DCFP	Dependent competing failure processes
${\tau }_1,{\tau }_2,\dots$	Shocks (mixed harmless, critical, extreme) arrival process in Model 1
${\sum }_{k=1}^{\infty }{\epsilon }_{{\tau }_k}$	The point process of shocks in the form of random measure
${\epsilon }_a$	Is Dirac point mass at point $a\in \mathbb{R}$
$W_1,W_2,\dots$	Shocks magnitudes at ${\tau }_1,{\tau }_2,\dots$
${\sum }_{k=1}^{\infty }{W}_k{\epsilon }_{{\tau }_k}$	Shocks process as marked random measure with ${\sum }_{k=1}^{\infty }{\epsilon }_{{\tau }_k}$ as the associated support counting measure
$H_1,H_2$	Shocks’ thresholds
$H_1\ge W$	Harmless shock
$H_1<W\le H_2$	Critical shock
$H_2<W$	Extreme shock
fatal shock	Nth critical or extreme shock
$\delta$-shock	When ${\tau }_{i+1}-{\tau }_i<\delta$
r.v.	Random variable
i.i.d.	Independent and identically distributed
$\left[W\right]$	Equivalence class of all r.v.s that are stochastically equivalent to r.v. W
$\left(\mathsf{\Omega },\mathcal{F},\mathbb{P}\right)$	Probability space
$E_1$	$\left\{W\le H_1\right\}$
$E_2$	$\left\{H_1<W\le H_2\right\}$
$E_3$	$\left\{H_2<W\right\}$
$1_A$	Indicator function parametrized by an event A
shocks identifiers	$X,Y,$ and later on $U,V$
Y	$1_{E_2}+N{\mathbf{1}}_{E_3}$
$\left\{a,b,c\right\}$	Probability distribution of r.v. Y
PGF	Probability generating function
$a\left(z\right)$	$\mathbb{E}{z}^Y=a{z}^N+bz+c$ PGF of r.v. Y
${\sum }_{k=1}^{\infty }{Y}_k{\epsilon }_{{\tau }_k}$	Auxiliary point process marked by the sequence $\left\{Y_k\right\}$ associated with sequence $\left\{W_k\right\}$ of shocks’ magnitudes
$B_n$	${\sum }_{k=1}^n{Y}_k,n=1,2,\dots$
$\nu$	Min$\left\{n:B_n\ge N\right\}$ nominal count of harmful shocks not counting $\delta$-shocks on system’s failure
$X_i$	$1_{\left\{\left|\left({\tau }_{i-1},{\tau }_i\right)\right|<\delta \right\}},i=1,2,\dots ,{\tau }_0=0,$ sequence of i.i.d. Bernoulli r.v.s counting $\delta$-shocks
$\left|A\right|$	Lebesgue measure of Borel set A
$\Delta$	An r.v. stochastically equivalent to r.v.s $\left|\left({\tau }_{i-1},{\tau }_i\right)\right|,i=1,2,\dots ,{\tau }_0=0$
$\alpha$	$\mathbb{P}\left\{\Delta <\delta \right\}$
LST	Laplace–Stieltjes transform
$\gamma \left(y,z,\theta \right)$	$\mathbb{E}{y}^X{z}^Y{e}^{-\theta \Delta }$ the joint transform of $\left(X_i,Y_i,{\Delta }_i\right),i=1,2,\dots$
${\gamma }_0\left(y,z,\theta \right)$	$\mathbb{E}{x}^{X_0}{z}^{Y_0}{e}^{-\theta {\Delta }_0}$
$g\left(y,\theta \right)$	$\mathbb{E}{y}^X{e}^{-\theta \Delta }=\mathbb{E}{y}^{{\mathbf{1}}_{\{\Delta <\delta \}}}{e}^{-\theta \Delta }=\alpha \left(\theta \right)y+\beta \left(\theta \right)$ the marginal transform of $\left(X_i,{\Delta }_i\right),i=1,2,\dots$
$\alpha \left(\theta \right)$	$\mathbb{E}{\mathbf{1}}_{\left\{\Delta <\delta \right\}}{e}^{-\theta \Delta }$
$\beta \left(\theta \right)$	$\mathbb{E}{\mathbf{1}}_{\left\{\Delta \ge \delta \right\}}{e}^{-\theta \Delta }$
$\alpha$	$\alpha \left(0\right)$
$\beta$	$\beta \left(0\right)$
