Stochastic modelling of operational quality of k-out-of-n systems

Abstract

In this paper, we study some operational quality measures of the k-out-of-n systems. Performance of these systems is characterized by a quality (output) function that is decreasing in the absence of components’ failures. Moreover, it decreases (downward jump) with each failure of a component in a system as well. We employ the point processes approach for description of the corresponding failure process and derivation of the main results. Expectations (unconditional and conditional on survival) and variability for the system’s quality function are derived and analyzed. The corresponding ‘future quality process’ is defined and described. Some illustrative examples are presented.

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Acknowledgements

The authors would like to thank referees for helpful comments and advices. The work of the first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2019R1A2B5B02069500). The work of the first author was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant Number: 2019R1A6A1A11051177).The work of the second author was supported by National Research Foundation (SA) (Grant No: 103613).

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Appendix

Appendix

Proof of Lemma 1

Let \(s_{0} \equiv 0\), \(s_{m + 1} \equiv t\), \(\Delta t_{0} \equiv 0\), and \(\Delta t_{j}\) is infinitesimally small so that \(s_{j} + \Delta t_{j} < s_{j + 1}\), \(j = 1,2, \ldots ,m\). Then, using the notion of stochastic intensity,

$$\begin{aligned} P(s_{j} \le S_{j} \le s_{j} + \Delta t_{j} ,j = 1, \ldots ,m,N(t) = m) = P(\{ {\text{No}}\,\,{\text{events}}\,\,{\text{in}}\,\,(s_{j - 1} + \Delta t_{j - 1} ,s_{j} ),\,\,1\,\,{\text{event}}\,\,{\text{in}}\,[s_{j} ,\,s_{j} + \Delta t_{j} ]\} ,j = 1,2, \ldots ,m,\, \hfill \\ \quad \quad {\text{no}}\,\,{\text{events}}{\kern 1pt} \,\,{\text{in}}\,\,(s_{m} + \Delta t_{m} ,t))\, \hfill \\ \quad \quad = \left[ {\exp \{ - \varLambda_{0} (0,s_{1} )\} (\lambda_{0} (s_{1} )\Delta t_{1} + o(\Delta t_{1} ))} \right. \times \exp \{ - \varLambda_{1} (s_{1} + \Delta t_{1} ,s{}_{2})\} (\lambda_{1} (s_{2} |s_{1} )\Delta t_{2} + o(\Delta t_{2} )) \times \cdots \hfill \\ \quad \quad \quad \times \exp \{ - \varLambda_{m - 1} (s_{m - 1} + \Delta t_{m - 1} ,s_{m} )\} (\lambda_{m - 1} (s_{m} |s_{1} ,s_{2} , \ldots ,s_{m - 1} )\Delta t_{m} + o(\Delta t_{m} ))\left. {\exp \{ - \varLambda_{m} (s_{m} + \Delta t_{m} ,t)\} } \right] \hfill \\ \quad \quad = \prod\limits_{j = 0}^{m} {\left( {\lambda_{j} (s_{j + 1} |s_{1} ,s_{2} , \ldots ,s_{j} )\Delta t_{j + 1} + o(\Delta t_{j + 1} )} \right)} \cdot \exp \left\{ { - \sum\limits_{j = 0}^{m} {\varLambda_{j} (s_{j} + \Delta t_{j} ,s_{j + 1} )} } \right\}. \hfill \\ \end{aligned}$$

Denote by \(f_{{S_{1} ,S_{2} , \ldots ,S_{N(t)} ,N(t)}} (s_{1} ,s_{2} , \ldots ,s_{m} ,m)\) the joint distribution of \((S_{1} = s_{1} ,S_{2} = s_{2} , \ldots ,S_{m} = s_{m} ,N(t) = m)\). Then

