Reliability analysis of a repairable system with geometric reneging and threshold-based recovery policy

Abstract

This article attempts to investigate the reliability measures of a repairable system with $M$ identical operating units, $S$ standby units, and a repairman. The repairman is subject to breakdowns and operates threshold-based recovery policy. As soon as the repairman breaks down, the repair does not start immediately until the number of failed units in the system reaches a specified threshold value $Q(0\le Q\le S)$ . When the repairman is working or broken down, failed units arriving for maintenance may renege and decide sequentially whether to leave the queue or not. The failed times of operating and standby units, maintenance times and reneging time of failed units, and breakdown and repair times of the repairman are all assumed to be exponential distributions. We develop the system reliability and mean time to system failure. Through numerical results, sensitivity analysis is carried out to assess the impact of various system parameters on the system reliability and mean time to system failure. Finally, an application example is given to evaluate the reliability measures of an electric power system.

Introduction

System reliability is an important concern in repairable systems, and it has been extensively investigated for applications in power plants, manufacturing systems, and industrial systems. This article focuses on reliability measures for a repairable system with standby units, geometric reneging, and threshold-based recovery policy. Threshold-based recovery policy was introduced by Efrosini and Semenova ¹ and further discussed by Efrosini and Winkler ² in the analysis of an M/M/1 retrial queue with constant retrial rate and unreliable server. Purohit et al. ³ utilized the Runge–Kutta method to compute the steady-state probabilities of an M/M/1 retrial queue with constant retrial policy, an unreliable server, threshold-based recovery, and state-dependent arrival rates. A cost analysis of a repairable M/M/1/N queueing system under a threshold-based recovery policy is conducted by Yang et al. ⁴ Recently, Yang and Chiang ⁵ dealt with a machine repair problem under a threshold-based recovery policy. They employed the particle swarm optimization algorithm to solve the cost optimization problem.

Considerable research has focused on repairable systems from the perspective of queueing; however, few studies have approached the problem from the perspective of reliability. Cao and Cheng ⁶ were the pioneers to incorporate the reliability concepts in the queueing systems. They investigated an M/G/1 queueing system with a repairable service station, where the lifetime and repair time of the service station are exponential and general distributions, respectively. Wang and Sivazlian ⁷ investigated the reliability characteristics of a repairable system with warm standbys and multi-repairmen. Later, Ke and Wang ⁸ extended the model of Wang and Sivazlian ⁷ to cases involving balking and reneging. Wang and Kuo ⁹ analyzed the reliability and availability characteristics of four different series system configurations with cold and warm standbys. Jain et al. ¹⁰ used the Laplace transform technique to obtain the reliability function and mean time to system failure (MTTF) of a machine repair system with spares and reneging, where the repairman operates the N-policy. Using the same technique, Ke et al. ¹¹ discussed the reliability measures of a repairable system with standby switching failures and reboot delay. Wang et al. ¹² carried out a comparison study of the reliability and availability in four different system configurations with warm standby components and standby switching failures. Lee et al. ¹³ employed a Bayesian method, using different prior distributions of the system parameters, to study a redundant repairable system with standby imperfect switching and reboot delay. Ke et al. ¹⁴ studied the reliability analysis of a repairable system with standby switching failures and reboot delay. Recently, Wang et al. ¹⁵ examined a repairable system with imperfect coverage under service pressure condition. They conducted a sensitivity analysis of the reliability characteristics to describe the impact of system parameters on the reliability characteristics.

Queueing models are generally investigated under the assumption of a reliable server. However, in most practical cases, servers are subject to breakdowns, making queueing systems with server breakdowns a topic worthy of further consideration. A comprehensive review of research on this topic is found in Krishnamoorthy et al. ¹⁶ From the perspective of reliability, Li et al. ¹⁷ discussed the reliability measures of the M/G/1 queue with Bernoulli vacation and server breakdowns. Li et al. ¹⁸ considered the PH/PH/1 queueing system with server breakdown and repair, in which the lifetime and repair time of the server are exponential and phase type distributions, respectively. In their work, they showed that the proposed reliability approximation can be applied to various Markovian queueing systems with server breakdowns. Using the supplementary variable method, Wang et al. ¹⁹ obtained explicit expressions for the availability, frequency of failure, and reliability function of the server for the retrial queues with server breakdowns and repairs. Wang et al. ²⁰ applied the Laplace transform technique to develop the reliability function and MTTF of a system with warm standbys and a repairable service station. Wang et al. ²¹ applied the same technique to examine the reliability characteristics of a repairable system with warm standby units and multiple unreliable service stations. Later, Ke et al. ²² expanded upon this work with the inclusion of imperfect switching mechanisms. Lin et al. ²³ constructed the membership functions of the reliability and availability characteristics of a repairable system with one primary machine, one standby machine, and an unreliable service station.

Dimou et al. ²⁴ considered a single vacation queueing model with geometric reneging, in which customers become impatient during the absence of the server. They applied a generating function approach to derive the equilibrium distribution for the number of customers in the system. In a recent work by Dimou and Economou, ²⁵ the generating function approach was applied in the analysis of single-server queues with catastrophes and geometric reneging. To the best of our knowledge, only little attention was paid to queueing models with geometric reneging. We therefore consider a Markovian repairable system with geometric reneging and threshold-based recovery policy from the perspective of reliability. The purpose of this article is threefold: (1) to develop the reliability measures of the repairable system with standbys, geometric reneging, and threshold-based recovery policy; (2) to examine the impact of system parameters on the reliability measures using numerical experiments; and (3) to present an application example for illustrating the potential applicability of the reliability model.

