MTSF AND COST-EFFECTIVENESS OF 2-OUT-OF-3 COLD STANDBY SYSTEM WITH PROBABILITY OF REPAIR AND INSPECTION

MTSF AND COST-EFFECTIVENESS OF

2-OUT-OF-3 COLD STANDBY SYSTEM

WITH PROBABILITY OF REPAIR AND

INSPECTION

*R.K. Bhardwaj, Assistant Professor

Department of Statistics, Punjabi University Patiala-147002, India. S.C. Malik, Professor

Department of Statistics, M.D. University Rohtak-124001, India.

Abstract: In this paper two stochastic models for a 2-out-of-3 redundant system are developed and cost-benefit analysis is carried out by using semi-Markov and regenerative processes. There are three identical units in each model and the system is considered in up-state if 2-out-of-3 units are functioning. A single repair facility is available which plays the dual role of inspection and repair. In model I server repairs the unit without inspection while in model II the unit is inspected at its failure to see the feasibility of repair. If repair of unit is not feasible it is replaced by new one in order to increase the reliability of the system. The failure time distribution follows negative exponential while that of inspection and repair are taken as arbitrary. The expressions for MTSF, steady state availability, busy period and expected number of visits of the server are derived. The profit function of each model is also estimated. A particular case is considered to derive the results in parametric form.

Keywords: MTSF, 2-out-of-3 system, cold standby, regenerative process and cost analysis.

Subject Classification: Primary 90B25 and Secondary 60K10.

1 Introduction

The provision of a spare unit is necessary to improve the reliability of a system. The stochastic models with spare units have widely been studied in the field of reliability by the scholars including Srinivasan and Gopalan [1984], Nakagawa [1989], Singh [1989], Gupta and Chaudhary [1994] and Chander [2005] as these are frequently used in modern technology. Dhillon [1992] has discussed stochastic models of k-out-of-n units family of a system. However there are systems of three units in which two units are sufficient to perform functions. Such a three unit redundant system is applied in real world to attain high reliability and performance. For example, consider a communication amplifier system in which redundancy is used as a means of increasing the reliability. The goal of any redundant amplifier system is to achieve a system reliability that is greater than the reliability of an individual amplifier. In many instances, satellite equipment is installed in remote locations which are not easily accessed for maintenance. Therefore it is imperative that any amplifier system used in satellite communication be equipped with some form of automatic backup or spare to make the system more reliable and available for use. Hence stochastic models for 2-out-of-3 modular amplifier system are under taken for study.

The expressions for some measures of system effectiveness such as transition probabilities, mean sojourn times, mean time to system failure, availability, busy period, expected number of visits by the server are derived by adopting semi-Markov process and regenerative point technique. The profit function is also evaluated to carry out the cost-benefit analysis. A particular case is considered to obtain the results for MTSF and profit of the models in parametric form.

2 Notations

E: Set of regenerative states

No: Unit in normal mode and operative

: Unit in normal mode but not working

Cs: Unit in normal mode and cold standby

a/b: Probability that repair is feasible / not feasible λ: Constant failure rate of an operative unit.

Mi(t): Probability that the system is up initially in state Si



E is up at time t without visiting

to any other regenerative state.

Wi(t): Probability that the server is busy in the state Si up to time t without making any

transition to any other regenerative state or returning to the same via one or more non-regenerative states.

qij(t) / Qij(t): pdf / cdf of first passage time from a regenerative state i to a regenerative state j or to a failed state without visiting any other regenerative state in (0,t].

qij.kr(t)/Qij.kr(t): pdf / cdf of first passage time from a regenerative state i to regenerative state j or to a failed state j visiting states k, r once in (0,t].

h(t)/H(t): pdf / cdf of inspection time

w(t)/W(t): pdf /cdf of waiting time of the server to arrive at the system.

g(t)/G(t): pdf / cdf of repair time of the server.

Fwi/FwI/Fui/FuI: Unit is completely failed and waiting for inspection / waiting for inspection continuouslyfrom previous state/ under inspection / under continuous inspection from previous state.

Fur/ FUR: Unit is completely failed and under repair / under repair continuously from previous state.

pij/ pij.kr: Probability of transition from regenerative state i to a regenerative state j without visiting any other state in (0,t] / visiting state k,r once in (0,t] i.e.

_∞ and _{. ∞ .}

i(t): cdf of first passage time from regenerative state i to a failed state.

Bi(t): Probability that the server is busy at an instant time t given that the system entered the regenerative state i at t=0.

Ni(t): Expected number of visits by the server in (0,t] given that the System entered the regenerative state i at t=0.

/ symbol for Stieltjes convolution/ Laplace convolution

~/* Symbols for Laplace Stieltjes transform (LST) / Laplace transform (LT).

 Symbol for derivative of the function.

For model I: E={ S0, S1}, For model II: E={S0, S1, S2, S3}. The possible transition states along with

transition rates for model I and model II are shown in fig. 1 and fig. 2 respectively.

