Behavioral Analysis of Two Unit System with Preventive...

International Journal of Engineering Technology, Management and Applied Sciences

www.ijetmas.com November 2015, Volume 3, Special Issue, ISSN 2349-4476

120 Vijay Goyal, Reenu Goel, Dr. Pardeep Goel

Behavioral Analysis of Two Unit System with Preventive

Maintenance in both units and Degradation in one unit

Vijay Goyal*, Reenu Goel**, Dr. Pardeep Goel*** *Asst. Prof., M.M. (P.G.) College, Fatehabad

**Assistant Professor, Maths Deptt., CDLU, Sirsa

** Asso. Prof. M.M. (PG) College, Fatehabad

Keywords: - Availability, Reliability, Primary Circuits, Tertiary Circuits, Preventive Maintenance, Degraded state,

Base-State, Regenerative Point Graphical Technique (RPGT), MTSF, Busy period of the Server, Expected No. of

Server’s Visits, Fuzzy Logic, Steady State.

Abstract:- In this paper Behavioral Analysis of a two units system with preventive maintenance on both units and

degradation in one unit after Complete Failure and the other unit with perfect repair using Regenerative Point

Graphical Technique (RPGT) is discussed.

Introduction: - In this paper Behavioral Analysis of a two units system with preventive maintenance

on both units and degradation in one unit after Complete Failure and the other unit with perfect

repair using Regenerative Point Graphical Technique (RPGT) is discussed. Such systems are

available in a number of process industries.Considering the importance of individual units in system

Kumar, J. & Malik, S. C. [1] have discussed the concept of preventive maintenance for a single unit

system. Liu., R. [2], Malik, S. C. [3], Nakagawa, T. and Osaki, S. [4] have discussed reliability

analysis of a one unit system with un-repairable spare units and its applications. Goel, P. & Singh J.

[5], Gupta, P., Singh, J. & Singh, I.P. [6], Kumar, S. & Goel, P. [7], Gupta, V. K. [8], Chaudhary,

Goel & Kumar [9] Sharma & Goel [10], Ritikesh & Goel [11] and Goyal & Goel [12] have discussed

behavior with perfect and imperfect switch-over of systems using various techniques. Both units A &

B have parallel sub units so it can work in reduced capacity as well as in full capacity. The unit A

can fail from normal mode to complete failure directly or via partial failure, whereas unit B can fail

can through partial mode only. Both units have preventive maintenance facility. Most of the systems

have a number of units and one of these units may be important for working of system, hence need

more care over other units. Further, there are units which may not be repaired to their original

capacity after complete failure. Since the capability of unit after repair depends on the repair

mechanism adopted and unit may have increased failure rate on subsequent failures. The system may

go under imperfect repair on complete failure and unit is degraded but operative state is obtained

again and again. Server inspects and repairs the unit as and when need arise. After a limiting

situation, when no further repair is possible, then system is replaced by new one.

Initially the two units A & B are working at full capacity in which unit A and B may have two types

of failures one is direct and second one is through partial failure mode, but unit B cannot fail directly.

There is a single server (repairman) who inspects and repairs the units as and when need arise. On

partial failure (before complete failure) server repairs the unit and on complete failure the unit A

cannot be restored to its original capacity i.e. it is in degraded state.Repair of Unit B is perfect. Fuzzy

concept is used to determine failure/working state of a unit. If the server report, that unit A is not

repairable, then it is replaced by a new one. Taking failure rates exponential, repair rates general and

taking into consideration various probabilities, a transition diagram of the system is developed to

determine Primary circuits, Secondary circuits & Tertiary circuits and Base state. Problem is

formulated and solved using RPGT. Repair and Failure are statistically independent. Expressions for

system parameters i.e. mean time to system failure, availability, number of server visits and busy




period of the server are evaluated to study the behavior of the system for steady state. Particular

cases are taken to study the effect of failure and repair rates on mean time to system failure,

availability, expected number of server visits and busy period of the server. Profit optimization is

also discussed. System behavior is discussed with the help of graphs and tables.