$$\begin{aligned} f_{{S_{1} ,S_{2} , \ldots ,S_{N(t)} ,N(t)}} (s_{1} ,s_{2} , \ldots ,s_{m} ,m) = \lim_{{\Delta t_{j} \to 0,j = 1,2, \ldots ,m}} \frac{{P(s_{j} \le S_{j} \le s_{j} + \Delta t_{j} ,j = 1, \ldots ,m,N(t) = m)}}{{\left( {\prod\nolimits_{j = 1}^{m} {\Delta t_{j} } } \right)}} \hfill \\ \quad = \left[ {\prod\limits_{j = 0}^{m - 1} {\lambda_{j} } (s_{j + 1} |s_{1} ,s_{2} , \ldots ,s_{j} )\exp \{ - \varLambda_{j} (s_{j} ,s_{j + 1} )\} } \right] \cdot \exp \{ - \varLambda_{m} (s_{m} ,t)\} ,\quad 0 \le s_{1} \le s_{2} \le \cdots \le s_{m} \le t,\quad m = 1,2, \ldots ,n - k. \hfill \\ \end{aligned}$$

Proof of Theorem 4

Denote by \(\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\}\) the realization of \(\{ S_{1} ,S_{2} , \ldots ,S_{N(s)} ,N(s)\}\). By definition,

$$\begin{aligned} Q_{E} & (t|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s) = E[\tilde{Q}(s + t)I(s + t)|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s] \hfill \\ & = E\left[ {Q(s + t)\prod\limits_{i = 1}^{n(s)} {\exp \{ - \psi_{i} (s_{i} )\} } \prod\limits_{j = N(s) + 1}^{N(s + t)} {\exp \{ - \psi_{j} (S_{j} )\} } \cdot I(s + t)|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s} \right] \hfill \\ & = Q(s + t)\prod\limits_{i = 1}^{n(s)} {\exp \{ - \psi_{i} (s_{i} )\} } E\left[ {\prod\limits_{j = N(s) + 1}^{N(s + t)} {\exp \{ - \psi_{j} (S_{j} )\} } \cdot I(s + t)|\{ t_{1} ,t_{2} , \ldots ,t_{n(s)} ,n(s)\} } \right] \hfill \\ & = Q(s + t)\prod\limits_{i = 1}^{n(s)} {\exp \{ - \psi_{i} (s_{i} )\} } E\left[ {\prod\limits_{j = 1}^{{N_{s} (t)}} {\exp \{ - \psi_{n(s) + j} (s + S_{j}^{*} )\} } \cdot I(s + t)|\{ t_{1} ,t_{2} , \ldots ,t_{n(s)} ,n(s)\} } \right], \hfill \\ \end{aligned}$$

where \(n(s) \le n - k\), \(N_{s} (t) = N(s + t) - N(s)\), \(S_{j}^{*}\) is the time from \(s\) to the \(j\) th component’s failure occurred in \((s,s + t]\), \(j = 1,2, \ldots ,N_{s} (t)\). Note that, given \(\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\}\), at time \(s\) the system is the \(k\)-out-of \(n - n(s)\) system with the common components failure rate \(\lambda (s + u)\), \(u \ge 0\). Thus, applying similar procedure as that in the proof of Theorem 1,

$$\begin{aligned} & E \left[ {\prod\limits_{j = 1}^{{N_{s} (t)}} {\exp \{ - \psi_{n(s) + j} (s + S_{j}^{*} )\} } \cdot I(s + t)|\{ t_{1} ,t_{2} , \ldots ,t_{n(s)} ,n(s)\} } \right] \hfill \\ & \quad = \exp \{ - (n - n(s))\varLambda_{s} (t)\} + \sum\limits_{m = 1}^{n - n(s) - k} {\int\limits_{0}^{t} {\int\limits_{0}^{{v_{m} }} \cdots } } \int\limits_{0}^{{v_{2} }} {\prod\limits_{i = 1}^{m} {\exp \{ - \psi_{n(s) + i} (s + v_{i} )\} } } \hfill \\ & \quad \;\; \times \left[ {\prod\limits_{j = 0}^{m - 1} {\lambda_{sj} } (v_{j + 1} |v_{1} ,v_{2} , \ldots ,v_{j} )\exp \{ - \varLambda_{sj} (v_{j} ,v_{j + 1} )\} } \right] \cdot \exp \{ - \varLambda_{sm} (v_{m} ,t)\} dv_{1} dv_{2} \ldots dv_{m} , \hfill \\ \end{aligned}$$

where \(\varLambda_{s} (t) = \int_{0}^{t} {\lambda (s + w)} dw\), \(\lambda_{sj} (t|v_{1} ,v_{2} , \ldots ,v_{j} ) = (n - n(s) - j)\lambda (s + t)\), \(j = 0,1,2, \ldots ,n - n(s) - k\), \(\varLambda_{sj} (u_{1} ,u_{2} ) = \int_{{u_{1} }}^{{u_{2} }} {\lambda_{sj} (w|v_{1} ,v_{2} , \ldots ,v_{j} )} dw\).

