# Semi-Markov Model of the System of Repairs and Preventive Replacements by Age of City Buses
## 4 Conditions for the Existence of a Maximum of the Criterion Function
### Maximum of the Criterion Function--General Analysis
* the coefficient \(\gamma\) is determined by the formula \[\gamma=\big{[}\mathrm{p}_{12}\mathrm{ET}_{2}(z_{2}-z_{5})+\mathrm{p}_{13} \mathrm{ET}_{3}(z_{3}-z_{5})+\big{(}\mathrm{p}_{13}\mathrm{P}_{34}+\mathrm{p} _{14}\big{)}\mathrm{ET}_{4}(z_{4}-z_{5})\big{]}\mathrm{ET}_{5}\] (18) The inequality \(\gamma<0\) is equivalent to the inequality \[\big{(}\mathrm{p}_{13}+\mathrm{p}_{34}+\mathrm{p}_{14}\big{)}\mathrm{ET}_{4}> \mathrm{p}_{12}\mathrm{ET}_{2}(z_{2}-z_{5})/(z_{5}-z_{4})+\mathrm{p}_{13} \mathrm{ET}_{3}(z_{3}-z_{5})/(z_{5}-z_{4})\] (19)  
In practice, it is difficult to unambiguously determine what is the relation between the state of repair by the Technical Service unit with course loss (state 3) and the state of preventive replacement (state 5), i.e., the relation between the average values of the times \(\mathrm{ET}_{3}\) and \(\mathrm{ET}_{5}\) and between the unit costs \(\mathrm{z}_{3}\) and \(z_{5}\) is unknown. With respect to the coefficient \(\gamma\), inequality (19) must be considered similarly to inequality (16) regarding the coefficient \(\alpha\). In this case, the right-hand sides of inequalities (16) and (19) are denoted by \(\delta_{1}\) and \(\delta_{2}\), respectively. Let \(\delta=\max\{\delta_{1}\), \(\delta_{2}\}\), then the condition \((\mathrm{p}_{13}\mathrm{P}_{34}+\mathrm{p}_{14})\)\(\mathrm{ET}_{4}>\delta\) and formulas (15), (16), (18) and (19) imply the inequalities \(\alpha<0\), \(\gamma<0\). From this, the following conclusion can be made:  
**Conclusion 1**. : _If \(p_{34}>\lceil\delta/(ET_{4}-p_{14})/p_{13}\rceil\), then the inequalities \(\alpha<0\), \(\gamma<0\) are true._  
The Maximum of the Criterion Function--The Distributions of the Random Variable of the IFR Classes and MTFR  
In this subsection of the paper, sufficient conditions for the existence of the maximum of the Criterion Function (10) will be formulated in two cases. In the first case, the considerations apply to a class of random variable distributions for which the time to failure of a technical object \(\mathrm{T}_{1}\) is assumed to be a random variable with increasing damage intensity function \(\lambda 1(t)\), i.e., \(\mathrm{T}_{1}\in\mathrm{IFR}\) (Increasing Failure Rate). In the second case, a class of random variable distributions with a unimodal failure intensity function, the \(\mathrm{T}_{1}\in\mathrm{MTFR}\) (Mean Time to Failure or Repair), is considered. The results of testing the properties of the random variable distributions of the MTFR class are presented in detail in the papers [32; 33; 34]. **Conclusion 2**. : _If \(T_{1}\in\mathrm{IFR}\), \(\lambda_{1}(t)\) is differentiable, \(\alpha<0\), \(\beta>0\), \(\gamma<0\), \(\beta+\gamma f_{1}(0+)>0\), \(\lambda_{1}(\infty)\)\(\alpha\)\(ET_{1}+\beta-\alpha<0\), then the criterion function \(g(x)\) reaches its maximum value._  
**Proof of Conclusion 2. ** The derivative of the criterion function \(\mathrm{g}(x)\) has the form  
\[\mathrm{g}^{\prime}(x)=\big{\{}\alpha[\mathrm{f}_{1}(x)\mathrm{ET}_{1}(x)- \mathrm{R}_{1}(x)\mathrm{F}_{1}(x)]+\beta\mathrm{R}_{1}(x)+\gamma\mathrm{f}_{1 }(x)\big{\}}/\mathrm{M}^{2}(x)\]where M(x) is the denominator of the criterion function g(x). It is known, that if the time to failure T\({}_{1}\) belongs to the class of distributions of the random variable MTFR, then the equality H(x) = \(\lambda_{1}\)(x) ET\({}_{1}\)(x) - F\({}_{1}\)(x) \(\geq\) 0 for x \(\geq\) 0 is true. The class of distributions of the random variable MTFR has been studied in the papers [33; 34]. Some lifetime distributions with unimodal damage intensity function belong to the class of MTFR [33; 34]. From the fact, that the derivative H\({}^{\prime}\)(x) = \(\lambda_{1^{\prime}}\)(x) ET\({}_{1}\)(x), it follows that if the damage intensity function \(\lambda_{1}\)(t) increases, the function H(x) also increases. The class of distributions of a random variable with a non-decreasing damage intensity function (IFR) is contained in the MTFR class.