Assumptions and Notations: - The following assumptions and notations are taken: -

1. A single repair facility is available.

2. The distributions of failure times and repair times are exponential and general respectively and

also different.

3. Repair of unit A is imperfect and repaired unit is not good as new on complete failure. Whereas

repair of unit B is perfect.

4. Both Units A and B are subjected to Preventive Maintenance.

5. Nothing can fail when the system is in failed state.

6. The system is discussed for steady-state conditions.

7. Replacement of Un-repairable unit and repair facility is immediate.

8. Unit A can fail directly and through partial failure but unit B can fail through partial failure only.

9. Failures and repairs are statistically independent.

10. The system fails and any of the units fails.

A circuit formed through un-failed states.

m-cycle : A circuit (may be formed through regenerative or non-regenerative / failed state)

whose terminals are at the regenerative state m.

m- : A circuit (may be formed through un-failed regenerative or non-regenerative state)

whose terminals are at the regenerative m state.

: r-th directed simple path from i-state to j-state; r takes positive integral values for

different paths from i-state to j-state.

: A directed simple failure free path from -state to i-state.

Vm,m : Probability factor of the state m reachable from the terminal state m of the m-cycle.

V :Probability factor of the state m reachable from the terminal state m of the m- .

Ri (t) : Reliability of the system at time t, given that the system entered the un-failed

regenerative state ‘i’ at t = 0.

Ai (t) : Probability of the system in up time at time ‘t’, given that the system entered

regenerative state ‘i’ at t = 0.

Bi (t) : Reliability that the server is busy for doing a particulars job at time ‘t’; given that

the system entered regenerative state ‘i’ at t = 0.

Vi (t) : The expected no. of server visits for doing a job in (o,t] given that the system

entered regenerative state ‘i’ at t = 0.

‘,’ : denote derivative

Wi (t) : Probability that the server is busy doing a particular job at time t without transiting

to any other regenerative state ‘i’ through one or more non-regenerative states, given

that the system entered the regenerative state ‘i’ at t = 0.

: Mean sojourn time spent in state i, before visiting any other states;

.




: The total un-conditional time spent before transiting to any other regenerative states,

given that the system entered regenerative state ‘i’ at t=0.

ni : Expected waiting time spent while doing a given job, given that the system entered

regenerative state ‘i’ at t=0;

ξ : Base state of the system.

: Fuzziness measure of the j-state. λn = (1+α) λn-1, n ≥ 6.

A/ /a : Unit in full capacity working / reduced state / failed state.

Taking into consideration the above assumptions and notations the Transition Diagram of the

system is given in Figure 1.

Figure-1

State Symbol

Up-state

Failed State

Reduced State/ Degraded State

Table 1

The system can be in any of the following states with respect to the above symbols.

S0 = AB S1 = 𝐴 B S2 = aB

S3 = A1B S4 = a1B S5 = A𝐵

S6 = 𝐴𝐵 S7 = 𝐴 b S8 = a𝐵

S9 = A1𝐵 S10 = Ab S10 = A1b

λ = The failure rate from A to a,

λ1 = The failure rate from A to 𝐴 . w1 = Repair rate from 𝐴 to A.

λ2 = The failure rate from 𝐴 to a. w2 = Repair rate from a to 𝐴 . Λn = The failure rate from A1 to a1. W3 = Repair rate from a1 to A1.

λ4 = The failure rate from B to 𝐵 . w4 = Repair rate from 𝐵 to B.

λ5 = The failure rate from 𝐵 to b. w5 = Repair rate from b to 𝐵 .




From this, we see that primary circuits in state ‘0’ and 3 are same. But state ‘0’ has lesser number of

secondary and tertiary circuits therefore base state’ ξ’ =’ 0.’