$\gamma \left(\theta \right)$	$\mathbb{E}{e}^{-\theta \Delta }=g\left(1,\theta \right)=\alpha \left(\theta \right)+\beta \left(\theta \right)$
$\frac{1}{\gamma }$	$\mathbb{E}\Delta$
$A_n$	${\sum }_{k=1}^n{X}_k,n=1,2,\dots ,$
$\mu$	Min$\left\{m:A_m=M\right\}$
$\rho$	$\mu \wedge \nu$ cumulative ruin indexalso total shocks count on failure in Model 1
${\tau }_{\rho }$	Time-to-failure or lifetime of the system in Model 1
$\left(\mathsf{\Omega },\mathcal{F},\left({\mathcal{F}}_t\right),\mathbb{P}\right)$	Filtered probability space
${\epsilon }_a$	Dirac (unit) point mass at a point a
$\left(A,B,\tau \right)$	${\sum }_{k=1}^{\infty }(X_k,Y_k){\epsilon }_{{\tau }_k}$ marked random measure representing the input stream of shocks
$\left(A_{\rho },B_{\rho },{\tau }_{\rho }\right)$	${\sum }_{k=1}^{\rho }(X_k,Y_k){\epsilon }_{{\tau }_k}$ input stream of shocks consolidated by ${\tau }_{\rho }$
$A_{\rho }$	$\delta$-shocks count by ${\tau }_{\rho }$
$B_{\rho }$	Critical/extreme shocks damage by ${\tau }_{\rho }$
${\Phi }_{\rho }(s,y,z,\theta ,u,v,$ $\vartheta ;M,N)$	$\mathbb{E}{s}^{\rho }{y}^{A_{\rho }}{z}^{B_{\rho }}{u}^{A_{\rho -1}}{v}^{B_{\rho -1}}{e}^{-\theta {\tau }_{\rho }-\vartheta {\tau }_{\rho -1}}$ comprehensive information on the system at time-to-failure ${\tau }_{\rho },$ including $\rho$ (total shock count on failure); $A_{\rho }$ $\left(\le M\right)$—the total number of $\delta$-shocks; $B_{\rho }$—the number of critical and fatal shocks combined; and at time ${\tau }_{\rho -1}$ preceding to failure, as their joint distribution
${\Phi }_{\mu >\nu }(s,y,z,\theta ,s,$ $u,v,\vartheta ;M,N)$	status of the system in the form of $\mathbb{E}{s}^{\rho }{y}^{A_{\rho }}{z}^{B_{\rho }}{u}^{A_{\rho -1}}{v}^{B_{\rho -1}}{e}^{-\theta {\tau }_{\rho }-\vartheta {\tau }_{\rho -1}}{\mathbf{1}}_{\mu >\nu }$ on the confined space $\left(\mathsf{\Omega },\mathcal{F}\cap \left\{\nu <\mu \right\},{\mathbb{P}}_{\left\{\nu <\mu \right\}}\right)$ that fails due to Nth critical or one extreme shock, but not due to $\delta$-shocks
${\mathcal{D}}_x^{k}F(x,y)$	$\mathcal{D}$-operator defined as ${lim}_{x\to 0}\frac{1}{k!}\frac{{\partial }^k}{\partial x^k}\left[\frac{1}{1-x}F(x,y)\right]$
${\phi }_{\rho }\left(s,y,z,\theta ;N\right)$	${\Phi }_{\rho }\left(s,y,z,\theta ,1,1,0;1,N\right)=\mathbb{E}{s}^{\rho }{y}^{A_{\rho }}{z}^{B_{\rho }}{e}^{-\theta {\tau }_{\rho }}$ marginal functional representing the joint probability distribution of $\left(\rho ,A_{\rho },B_{\rho },{\tau }_{\rho }\right)$ pertaining to the status of the system predicted by the time-to-failure at ${\tau }_{\rho }$ in Model 1
$T_j$	${\tau }_{{\eta }_j}=$ min$\left\{{\tau }_n>T_{j-1}:W_n>H_1\right\},j=1,2,\dots ,T_0={\tau }_0=0$ in Model 2
${\eta }_j$	Min$\left\{n>{\eta }_{j-1}:W_n>H_1\right\},j=1,2,3,\dots ,{\eta }_0=0$
$\left\{T_j\right\}$	Embedded sequence of consecutive harmful shocks in Model 2
$V_j$	$1_{\left\{T_j-T_{j-1}<\delta \right\}},j=1,2,\dots$
