Reliability analysis of multi-state k-out-of-n: G system with common bus performance sharing

Highlights

• The structure of system with common bus is extended to k-out-of-n pattern.

• The common bus system sate is first divided into multiple levels.

• Distributed rules of redistributed performance are considered.

Abstract

In this paper, a multi-state k-out-of-n: G system with common bus performance sharing is proposed, in which k-out-of-n: G structure is considered, and the system state is divided into multiple levels for the first time. Such a system consists of a common bus and n multi-state units, where each unit has several performance levels to satisfy its individual random demands. If one unit has surplus performance, the performance surplus can be transmitted to other units which are experiencing performance deficiency via the common bus. The state of the system is divided into multiple levels according to the number of the units whose demands are satisfied. An algorithm based on the universal generating function (UGF) technique and the stochastic process method is proposed to evaluate the system reliability. Analytical and numerical examples are presented for demonstrating the application of the proposed method finally.

Introduction

Performance sharing systems are attracting great attention from both industry and academia in recent years. Some engineering systems, such as power supply systems and distributed computing systems, can be modelled by performance sharing systems (Barrero et al., 2014, Lisnianski and Ding, 2009, Piris-Botalla et al., 2014, Wang et al., 2018).

Lisnianski and Ding (2009) firstly proposed the modeling of the performance sharing system where two multi-state units are considered, the main unit and the reserve unit, and performance surplus can only be transmitted from the reserve unit to the main unit. Then Levitin (2011) extended the performance sharing system to a series system which consists of several multi-state units and can transmit the surplus performance in any direction by a common bus. Furthermore, this system has been extended by many researchers. Yu, Yang, and Mo (2014) proposed a series-parallel repairable binary-state system consisting of multi-state units with common bus performance sharing. Peng, Xiao, and Liu (2017) proposed a series system with a performance sharing group of limited size. Yu, Yang, Lin, and Zhao (2017) proposed a phased-mission common bus system consisting of service elements, nodes and a common bus. Wang et al., 2018, Wang et al., 2018 proposed a series system with performance sharing where the surplus performance of each unit is firstly transmitted to the units adjacent to it and then further shared by other non-adjacent units.

Based on the abovementioned systems, a series of optimal problems are researched. Xiao and Peng (2014) developed a model to optimize the allocation and maintenance of multi-state units in series-parallel binary-state systems with common bus performance sharing. Zhai, Ye, Peng, and Wang (2017) put forward the defense and attack strategies for a system which is subject to intentional attacks and has a common bus performance sharing mechanism. Yu et al. (2017) optimized allocation strategies of service elements for the phased-mission common bus system. Algorithms based on universal generating function (UGF) (Levitin, 2005) are used to obtain the system reliability in all above works except Yu et al. (2017). A recursive algorithm is used to evaluate the reliability of the system with common cause failures and a genetic algorithm is used to search the optimal allocation strategies of service elements in Yu et al. (2017).

For the system with common bus performance sharing, the previous researches only focus on series or series-parallel structure, and the binary-state system in which units have multiple states. However, k-out-of-n: G structure is also important in the study of reliability, and multi-state systems are common in practice. A k-out-of-n: G system consists of n units and works if and only if at least k units operate at the same time. Such systems are widely used in both industrial and military fields (Griffith, 2004). Most studies focus on the reliability analysis (Barlow and Heidtmann, 1984, Behr and Camarinopoulos, 1997, Boland and Proschan, 1983, Li et al., 2014, Zhai et al., 2015) and the modeling of extensions of k-out-of-n: G system (Griffith, 2004, Kuo et al., 1990, Yeh, 2006, Zuo, 1993). Furthermore, systems having multiple performance states are called multi-state system (MSS). The MSS was introduced in the middle of the 1970s (Caldarola, 1980, El-Neweihi, 1978, Murchland, 1975, Ross, 1979). Lisnianski and Levitin (2003) summarized the theory of multi-state system reliability. UGF methods (Levitin, 2005), Markov process methods (Zhao et al., 2018, Zhao et al., 2018, Zhao et al., 2018, Zhao et al., 2018) and decision diagrams methods (Jia et al., 2017, Mo et al., 2018, Mo et al., 2017, Shrestha and Xing, 2008, Shrestha et al., 2010) are usually used to evaluate the multi-state system reliability. Multi-state k-out-of-n system was first defined by El-Neweihi (1978). Then extensions of multi-state k-out-of-n system and calculations of the system reliability were widely investigated in recent years (Eryilmaz and Bozbulut, 2014, Huang and Zuo, 2000, Huang et al., 2000, Jenab and Dhillon, 2006, Li et al., 2016, Li et al., 2016, Mo et al., 2015, Sahba et al., 2013, Tian et al., 2008, Zhao and Cui, 2010, Zhao et al., 2012).