The model

We consider a reliability model with $M$ identical operating units, $S$ standby units, and an unreliable repairman. Each operating unit fails according to an exponential distribution with parameter $\lambda$ . When an operating unit fails, it is replaced by an available standby unit with perfect switch. Each available standby unit may fail independent of each other and has an exponential distribution with parameter $\eta$ . Moreover, the switchover times are assumed to be instantaneous. As soon as a standby unit becomes an operating unit, its failure rate will be transformed by that of an operating unit. According to different failure rates of a standby unit, there are three types of standby units: cold, warm, and hot standby units. A standby unit is called as a cold standby, if its failure rate is 0 ( $\eta =0$ ). A hot standby unit implies that the failure rate of a standby unit is equal to that of an operating unit ( $\eta =\lambda$ ). When the failure rate of a standby unit is between that of cold and hot standby ones, it is said to be a warm standby unit ( $0<\eta <\lambda$ ). Whenever an operating unit or a standby unit fails, it is maintained by an unreliable repairman based on the order of their breakdowns. The repairman can maintain only one failed unit at a time, and the maintenance times of failed units by the repairman are exponentially random variables with mean $1/\mu$ .

When the repairman is maintaining the failed units, he is subject to breakdowns and repairs. If the repairman breaks down, he operates the threshold-based recovery policy. This means that the breakdown repairman cannot be repaired until the number of failed units in the system reaches to a specified threshold value $Q(0\le Q\le S)$ . Breakdown and repair times of the repairman are assumed to be exponential distributions with rates $\alpha$ and $\beta$ , respectively. While the repairman breaks down, the stream of new failed units continues. After the broken repairman is repaired, the maintenance of failed units is immediately started. When the repairman is working or broken down, the failed unit waits in the queue will be maintained after a certain length of time $t$ , and it may conduct abandonments. The time $t$ is a random variable, which follows exponential distribution with parameter $r$ . Failed units are considered one by one sequentially rather than simultaneously. Each failed unit leaves the queue with probability $\delta$ or remains in the queue with complementary probability $q=(1-\delta )$ . As soon as the failed units begin sequentially to leave the queue, the number of failed units in the system is reduced until the first one stays in the queue or all failed units leave the queue. Thus, the number of failed units in the system is reduced following a geometric distribution. Various stochastic processes involved in the system are mutually independent of each other. Furthermore, we will refer this model as the repairable system with standbys, geometric reneging, and threshold-based recovery policy.

The system failure is defined to be less than $K$ units in operative state, where $K=1,2,\dots ,M$ . Therefore, the system failure occurs if and only if the number of failed units in the system is larger or equal to $L=M+S-K+1$ .

Reliability measures

In this section, we develop the reliability function of the repairable system with standbys, geometric reneging, and threshold-based recovery policy. Then, the MTTF is derived by using the reliability function.

Differential-difference equations

We first define the mean failure rate ${\lambda }_n$ as follows

$${\lambda }_n=\{\begin{array}{c}\hfill M\lambda +(S-n)\eta ,0\le n\le S-1\hfill \\ \multicolumn{1}{c}{(M+S-n)\lambda ,S\le n\le L-1}\\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$$

(1)

The state-transition-rate diagram for the reliability system with standbys, geometric reneging, and threshold-based recovery policy is shown in Figure 1. Referring to Figure 1, we set the differential-difference equations governing the reliability model as follows

$$\frac{d{P}_{0,0}(t)}{\mathrm{dt}}=-({\lambda }_0+\alpha )P_{0,0}(t)+u{P}_{0,1}(t)+{\displaystyle \sum _{i=1}^{L-1}}{\delta }^{i}r{P}_{0,i}(t)$$

(2)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{d{P}_{0,1}(t)}{\mathrm{dt}}}& =-({\lambda }_1+\alpha +u+\delta r)P_{0,1}(t)\hfill \\ \multicolumn{1}{c}{}& +{\lambda }_0{P}_{0,0}(t)+u{P}_{0,2}(t)+{\displaystyle \sum _{i=2}^{L-1}}{\delta }^{i-1}\mathrm{qr}{P}_{0,i}(t)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{d{P}_{0,n}(t)}{\mathrm{dt}}=}& -({\lambda }_n+\alpha +u+{\delta }^{n}r+{\displaystyle \sum _{i=1}^{n-1}}{\delta }^i\mathrm{qr})P_{0,n}(t)+{\lambda }_{n-1}\hfill \\ \multicolumn{1}{c}{}& P_{0,n-1}(t)+u{P}_{0,n+1}(t)\hfill \\ \multicolumn{1}{c}{}& +{\displaystyle \sum _{i=n+1}^{L-1}}{\delta }^{i-n}\mathrm{qr}{P}_{0,i}(t),2\le n\le Q-1\hfill \end{array}$$

(4)

$$\begin{array}{cc}\multicolumn{1}{c}{}& \frac{d{P}_{0,n}(t)}{\mathrm{dt}}=-({\lambda }_n+\alpha +u+{\delta }^{n}r+{\displaystyle \sum _{i=1}^{n-1}}{\delta }^i\mathrm{qr})\hfill \\ \multicolumn{1}{c}{}& P_{0,n}(t)+{\lambda }_{n-1}{P}_{0,n-1}(t)+u{P}_{0,n+1}(t)\hfill \\ \multicolumn{1}{c}{}& +{\displaystyle \sum _{i=n+1}^{L-1}}{\delta }^{i-n}\mathrm{qr}{P}_{0,i}(t)+\beta P_{1,n}(t),Q\le n\le L-2\hfill \end{array}$$

(5)

$$\frac{d{P}_{0,L-1}(t)}{\mathrm{dt}}=-({\lambda }_{L-1}+\alpha +u+{\delta }^{L-1}r+{\displaystyle \sum _{i=1}^{L-2}}{\delta }^i\mathrm{qr})P_{0,L-1}(t)+{\lambda }_{L-2}{P}_{0,L-2}(t)+\beta P_{1,L-1}(t)$$

(6)

$$\frac{d{P}_{0,L}(t)}{\mathrm{dt}}={\lambda }_{L-1}{P}_{0,L-1}(t)$$

(7)

$$\frac{d{P}_{1,0}(t)}{\mathrm{dt}}=-{\lambda }_0{P}_{1,0}(t)+\alpha P_{0,0}(t)+{\displaystyle \sum _{i=1}^{L-1}}{\delta }^{i}r{P}_{1,i}(t)$$