$U_j$	$N{\mathbf{1}}_{\left\{W_{{\eta }_j}>H_2\right\}}+{\mathbf{1}}_{\left\{H_1<W_{{\eta }_j}\le H_2\right\}}$
$\left(\mathcal{V},\mathcal{U},\mathcal{T}\right)$	${\sum }_{k=0}^{\infty }\left(V_j,U_j\right){\epsilon }_{T_k}$delayed marked renewal process representing only harmful shocks in Model 2
V	$1_{\left\{{\tau }_{\eta }<\delta \right\}}$
p	$a+b$
$a_0$	$\frac{a}{a+b}$
$b_0$	$\frac{b}{a+b}$
$a_0\left(z\right)$	$a_0{z}^N+b_{0}z=\frac{1}{p}\left[a{z}^N+bz\right]=\mathbb{E}\left[z^Y{\mathbf{1}}_{{\mathsf{\Omega }}_0}\right]\frac{1}{\mathbb{P}\left({\mathsf{\Omega }}_0\right)}=\frac{1}{\mathbb{P}\left({\mathsf{\Omega }}_0\right)}{\int }_{{\mathsf{\Omega }}_0}{z}^{Y}d\mathbb{P}$
${\mathsf{\Omega }}_0$	$\left\{W>H_1\right\}=\left\{Y\in \left\{1,N\right\}\right\}=E_2\cup E_3$
$\Gamma \left(y,z,\theta \right)$	$\mathbb{E}{y}^V{z}^U{e}^{-\theta {\tau }_{\eta }}=a_0\left(z\right)G\left(y,\theta \right)=\left(a_0{z}^N+b_{0}z\right)\left[{\alpha }_0\left(\theta \right)y+{\beta }_0\left(\theta \right)\right]$
$\Gamma \left(\theta \right)$	$\Gamma \left(1,1,\theta \right)=\mathbb{E}{e}^{-\theta {\tau }_{\eta }}=\mathbb{E}{e}^{-\theta T_1}$
$G\left(y,\theta \right)$	$\mathbb{E}{e}^{-\theta {\tau }_{\eta }}{y}^V=y\mathbb{E}{e}^{-\theta {\tau }_{\eta }}{\mathbf{1}}_{\left\{{\tau }_{\eta }<\delta \right\}}+\mathbb{E}{e}^{-\theta {\tau }_{\eta }}{\mathbf{1}}_{\left\{{\tau }_{\eta }\ge \delta \right\}}={\alpha }_0\left(\theta \right)y+{\beta }_0\left(\theta \right)$
${\alpha }_0\left(\theta \right)$	$\mathbb{E}{e}^{-\theta {\tau }_{\eta }}{\mathbf{1}}_{\left\{{\tau }_{\eta }<\delta \right\}}$
${\beta }_0\left(\theta \right)$	$\mathbb{E}{e}^{-\theta {\tau }_{\eta }}{\mathbf{1}}_{\left\{{\tau }_{\eta }\ge \delta \right\}}$
${\alpha }_0$	${\alpha }_0\left(0\right)=\mathbb{P}\left\{{\tau }_{\eta }<\delta \right\}$
${\beta }_0$	${\beta }_0\left(0\right)=\mathbb{P}\left\{{\tau }_{\eta }\ge \delta \right\}$
${\mathcal{U}}_n$	${\sum }_{k=1}^n{U}_k,n=1,2,\dots$
$\zeta$	Min$\left\{n:{\mathcal{U}}_n\ge N\right\}$
$\chi$	Min$\left\{m:{\mathcal{A}}_m={\sum }_{k=1}^m{V}_k=1\right\}$
$\rho$	$\chi \wedge \zeta$ ruin index in Model 2, also total count of harmful shocks (critical/extreme/$\delta$-shocks) until failure in Model 2
$T_{\rho }$	Time-to-failure in Model 2
${\left(\mathcal{V},\mathcal{U},\mathcal{T}\right)}_{\rho }$	${\sum }_{k=0}^{\rho }\left(V_j,U_j\right){\epsilon }_{T_k}$ input stream of harmful shocks consolidated by $T_{\rho }$
${\mathcal{A}}_{\rho }$	$\delta$-shocks count by $T_{\rho }$
${\mathcal{B}}_{\rho }$	Impact of critical/extreme shocks in Model 2
${\psi }_{\rho }\left(s,y,z,\theta ;N\right)$	$\mathbb{E}{s}^{\rho }{y}^{{\mathcal{A}}_{\rho }}{z}^{{\mathcal{B}}_{\rho }}{e}^{-\theta T_{\rho }}$ marginal functional representing the joint probability distribution pertaining to the status of the system driven by $\left(\rho ,{\mathcal{A}}_{\rho },{\mathcal{B}}_{\rho },T_{\rho }\right)$ predicted at the time-to-failure at $T_{\rho }$ in Model 2