3 Transition Probabilities and Mean Sojourn Times

Simple probabilistic considerations yield the following expressions for the non-zero elements

∞ as:

For Model I

p01=1, p10=g*(2), p12=1-g*(2), p11.2=1-g*(2)

Clearly, p01=p10+p12=p10+p11.2=1

For Model II

, , , , , ,

. , . , . (3.1) It can be easily verified that

+ _{. . .}

The unconditional mean time taken by the system to transit to any regenerative state Sj when it (time) is counted

from epoch of entrance into that state Si, is given by ,

∞

(3.2) and the mean Sojourn time in the state Si is given by

∞

(3.3) where T denotes the time to system failure. Using these, we have following expressions

For model I

0=m01,1=m10+m12, '1=m10+m11.2

For model II:

, , _, , _{, ,} ,

. (3.4) 4 Analysis for Model I

4.1 Mean time to system failure (MTSF)

On the basis of arguments used for regenerative processes, we obtain the following recursive relations for i(t):

(4.1) Taking LST of above relations (4.1) and solving for





₀(s), we get MTSF as

MTSF (T1)=lim ∞ ~

, where, and

4.2 Steady state availability

The recursive relations for Ai(t) are given as:

A0(t) = M0(t)+q01(t)  A1(t)

A1(t) = M1(t)+q10(t)  A0(t)+q11.2(t) A1(t) (4.2)

M0(t) = e-2t , M1(t) = e-2t G(t)

Taking LT of above relations (4.2) and solving for

A

*₀(s), we get steady state availability of the system as:

lim (4.3)

Where and D12 =



2

1

- {1-g*(2)}.g*'(0)

4.3 Busy period of the server

The recursive relations for Bi(t) are given as :

B0(t) = q01(t)  B1(t)

B1(t) = W1(t)+q10(t)  Bo(t) + q11.2(t)  B1(t) (4.5)

W1(t) = {e-2t + (2e-2t 1)}G(t)

Taking L.T. of relations (4.5) and solving for

B (s)

*₀ , we get in the long run the time for which the system is under repair as:

lim (4.6)

Where N13 = -g*'(0) and D12 is already specified 4.4 Expected number of visits by the server

The recursive relations of Ni(t) are given as

N0(t)=Q01(t) [1+N1(t)] (4.5)

N1(t) = Q10(t) N0(t)+Q11.2(t) N1(t)

Taking LST of above relations (4.5) and solving for

N



₀(s), we get the expected number of visits per unit time as:

5 Analysis for model II

5.1 Mean time to system failure (MTSF)

On the basis of arguments used for regenerative processes, we obtain the following recursive relations for i(t):

(5.1) Taking LST of above relations (5.1) and solving for , we get MTSF as

lim ∞

~

where, and

5.2 Steady state availability

The recursive relations for Ai(t) are given as

©

A t M t q t ©A t q _. t q _. t ©A t q t©A ©A t q _. t ©A t (5.2)

Where , , and

Taking LT of above relations (5.2) and solving for

A

*₀(s), we get steady state availability of the system as:

lim (5.3)

Where and

, ,

(2λ)

5.3 Busy Period of the Server

The recursive relations for Bi(t) are given as

©

© _{. .} © ©

© _. © (5.4)

Where © λe λ© © and

©

Taking L.T. of relations (5.4) and solving for

B (s)

*₀ , we get in the long run the time for which the system is under repair as

lim (5.5)

where , , λ

λ

∞

5.4 Expected number of visits by the server.

The recursive relations of Ni(t) are given as:

. . (5.6)

.

Taking LST of above relations (18) and solving for , we get the expected number of visits per unit time as

lim (5.7)

Where λ λ and D22 is already specified.

6 Cost- analysis

1. The expected up time of the system in (0,t] is given by

, , (6.1) Taking Laplace transform, we get

, , (6.2) 2. Expected down time of the system in (0, t]

Di(t) = t- Ui(t), i=1,2 (6.3)

Taking Laplace transform, we get D*(s) = (6.4) 3. Expected duration of busy time of server in (0, t] is given by

, , (6.5)

0r , , (6.6) 4. The expected profit gained in (0, t] is defined as the difference between total revenue and total

expenditure incurred in (0, t]. Hence the total profit in (0, t] is given by . ) Where k1= Revenue per unit up time for the system

k2= Cost per unit time for which the system is under repair

k3= Cost per visit by the server

Expected profit per unit time in steady state is given by

lim ∞ lim , for i=1,2 (6.7) (6.8)

7. Particular Case (Results in Parametric Form)

Let us take , , , we can obtain the following results

Expected number of visits ( N10) =

For Model 2: MTSF(T2)= , Availability (A20) = , Busy period (B20) = ,

Expected number of visits( N20) = . Where 4

, 4 , , 4 , , , ,

4 , , 4 , ,

,

, ,

, , ,

,

8. State transition diagrams

 

Figure 1: Model – I 

 

 

 

 

 

Figure 2: Model – II

 Regenerative point ---- Up-state ---- Down state

Acknowledgements: The authors are very grateful to the editor and anonymous referees for their valuable suggestions which helped to improve the manuscript.

      

 

  S ₀             S₁           S ₂  

        

       2   2

 

      g(t)        g(t)

 

N _o ,N_o,   C_s   

N_o,N_o, 

Fur 

  

N _o , F_uR,    F _wr 

 

0 0 ,

N , N C s 

0 0 ,

N , N C s

0 0 , u i

N , N F  

0 0 , ur

N , N F  

0 w i, u I

N , F F

0 w i, uR

N , F F   _{0 w I ur}

,

N , F F  

2 

2 

2 

bh(t)  bh(t)

g(t) 

g(t)  g(t)  _ah(t) 

ah(t) 

S0  S1 

S2 

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