(8)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{d{P}_{1,1}(t)}{\mathrm{dt}}=}& -({\lambda }_1+\delta r)P_{1,1}(t)+{\lambda }_0{P}_{1,0}(t)+\alpha P_{0,1}(t)\hfill \\ \multicolumn{1}{c}{}& +{\displaystyle \sum _{i=2}^{L-1}}{\delta }^{i-1}\mathrm{qr}{P}_{1,i}(t)\hfill \end{array}$$

(9)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{d{P}_{1,n}(t)}{\mathrm{dt}}=}& -({\lambda }_n+{\delta }^{n}r+{\displaystyle \sum _{i=1}^{n-1}}{\delta }^i\mathrm{qr})P_{1,n}(t)\hfill \\ \multicolumn{1}{c}{}& +{\lambda }_{n-1}{P}_{1,n-1}(t)+\alpha P_{0,n}(t)\hfill \\ \multicolumn{1}{c}{}& +{\displaystyle \sum _{i=n+1}^{L-1}}{\delta }^{i-n}\mathrm{qr}{P}_{1,i}(t),2\le n\le Q-1\hfill \end{array}$$

(10)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{d{P}_{1,n}(t)}{\mathrm{dt}}=}& -({\lambda }_n+{\delta }^{n}r+{\displaystyle \sum _{i=1}^{n-1}}{\delta }^i\mathrm{qr}+\beta )P_{1,n}(t)\hfill \\ \multicolumn{1}{c}{}& +{\lambda }_{n-1}{P}_{1,n-1}(t)+\alpha P_{0,n}(t)\hfill \\ \multicolumn{1}{c}{}& +{\displaystyle \sum _{i=n+1}^{L-1}}{\delta }^{i-n}\mathrm{qr}{P}_{1,i}(t),Q\le n\le L-2\hfill \end{array}$$

(11)

$$\frac{d{P}_{1,L-1}(t)}{dt}=-({\lambda }_{L-1}+{\delta }^{L-1}r+{\displaystyle {\displaystyle \sum _{i=1}^{L-2}}{\delta }^{i}qr+\beta })P_{1,L-1}(t)+{\lambda }_{L-2}{P}_{1,L-2}(t)+\alpha P_{0,L-1}(t)$$

(12)

$$\frac{d{P}_{1,L}(t)}{\mathrm{dt}}={\lambda }_{L-1}{P}_{1,L-1}(t)$$

(13)

Figure 1.

State-transition-rate diagram for the reliability system with standbys, geometric reneging, and threshold-based recovery policy.

Laplace transform equations

The Laplace transform of $P_{0,n}(t)$ and $P_{1,n}(t)$ are given by

$$P_{j,n}^{*}(s)={\displaystyle \underset{0}{\overset{\infty }{\int }}}{e}^{-\mathrm{st}}{P}_{j,n}(t)\mathrm{dt},j=0,1\mathrm{and}n=0,1,\dots ,L$$

(14)

Taking the Laplace transform on both sides of equations (2)–(13) and using equation (14), we obtain

$$({\lambda }_0+\alpha +s)P_{0,0}^{*}(s)-\mu P_{0,1}^{*}(s)-{\displaystyle \sum _{i=1}^{L-1}}{\delta }^{n}r{P}_{0,i}^{*}(s)=P_{0,0}(0)$$

(15)

$$\begin{array}{cc}\multicolumn{1}{c}{}& ({\lambda }_n+\alpha +\mu +\delta r+s)P_{0,n}^{*}(s)-{\lambda }_{n-1}{P}_{0,n-1}^{*}(s)\hfill \\ \multicolumn{1}{c}{}& -\mu P_{0,n+1}^{*}(s)-{\displaystyle \sum _{i=n+1}^{L-1}}{\delta }^{i-n}\mathrm{qr}{P}_{0,i}^{*}(s)=P_{0,n}(0),\hfill \\ \multicolumn{1}{c}{}& 1\le n\le Q-1\hfill \end{array}$$

(16)

$$({\lambda }_n+\alpha +\mu +\delta r+s)P_{0,n}^{*}(s)-{\lambda }_{n-1}{P}_{0,n-1}^{*}(s)-\mu P_{0,n+1}^{*}(s)-{\displaystyle {\displaystyle \sum _{i=n+1}^{L-1}}{\delta }^{i-n}qr}{P}_{0,i}^{*}(s)-\beta P_{1,n}^{*}(s)=P_{0,n}(0),Q\le n\le L-2$$

(17)

$$\begin{array}{cc}\multicolumn{1}{c}{}& ({\lambda }_{L-1}+\alpha +\mu +\delta r+s)P_{0,L-1}^{*}(s)\hfill \\ \multicolumn{1}{c}{}& -{\lambda }_{L-2}{P}_{0,L-2}^{*}(s)-\beta P_{1,L-1}^{*}(s)=P_{0,L-1}(0)\hfill \end{array}$$

(18)

$${\mathrm{sP}}_{0,L}^{*}(s)-{\lambda }_{L-1}{P}_{0,L-1}^{*}(s)=P_{0,L}(0)$$

(19)

$$({\lambda }_0+s)P_{1,0}^{*}(s)-\alpha P_{0,0}^{*}(s)-{\displaystyle \sum _{i=1}^{L-1}}{\delta }^{i}r{P}_{1,i}^{*}(s)=P_{1,0}(0)$$

(20)

$$\begin{array}{cc}\multicolumn{1}{c}{}& ({\lambda }_n+\delta r+s)P_{1,n}^{*}(s)-\alpha P_{0,n}^{*}(s)-{\lambda }_{n-1}{P}_{1,n-1}^{*}(s)\hfill \\ \multicolumn{1}{c}{}& -{\displaystyle \sum _{i=n+1}^{L-1}}{\delta }^{i-n}\mathrm{qr}{P}_{1,i}^{*}(s)=P_{1,n}(0),1\le n\le Q-1\hfill \end{array}$$