In this paper, the system structure is extended to k-out-of-n pattern, and the system state is divided into multiple levels for the first time in the study of common bus performance sharing. This extended system is called multi-state k-out-of-n: G system with common bus performance sharing, which consists of n multi-state units. Each unit has some different performance levels to satisfy its random demands. A common bus connects all units. If one unit has performance surplus, it can be transmitted to other units which are experiencing performance deficiency through the common bus. The system has multiple states which rely on the number of the units whose demands are satisfied. This system is a generalization of some other earlier models and it can be degenerated to the series system with common bus performance sharing, the parallel system with common bus performance sharing and the multi-state k-out-of-n: G system. The relationships are shown in Fig. 1.

The multi-state k-out-of-n: G system with common bus performance sharing is motivated by the practical demands in the engineering fields such as solar lighting systems, downhole sensor systems and ground-based augmentation systems. A solar lighting system consists of n solar LED lighting subsystems which are connected through a common bus in a work space. A solar LED subsystem can obtain electricity from its own solar cell which converts solar energy into electricity (Zajkowski, 2003). If one solar LED subsystem has electricity deficiency, it can obtain electricity from other subsystems whose electricity exceed their own demands via the common bus. A subsystem works normally when its required electricity is satisfied. Otherwise, it shuts down. The lighting level is determined by the number of working subsystems in the work space. For example, if the number of working subsystems is more than $k_1$, the lighting level in the work space is at 1000 Lux. If the number of working subsystems is no more than $k_2(k_2<k_1)$, the lighting level is at less than 500 Lux. If the number of working subsystems is more than $k_2$ but less than $k_1$, the lighting level is at between 500 Lux and 1000 Lux.

A downhole sensor system consists of n subsystems which are connected by a common bus in a mining area. Each subsystem consists of several downhole sensors which are used to measure environmental indicators such as temperature, pressure, gas concentration and so on, and works by obtaining electricity from its own flameproof wind self-powered device. The flameproof wind self-powered device uses downhole wind to generate electricity (Chen, Zhuo, & Wu, 2017). If the required electricity of a subsystem is not satisfied, the subsystem shuts down. If one subsystem has surplus electricity, the surplus electricity can be transmitted to other subsystems which are experiencing electricity deficiency via the common bus. The monitoring error level of the whole downhole sensor system is determined by the number of working subsystems. For instance, if the number of subsystems is no more than $k_1^{\prime }$, the downhole sensor system cannot measure any information. If the number of working subsystems is more than $k_2^{\prime }(k_2^{\prime }>k_1^{\prime })$, the monitoring error is less than 0.1%. If the number of working subsystems is no more than $k_1^{\prime }$ but less than $k_2^{\prime }$, the monitoring error is between 0.1% and 0.5%.

A ground-based augmentation system consists of several base stations, one data-processing system, one operation service platform, one data broadcasting system and a number of subscriber terminal equipment. The base station is used to receive satellite navigation signals and obtain the information of local positions. However, the information of local positions may exist some deviations and the deviation level of the position information is determined by the number of working base stations. The base station works on electricity and a solar battery board is responsible for recharging the base station when it is out of power. The base station shuts down when the required electricity cannot be satisfied. Generally, if the number of the working base stations is larger in a certain region, the deviation level of the position information is lower. A common bus is installed to connect all solar battery boards and transmit the electricity. For example, there are n base stations in a region and each base station is recharged by a solar battery board. A common bus connects the n solar battery boards. The base station whose electricity is satisfied can receive satellite navigation signals. If a solar battery board has surplus electricity, the surplus electricity can be transmitted to other base stations which are experiencing electricity deficiency through the common bus. If the number of working base stations is $k_1^{″},k_2^{″}$ and $k_3^{″}$, the whole system of base stations provides meter-level position accuracy, decimeter-level position accuracy and centimeter-level position accuracy respectively, where $k_1^{″}<k_2^{″}<k_3^{″}$.

In this paper, two cases are considered for the multi-state k-out-of-n: G system with common bus performance sharing: (1) The system state is steady. (2) The system state is time-varied.

The organization of this paper is as follows. Section 2 presents the model for the multi-state k-out-of-n: G system with common bus performance sharing. Section 3 evaluates the reliability of the system using UGF-based algorithm and stochastic process method. Section 4 presents analytical and numerical examples. The conclusions and our future research are presented in Section 5.