Load Dependent Series-Parallel Systems with Common Bus Performance Sharing Mecha

XIAO Hui (肖 辉), PENG Rui (彭 锐)

1 School of Statistics, Southwestern University of Finance and Economics, Chengdu 611130, China 2 Dongling School of Economics and Management, University of Science and Technology Beijing, Beijing 100083, China

Load Dependent Series-Parallel Systems with Common Bus Performance Sharing Mechanism

XIAO Hui (肖 辉)1*, PENG Rui (彭 锐)2

1SchoolofStatistics,SouthwesternUniversityofFinanceandEconomics,Chengdu611130,China2DonglingSchoolofEconomicsandManagement,UniversityofScienceandTechnologyBeijing,Beijing100083,China

A series-parallel system was proposed with common bus performance sharing in which the performance and failure rate of the element depended on the load it was carrying. In such a system, the surplus performance of a sub-system can be transmitted to other deficient sub-systems. The transmission capacity of the common bus performance sharing mechanism is a random variable. Effects of load on element performance and failure rate were considered in this paper. A reliability evaluation algorithm based on the universal generating function technique was suggested. Numerical experiments were conducted to illustrate the algorithm.

multi-statereliability;series-parallelsystem;loaddependentfailurerate;commonbusperformancesharing

Introduction

Most of the engineering systems are designed to support the load, and the performance of the system depends on how much load it undertakes. Examples of such loading-carrying systems include coal conveyors, cargo trucks, and power generating units. In the literatures, various studies have empirically shown that the component failure rate is strongly affected by the workload[1-2]. Therefore, it is important to consider the effect of load when analyzing reliability of the multi-state system. Recent studies considering the effect of load on the failure rate of the system components can be found in Refs. [3-4].

Series-parallel systems have been well studied in Refs.[4-8]. However, most of the studies assume that the sub-systems connected in series are independent and support their own demands individually. However, practical examples in power, communication, data processing, and production systems indicate that each unit meets its individual demand first. After that, some of the unconsumed performance can be shared by other units which are experiencing performance deficiency. This type of performance sharing was first studied by Lisnianski and Ding[9], where the performance could only be transmitted in one direction. The idea of performance sharing was extended to multi-directional performance transmission in a series system[10]and a series-parallel system[11].

The series-parallel system fails if and only if at least one of the sub-systems is not able to meet its demand after the performance redistribution. One may be interested in finding the optimal load distribution among theMMEs such that the reliability of the system can be maximized. The organization of the paper is as follows. Section 1 defines the reliability of series-parallel systems with common bus performance sharing. Section 2 discusses the effects of load on the failure rate and performance of the elements. Section 3 suggests a reliability evaluation algorithm to compute the reliability of the proposed system based on universal generating function technique. Numerical examples are provided in Section 4. Finally, we conclude this paper in Section 5.

1 Reliability of Series-Parallel Systems with Common Bus Performance Sharing

The surplus performance comes from the sub-system withGj>Wjand the total system performance deficiency resulteds from the sub-systems withGj

(1)

Similarly, the total system deficiency can be written as,

(2)

whereEjis the set of elements located at sub-systemj.

By definition, the amount that can be transmitted through the common bus sharing model must be the minimum ofSandD,i.e., min(S,D). In addition, the amount that can be redistributed is further constrained by the transmission capacity. LetZbe the performance that can be redistributed throughout the series-parallel system.

(3)

whereSandDare statistically dependents, whileSandCas well asDandCare statistically independents.

The total system performance deficiency remaining after redistribution is,

(4)

By definition of the series-parallel system, the reliability of the system is the probability that there is no performance deficiency in any of the sub-systems.

(5)

2 Load Dependent Failure Rate and Performance

In order to evaluate the reliability of the system with load dependent failure rate, one needs to know the load-failure relationship of each ME. The reliability of each ME can be derived based on the failure rate function. The accelerated life test models provide useful information in determining the load failure rate relationships. A comprehensive literature review of the accelerated life test models can be found in Ref. [12].

Proportional hazard model (PHM) was first proposed by Cox[13]. It has received popularity in the field of reliability engineering in recent years[14-16]. PHM states that the failure rate of a component is the product of the baseline hazard rate and factors based on the conditions. In general, PHM can be expressed as

(6)

In this paper, we assume that the load is the only factor that affects the failure rate and the baseline failure rate is constant across time, therefore, the PHM can be reduced to the following single factor model.

(7)

There are many other models existing in the literatures. Though the models take different forms, they can be applied in our problem similarly as PHM.

The load dependent performancegi(Li) can take various forms depending on the situations. For simplicity, we assume the function measures relationship between the performance and the load can be expressed as

gi(Li)=ai+ciLi,

(8)

whereaiandciare the coefficients of the linear equation. Other types of relationship between the performance and the load can be applied similarly.

3 Reliability Evaluation Algorithms

The reliability evaluation algorithm is proposed based on universal generating function (UGF) technique, first introduced by Ushakov[17]and has been proven to be very efficient for evaluating reliability of complex systems in different structures[18].

(9)

Therefore, the UGF ofeicarrying loadLican be denoted as

(10)

(1-pf(Lf))z0+pf(Lf)zgf(Lf))=

(11)

(12)

where

On the other hand, the UGF for the demand of the sub-systemjcan be written as,

(13)

Similarly, the transmission capacity UGF of the common bus performance sharing system is,

(14)

(15)

(16)

(17)

(18)

4 Numerical Experiments

Consider a series-parallel system with common bus performance sharing model. The system consists of 3 sub-systems with 7 multi-state elements in total. Each sub-system is subjected to a random demand. The surplus performance of each sub-system can be transmitted to other deficient sub-systems through the common bus performance sharing model with capacityC. The structure of the system and the allocation of the multi-state elements are shown in Fig.1.

