1 Introduction

The activity of evaluation, whether of a given concept, theory, value, parameter or set of parameters, does not escape the daily lives of men. Sometimes, even unconsciously, we evaluate. What changes is the domain or context. In a family or industrial setting, for example, we can estimate the amount of resources needed or gains in terms of finances, materials, consumables, etc. for a period of time. We will be able to estimate what we need to consume to maintain a good physical fitness for a day or even longer; a head of household will be able to assess what he needs for his children’s schooling for one or more years; A civil engineer will be able to estimate the amount of money needed to construct a building, or what he will be able to earn after the construction of a building, and/or what he has earned after the construction of said building. As for the mechanical engineering designer, he will be able to evaluate, as in the latter case, the costs related to the design of a piece of equipment or a production system, but also several other parameters such as time, quality and operational safety, which are constraints that should not be neglected nowadays with the competitiveness of the global market and the complexity of the systems designed due to the integration of various technologies (Guillerm 2011). It is in this context that the notion of robustness takes place. This is because a robust production system must be able to cope with disruptions in order to maintain its performance at a high level (Ntricker and Lanza 2014). However, its disruptions take place throughout the life cycle of the system, and therefore both when it is in operation and when it is down (the quality of maintenance has an influence on the lifespan of a system). For this reason, the operational safety can be seen as an indicator or parameter to validate the robustness of a system.

Basically, there are three main concepts that help to understand the concept of operational safety (Avizienis et al. 2000). These are the attributes of operational safety, the impediments to operational safety, and the means by which operational safety is achieved. Operational safety attributes refer to the quantifiable, evaluable properties that characterize the performance of the system. These are reliability, maintainability, availability, and security. Reliability refers to continuity of service, i.e., “the ability of an entity to perform a required function, under given conditions, during a given period of time” (NF X 60–500). Maintainability refers to the ability to be repaired and evolved. It is “the ability of an entity to be maintained or restored, within a given period of time, to a state in which it can perform a required function, when maintenance is performed under given conditions, with prescribed procedures and means” (NF X 60–500). Availability is “the ability of a good to perform a required function under given conditions, at a given time during a given time interval, assuming that the supply of external needs is assured” (Nzié 2006). Safety is defined for a machine as its ability to perform a function without harming the health and integrity of the operator (Nzié 2006). Obstacles to operational safety refer to undesirable and unexpected causes or results of operation (Ciame 2009). It can be a failure, a mistake or a mistake (Avizienis et al. 2000). Failure is the “cessation of an entity’s ability to perform a required function” (DIN EN 13306 2010); Fault is the supposed cause of an error (Villemeur 1991); The error is the part of a system’s state that can cause a failure (Laprie 1995). As far as the means of operational safety are concerned, they refer to all the techniques used in order to avoid the presence of faults and thus avoid potential failures. It can be measures to prevent, tolerate, eliminate or predict faults (Avizienis et al. 2000). Fault prevention refers to actions aimed at preventing the occurrence or introduction of faults; The purpose of the fault tolerance action is to give the system the ability to deliver an acceptable service, even in the presence of faults; the elimination of faults refers to the reduction of the presence of faults; Defect prediction refers to the estimation of the presence, creation, and consequences of defects. Although operational safety is considered to be the science of failures (Villemeur 1991; Zwingelstein 1995), It can also be understood on the one hand as the ability of a system to have its functional performance (reliability, maintainability, availability) and not to generate risk (security), and on the other hand as the set of activities for assessing the suitability of a system. As a result, the assessment of operational safety involves the estimation of its attributes of reliability, maintainability, availability and safety.

A number of recent studies relating to operational safety studies of identifiable systems can be mentioned. Among them, we have the work related to computer networking systems such as, modeling the work reliability of a computer system with priority to hardware repair over hardware replacement, and upgrading software subject to technical control and MRT from the semi-Markov regenerative point technique (Kumar and Malik 2014); Availability study and analysis of power systems in the sugar industry based on variables under different types of failure and general repair (Kadyan and Kumar 2015); Analysis of the reliability and performance of an industrial system on the basis of a free warranty policy under a period of Niwas and Garg (2018). For the studies of the systems based on the copula approach (joint probability distribution), we can mention, the analysis of the availability, the MTTF (Mean time to Failure) and the costs of a system with two units in a series configuration with the failure of the controller and the man with different types of failure and two types of repair of the Gumbel-Hougaard family (Singh et al. 2013); the study of the cost evaluation based on the performance of a system composed of two subsystems in series Variable and priority additional repair of the first defective unit (Lado et al. 2018); the study of some reliability characteristics of a 2 out of 4 consecutive linear system connected to a support device for operation using the Kolmogorov method (Yusuf et al. 2018); Analysis of the network environment using a reliability approach using the Markov process that converts the Markov process to a non-Markov process (Kuldeep et al. 2017); the study of a complex repairable system composed of two subsystems in serial configurations under k-out-of-n: G/F diagram with different types of failure and two types of repair using Gumbel-Hougaard (Gahlot et al. 2020). reliability analysis of distributed hardware-software systems (Vijayalakshmi 2015); Fault tolerance study (Sari and Akkaya 2015); the study of reliability measurements, sensitivity analysis of a coal handling unit for a thermal power plant consisting of two subsystems in series configuration using additional variable techniques (Kumar and Ram 2013); Analysis of a system comprising two subsystems in serial configuration with different types of failures and copula repair approach (Abdul and Singh 2019); Prediction of the reliability of a distributed system with homogeneity in the software and the server using a joint probability distribution via the copula approach (Raghav et al. 2021).