(21)

$$\begin{array}{cc}\multicolumn{1}{c}{}& ({\lambda }_n+\delta r+\beta +s)P_{1,n}^{*}(s)-\alpha P_{0,n}^{*}(s)-{\lambda }_{n-1}{P}_{1,n-1}^{*}(s)\hfill \\ \multicolumn{1}{c}{}& -{\displaystyle \sum _{i=n+1}^{L-1}}{\delta }^{i-n}\mathrm{qr}{P}_{1,i}^{*}(s)=P_{1,n}(0),Q\le n\le L-2\hfill \end{array}$$

(22)

$$\begin{array}{cc}\multicolumn{1}{c}{}& ({\lambda }_{L-1}+\delta r+\beta +s)P_{1,L-1}^{*}(s)-{\lambda }_{L-2}{P}_{1,L-2}^{*}(s)\hfill \\ \multicolumn{1}{c}{}& -\alpha P_{0,L-1}^{*}(s)=P_{1,L-1}(0)\hfill \end{array}$$

(23)

$${\mathrm{sP}}_{1,L}^{*}(s)-{\lambda }_{L-1}{P}_{1,L-1}^{*}(s)=P_{1,L}(0)$$

(24)

Equations (15)–(24) can be written in matrix form as

$$\mathbf{D}(s){\mathbf{W}}^{\mathbf{*}}(s)=\mathbf{W}(0)$$

(25)

where $\mathbf{D}(s)=$

$$\begin{array}{cc}\begin{array}{c}\\ (0,0)\\ (0,1)\\ (0,2)\\ ⋮\\ (0,Q-1)\\ (0,Q)\\ (0,Q+1)\\ ⋮\\ (0,L-1)\\ (0,L)\\ (1,0)\\ (1,1)\\ (1,2)\\ ⋮\\ (1,Q-1)\\ (1,Q)\\ (1,Q+1)\\ ⋮\\ (1,L-1)\\ (1,L)\end{array}& \begin{array}{c}\begin{array}{cccccccccccccccccccc}(0,0)& (0,1)& (0,2)& \cdots & (0,Q-1)& (0,Q)& (0,Q+1)& \cdots & (0,L-1)& (0,L)& (1,0)& (1,1)& (1,2)& \cdots & (1,Q-1)& (1,Q)& (1,Q+1)& \cdots & (1,L-1)& (1,L)\end{array}\\ \left[\begin{array}{cccccccccccccccccccc}s+{\lambda }_0+\alpha & -\mu -\delta r& -{\delta }^{2}r& \cdots & -{\delta }^{Q-1}r& -{\delta }^{Q}r& -{\delta }^{Q+1}r& \cdots & -{\delta }^{L-1}r& 0& 0& 0& 0& \cdots & 0& 0& 0& \cdots & 0& 0\\ -{\lambda }_0& \begin{array}{l}s+{\lambda }_1+\alpha \\ +\mu +\delta r\end{array}& -\mu -\delta qr& \cdots & -{\delta }^{Q-2}qr& -{\delta }^{Q-1}qr& -{\delta }^{Q}qr& \cdots & -{\delta }^{L-2}qr& 0& 0& 0& 0& \cdots & 0& 0& 0& \cdots & 0& 0\\ 0& -{\lambda }_1& \begin{array}{l}s+{\lambda }_2+\alpha \\ +\mu +\delta r\end{array}& \cdots & -{\delta }^{Q-3}qr& -{\delta }^{Q-2}qr& -{\delta }^{Q-1}qr& \cdots & -{\delta }^{L-3}qr& 0& 0& 0& 0& \cdots & 0& 0& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & \begin{array}{l}s+{\lambda }_{Q-1}+\alpha \\ +\mu +\delta r\end{array}& -\mu -\delta qr& -{\delta }^{2}qr& \cdots & -{\delta }^{L-Q}qr& 0& 0& 0& 0& \cdots & 0& 0& 0& \cdots & 0& 0\\ 0& 0& 0& \cdots & -{\lambda }_{Q-1}& \begin{array}{l}s+{\lambda }_Q+\alpha \\ +\mu +\delta r\end{array}& -\mu -\delta qr& \cdots & -{\delta }^{L-Q-1}qr& 0& 0& 0& 0& \cdots & 0& -\beta & 0& \ddots & 0& 0\\ 0& 0& 0& \cdots & 0& -{\lambda }_Q& \begin{array}{l}s+{\lambda }_{Q+1}+\alpha \\ +\mu +\delta r\end{array}& \cdots & -{\delta }^{L-Q-2}qr& 0& 0& 0& 0& \cdots & 0& 0& -\beta & \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 0& 0& \cdots & \begin{array}{l}s+{\lambda }_{L-1}+\alpha \\ +\mu +\delta r\end{array}& 0& 0& 0& 0& \cdots & 0& 0& 0& \cdots & -\beta & 0\\ 0& 0& 0& \cdots & 0& 0& 0& \cdots & -{\lambda }_{L-1}& s& 0& 0& 0& \cdots & 0& 0& 0& \cdots & 0& 0\\ -\alpha & 0& 0& \cdots & 0& 0& 0& \cdots & 0& 0& s+{\lambda }_0& -\delta r& -{\delta }^{2}r& \cdots & -{\delta }^{Q-1}r& -{\delta }^{Q}r& -{\delta }^{Q+1}r& \cdots & -{\delta }^{L-1}r& 0\\ 0& -\alpha & 0& \cdots & 0& 0& 0& \cdots & 0& 0& -{\lambda }_0& s+{\lambda }_1+\delta r& -\delta qr& \cdots & -{\delta }^{Q-2}qr& -{\delta }^{Q-1}qr& -{\delta }^{Q}qr& \cdots & -{\delta }^{L-2}qr& 0\\ 0& 0& -\alpha & \cdots & 0& 0& 0& \cdots & 0& 0& 0& -{\lambda }_1& s+{\lambda }_2+\delta r& \cdots & -{\delta }^{Q-3}qr& -{\delta }^{Q-2}qr& -{\delta }^{Q-1}qr& \cdots & -{\delta }^{L-3}qr& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & -\alpha & 0& 0& \cdots & 0& 0& 0& 0& 0& \cdots & s+{\lambda }_{Q-1}+\delta r& -\delta qr& -{\delta }^{2}qr& \cdots & -{\delta }^{L-Q}qr& 0\\ 0& 0& 0& \cdots & 0& -\alpha & 0& \cdots & 0& 0& 0& 0& 0& \cdots & -{\lambda }_{Q-1}& s+{\lambda }_Q+\beta +\delta r& -\delta qr& \cdots & -{\delta }^{L-Q-1}qr& 0\\ 0& 0& 0& \cdots & 0& 0& -\alpha & \cdots & 0& 0& 0& 0& 0& \cdots & 0& -{\lambda }_Q& s+{\lambda }_{Q+1}+\beta +\delta r& \cdots & -{\delta }^{L-Q-2}qr& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 0& 0& \cdots & -\alpha & 0& 0& 0& 0& \cdots & 0& 0& 0& \cdots & s+{\lambda }_{L-1}+\beta +\delta r& 0\\ 0& 0& 0& \cdots & 0& 0& 0& \cdots & 0& 0& 0& 0& 0& \cdots & 0& 0& 0& \cdots & -{\lambda }_{L-1}& s\end{array}\right]\end{array}\end{array}$$