Primary, Secondary & Tertiary Circuits with simple path (‘0’ as base state)

Vertex (0 j):P0

P1 P2

0 (0

𝑆1 0): (0,1,0)

(0𝑆2 0): (0,5,0)

(0𝑆3 0): (0,1,6,5,0)

(0𝑆4 0): (0,5,6,1,0)

(0𝑆5 0): (0,1,2,3,4,0)

(0𝑆6 0): (0,2,3,4,0)

(0𝑆7 0): (0,1,6,8,9,3,4,0)

(0𝑆8 0): (0,5,8,9,3,4,0)

(0𝑆9 0): (0,5,6,8,9,3,4,0)

(1,6,1)

(5,6,5)

(5,10,5)

(6,7,6)

(5,10,5)

(5,10,5)

(6,7,6)

(1,6,1)

(3,9,3)

(3,9,3)

(6,7,6)

(9,11,9)

(9,11,9)

(5,6,5)

(5,10,5)

(5,10,5)

(6,7,6)

(9,11,9)

(6,5,6), (6,7,6)

(6,7,6)

Nil

Nil

(6,7,6), (6,5,6)

(9,11,9)

(9,11,9)

Nil

(6,7,6)

(6,1,6)

(1,6,1)

1 (0

𝑆1 1): (0,1)

(1,6,1) (6,7,6)

(6,5,6)

2 (0

𝑆1 2): (0,2)

(0𝑆2 2): (0,1,2)

Nil

(1,6,1)

Nil

(6,7,6)

(6,5,6)

3 (0

𝑆1 3): (0,1,2,3)

(0𝑆2 3): (0,2,3)

(0𝑆3 3): (0,5,8,9,3)

(0𝑆4 3): (0,1,6,8,9,3)

(0𝑆5 3): (0,5,6,8,9,3)

(0𝑆6 3): (0,1,6,5,8,9,3)

(3,9,3)

(1,6,1)

(3,4,3)

(3,9,3)

(3,4,3)

(5,6,5)

(5,10,5)

(9,11,9)

(3,4,3)

(6,7,6)

(9,11,9)

(3,4,3)

(5,10,5)

(6,7,6)

(9,11,9)

(3,4,3)

(6,7,6)

(5,10,5)

(9,11,9)

(3,4,3)

(6,7,6)

(6,5,6)

(9,11,9)

(6,7,6)

(5,10,5)

Nil

Nil

Nil

4 (0

𝑆1 4): (0,1,2,3,4)

(1,6,1)

(3,9,3)

(3,9,3)

(6,7,6)

Nil




(0𝑆2 4): (0,2,3,4)

(0𝑆3 4): (0,5,8,9,3,4)

(0𝑆4 4): (0,1,6,8,9,3,4)

(0𝑆5 4): (0,5,6,8,9,3,4)

(0𝑆6 4): (0,1,6,5,8,9,3,4)

(5,6,5)

(5,10,5)

(9,11,9)

(3,9,3)

(6,7,6)

(9,11,9)

(3,9,3)

(5,10,5)

(6,7,6)

(9,11,9)

(6,7,6)

(5,10,5)

(9,11,9)

(6,7,6)

(9,11,9)

(5,10,5)

Nil

Nil

Nil

5 (0

𝑆1 5): (0,5)

(0𝑆2 5): (0,1,6,5)

(5,10,5)

(5,6,5)

(6,7,6)

(5,10,5)

(6,7,6)

Nil

6 (0

𝑆1 6): (0,1,6)

(0𝑆2 6): (0,5,6)

(6,7,6)

(5,10,5)

Nil

Nil

7 (0

𝑆1 7): (0,1,6,7)

(0𝑆2 7): (0,5,6,7)

Nil

(5,10,5)

Nil

Nil

8 (0

𝑆1 8): (0,5,8)

(0𝑆2 8): (0,1,6,8)

(0𝑆3 8): (0,1,6,5,8)

(5,10,5)

(5,6,5)

(6,7,6)

(6,7,6)

(5,10,5)

(6,7,6)

Nil

Nil

9 (0

𝑆1 9): (0,5,8,9)