From the above, we can again notice a strong mark of the attributes of dependability. However, the evaluation of its parameters often involves the use of probability distributions. Discrete probability distributions (binomial distribution and Poisson distribution) are used to quantify the failure under stress (Lyonnet 2006), that occurs when an entity refuses to change state when asked to do so (Villemeur 1991). Continuous probability distributions (Weibull’s distribution, exponential distribution, normal distribution, lognormal distribution, Gamma’s distribution, and Gumbel’s distribution) are associated with continuous random variables, and are used to quantify the duration of a feature’s operation (Villemeur 1991). However, the assumption made for the statistical distribution of technical repair times and operating times assumes that there is an estimator for each of the parameters. The hypothesis of invariability of the repair rate imposes a law of exponential maintainability, which is not, however, the most frequent in practice (Dhillon 2006). In addition, the lognormal distribution, which is the most common (Bellaouar 2014; Morice 1968), can be modeled by a two-parameter Weibull distribution. The two-parameter Weibull reliability distribution is also modelled, using the maximum likelihood method (Perreault and Bobee 1992; Lannoy 1994), which consists in establishing an analytic function called likelihood, based on the probability density function, whose variables are the parameters that we want to estimate, so that the maximum of said function is obtained at the estimated estimators. The maximum likelihood method has the advantage of providing an unbiased, efficient, consistent, and robust estimator (Lloyd and Lipow 1962). Two distributions allow a study of maintainability according to the size \(N\) of the sample. These are the lognormal distribution for large samples and the distribution of extreme values (LVE), including Gumbel, for small samples. This paper proposes a model for assessing operational safety and analysis in the case where operation and repair times are variable over time, using the maximum likelihood analytical method and the Gumbel graphical method as parameter estimation methods.

2 Methods

Safety analyses are important to ensure proper design and monitoring of production systems. The probabilistic approach is an effective way to account for uncertainties, and has so far been addressed for reliability, maintainability and safety studies. We therefore propose in the following paragraphs, a model to consider them together.

2.1 Reliability and maintenability: operational safety fundamental parameters

The reliability function over a period of time t is the difference between the cumulative distribution function where \(t \to \infty\) and the cumulative distribution function in the period of time t or, alternately, it is the subtraction of the cumulative distribution function of failure over a period of time t from unity.

$$R\left(t\right)=P\left({T}_{f}>t\right)=1-F\left(t\right)$$
(1)

The hazard rate function is a representation of the failure rate pattern of the ratio between a particular probability density function (p. d. f.), and its cumulative distribution function (c. d. f.) or its reliability function. For continuous random variables, the cumulative distribution function is defined by

$$F(t)={\int }_{0}^{t}f(t)dt$$
(2)

where: f (t) = probability density function of the distribution of value t over the interval \(0\) to \(t\). The hazard rate function is then defined as

$$\lambda \left(t\right)=\frac{f(t)}{R(t)}$$
(3)

Thus the reliability function is defined by:

$$R\left(t\right)={e}^{-{\int }_{0}^{t}\lambda \left(t\right)dt}$$
(4)

The maintainability function \(M(t)\), for any distribution, is expressed by the following relationship

$$M\left(t\right)=P\left({T}_{r}<t\right)= {\int}_{0}^{t}g(t)dt$$
(5)

g(t) is the probability density function of the maintenance (repair) time. Obviously maintainability is as a cumulative distribution function as the complementary of reliability F(t). Thus the maintainability function is defined also by the following expression, thanks to (1) and (4):

$$M\left(t\right)=1-{e}^{-{\int}_{0}^{t}\mu \left(t\right)dt}$$
(6)