is a $2(L+1)\times 2(L+1)$ matrix. ${\mathbf{W}}^{*}(s)={[{\mathbf{P}}_0^{*}(s),{\mathbf{P}}_1^{*}(s)]}^T$ is a column vector, where ${\mathbf{P}}_0^{*}(s)=[P_{0,0}^{*}(s),P_{0,1}^{*}(s),\dots ,P_{0,L-1}^{*}(s),P_{0,L}^{*}(s)]$ , ${\mathbf{P}}_1^{*}(s)=[P_{1,0}^{*}(s),P_{1,1}^{*}(s),\dots ,P_{1,L-1}^{*}(s),P_{1,L}^{*}(s)]$ , and the superscript $T$ denotes the transpose. $\mathbf{W}(0)={[{\mathbf{P}}_0(0),{\mathbf{P}}_1(0)]}^T$ is a column vector, where ${\mathbf{P}}_0(0)=[P_{0,0}(0),P_{0,1}(0),\dots ,P_{0,L-1}(0),P_{0,L}(0)]$ and ${\mathbf{P}}_1(0)=[P_{1,0}(0),P_{1,1}(0),\dots ,P_{1,L-1}(0),P_{1,L}(0)]$ . We assume that initially there are no failed units in the system when the repairman is in working status, that is, the initial conditions are given by $P_{0,0}(0)=1$ ; $P_{0,n}(0)=0(n=1,2,\dots ,L)$ ; and $P_{1,n}(0)=0(n=0,1,\dots ,L)$ . Thus, it follows $\mathbf{W}(0)={[1,0,\dots ,0,0]}^T$ .

Using Cramer’s rule to solve equation (25), $P_{0,L}^{*}(s)$ and $P_{1,L}^{*}(s)$ are given by

$$P_{0,L}^{*}(s)=\frac{det[{\mathbf{N}}_{L+1}(s)]}{det[\mathbf{D}(s)]}$$

(26)

and

$$P_{1,L}^{*}(s)=\frac{det[{\mathbf{N}}_{2(L+1)}(s)]}{det[\mathbf{D}(s)]}$$

(27)

where $det[\mathbf{D}(s)]$ denotes the determinant of the matrix $\mathbf{D}(s)$ . The determinants $det[{\mathbf{N}}_{L+1}(s)]$ and $det[{\mathbf{N}}_{2(L+1)}(s)]$ are obtained from $\mathbf{D}(s)$ by the replacement of the $(L+1)\mathrm{th}$ and $2(L+1)\mathrm{th}$ column with the initial vector $\mathbf{W}(0)={[1,0,0,\dots ,0]}^T$ , respectively. Since it is quite difficult to derive the explicit expressions of $det[\mathbf{D}(s)]$ , $det[{\mathbf{N}}_{L+1}(s)]$ , and $det[{\mathbf{N}}_{2(L+1)}(s)]$ , we use a Maple software package to compute $P_{0,L}^{*}$ and $P_{1,L}^{*}$ numerically.

Special case

Let $Q=0$ , $\delta =0$ , and $0<\eta <\lambda$ , our model becomes the system with warm standbys and a repairable service station considered by Wang et al. ²⁰

Reliability function and MTTF

Let $Y$ be the random variable denoting the time to failure of the system. Since $P_{0,L}(t)$ represents the probability that the system is failed on or before time $t$ when the repairman is working, and $P_{1,L}(t)$ represents the probability that the system is failed on or before time $t$ when the repairman is broken down. Thus, the reliability function is given by

$$R_Y(t)=1-P_{0,L}(t)-P_{1,L}(t),t\ge 0$$

(28)

where $P_{0,L}(t)$ and $P_{1,L}(t)$ can be obtained by taking the inverse Laplace transformation of $P_{0,L}^{*}(s)$ and $P_{1,L}^{*}(s)$ in equations (26) and (27).