Observe that

$$\begin{aligned} & Q_{E} (t|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s) = E[\tilde{Q}(s + t)I(s + t)|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s] \hfill \\ & = E[\tilde{Q}(s + t)I(s + t)|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s,T > s + t]P(T > s + t|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s) \hfill & \\ + E[\tilde{Q}(s + t)I(s + t)|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s,T \le s + t]P(T \le s + t|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s) \hfill & \\ = E[\tilde{Q}(s + t)I(s + t)|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s + t]P(T > s + t|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s) \hfill & \\ = Q_{ES} (t|\{ t_{1} ,t_{2} , \ldots ,t_{n(s)} ,n(s)\} ,T > s)P(T > s + t|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ). \hfill \\ \end{aligned}$$

Again, given \(\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\}\) at time \(s\), the system is the \(k\)-out-of \(n - n(s)\) system with the common components failure rate \(\lambda (s + u)\), \(u \ge 0\). Thus,

$$P(T > s + t|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ) = \sum\limits_{l = k}^{n - n(s)} {} \left( {\begin{array}{*{20}c} {n - n(s)} \\ l \\ \end{array} } \right)(\exp \{ - \varLambda_{s} (t)\} )^{l} (1 - \exp \{ - \varLambda_{s} (t)\} )^{n - n(s) - l} ,$$

and

$$Q_{ES} (t|\{ t_{1} ,t_{2} , \ldots ,t_{n(s)} ,n(s)\} ,T > s) = \frac{{Q_{E} (t|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} ,T > s)}}{{P(T > s + t|\{ s_{1} ,s_{2} , \ldots ,s_{n(s)} ,n(s)\} )}}.$$

Proof of Theorem 5

Observe that

$$\begin{aligned} Q_{E} (t|T > s) & = E[\tilde{Q}(s + t)I(s + t)|T > s] \hfill \\ & = E_{{(\{ S_{1} ,S_{2} , \ldots ,S_{N(s)} ,N(s)\} |T > s)}} [E[\tilde{Q}(s + t)I(s + t)|\{ S_{1} ,S_{2} , \ldots ,S_{N(s)} ,N(s)\} ,T > s]], \hfill \\ \end{aligned}$$

where \(E_{{(\{ S_{1} ,S_{2} , \ldots ,S_{N(s)} ,N(s)\} |T > s)}} [ \cdot ]\) stands for the expectation with respect to the conditional joint distribution \((\{ S_{1} ,S_{2} , \ldots ,S_{N(s)} ,N(s)\} |T > s)\). From Theorem 4

$$\begin{aligned} & E[\tilde{Q}(s + t)I(s + t)|\{ T_{1} ,T_{2} , \ldots ,T_{N(s)} ,N(s)\} ,T > s] \hfill \\ & = Q(s + t)\prod\limits_{i = 1}^{n(s)} {\exp \{ - \psi (s_{i} )\} } \times \left\{ {\exp \{ - (n - n(s))\varLambda_{s} (t)\} } \right. \hfill \\ & \quad + \sum\limits_{m = 1}^{n - n(s) - k} {\int\limits_{0}^{t} {\int\limits_{0}^{{v_{m} }} \cdots } } \int\limits_{0}^{{v_{2} }} {\prod\limits_{i = 1}^{m} {\exp \{ - \psi_{n(s) + i} (s + v_{i} )\} } } \hfill \\ & \;\; \times \left. {\left[ {\prod\limits_{j = 0}^{m - 1} {\lambda_{sj} } (v_{j + 1} |v_{1} ,v_{2} , \ldots ,v_{j} )\exp \{ - \varLambda_{sj} (v_{j} ,v_{j + 1} )\} } \right] \cdot \exp \{ - \varLambda_{sm} (v_{m} ,t)\} dv_{1} dv_{2} \ldots dv_{m} } \right\}, \hfill \\ \end{aligned}$$