The repair rate function is analogically defined by:

$$\mu \left(t\right)=\frac{g\left(t\right)}{1-M\left(t\right)}$$
(7)

Given that many workplace accidents occur during maintenance operations or as a result of a reliability problem (Nzié 2006), guaranteeing the security of equipment in operation implies taking the necessary measures to ensure reliability and maintainability (Nzié et al. 2017). In this case, security can be expressed as follows (Ciame 2009):

$$\begin{array}{*{20}{c}}{S\left( {{t_f},{t_r}} \right) = P\left[ {\left( {{T_f} > {t_f}} \right) \cup \left( {{T_r} < {t_r}} \right)} \right] = R\left( {{t_f}} \right) + M\left( {{t_r}} \right) - R\left( {{t_f}} \right)M\left( {{t_r}} \right)}\\ \qquad{ = {e^{ - {\int}_0^{{t_f}} \lambda \left( t \right)dt}} + 1 - {e^{ - {\int}_0^{{t_r}} \mu \left( t \right)dt}} - {e^{ - {\int}_0^{{t_f}} \lambda \left( t \right)dt}}\left( {1 - {e^{ - {\int}_0^{{t_r}} \mu \left( t \right)dt}}} \right)}\end{array}$$
(8)
$$S\left({t}_{f},{t}_{r}\right) =1+{e}^{-{\int }_{0}^{{t}_{r}}\mu \left(t\right)dt}\left({e}^{-{\int }_{0}^{{t}_{f}}\lambda \left(t\right)dt}-1\right)$$
(9)

\({t}_{f}\) and \({t}_{r}\) are respectively the variables operational and repair time of the production system

2.1.1 Probabilistic operational safety assessment model

Let us name \(OpS({t}_{f},{t}_{r})\) the operational safety function. From the equations of its fundamental parameters, reliability and maintainability given in (4) and (6), and considering the expression of safety given to Eq. (8) we express it as follows:

$$OpS\left({t}_{f},{t}_{r}\right)=P\left[\left({T}_{f}>{t}_{f}\right)\cup \left({T}_{r}<{t}_{r}\right)\cup \left({T}_{f}>{t}_{f}\right)\cup \left({T}_{r}<{t}_{r}\right)\right]=1-{\left[\left(R\left({t}_{f}\right)-1\right)\left(1-M\left({t}_{r}\right)\right)\right]}^{2}$$
(10)

Thus is deduced after some transformations the mathematical expression of operational safety.

$$OpS\left({t}_{f},{t}_{r}\right)=1-{\left[({e}^{-{\int }_{0}^{{t}_{f}}\lambda \left(t\right)dt}-1){e}^{-{\int }_{0}^{{t}_{r}}\mu \left(t\right)dt}\right]}^{2}$$
(11)

By analogy with Eqs. (3) and (7), the risk rate function can be deduced, after calculating the probability density of the functional safety function, which also makes it possible to calculate the function of the confidence rate.

Let us name now \(l\left({t}_{f},{t}_{r}\right)\), \(\varphi \left({t}_{f},{t}_{r}\right)\) and \(\psi ({t}_{f},{t}_{r})\), the functions of probability density, risk rate and confidence rate, respectively. We can express them as follows

$$l\left({t}_{f},{t}_{r}\right)=\frac{{\partial }^{2}OpS\left({t}_{f},{t}_{r}\right)}{\partial {t}_{f}\partial {t}_{r}}=4f\left({t}_{f}\right)\left[R\left({t}_{f}\right)-1\right]g\left({t}_{r}\right)\left[1-M\left({t}_{r}\right)\right]$$
(12)
$$\varphi \left({t}_{f},{t}_{r}\right)=\frac{l\left({t}_{f},{t}_{r}\right)}{OpS\left({t}_{f},{t}_{r}\right)}$$
(13)
$$\psi \left({t}_{f},{t}_{r}\right)=1-\varphi \left({t}_{f},{t}_{r}\right)$$
(14)

3 Operational safety analysis by the maximum likelihood method

3.1 Maximum likelihood method

Either to estimate a parameter \(\theta\) of a statistical distribution law on the basis of observations \({t}_{1}, {t}_{2}, ..., {t}_{n}\) of the random variable time. The likelihood \(L({t}_{1}, {t}_{2}, ... , {t}_{n}; \theta )\) is defined as the probability of observing the durations \({t}_{1},{ t}_{2}, ... , {t}_{n}\) for a given value of the parameter \(\theta\). The maximum likelihood method consists in taking as estimator of the parameter \(\theta\), the value of \(\theta\) which maximizes the likelihood.

