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In considering Composite Material Systems, the Markov Model is important for studying the behavior of composite materials. The monitoring of crack growth is suggested as the basis for this study. In fact, crack growth strongly impacts Composite Material Systems. Crack growth may lead to system failure, especially if we cannot prevent the various kinds of risk states and if we do not take necessary actions to maintain this system while in operation. In order to analyze risk states for steel materials, in the Moroccan National Railway Office, the Markov Model of a unit jump is chosen to analyze the crack growth of a composite material. This model is defined by a transition vector and a state vector, with a calculation of the averages and the extensions of the crack. Using these parameters, the jump of each extension of the crack and the number of the crack extensions are considered. A mathematical calculation helps us to find the formula for the transition probability, based on the average. An algorithm allows us to estimate the value of the crack jump. These estimations indicate the level of risk for each system state and values of the crack extension. The obtained results show that more the unit jump approximates to zero, the more the system is maintained in an acceptable operation, despite any disruptions that may influence the results.

An urban system is influenced by many disruptions that may cause failures for it, in the end. In order to maintain continuity of its operations, analysis of its components operation becomes very necessary. To do this, water infrastructure is chosen from its components to analyze the evolution of the water flow, when the population consumes the drinking water. This infrastructure is essential for the urban system and it is used daily by the population. For examining how to maintain water consumption, the evolution of the discharge head (the maximum height reached by the pipe after the pump) is analyzed and monitored. This height is strongly linked to the drinking water rate. Using water is estimated by a Markov model and the futures heights are prevented. This prevention requires the calculation of the transition probability of the water flow used by the population. An example is provided, where it is determined the level of risk. Under this one, the urban system operates in security, against failures.

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