Journal of Applied Mathematics and Physics, 2013, 1, 21-27

http://dx.doi.org/10.4236/jamp.2013.13005 Published Online August 2013 (http://www.scirp.org/journal/jamp)

Assessment of Profit of a Two-Stage Deteriorating Linear

Consecutive 2-out-of-3 Repairable System

Ibrahim Yusuf1*, Fatima Salman Koki2

1Department of Mathematical Sciences, Bayero University, Kano, Nigeria

2Department of Physics, Bayero University, Kano, Nigeria

Email: *Ibrahimyusuffagge@gmail.com, FatimaSK2775@gmail.com

Received June 5, 2013; revised July 6, 2013; accepted August 15, 2013

Copyright © 2013 Ibrahim Yusuf, Fatima Salman Koki. This is an open access article distributed under the Creative Commons At-

tribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-

erly cited.

ABSTRACT

Most of the researches on profit and cost evaluation of redundant system focus on the effect of failure and repair on

revenue generated. However, as these systems continue to work, their strength gradually deteriorates. Where such dete-

rioration occurs, minor an d major maintenance is employed to remedy the deterioration . Little o r no attentio n is paid on

the effect of deterioration on the impact of deterioration and their maintenance on the revenue generated. In this paper,

we study the profit generated of two-stage deteriorating linear consecutive 2-out-of-3 system. Failure, repair and dete-

rioration time are assumed exponential. The explicit expressions of availability, busy period of a repairman and profit

function are derived using Kolmogorov’s forward equations method. Various cases are analyzed graphically to investi-

gate the effect of deterioration parameters such as slow deterioration, fast deterioration, and their maintenance such as

minor and major minimal maintenance on profit generated.

Keywords: Reliability; Availability; Profit; Deterioration

1. Introduction

During operation, the strengths of systems are gradually

deteriorated, until some point of deterioration failure, or

other types of failures. Minor and major maintenance

policies are vital in the analysis of deterioration and d ete-

riorating systems as they help in improving reliability,

availability and the overall revenue generated. Both mi-

nor and major minimal maintenance are employed to

check the effect of slow and fast deterioration and return

the system to its state prior to slow and fast deterioration.

Maintenance models assume perfect repair (as good as

new), minimal repair (as bad as old) and imperfect repair

which is between perfect and minimal repair. Many re-

search results have been reported on the reliability of

2-out-of-3 redu nd ant systems. For example, [1], analyzed

reliability models for 2-out-of-3 redundant system are

subject to conditional arrival time of the server. Refer-

ence [2] presented reliability and economic analysis of

2-out-of-3 redundant system with priority to repair and [3]

studied MTSF and cost effectiveness of 2-out-of-3 cold

standby system with probability of rep air and inspection,

while [4] examined the cost benefit analysis of series

systems with cold standby components and repairable

service station. Reference [5,6] examined the cost analy-

sis of two unit cold standby system involving preventive

maintenance respectively. Reference [7] studied the cost

and probabilistic analysis of series system with mixed

standby componen ts while [8] studied cost benefit analy-

sis of series systems with warm standby components

involving general repair time where the server is not

subject to breakdowns. The failure time and repair time

are assumed to have exponential distribution. Measures

of system effectiveness such as MTSF, steady-state

availability, busy p eriod and profit fun ction are obtained.

[9] studied availability of a system with different repair

options, while [10] evaluated the reliability of network

flows with stochastic capacity and cost constraint. The

problem considered in this paper is different from the

work of [4-6]. In this paper, a linear consecutive 2-out-

of-3 repairable system with two consecutive deterioration

stages (slow and fast) is studied with minor and major

minimal maintenance at slow and fast deterioration re-

spectively. In this paper, a two-stage deteriorating linear

consecutive 2-out-of-3 system was constructed and de-

rived its corresponding mathematical models. The main

*Corresponding author.

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