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A Transition Matrix Analysis of the Hinge Model
Abstract
We adapt the transition matrix procedure to time-dependent effects in communication systems and subsequently calculate the outage-time statistics resulting from polarization mode dispersion (PMD) in the hinge model.
In transparent optical networks, the signal transmission is degraded by optical layer physical impairments. Therefore, lightpaths may be blocked due to unacceptable quality of transmission (QoT). Among physical impairments, polarization mode dispersion (PMD) is a detrimental effect which has stochastic characteristics. Moreover, PMD depends on time-variant factors, such as the temperature and the fiber stress. When implementing a dynamic GMPLS-controlled transparent optical network, the GMPLS protocol suite must take into account physical impairment information in order to establish lightpaths while guaranteeing the required QoT. In the literature, solutions for QoT-aware GMPLS control plane commonly consider that the effects of PMD on QoT are not detrimental when the average differential group delay (DGD) does not exceed a threshold. However, even with a high average DGD, it may happen that the instantaneous DGD is not detrimental. Additionally, given PMD temporal correlation properties, once that the instantaneous DGD is not detrimental, it continues to be not detrimental within considerable time ranges. Therefore, more accurate models can be implemented in the GMPLS control plane to account for PMD. In this paper we propose a novel lightpath provisioning scheme based on a PMD prediction model which accounts for PMD temporal correlation properties. The proposed PMD-temporal- correlation (PTC) based lightpath provisioning scheme is compared with a scheme based on a classical PMD model. Simulation results show that PTC scheme significantly reduces the lightpath blocking probability with respect to the classical scheme. Moreover, PTC demonstrates that, by considering PMD temporal correlation, the transparency domain size can be increased, since paths that would be rejected by a classical model can be actually accepted within specific time ranges.
System outage due to first-order polarization-mode dispersion of links obeying the hinge model is analyzed using outage maps. We find that some fraction of the wavelength-division-multiplexed fiber capacity does not meet any outage specification.
In this paper, we employ measurements of transponder tolerance to both differential group delay (DGD) and second-order polarization mode dispersion (SOPMD) and of the temporal evolution of DGD and SOPMD in installed transmission systems to predict the influence of PMD on the rate and duration of PMD-induced system outages. An empirical 2-D random-walk model predicts that the outage rate and duration depends solely on the mean fiber DGD. We find that the step size of the random walk is nearly uncorrelated with the instantaneous value of the PMD. We then justify the assumptions of this procedure with a full numerical simulation and employ a biased Markov chain algorithm to generate highly accurate results for system outages where simplified models fail.
Modeling of the polarization-mode dispersion (PMD) temporal changes in long fiber routes represents a critical aspect of system outage characterization. In this paper, an analysis of the PMD temporal statistics in links obeying the recently proposed hinge model is presented. The analytical expression of the differential group delay probability density function is derived and used to show that the outage probability is channel specific in a wavelength-division-multiplexed system. The existence of outage-free channels, which represents one of the most important implications of the hinge model, is demonstrated and quantified. The frequency dependence of the temporal statistics is also characterized, and an expression for PMD-induced pulse broadening is derived
We present a formalism of the transition matrix Monte Carlo method. A
stochastic matrix in the space of energy can be estimated from Monte Carlo
simulation. This matrix is used to compute the density of states, as well as to
construct multi-canonical and equal-hit algorithms. We discuss the performance
of the methods. The results are compared with single histogram method,
multi-canonical method, and other methods. In many aspects, the present method
is an improvement over the previous methods.
PACS numbers: 02.70.Tt, 05.10.Ln, 05.50.+q. Keywords: Monte Carlo method,
flat histogram, multi-canonical ensemble.
We observe distinct variations between the DGD temporal statistics for different channels in a field installed system. This phenomenon is confirmed with statistical analysis, using a model in which DGD dynamics are due only to a finite number of active points along the link.
This paper reviews the fundamental concepts and basic theory of polarization mode dispersion (PMD) in optical fibers. It introduces a unified notation and methodology to link the various views and concepts in Jones space and Stokes space. The discussion includes the relation between Jones vectors and Stokes vectors, rotation matrices, the definition and representation of PMD vectors, the laws of infinitesimal rotation, and the rules for PMD vector concatenation.
We adapt transition matrix methods to time-dependent communication systems. We then calculate the distribution of the outage times of an optical fiber system impaired by stochastically varying polarization-mode dispersion.
We numerically study polarization-mode dispersion in fiber routes with a finite number of degrees of freedom, show that individual channels in such systems exhibit a non-Maxwellian differential group delay distribution, and analyze the resulting statistics of outage probabilities. We show that a significant number of channels in such systems will be outage-free for long time periods.
We demonstrate that probability distribution functions (pdfs) such as those of interest in communication theory can be efficiently calculated through a random walk algorithm. To illustrate the procedure, we determine the pdf of polarization-mode dispersion in optical fibers. We find that the technique efficiently samples configurations with extremely low probability for which it converges far more rapidly than the multicanonical procedure.