causes a link Bit Error Rate (BER) to exceed a design specification, e.g. BER>10^-12 [1]. Optical impairments commonly vary with time. For example, the PMD differential group delay (DGD) is often characterized as a Maxwellian distribution [9] due to the random birefringence of the fiber medium (there has also been work that points to small long-term reversible variations [10]). Fiber laid in long haul all-optical networks, without optical regeneration or dispersion compensation, can be subject to long term seasonal soil temperature variations (on the timescale of months) in different geographical locations of between -40°C and 40°C that can cause a change in the dispersion and dispersion slopes of buried fibers [11]. In EDFAs, ASE is considered to be a stable quantity, characterized by the amplifier noise factor. However, the addition or removal of channels can cause OSNR and ASE transients on millisecond timescales that result in transmission penalties [12]. These events are driven by user traffic demands as light-paths are set up or torn down in a network that could be on the timescale ranging from minutes to months.

Using techniques such as the recently proposed optical multi-impairment monitor [3], it is possible to build histograms and consequently probability mass functions (PMF) of the values of various optical impairment mechanisms over sliding intervals of time. These impairment PMFs can be transformed into ECP PMFs for each of the impairments (in dB). For PMFs, this is a simple matter of a “transformation of scale”, where the discrete values of the PMF bins are transformed, but the probabilities unchanged. For continuous probability density functions, this is still possible with some mathematical manipulation, covered in [13].

The individual ECP PMFs can then be combined using the convolution sum (corresponding to the addition of the individual ECP dB values, because we are, in effect, computing the PMF of a random variable which is a sum of multiple independent random variables) into a PMF for the combined ECP. We then apply the “transformation of scale” with (8) to obtain the PMF of the Q-factor and finally again apply a “transformation of scale” with:

to compute the corresponding BER PMF and obtain an estimate of the outage probability for a specified required BER using numerical integration over the range where BER>10^-12. For clarity, Fig. 1 gives an overview of this multi-step process.

 

Fig. 1. Outage probability computation process from impairment PMFs.

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4. Examples

To demonstrate the outage probability computation process, simulations of a single link were conducted, using a commercial package [14], with NRZ modulation at bit rates of 10 Gb/s, 40 Gb/s and 100 Gb/s in the presence of PMD, RCD and ASE. These results then provided the constants described in the previous section to enable computation of the ECP from each of the impairments.

The PMF of PMD was characterized by a Maxwellian distribution, while the PMFs of RCD and ASE were assumed to follow Gaussian relationships with a standard deviation of 5 percent of the mean, for simplicity and illustration of the mathematical concept. Note also that the concept is applicable for arbitrary measured statistics which are not Maxwellian or Gaussian. The process described in section 3 was used to compute the outage probability (for a BER>10^-12 specification) due to the individual impairment PMFs, the results of which are shown in Fig. 2 and various combinations of impairments (at 10Gb/s), shown in Fig. 3.

 

Fig. 2. (a) Outage probability due to DGD for various bit rates. (b) Outage probability due to RCD for various bit rates. (c) Outage probability due to OSNR for various bit rates.

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We observe in Fig. 2 that for all impairments, as the bit rate increases, so does the outage probability for a specified impairment level. This has obvious implications in network planning, as optical impairments must be more stringently reduced and controlled as bit rates increase [15]. Conversely, this methodology can assist network planners to determine whether a bit rate upgrade on a specific link will be possible, at a target outage probability, with the currently installed fiber span and its associated impairments.

Figure 3 is a movie of a 3D plot with mean impairment values on each axis, and computed outage probability encoded as the color of the point. This figure gives an indication of the relative contributions of each of the impairments. Outage probability contributions from OSNR and RCD have a very steep “knee” whilst outage probability contribution from PMD is more gradual.

Conceptually, it is possible to implement the described techniques to use the outage probability as a cost metric within algorithms designed for routing and wavelength assignment [4]. To reduce computation requirements, this method can produce lookup tables associating impairment levels with outage probabilities.

 

Fig. 3. (3.75 MB) Movie of outage probability due to multiple impairments (marker color represents magnitude of outage probability). Music copyright by Divine Mechanism. (www.divinemechanism.com.au). [Media 1]

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5. Conclusion

We have presented a new process enabling the computation of the outage probability of an optic fiber link or all-optical path with respect to multiple linear optical impairments, such as PMD, RCD and ASE. This method can be applied to assist in network planning, and could be used as a cost metric in routing algorithms.