and the conditional joint distribution \((\{ S_{1} ,S_{2} , \ldots ,S_{N(s)} ,N(s)\} |T > s)\) is given by

$$\begin{aligned} & \frac{1}{P(T > s)}\left[ {\prod\limits_{j = 0}^{m - 1} {\lambda_{j} } (s_{j + 1} |s_{1} ,s_{2} , \ldots ,s_{j} )\exp \{ - \varLambda_{j} (s_{j} ,s_{j + 1} )\} } \right] \cdot \exp \{ - \varLambda_{m} (s_{m} ,t)\} \hfill \\ & \quad 0 \le s_{1} \le s_{2} \le \cdots \le s_{m} \le s,m = 1,2, \ldots ,n - k. \hfill \\ \end{aligned}$$

Let

$$\begin{aligned} \varPhi (r) & \equiv \left\{ {\exp \{ - (n - r)\varLambda_{s} (t)\} + \sum\limits_{m = 1}^{n - r - k} {\int\limits_{0}^{t} {\int\limits_{0}^{{v_{m} }} \cdots } } \int\limits_{0}^{{v_{2} }} {\prod\limits_{i = 1}^{m} {\exp \{ - \psi_{r + i} (s + v_{i} )\} } } } \right. \hfill \\ & \quad \left. { \times \left[ {\prod\limits_{j = 0}^{m - 1} {\lambda_{sj} } (v_{j + 1} |v_{1} ,v_{2} , \ldots ,v_{j} )\exp \{ - \varLambda_{sj} (v_{j} ,v_{j + 1} )\} } \right] \cdot \exp \{ - \varLambda_{sm} (v_{m} ,t)\} dv_{1} dv_{2} \ldots dv_{m} } \right\}. \hfill \\ \end{aligned}$$

Then

$$\begin{aligned} Q_{E} (t|T > s) & = \frac{{\exp \{ - n\varLambda (s)\} }}{{\sum\nolimits_{l = k}^{n} {\left( {\begin{array}{*{20}c} n \\ l \\ \end{array} } \right)} (\exp \{ - \varLambda (s)\} )^{l} (1 - \exp \{ - \varLambda (s)\} )^{n - l} }}Q(s + t)\varPhi (0) \hfill \\ & \; + \frac{Q(s + t)}{{\sum\nolimits_{l = k}^{n} {\left( {\begin{array}{*{20}c} n \\ l \\ \end{array} } \right)} (\exp \{ - \varLambda (s)\} )^{l} (1 - \exp \{ - \varLambda (s)\} )^{n - l} }} \hfill \\ & \times \sum\limits_{r = 1}^{n - k} {\int\limits_{0}^{s} {\int\limits_{0}^{{s_{r} }} \cdots \int\limits_{0}^{{s_{2} }} {\left[ {\prod\limits_{i = 1}^{r} {\exp \{ - \psi (s_{i} )\} } } \right]\varPhi (r)} } } \hfill \\ & \times \left[ {\prod\limits_{j = 0}^{r - 1} {\lambda_{j} } (s_{j + 1} |s_{1} ,s_{2} , \ldots ,s_{j} )\exp \{ - \varLambda_{j} (s_{j} ,s_{j + 1} )\} } \right] \cdot \exp \{ - \varLambda_{r} (s_{r} ,t)\} ds_{1} \ldots ds_{r - 1} ds_{r} . \hfill \\ \end{aligned}$$

On the other hand, by definition, \(Q_{ES} (t|T > s) = E[\tilde{Q}(s + t)|T > s + t] = Q_{ES} (s + t)\)

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Cha, J.H., Finkelstein, M. Stochastic modelling of operational quality of k-out-of-n systems. TOP 28, 424–441 (2020). https://doi.org/10.1007/s11750-019-00536-y

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Keywords

  • Quality function
  • k-out-of-n system
  • Point process
  • Stochastic intensity
  • Residual lifetime

Mathematics Subject Classification

  • 90B25
  • 60K10