For a reliability analysis, the likelihood function is given by the following expression (Kumar and Malik 2014):

$$L\left({t}_{1}, {t}_{2}, ... , {t}_{n};\theta \right)={\prod }_{i}^{n}f({t}_{i},\theta )$$
(15)

With \({t}_{1}, {t}_{2}, ..., {t}_{n}\): observed failure times; \(\theta = ({\theta }_{1},{ \theta }_{2}, .. , {\theta }_{r})\): the vector of parameters sought; \(f({t}_{i},\theta )\): the failure probability density function.

If \(L(t, \theta )\) is differentiable and if the maximum of the likelihood function exists at \(\widehat{\theta }\), then it satisfies the following equation:

$${\left.\frac{\partial L\left({t}_{1}, ... , {t}_{n};{\theta }_{1},\dots ,{\theta }_{r}\right)}{\partial {\theta }_{i}}\right|}_{{\theta }_{i}={\widehat{\theta }}_{i}}=0$$
(16)

Or in logarithmic form by the following equation:

$${\left.\frac{\partial ln(L\left({t}_{1}, ... , {t}_{n};{\theta }_{1},\dots ,{\theta }_{r}\right))}{\partial {\theta }_{i}}\right|}_{{\theta }_{i}={\widehat{\theta }}_{i}}=0$$
(17)

With \(i = 1, ..., r\)

3.2 Probabilistic evaluation model

For a variable failure rate, the following relation may describe Weibull’s law for the reliability function (Dhillon 2006):

$$R\left(t\right)={e}^{-{(\frac{t}{\alpha })}^{\theta }}$$
(18)

With \(\theta\), the shape parameter and \(\alpha\), the scale parameter.

The maintainability function for the law of extreme values is expressed by:

$$M\left(t\right)={e}^{{-e}^{-a(t-u)}}$$
(19)

With a, the inverse of the slope and u, the location parameter read from the Gumbel graph paper.

Thus, the operational safety expression given to Eq. (10) takes the following form:

$$OpS\left({t}_{f},{t}_{r}\right)=1-{\left[\left({e}^{-{\left(\frac{t}{\alpha }\right)}^{\theta }}-1\right)\left(1-{e}^{{-e}^{-a\left({t}_{r}-u\right)}}\right)\right]}^{2}$$
(20)

3.3 Application of the maximum likelihood method to weibull’s law of operational safety

Relation (15) assumes knowledge of the probability density function, which is defined as the time derivative of the dependability function. Starting from relation (12) we can deduce the Weibull operational safety law given in relation (21) below:

$$l\left({t}_{f},{t}_{r}\right)=4\left[-\frac{\theta }{\alpha }{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta -1}{e}^{-{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta }}\left({e}^{-{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta }}-1\right)\right]*\left[a{e}^{-a\left({t}_{r}-u\right)}{e}^{{-e}^{-a\left({t}_{r}-u\right)}}\left(1-{e}^{{-e}^{-a({t}_{r}-u)}}\right)\right]$$
(21)

The application of relation (15) for the likelihood on the probability density function given in relation (21) leads to the following relation:

$$L\left({t}_{1}, {t}_{2}, \dots , {t}_{n};\theta \right)={\prod }_{i=1}^{n}l\left({t}_{i},\theta ,\alpha \right)={\prod }_{i=1}^{n}l\left({t}_{fi},{t}_{ri},\theta ,\alpha \right)$$
$$={(4)}^{n}{\prod }_{i=1}^{n}\left[-\frac{\theta }{\alpha }{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta -1}{e}^{-{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta }}\left({1-e}^{-{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta }}\right)\right]*$$
$${\prod }_{i=1}^{n}\left[a{e}^{-a\left({t}_{r}-u\right)}{e}^{{-e}^{-a\left({t}_{r}-u\right)}}\left(1-{e}^{{-e}^{-a({t}_{r}-u)}}\right)\right]$$
(22)

By introducing the natural logarithm in each member of the relation (22), we obtain the following relation:

$$lnL\left({t}_{1}, {t}_{2}, \dots , {t}_{n};\theta \right)=nln4+nln\theta -nln\alpha +\left(\theta -1\right){\sum\limits_{i=1}^{n}}ln\left(\frac{{t}_{fi}}{\alpha }\right)+{\sum\limits_{i=1}^{n}}ln\left({1-e}^{-{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta }}\right)+n{\text{ln}}a+{\sum\limits_{i=1}^{n}}\left[-a\left({t}_{ri}-u\right)\right]-{\sum\limits_{i=1}^{n}}\left[{-e}^{-a\left({t}_{ri}-u\right)}\right]+{\sum\limits_{i=1}^{n}}ln\left(1-{e}^{{-e}^{-a({t}_{r}-u)}}\right)$$
(23)