The MTTF is defined as

$$\mathrm{MTTF}={\displaystyle \underset{0}{\overset{\infty }{\int }}}{R}_Y(t)\mathrm{dt}=\underset{s\to 0}{lim}[{\displaystyle \underset{0}{\overset{\infty }{\int }}}{R}_Y(t)e^{-\mathrm{st}}\mathrm{dt}]$$

(29)

Substitution of equation (28) into equation (29) and after some algebraic manipulations, it yields

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{MTTF}}& =\underset{s\to 0}{lim}\{{\displaystyle \underset{0}{\overset{\infty }{\int }}}[1-P_{0,L}(t)-P_{1,L}(t)]e^{-\mathrm{st}}\mathrm{dt}\}\hfill \\ \multicolumn{1}{c}{}& =\underset{s\to 0}{lim}[\frac{1}{s}-P_{0,L}^{*}(s)-P_{1,L}^{*}(s)]\hfill \\ \multicolumn{1}{c}{}& =\underset{s\to 0}{lim}\left[\frac{\frac{det[\mathbf{D}(s)]}{s}-det[{\mathbf{N}}_{L+1}(s)]-det[{\mathbf{N}}_{2(L+1)}(s)]}{det[\mathbf{D}(s)]}\right]\hfill \end{array}$$

(30)

Numerical results

This section presents extensive numerical results to study the impact of various system parameters on $R_Y(t)$ and MTTF. We select the number of operating units $M=10$ , the number of standby units $S=6$ , the threshold value $Q=4$ , the failure rate of an operating unit $\lambda =0.4$ , the failure rate of a standby unit $\eta =0.05$ , the maintenance rate of a failed unit $\mu =1.0$ , the breakdown rate of the repairman $\alpha =0.2$ , the repair rate of the repairman $\beta =3.0$ , the probability that each failed unit remains in the queue $q=0.95$ , the reneging rate $r=0.1$ , and $K=1$ as the default parameters. Figures 2–12 plot the curves of $R_Y(t)$ versus various system parameters for 11 cases shown in Table 1.

Figure 2.

The system reliability with different threshold values of $Q$ (Case 1).

Figure 3.

The system reliability with different number of operating units (Case 2).

Figure 4.

The system reliability with different number of standby units (Case 3).

Figure 5.

The system reliability with different values of $K$ (Case 4).

Figure 6.

The system reliability with different values of $\lambda$ (Case 5).

Figure 7.

The system reliability with different values of $\eta$ (Case 6).

Figure 8.

The system reliability with different values of $\mu$ (Case 7).

Figure 9.

The system reliability with different values of $\alpha$ (Case 8).

Figure 10.

The system reliability with different values of $\beta$ (Case 9).

Figure 11.

The system reliability with different values of $q$ (Case 10).

Figure 12.

The system reliability with different values of $r$ (Case 11).

Table 1.

Parameters setting for Cases 1–11.

Case	$Q$	$M$	$S$	$K$	$\lambda$	$\eta$	$\mu$	$\alpha$	$\beta$	$q$	$r$
1	1, 2, 4, 6	10	6	1	0.4	0.05	1.0	0.2	3.0	0.95	0.1
2	4	8, 10, 12, 14	6	1	0.4	0.05	1.0	0.2	3.0	0.95	0.1
3	4	10	6, 7, 8, 9	1	0.4	0.05	1.0	0.2	3.0	0.95	0.1
4	4	10	6	1, 2, 3, 4	0.4	0.05	1.0	0.2	3.0	0.95	0.1
5	4	10	6	1	0.4–1.0	0.05	1.0	0.2	3.0	0.95	0.1
6	4	10	6	1	0.4	0.0–0.4	1.0	0.2	3.0	0.95	0.1
7	4	10	6	1	0.4	0.05	0.8–1.4	0.2	3.0	0.95	0.1
8	4	10	6	1	0.4	0.05	1.0	0.1–0.4	3.0	0.95	0.1
9	4	10	6	1	0.4	0.05	1.0	0.2	3.0–12.0	0.95	0.1
10	4	10	6	1	0.4	0.05	1.0	0.2	3.0	0.2–0.95	0.1
11	4	10	6	1	0.4	0.05	1.0	0.2	3.0	0.95	0.05–0.5

It is apparent from Figure 2 that the threshold value $Q$ does not affect the system reliability significantly. From Figures 3 and 4, we see that the system reliability increases as either the number of operating units or the number of standby units increases. Moreover, increasing the number of standby units affects the system reliability significantly more than increasing the number of operating units. Figure 5 shows that the system reliability increases with decreasing values of $K$ . Note that the influence of the system reliability on $K$ is significant. One observes from Figures 6, 7, and 9 that the system reliability increases as one of $\lambda$ , $\eta$ , and $\alpha$ decreases. Among these three parameters, $\lambda$ has the most significance for the system reliability. As can be seen from Figures 8 and 10, it reveals that the system reliability increases as either $\mu$ or $\beta$ increases. Obviously, $\mu$ affects the system reliability significantly. From Figure 11, it is found that the system reliability decreases with increasing values of $q$ . As shown in Figure 12, the system reliability increases as $r$ increases. Moreover, the system reliability is insensitive to changes in $r$ .

In order to examine the impact of the system parameters on the behavior of the system reliability, Figure 13 exhibits the sensitivity analysis of the system reliability with respect to $\lambda$ , $\eta$ , $\mu$ , $\alpha$ , $\beta$ , $q$ , and $r$ . It is noted that the sign of sensitivity indicates an increase or decrease in the system reliability by changing the values of system parameters. In Figure 13, we observe that $\partial R_Y(t)/\partial \lambda$ , $\partial R_Y(t)/\partial \eta$ , $\partial R_Y(t)/\partial \alpha$ and $\partial R_Y(t)/\partial q$ are negative, which means that an incremental change in $\theta$ reduces the system reliability when $\theta =\lambda ,\eta ,\alpha ,\mathrm{and}q$ . On the contrary, $\mu$ , $\beta$ and $r$ have positive signs, which indicates that incremental changes in $\mu$ , $\beta$ and $r$ improve the system reliability. Moreover, the impact of $\lambda$ and $\mu$ on the system reliability is more significant than that of other parameters. The system reliability is relatively insensitive to changes in $\eta$ , $\beta$ , $q$ , and $r$ . The above results could be helpful for system managers to improve the system reliability.

Figure 13.

The sensitivity analysis of system parameters on the system reliability.

