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Modeling the influence of the optical fiber strain on the readiness performance of fiber-optic communication lines

A mathematical model for determining the readiness performance of the fiber-optic communication lines depending on the degree of optical fiber strain with different methods of influence. The model is based on the theory of Markov chains. The adequacy of the model is achieved by the false positive and false negative errors. The application of the developed model enables us to determine the allowable maintenance interval for the fiber-optic communication lines.


Introduction
The field experience of the fiber-optic communication lines (FOCLs) has shown that a life of optic cables (OCs) depends on mechanical strains in optic fibers (OFs) [1 -6]. The early diagnostics and monitoring of the optical fibers (OFs) in the OCs are important at present due to the mass distribution of the FOCL. The use of the automatic monitoring systems based on optical time domain reflectometers (OTDRs) is one of the solutions for the FOCL. Reflectometers well detect defects and troubles of the OF and allow attenuation at any point of the FOCL to be determined. Defects and troubles of the OFs include sections with high strain or fall of temperature, bends or microbends, violation of the OF integrity due to unauthorized access to it. The underground OCs are subjected to high mechanical loads because of the ground deformations. The aerial optic cables also can be subjected to mechanical loads in case of icing the cable. This can lead to irreversible changes associated with high mechanical strains within the OF of the FOCL and reduce the OC life [2 -5].
To provide a trouble-free operation of the FOCL, an ongoing monitoring of the OFs is needed for timely detection of sections with high OF strain [1 -4]. The article describes the influence of tensile loads on the OC life.

Statement of the problem
We need to have the right information on the OF state in the FOCL to estimate the reliability of the FOCL. In addition, not every optical reflectometer detects OF sections with high strain. A method of Brillouin reflectometry should be applied to find such sections in the FOCL. Due to this, the monitoring system for the FOCL should necessarily include the Brillouin optical time-domain reflectometers (BOTDR).
The absence of mechanical strains in the OFs is necessary to achieve a trouble-free operation of the FOCL during the life. The reliability of the FOCL and its durability are directly influenced by the increase in the OF strain. There are several ranges of the OF strain at which various periods of trouble-free operation are possible according to the data from the "Fujikura" Ltd. A strain of less than 0.2% is a safe range which has no influence on the OF life. A strain range of more than 0.6 % is an unacceptable because it leads to destruction of the OF. The life of the optic cable is reduced in the range of tensile loads of 0.2-0.6 %. A guaranty period of the optic cable is 25 years. Tensile load should be less than 0.2% to maintain this guaranty period. If the OF strain is 0.45%, the optical fiber break will occur with a probability of 50%. Detection and analysis of the distribution of the Mandelstam -Brillouin backscattering spectrum (MBBS) along the OF is carried out in the BOTDR [1 -6]. We analyze the BOTDR reflectograms for OF sections with modified tensile loads. Fig.1 -Fig. 4 show the examples of the BOTDR reflectograms obtained by investigation tests of the OFs containing sections with modified tensile loads [4 -6]. The light pipe is composed of single mode optic fibers: G.652 OF (usual OF), welded with G.657 OF, which in turn is welded with G.653 (DSF-dispersion-shifted fiber) in the investigation tests, the data of which are given below [5,6]. A BOTDR reflectogram with a modified OF section subjected to a strong tensile load is shown in Fig.  1. The place of the OF strain is clearly noticeable by the shift of the MBBS maximum (f Bs ) from 10.84 GHz (f B0 ) to 11.08 GHz towards a higher frequency of f B (F2). The comprehensive picture of the OF strain conforming to the distribution of the MBBS (Fig. 1) is shown in Fig. 2. The graph demonstrates that the strain increases in the place of mechanical load on the OC by more than 0.45 %, which is dangerous for the OF.    Fig. 4). Evaluation of the influence of optic fiber strain on the reliability of the FOCL is of particular interest. We analyze the dependence of the readiness performance of the FOCL on the OF strain.

The theory
The transmission path is ready to operate if both of its ways are in readiness. There are the following performances of readiness in the G.821 recommendation: a readiness performance of K AF and an unreadiness performance of К U . The readiness performance is the ratio of time during which the transmission path is in readiness to the overall observation time. The unreadiness performance is the ratio between the time during which the transmission path is in unreadiness and the overall observation time. The Markov chain theory and probabilistic modeling are used to estimate the readiness performance [4 -9]. The development of the mathematical model was carried out in several stages. These stages are described in [4 -9]. The investigated system (FOCL) has the following states:  The system transition into the nondetectable failure state is possible because of the false positive and false negative errors (type I error and type II error respectively). A possible reason for the system transition from the S 1ТО to the S 3 is an incorrect choice of performances in the test module (in reflectometer) of the OF monitoring system, leading to a formation of non-fixed irregularities. If there is no failure in the S 1ТО state, the system passes into the operational state of S 0. Then the operating cycle is repeated. If the system is in the state of S 0 for a time no more than T, the system failure may occur. It is characterized by the transition of the system to the state of S 2 . If the system operates without troubles the time of T in the state of S 0 , it will pass into the technical maintenance of S ТО . The system is checked in this state. The duration of the test is t p . The system goes into the state of S 3 only during the check and stays here until the next system control. Unconditional transition of the system to the state of S 3ТО occurs from the S 3 state. In case of transition from the state of S 3ТО to the failure state due to failure detection, the system returns into the operational state. If the failure is unfound, the system will return into the nondetectable failure state (S 3 ). The return cause of the system to the state S 3 is false negative error (β 2 ). The system can go to a fictitious failure state of S 4 . Fictitious failure occurs because of the false positive error of built-in diagnostic equipment. Timely detection of the fictitious failure returns the system to its operational state. The operating cycle is repeated each time when the system returns to the state of S 0 .
The detailed procedure of determination of the readiness performance is given in [7 -11]. The readiness performance is determined by the following algorithm:  6 1. The matrix of transition probabilities (P) is written in accordance with the graph of system states (Fig. 5). The matrix of transition probabilities is given in [7 -9]. We wrote the probabilities of the system transition from the state of S i to the state of S j . The probabilities are presented in table 1.
A row matrix of final probabilities is written [7 -9].
π π π π π π π π π = , 3. The final probabilities of the system are determined in each state. To find ones we need to multiply the matrix of transition probabilities by the row matrix of final probabilities and execute the necessary transformations [9 -12]. . To find the readiness performance of the FOCL is necessary to identify the real time of ω i (T) and the current time of υ i (T) for the system in specific states. The real time is determined for the following states: S 0 , S 1 and S 4 .
where p ij is the transition probability from this state, τ ij is the time for a system in this state, F ij (τ ij ) is the distribution function for this stage of process [8 -10].
As an example, we write an equation to find the real time for the state of  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  K π ω π ω π ω π ν π ν π ν π ν π ν + + = + + + + .
To solve this task is possible with the help of mathematical programs (MathCAD), because they are suited both for numerical and symbolic solutions to many problems. Ones also have a mathematical apparatus to find solutions to linear algebraic equations [7 -12].

Experimental results
The achieved results are applied to construct the graph of dependence of the OF durability on the magnitude of the tensile load. The graph is shown in Fig. 6.