By performing the member-to-member partial derivatives of expression (23) with respect to the scale parameter \(\alpha\) and the shape parameter \(\theta\), we successively obtain the following relations:

$$\frac{\partial lnL}{\partial \alpha }=-\frac{n\theta }{\alpha }+{\sum\limits_{i=1}^{n}}\frac{\theta }{{(\alpha )}^{\theta +1}}{{(t}_{fi})}^{\theta }-{\sum\limits_{i=1}^{n}}\frac{\left(\frac{\theta }{\alpha }\right)\left(\frac{{t}_{fi}}{\alpha }\right){e}^{-{\left(\frac{{t}_{fi}}{\alpha }\right)}^{\theta }}}{\left({1-e}^{-{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta }}\right)}$$
(24)
$$\frac{\partial lnL}{\partial \theta }=\frac{n}{\theta }+{\sum\limits_{i=1}^{n}}ln\left(\frac{{t}_{fi}}{\alpha }\right)-{\sum\limits_{i=1}^{n}}{\left(\frac{{t}_{fi}}{\alpha }\right)}^{\theta }ln\left(\frac{{t}_{fi}}{\alpha }\right)-{\sum\limits_{i=1}^{n}}\frac{\left(\frac{{t}_{fi}}{\alpha }\right)ln\left(\frac{{t}_{fi}}{\alpha }\right){e}^{-{\left(\frac{{t}_{fi}}{\alpha }\right)}^{\theta }}}{\left({1-e}^{-{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta }}\right)}$$
(25)

The estimators of the scale parameters \(\alpha\) and form \(\theta\) are those, which maximize the likelihood function in its form in relation (23). This means that they are the solutions of the system of equations below:

$$\left\{\begin{array}{c}\frac{\partial lnL}{\partial \alpha }=0\\ \frac{\partial lnL}{\partial \theta }=0\end{array}\right.$$
(26)

By carrying the expressions (24) and (25) in the system of Eqs. (26), the latter takes the following form:

$$\left\{\begin{array}{c}-\frac{n\theta }{\alpha }+{\sum\limits_{i=1}^{n}}\frac{\theta }{{\left(\alpha \right)}^{\theta +1}}{{(t}_{fi})}^{\theta }-{\sum\limits_{i=1}^{n}}\frac{\left(\frac{\theta }{\alpha }\right)\left(\frac{{t}_{fi}}{\alpha }\right){e}^{-{\left(\frac{{t}_{fi}}{\alpha }\right)}^{\theta }}}{\left({1-e}^{-{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta }}\right)}=0\\ \frac{n}{\theta }+{\sum\limits_{i=1}^{n}}ln\left(\frac{{t}_{fi}}{\alpha }\right)-{\sum\limits_{i=1}^{n}}{\left(\frac{{t}_{fi}}{\alpha }\right)}^{\theta }ln\left(\frac{{t}_{fi}}{\alpha }\right)-{\sum\limits_{i=1}^{n}}\frac{\left(\frac{{t}_{fi}}{\alpha }\right)ln\left(\frac{{t}_{fi}}{\alpha }\right){e}^{-{\left(\frac{{t}_{fi}}{\alpha }\right)}^{\theta }}}{\left({1-e}^{-{\left(\frac{{t}_{f}}{\alpha }\right)}^{\theta }}\right)}=0\end{array}\right.$$
(27)

The system of Eqs. (27) can be solved by successive approximations following indications with starting values (Lannoy 1994; Lloyd and Lipow 1962).

4 Results and discussion: application to the equipment of the CCPD in garoua

The FMEA studies of the equipment of the CCPD of Garoua given in Tables 1, 2, 3 and 4 show that the feed pump, the fuel and oil filter, the injection pump, all the bearings of the diesel and gasoline engines have criticalities greater than 12. They are therefore considered critical according to the company.

Table 1 FMEA centrifugal pump
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Table 2 FMEA of the lubrication circuit
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Table 3 FMEA of the air cooling circuit
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Table 4 FMECA of diesel engine auxiliary elements
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Also, the criticality values observed in the FMEA tables allow us to note that diesel and gasoline engine assemblies are critical equipment for the fire protection room; pumps and its various auxiliaries in general are critical equipment for the pump; and the diesel engine of the emergency generator is the critical equipment of this room. Figures 1 and 2 give a view of the different fire pump unit.

Fig. 1
figure 1

View of fire pump unit No. 1 and 3

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Fig. 2
figure 2

View of fire pump unit No. 2

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Table 5 provides information on the time between failures and technical times to repair for its equipment over a two-year period.