Next, we investigate the impact of various combinations of the threshold value $Q$ and system parameters on the MTTF. Numerical results are summarized in Tables 2–11. Fix the threshold value $Q$ , Tables 2–11 show that (1) the MTTF increases as one of $M$ , $S$ , $\mu$ , $\beta$ and $r$ increases; (2) the MTTF decreases as one of $K$ , $\lambda$ , $\eta$ , $\alpha$ and $q$ increases; and (3) $K$ , $\lambda$ and $\mu$ have larger influences on the MTTF than other parameters. On the other hand, we fix the system parameters and find that the MTTF decreases with increase in the value of $Q$ . It also can be seen that the threshold value $Q$ rarely affects the MTTF. Before closing this section, we perform sensitivity analysis of the MTTF with respect to parameters $\lambda$ , $\eta$ , $\mu$ , $\alpha$ , $\beta$ , $q$ , and $r$ . From Table 12, one can easily see that $\partial \mathrm{MTTF}/\partial \theta >0$ , if $\theta =\mu$ , $\beta$ , and $r$ , and $\partial \mathrm{MTTF}/\partial \theta <0$ , if $\theta =\lambda$ , $\eta$ , $\alpha$ , and $q$ . It leads to the same results as shown in Tables 5–11 when fixing the threshold value $Q$ . Based on the magnitude of $|\partial \mathrm{MTTF}/\partial \theta |$ , the order of influence of these seven parameters on the MTTF is as follows: $\lambda >\mu >\alpha >q>\eta >r>\beta$ .

Table 2.

MTTF for different values of $Q$ and $M$ ( $S=6$ , $K=1$ , $\lambda =0.4$ , $\eta =0.05$ , $\mu =1.0$ , $\alpha =0.2$ , $\beta =3.0$ , $q=0.95$ , $r=0.1$ ).

$Q$	$M$
	8	10	12	14
0	20.685	20.797	20.989	21.199
1	20.681	20.795	20.987	21.198
2	20.668	20.788	20.984	21.196
3	20.646	20.777	20.977	21.192
4	20.614	20.762	20.969	21.186
5	20.574	20.742	20.958	21.180
6	20.524	20.719	20.945	21.172

Table 3.

MTTF for different values of $Q$ and $S$ ( $M=10$ , $K=1$ , $\lambda =0.4$ , $\eta =0.05$ , $\mu =1.0$ , $\alpha =0.2$ , $\beta =3.0$ , $q=0.95$ , $r=0.1$ ).

$Q$	$S$
	6	7	8	9
0	20.797	21.091	21.382	21.668
1	20.795	21.089	21.380	21.666
2	20.788	21.083	21.374	21.660
3	20.777	21.072	21.364	21.650
4	20.762	21.058	21.350	21.637
5	20.742	21.040	21.332	21.620
6	20.719	21.017	21.311	21.600

Table 4.

MTTF for different values of $Q$ and $K$ ( $M=10$ , $S=6$ , $\lambda =0.4$ , $\eta =0.05$ , $\mu =1.0$ , $\alpha =0.2$ , $\beta =3.0$ , $q=0.95$ , $r=0.1$ ).

$Q$	$K$
	1	2	3	4
0	20.797	10.771	7.5484	5.8639
1	20.795	10.770	7.5465	5.8620
2	20.788	10.763	7.5398	5.8553
3	20.777	10.752	7.5288	5.8443
4	20.762	10.737	7.5136	5.8292
5	20.742	10.717	7.4943	5.8099
6	20.719	10.693	7.4705	5.7862

Table 5.

MTTF for different values of $Q$ and $\lambda$ ( $M=10$ , $S=6$ , $K=1$ , $\eta =0.05$ , $\mu =1.0$ , $\alpha =0.2$ , $\beta =3.0$ , $q=0.95$ , $r=0.1$ ).

$Q$	$\lambda$
	0.4	0.6	0.7	0.8	1.0
0	20.797	9.8633	7.7537	6.3774	4.6981
1	20.795	9.8627	7.7532	6.3771	4.6980
2	20.788	9.8607	7.7520	6.3762	4.6975
3	20.777	9.8576	7.7500	6.3749	4.6968
4	20.762	9.8533	7.7473	6.3731	4.6959
5	20.742	9.8478	7.7438	6.3707	4.6946
6	20.719	9.8409	7.7395	6.3678	4.6931

Table 6.

MTTF for different values of $Q$ and $\eta$ ( $M=10$ , $S=6$ , $K=1$ , $\lambda =0.4$ , $\mu =1.0$ , $\alpha =0.2$ , $\beta =3.0$ , $q=0.95$ , $r=0.1$ ).

$Q$	$\eta$
	0.0	0.05	0.1	0.2	0.4
0	20.894	20.797	20.710	20.564	20.341
1	20.891	20.795	20.709	20.562	20.340
2	20.883	20.788	20.703	20.558	20.337
3	20.870	20.777	20.694	20.551	20.333
4	20.853	20.762	20.681	20.541	20.326
5	20.831	20.742	20.663	20.526	20.316
6	20.805	20.719	20.641	20.507	20.300

Table 7.

MTTF for different values of $Q$ and $\mu$ ( $M=10$ , $S=6$ , $K=1$ , $\lambda =0.4$ , $\eta =0.05$ , $\alpha =0.2$ , $\beta =3.0$ , $q=0.95$ , $r=0.1$ ).

$Q$	$\mu$
	0.8	1.0	1.1	1.2	1.4
0	16.845	20.797	23.285	26.204	33.679
1	16.843	20.795	23.283	26.201	33.675
2	16.839	20.788	23.275	26.192	33.664
3	16.831	20.777	23.262	26.178	33.645
4	16.820	20.762	23.245	26.157	33.618
5	16.806	20.742	23.222	26.131	33.584
6	16.789	20.719	23.195	26.099	33.541

Table 8.

MTTF for different values of $Q$ and $\alpha$ ( $M=10$ , $S=6$ , $K=1$ , $\lambda =0.4$ , $\eta =0.05$ , $\mu =1.0$ , $\beta =3.0$ , $q=0.95$ , $r=0.1$ ).

