An N-Component Series Repairable System with Repairman Doing Other Work and Priority in Repair

Jianying Yang Department of Science, Yanshan University Qinhuangdao 066004, China E-mail: jianying_yang27@163.com Xianyun Meng Department of Science, Yanshan University Qinhuangdao 066004, China Weiyan Guo Acheng collage, Harbin Normal University Yanqin Guan Department of Science, Yanshan University Qinhuangdao 066004, China Taotao Wang Department of Science, Yanshan University Qinhuangdao 066004, China


Introduction
In order to improve the interest of the system, the supervisor always arranges the repairman service for the customers out of system.Under this condition and assumptions that the working time distributions are exponential and the repair time distributions and the time distribution of the repairman doing other work are both general continuous distributions, by using supplementary technique and vector Markov process, the reliability indices have been obtained and the benefits of the model discussed (Su, B.H., 1994, pp.34-39).Later the number of the customers has been introduced.LIU R. B. and TANG Y.H. (2005, pp.493-496) assumed that the working time and the arrival interval time have exponential distribution while others to be general continuous distributions.By using the supplementary variable method, the vector Markov process and the tool of the Laplace transform, some reliability indexes of the system have been derived and the benefit of the system discussed.HU L.M., WU J.B. and TIAN R.L. (2007, pp.47-50) have been introduced the assumption that each unit had two types of failure.They have obtained the reliability indices of the system.Zhang Y.L. and Wang G. J. (2006,pp.278 295) introduced the priority in use in a deteriorating cold standby repairable system.
While it isn't always maximizing the interest of system that repairman is servicing for customers out of system, the repair work of the components maybe delayed for the repairman doing other work.The purpose of this paper is to apply the priority repair model to an n-component series repairable system with the repairman doing other work.Now we may assume that the units after repair is "as good as new" and the units have priority in repair.Furthermore, we assume that the working time of the components and the arrival interval time customers are both exponentially distributed and others generally distributed.

Model
We study an n-component series repairable system with repairman doing other work and priority in repair by making the following assumptions: Assumption 1.Initially, the n components are all new and work in double harness while the repairman is idle.
Assumption 2. When one component is on repairing, the others will stop working and not go wrong.The system is failing now.Assumption 3. If and only if when the n components are all working and the customer doesn't arrive, the repairman is idle.If and only if when the n components are all working, the customers will be likely to arrival.The customer won't arrive, when one component is repaired or the repairman is serving for one customer and another customer is waiting for service.When one customer is waiting for service and one component goes wrong, the customer will leave.And the component is repaired immediately, while the customer served will be waiting for service.The customer will be served after the repair over and the service before is valid.

Let i
X and i Y be, respectively, the working time and the repair time of the component i, Let H be the service time of the customer.We assume that the distribution of H is Let V be the time between repairman start idle and the arrival of the first customer and between the arrival of the first customer and the second one.We assume that the distribution ofV is ), H and V are independent.

The system analysis
be a stochastic process characterized by the following mutually exclusive events: : n components are all working and the repairman is idle.
: n components are all working, the repairman is serving for the customer.

{ }
3 ) ( = t S : n components are all working, the repairman is serving for one customer and the anther waiting for service.
is a stochastic process with state space . The set of working states is is not a Markov process.However, it can be extended to a generalized Markov process by introducing a supplementary variable.Let ) (t X i be the repair time of the component i used at time t , The state marginal probability of the system at time t are defined by According to the model assumptions and the supplementary variable technique, we can obtain the following differential equations for the system.By straightforward probability arguments, for example, we have (1) In the same way, we have (3) The boundary conditions are The system differential equations using Laplace transforms are obtained as follows: ( ) Together with the borderline conditions and the initial conditions, this system of equations can be solved to yield:

system availability and system rate of occurrence of failure
By the definition, the system availability at t is given by A(t) = P (the system is working at time Where, according to (15), ( 17), ( 18) and (20), thus ( ) . Using Tauberian theorem, the stay state availability or the limiting availability of the system is given by be the rate of occurrence of failure or the failure frequency of the system at time t.
Using Tauberian theorem, the stay state failure frequency of the system is given by

The idle time probability of the repairman
Clearly, the repairman will be idle if and only if the components are all working and no customer is arrival.Thus, the idle time probability of the repairman at time t is given by Using Tauberian theorem, the stay state failure frequency of the system is given by Using Tauberian theorem, the stay state number of the customer served by the repairman per unit time is given by

The system benefit analysis
In this section, our objective is to determine whether or not given the priority to components in repairing such that the benefit of the system is maximized.Let 1 x′ be the working reward per unit time of the system, 2 x′ be the average cost each time of the system, and 3 x′ be the average reward of serving for one customer.Based on the assumptions, the stay state average reward per unit time of the system is In order to solve the our problem, it is necessary to introduce some conclusion and assumption that LIU,Ren -bin, TANG,Ying -hui &LUO Chuan -yi(2005) have studied.In this paper, let 1 x be the working reward per unit time of the system, 2 x be the average cost each time failure of the system, and 3 x be the average reward of serving for one customer.Clearly, for one system, the 1 x′ and 3 x′ will be, respectively, same to the 1 x and 3 x while the 2 x′ will be different from 2 x , because the 2 x include the cost produced by waiting repair.So the equation will be changed to y y − is negative.In this way the benefit of the system will be maximized.
The stay state average reward per unit time of the system in Lam [2the components given the priority in repairing will depend on the result of the equation 0 1 y y − : the component will be given priority in repair when the result of equation 0 1 y y − is positive number, while the component will not be given priority in repair when the result of equation 0 1