Table 5 TBF and TTR readings for each module
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The following expression is used to estimate the initial values of \(\theta\) (Niwas and Garg 2018). The mean of each sample considered as the initial value for \(\alpha\).

$$\frac{\sum {t}_{i}}{n}={\left(\frac{\sum {{t}_{i}}^{\theta }}{n}\right)}^{\frac{1}{\theta }}*\Gamma \left(1+\frac{1}{\theta }\right)$$
(28)

For the fire pump unit No.1, we have:

$$\frac{\sum {t}_{i}}{n}=35.45$$
(29)

For \(\theta =1.2\) we have:

$${\left(\frac{\sum {{t}_{i}}^{\theta }}{n}\right)}^{\frac{1}{\theta }}=36.70261$$
(30)
$$\Gamma \left(1+\frac{1}{\theta }\right)=0.9664$$
(31)

And,

$${\left(\frac{\sum {{t}_{i}}^{\theta }}{n}\right)}^{\frac{1}{\theta }}*\Gamma \left(1+\frac{1}{\theta }\right)=35.4694$$
(32)

The values given by the expressions (29) and (32) being close, we can take 1.2 as the initial value for FPU No. 1. For the FPUs No. 2 and 3, we obtain as initial values 1.5 and 1.2 respectively. With R under calculator by successive approximations, we obtain the values of Table 6 for \(\theta\) and \(\alpha\).

Table 6 Different parameters for each FPU
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The different \(\theta\) values obtained for each FPU allow us to realize that its equipment is already in the aging phase of its life cycles \((\theta >1)\), with a slightly advanced state compared to the others for FPU n°2. However, it is more difficult to predict a failure at this stage of a system’s life cycle because failures are random, hence the need to pay more attention to them to ensure better maintenance.

According to Table 7, we have: \({D}_{n,\alpha }={D}_{\mathrm{14,005}}=0.3489\) et \({D}_{max}=0.179\). Since \({D}_{max}<{D}_{\mathrm{14,005}}\), the hypothesis of a Weibull distribution with a 5% risk of being wrong is therefore accepted. \({D}_{max}=0.154\) For FPU No. 2 and \({D}_{max}=0.065\) for the FPU No. 3, which also makes it possible to validate for its two other FPUs the hypothesis of a Weibull distribution with a risk of error of 5%. Therefore, we can write the following expressions for each FPU. We can therefore have the expressions of \(OpS\left({t}_{f},{t}_{r}\right)\),\(\varphi \left({t}_{f},{t}_{r}\right)\) and \(\psi \left({t}_{f},{t}_{r}\right)\) for each FPU. Their different curves are given in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12.

Table 7 Kolmogorov–Smirnov test for FPG No. 1
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Fig. 3
figure 3