$Q$	$\alpha$
	0.1	0.2	0.25	0.3	0.4
0	21.631	20.797	20.418	20.063	19.413
1	21.630	20.795	20.416	20.060	19.409
2	21.626	20.788	20.408	20.051	19.398
3	21.620	20.777	20.395	20.036	19.380
4	21.611	20.762	20.377	20.016	19.356
5	21.600	20.742	20.354	19.990	19.326
6	21.586	20.719	20.327	19.959	19.290

Table 9.

MTTF for different values of $Q$ and $\beta$ ( $M=10$ , $S=6$ , $K=1$ , $\lambda =0.4$ , $\eta =0.05$ , $\mu =1.0$ , $\alpha =0.2$ , $q=0.95$ , $r=0.1$ ).

$Q$	$\beta$
	3.0	6.0	7.0	9.0	12.0
0	20.797	21.667	21.796	21.970	22.123
1	20.795	21.665	21.795	21.969	22.122
2	20.788	21.659	21.788	21.962	22.116
3	20.777	21.648	21.777	21.951	22.104
4	20.762	21.632	21.761	21.935	22.089
5	20.742	21.613	21.742	21.916	22.069
6	20.719	21.589	21.718	21.892	22.045

Table 10.

MTTF for different values of $Q$ and $q$ ( $M=10$ , $S=6$ , $K=1$ , $\lambda =0.4$ , $\eta =0.05$ , $\mu =1.0$ , $\alpha =0.2$ , $\beta =3.0$ , $r=0.1$ ).

$Q$	$q$
	0.2	0.4	0.6	0.8	0.95
0	26.404	23.475	22.089	21.248	20.797
1	26.401	23.473	22.087	21.246	20.795
2	26.390	23.465	22.080	21.239	20.788
3	26.373	23.453	22.068	21.228	20.777
4	26.349	23.436	22.052	21.213	20.762
5	26.317	23.414	22.032	21.193	20.742
6	26.276	23.386	22.007	21.169	20.719

Table 11.

The MTTF for different values of $Q$ and $r$ ( $M=10$ , $S=6$ , $K=1$ , $\lambda =0.4$ , $\eta =0.05$ , $\mu =1.0$ , $\alpha =0.2$ , $\beta =3.0$ , $q=0.95$ ).

$Q$	$r$
	0.05	0.1	0.25	0.4	0.5
0	20.733	20.797	20.990	21.187	21.319
1	20.731	20.795	20.988	21.185	21.317
2	20.724	20.788	20.982	21.178	21.310
3	20.713	20.777	20.971	21.167	21.299
4	20.698	20.762	20.955	21.151	21.283
5	20.679	20.742	20.936	21.132	21.264
6	20.655	20.719	20.912	21.108	21.240

Table 12.

Sensitivity analysis for the MTTF ( $M=10$ , $S=6$ , $K=1$ , $Q=4$ , $\lambda =0.4$ , $\eta =0.05$ , $\mu =1.0$ , $\alpha =0.2$ , $\beta =3.0$ , $q=0.95$ , $r=0.1$ ).

	$\theta =\lambda$	$\theta =\eta$	$\theta =\mu$	$\theta =\alpha$	$\theta =\beta$	$\theta =q$	$\theta =r$
$\frac{\partial \mathrm{MTTF}}{\partial \theta }$	−109.527	−1.716	22.924	−7.944	0.557	−2.650	1.281

An application example

To illustrate the applicability of the proposed reliability model to real-world systems, we consider an electric power system consisting of eight primary generators and six standby generators. The system distributes electric power and maintains sufficient power to meet the requirements of its customers. When a primary generator fails, it is immediately replaced by an available standby generator. The system is considered to have failed when all generators (primary and standby) fail. The time-to-failure of the primary and standby generators follow exponential distributions with rates $\lambda =0.6$ generator/h and $\eta =0.3$ generator/h, respectively. When a generator breaks down, it is repaired by an electrical technician with exponentially distributed repair times of rate $\mu =1.2$ generator/h. In addition, the electrical technician breaks down according to a Poisson process at a rate of $\alpha =0.1$ technician/h. Since the maintenance of the breakdown technician is rather expensive, it is only repaired when at least three generators have failed. Repair times for the technician are exponentially distributed with rate $\beta =9.0$ technician/h. An external repair facility is organized in order to avoid power outages. The time that failed generator spent in the external repair facility is exponentially distributed at a rate of $r=0.5$ generator/h. Each failed generator decides sequentially to be repaired by the external repair facility with probability $\delta =0.2$ or by the electrical technician with probability $q=0.8$ . When the failed generators begin sequentially to be repaired by the external repair facility, the number of failed generators is reduced following a geometric distribution. Thus, this system is accurately described using the proposed reliability model.

Reliability is an important aspect of performance assessment for electric power systems, making reliability measures an essential tool. Equations (28) and (30) are used to compute system reliability and MTTF with the aid of Maple software, respectively. The reliability function is then plotted with respect to time (Figure 14). From this figure, it can be observed that (1) system reliability decreases by approximately 50% as time increases from 0 to 10 h and (2) if $t$ is greater than 45 h, system reliability is less than 0.004. That is, the probability that the electric power system will remain functional tends to 0 when $t>45$ . Moreover, the MTTF is calculated as 12.448 h. Based on the above result, the system manager can judge whether the reliability is at an acceptable level.

Figure 14.

System reliability of the electric power system with respect to $t$ .

Conclusion

In this article, we investigated a repairable system with standby units, geometric reneging, and threshold-based recovery policy. We first presented the differential-difference equations of the reliability model. Then, the Laplace transform and matrix methods were used to obtain the reliability function and MTTF. Numerical results were given to study the impact of various system parameters on the system reliability and MTTF. It showed that $K$ , $\lambda$ , and $\mu$ affect the system reliability and MTTF significantly. Moreover, the order of influence of seven parameters on the MTTF is $\lambda >\mu >\alpha >q>\eta >r>\beta$ . Finally, the applicability of the proposed reliability model was demonstrated using an example of an electric power system.