Fig.1 A series-parallel system with common bus performance sharing

The characteristics of the multi-state elements are shown in Table 1. The demand of each sub-system is shown in Table 2. The system availabilities under the minimum, the mean and the maximum loads are shown in Table 3, where the mean load refers to the average of the minimum and the maximum loads. Based on the numerical results, the best load on each element is not the minimum load or the maximum load. Hence, one may be interested in finding the optimal load given to each element.

Table 1 Parameters of multi-states elements

Table 2 Demand distribution of each sub-system

Table 3 System availability under different load

5 Conclusions

In this paper, a series-parallel system with common bus performance sharing is studied with the consideration of the effect of load on the performance and failure rate of the system elements.The surplus performance of a sub-system can be transmitted to other deficient sub-systems subjected to the constraint of the transmission capacity. Using the universal generating function technique, we suggested a reliability evaluation algorithm for the load dependent systems. Future research on this problem will consider the optimal load allocation on all the system elements so that the system reliability can be maximized.

[1] Iyer R K, Rossetti D J. A Measurement-Based Model for Workload Dependence of CPU Errors [J].IEEETransactionsonComputers, 1986, 35(6): 511-519.

[2] Levitin G, Amari S. Optimal Load Distribution in Series-Parallel Systems [J].ReliabilityEngineering&SystemSafety, 2009, 94(2): 254-260.

[3] Hu L M, Yue D Q, Li J D. Availability Analysis and Design Optimization for a Repairable Series-Parallel System with Failure Dependence [J].InternationalJournalofInnovativeComputingInformationandControl, 2012, 8(10A): 6693-6705.

[4] Nourelfath M, Yalaoui F. Integrated Load Distribution and Production Planning in Series-Parallel Multi-state Systems with Failure Rate Depending on Load [J].ReliabilityEngineering&SystemSafety, 2012, 106: 138-145.

[5] Tian Z, Levitin G, Zuo M J. A Joint Reliability-Redundancy Optimization Approach for Multi-state Series-Parallel Systems [J].ReliabilityEngineering&SystemSafety, 2009, 94(10): 1568-1576.

[6] Yalaoui A, Chu C, Chatelet E. Reliability Allocation Problem in a Series-Parallel System [J].ReliabilityEngineering&SystemSafety, 2005, 90(1): 55-61.

[7] Safari J. Multi-objective Reliability Optimization of Series-Parallel Systems with a Choice of Redundancy Strategies [J].ReliabilityEngineering&SystemSafety, 2012, 108: 10-20.

[8] Zhou Y, Zhang Z, Lin T R,etal. Maintenance Optimization of a Multi-state Series-Parallel System Considering Economic Dependence and State-Dependent Inspection Intervals [J].ReliabilityEngineering&SystemSafety, 2013, 111: 248-259.

[9] Lisnianski A, Ding Y. Redundancy Analysis for Repairable Multi-state System by Using Combined Stochastic Processes Methods and Universal Generating Function Technique [J].ReliabilityEngineering&SystemSafety, 2009, 94(11): 1788-1795.

[10] Levitin G. Reliability of Multi-state Systems with Common Bus Performance Sharing [J].IIETransactions, 2011, 43(7): 518-524.

[11] Xiao H, Peng R. Optimal Allocation and Maintenance of Multi-state Elements in Series-Parallel Systems with Common Bus Performance Sharing[J].Computers&IndustrialEngineering, 2014, 72: 143-151.

[12] Escobar L A, Meeker W Q. A Review of Accelerated Test Models [J].TatisticalScience, 2006, 21(4): 552-577.

[13] Cox D R. Regression Models and Life Tables (with Discussion) [J].JournaloftheRoyalStatisticSociety:SeriesB, 1972, 34: 187-220.

[14] Jozwiak I J. An Introduction to the Studies of Reliability of Systems Using the Weibull Proportional Hazards Model [J].MicroelectronicsandReliability, 1997, 37(6): 915-918.

[15] Li Z G, Zhou S Y, Sievenpiper C,etal. Change Detection in the Cox Proportional Hazards Models from Different Reliability Data [J].QualityandReliabilityEngineeringInternational, 2010, 26(7): 677-689.

[16] Tiwari A, Roy D. Estimation of Reliability of Mobile Handsets Using Cox-Proportional Hazard Model [J].MicroelectronicsandReliability, 2013, 53(3): 481-487.

[17] Ushakov I. Optimal Standby Problems and a Universal Generating Function [J].SovietJournalofComputerSystemsScience, 1987, 25: 79-82.

[18] Levitin G. The Universal Generating Function in Reliability Analysis and Optimization [M]. New Delhi, India: CBS Publishers, 2005.

National Natural Science Foundations of China (Nos. 71231001, 11001005, 71301009); China Postdoctoral Science Foundation (No. 2013M530531); the Fundamental Research Funds for the Central Universities of China (Nos. FRF-MP-13-009A, FRF-TP-13-026A); and the MOE PhD Supervisor Fund of China (No. 20120006110025)

1672-5220(2014)06-0770-04

Received date: 2014-08-08

* Correspondence should be addressed to XIAO Hui, E-mail: msxh@swufe.edu.cn

CLC number: N945.17 Document code: A