Operational safety curve for FPU No. 1

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Fig. 4
figure 4

Probability density curve for FPU No. 1

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Fig. 5
figure 5

Risk rate curve for FPU No. 1

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Fig. 6
figure 6

Confidence rate curve for FPU No. 1

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Fig. 7
figure 7

Operational safety curve for FPU No. 2

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Fig. 8
figure 8

Probability density curve for FPU No. 2

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Fig. 9
figure 9

Risk rate curve for FPU No. 2

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Fig. 10
figure 10

Confidence rate curve for FPU No. 2

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Fig. 11
figure 11

Operating safety curve for FPU No. 3

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Fig. 12
figure 12

Probability density curve for FPU No. 3

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$${OpS}_{1}\left({t}_{f},{t}_{r}\right)=1-{\left[\left({e}^{-{\left(\frac{{t}_{f}}{35.344}\right)}^{1.312}}-1\right)\left({1-e}^{{-e}^{-0.683\left({t}_{r}-1\right)}}\right)\right]}^{2}$$
(33)
$${l}_{1}\left({t}_{f},{t}_{r}\right)=-4\left[0.037{\left(\frac{{t}_{f}}{35.344}\right)}^{0.312}{e}^{-{\left(\frac{{t}_{f}}{35.344}\right)}^{1.312}}\left({e}^{-{\left(\frac{{t}_{f}}{35.344}\right)}^{1.312}}-1\right)\right]\left[0.683{e}^{-0.683\left({t}_{r}-1\right)}{e}^{{-e}^{-0.683\left({t}_{r}-1\right)}}\left(1-{e}^{{-e}^{-0.683({t}_{r}-1)}}\right)\right]$$
(34)
$${\varphi }_{1}\left({t}_{f},{t}_{r}\right)=-4\left[\frac{0.037{\left(\frac{{t}_{f}}{35.344}\right)}^{0.312}{e}^{-{\left(\frac{{t}_{f}}{35.344}\right)}^{1.312}}\left({e}^{-{\left(\frac{{t}_{f}}{35.344}\right)}^{1.312}}-1\right)}{1-{\left[\left({e}^{-{\left(\frac{{t}_{f}}{35.344}\right)}^{1.312}}-1\right)\left({1-e}^{{-e}^{-0.683\left({t}_{r}-1\right)}}\right)\right]}^{2}}\right]*$$
$$\left[0.683{e}^{-0.683\left({t}_{r}-1\right)}{e}^{{-e}^{-0.683\left({t}_{r}-1\right)}}\left(1-{e}^{{-e}^{-0.683({t}_{r}-1)}}\right)\right]$$
(35)
$${\psi }_{1}\left({t}_{f},{t}_{r}\right)=1+4\left[\frac{0.037{\left(\frac{{t}_{f}}{35.344}\right)}^{0.312}{e}^{-{\left(\frac{{t}_{f}}{35.344}\right)}^{1.312}}\left({e}^{-{\left(\frac{{t}_{f}}{35.344}\right)}^{1.312}}-1\right)}{1-{\left[\left({e}^{-{\left(\frac{{t}_{f}}{35.344}\right)}^{1.312}}-1\right)\left({1-e}^{{-e}^{-0.683\left({t}_{r}-1\right)}}\right)\right]}^{2}}\right]*$$
$$\left[0.683{e}^{-0.683\left({t}_{r}-1\right)}{e}^{{-e}^{-0.683\left({t}_{r}-1\right)}}\left(1-{e}^{{-e}^{-0.683({t}_{r}-1)}}\right)\right]$$
(36)
$${OpS}_{2}\left({t}_{f},{t}_{r}\right)=1-{\left[\left({e}^{-{\left(\frac{t}{62.102}\right)}^{1.643}}-1\right)\left(1-{e}^{{-e}^{-0.725\left({t}_{r}-0.825\right)}}\right)\right]}^{2}$$
(37)
$${l}_{2}\left({t}_{f},{t}_{r}\right)=-4\left[0.026{\left(\frac{{t}_{f}}{62.102}\right)}^{0.643}{e}^{-{\left(\frac{{t}_{f}}{62.102}\right)}^{1.643}}\left({e}^{-{\left(\frac{{t}_{f}}{62.102}\right)}^{1.643}}-1\right)\right]\left[0.725{e}^{-0.725\left({t}_{r}-0.825\right)}{e}^{{-e}^{-0.725\left({t}_{r}-0.825\right)}}\left(1-{e}^{{-e}^{-0.725({t}_{r}-0.825)}}\right)\right]$$
(38)
$${\varphi }_{2}\left({t}_{f},{t}_{r}\right)=-4\left[\frac{0.026{\left(\frac{{t}_{f}}{62.102}\right)}^{0.643}{e}^{-{\left(\frac{{t}_{f}}{62.102}\right)}^{1.643}}\left({e}^{-{\left(\frac{{t}_{f}}{62.102}\right)}^{1.643}}-1\right)}{1-{\left[\left({e}^{-{\left(\frac{t}{62.102}\right)}^{1.643}}-1\right)\left(1-{e}^{{-e}^{-0.725\left({t}_{r}-0.825\right)}}\right)\right]}^{2}}\right]*$$
$$\left[0.725{e}^{-0.725\left({t}_{r}-0.825\right)}{e}^{{-e}^{-0.725\left({t}_{r}-0.825\right)}}\left(1-{e}^{{-e}^{-0.725({t}_{r}-0.825)}}\right)\right]$$
(39)
$${\psi }_{2}\left({t}_{f},{t}_{r}\right)=1+4\left[\frac{0.026{\left(\frac{{t}_{f}}{62.102}\right)}^{0.643}{e}^{-{\left(\frac{{t}_{f}}{62.102}\right)}^{1.643}}\left({e}^{-{\left(\frac{{t}_{f}}{62.102}\right)}^{1.643}}-1\right)}{1-{\left[\left({e}^{-{\left(\frac{t}{62.102}\right)}^{1.643}}-1\right)\left(1-{e}^{{-e}^{-0.725\left({t}_{r}-0.825\right)}}\right)\right]}^{2}}\right]*$$
$$\left[0.725{e}^{-0.725\left({t}_{r}-0.825\right)}{e}^{{-e}^{-0.725\left({t}_{r}-0.825\right)}}\left(1-{e}^{{-e}^{-0.725({t}_{r}-0.825)}}\right)\right]$$
(40)
$${OpS}_{3}\left({t}_{f},{t}_{r}\right)=1-{\left[\left({e}^{-{\left(\frac{{t}_{f}}{46.695}\right)}^{1.223}}-1\right)\left(1-{e}^{{-e}^{-0.793\left({t}_{r}-1.07\right)}}\right)\right]}^{2}$$
(41)
$${l}_{3}\left({t}_{f},{t}_{r}\right)=-4\left[0.026{\left(\frac{{t}_{f}}{46.695}\right)}^{0.223}{e}^{-{\left(\frac{{t}_{f}}{46.695}\right)}^{1.223}}\left({e}^{-{\left(\frac{{t}_{f}}{46.695}\right)}^{1.223}}-1\right)\right]\left[0.793{e}^{-0.793\left({t}_{r}-1.07\right)}{e}^{{-e}^{-0.793\left({t}_{r}-1.07\right)}}\left(1-{e}^{{-e}^{-0.793({t}_{r}-1.07)}}\right)\right]$$
(42)
$${\varphi }_{3}\left({t}_{f},{t}_{r}\right)=-4\left[\frac{0.026{\left(\frac{{t}_{f}}{46.695}\right)}^{0.223}{e}^{-{\left(\frac{{t}_{f}}{46.695}\right)}^{1.223}}\left({e}^{-{\left(\frac{{t}_{f}}{46.695}\right)}^{1.223}}-1\right)}{1-{\left[\left({e}^{-{\left(\frac{{t}_{f}}{46.695}\right)}^{1.223}}-1\right)\left(1-{e}^{{-e}^{-0.793\left({t}_{r}-1.07\right)}}\right)\right]}^{2}}\right]*$$
$$\left[0.793{e}^{-0.793\left({t}_{r}-1.07\right)}{e}^{{-e}^{-0.793\left({t}_{r}-1.07\right)}}\left(1-{e}^{{-e}^{-0.793({t}_{r}-1.07)}}\right)\right]$$
(43)
$${\psi }_{3}\left({t}_{f},{t}_{r}\right)=1+4\left[\frac{0.026{\left(\frac{{t}_{f}}{46.695}\right)}^{0.223}{e}^{-{\left(\frac{{t}_{f}}{46.695}\right)}^{1.223}}\left({e}^{-{\left(\frac{{t}_{f}}{46.695}\right)}^{1.223}}-1\right)}{1-{\left[\left({e}^{-{\left(\frac{{t}_{f}}{46.695}\right)}^{1.223}}-1\right)\left(1-{e}^{{-e}^{-0.793\left({t}_{r}-1.07\right)}}\right)\right]}^{2}}\right]*$$
$$\left[0.793{e}^{-0.793\left({t}_{r}-1.07\right)}{e}^{{-e}^{-0.793\left({t}_{r}-1.07\right)}}\left(1-{e}^{{-e}^{-0.793({t}_{r}-1.07)}}\right)\right]$$
(44)

By observing the different curves obtained, we can see that the different operational safety curves all have the same look. The same applies to probability densities, risk rates and confidence rates. For operational safety curves, we have an area where the probability that the system is safe presents uncertainties, relatively between 0.93 and 0.98 with repair times between 0.5 and 2 h, and operating times greater than or equal to 45. The other part of the different operational safety curves gives probabilities relatively close to 1. It can also be seen that for the different safety curves, in the area where the probability of the system being safe is low, the probability density and the rate of risk of non-security are significantly greater, and the confidence rate is greater. The reverse also occurs when we are in the zone where this probability is close to 1. It is practically zero for FPUs No. 2 and 3, with values close to 10−3. The fact that the different FPUs have an overall probability of being 1 and therefore 100% safe is also indicative of a good design of its equipment, guaranteeing good reliability and maintainability. This fact can be perceived by observing the confidence rate curves, which have values that oscillate around 99.65% and 100% (see Figs. 13 and 14).

Fig. 13
figure 13

Risk rate curve for FPU No. 3

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Fig. 14
figure 14

Confidence rate curve for FPU No. 3

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5 Conclusion

Setting up a functioning production system is the challenge of any designer. In this document, our concern was to propose a probabilistic model for assessing operational safety and to show how it can be exploited, supported by the practical case of the equipment of the CCPD of Garoua. The exploitation of probability distributions allowed us to implement a model. The equations and curves obtained for each pumping unit considered in the practical case allow us to consider this model in a very positive way, insofar as they sufficiently describe what is real. Although it is not easily exploitable and requires knowledge of the times between failure and technical times to repair of the system, we propose in perspective, the implementation of techniques allowing the exploitation of the model at the design stage.