(0𝑆2 9): (0,1,6,8,9)

(0𝑆3 9): (0,1,6,5,8,9)

(0𝑆4 9): (0,1,2,3,9)

(0𝑆5 9): (0,2,3,9)

(5,10,5)

(5,6,5)

(9,11,9)

(6,7,6)

(9,11,9)

(6,7,6)

(5,10,5)

(9,11,9)

(1,6,1)

(3,4,3)

(3,4,3)

(6,7,6)

Nil

Nil

(6,7,6)

Nil

10 (0

𝑆1 10): (0,5,10)

(0𝑆2 10): (0,1,6,5,10)

(5,6,5)

(6,7,6)

(6,7,6)

Nil

11 (0

𝑆1 11): (0,5,8,9,11)

(0𝑆2 11): (0,1,6,8,9,11)

(0𝑆3 11): (0,1,6,5,8,9,11)

(0𝑆4 11): (0,1,2,3,9,11)

(0𝑆5 11): (0,2,3,9,11)

(5,10,5)

(5,6,5)

(6,7,6)

(6,7,6)

(5,10,5)

(1,6,1)

(3,4,3)

(3,4,3)

(6,7,6)

Nil

Nil

(6,7,6)

Nil

Table 2




Transition Probabilities and the Mean sojourn times.

qi,j(t) : Probability density function (p.d.f.) of the first passage time from a regenerative state ‘i’

to a regenerative state ‘j’ or to a failed state ‘j’ without visiting any other regenerative state

in (0,t].

pi,j : Steady state transition probability from a regenerative state ‘i’ to a regenerative state ‘j’

without visiting any other regenerative state. pi,j = ; where denotes Laplace

transformation.

Transition Probabilities Mean Sojourn Times

qi,j(t)

Pij = q*i,j(t)

𝑞0,1= λ1𝑒− 𝜆1+𝜆4+𝜆 𝑡

𝑞0,2= λ𝑒− 𝜆1+𝜆4+𝜆 𝑡

𝑞0,5= λ4𝑒− 𝜆1+𝜆4+𝜆 𝑡

𝑝0,1= λ1/(λ1+λ4+λ)

𝑝0,2= λ/(λ1+λ4+λ)

𝑝0,5= λ4/(λ1+λ4+λ)

𝑞1,0= 𝑤1𝑒− 𝑤1+𝜆2+𝜆4 𝑡

𝑞1,2= 𝜆2𝑒− 𝑤1+𝜆2+𝜆4 𝑡

𝑞1,6= 𝜆4𝑒− 𝑤1+𝜆2+𝜆4 𝑡

𝑝1,0= w1/(w1+λ2+λ4)

𝑝1,2= λ2/(w1+λ2+λ4)

𝑝1,6= λ4/(w1+λ2+λ4)

𝑞2,3= 𝑤2𝑒−𝑤2𝑡 𝑝2,3= w2/w2 = 1

𝑞3,4= 𝜆𝑛𝑒−(𝜆𝑛+𝜆4)𝑡

𝑞3,9= 𝜆4𝑒−(𝜆𝑛+𝜆4)𝑡

𝑝3,4= λn/(λn+λ4)

𝑝3,9= λ4/(λn+λ4)

𝑞4,0= 𝑔 𝑡 𝑒−𝑤3𝑡

𝑞4,3= 𝑤3𝑒−𝑤3𝑡𝑔(𝑡)

𝑝4,0= g*w3

𝑝4,3= 1-g*w3

𝑞5,0= 𝑤4𝑒− 𝑤4+𝜆1+𝜆5+𝜆 𝑡

𝑞5,6= 𝜆1𝑒− 𝑤4+𝜆1+𝜆5+𝜆 𝑡

𝑞5,10= 𝜆5𝑒− 𝑤4+𝜆1+𝜆5+𝜆 𝑡

𝑞5,8= 𝜆𝑒− 𝑤4+𝜆1+𝜆5+𝜆 𝑡

𝑝5,0= w4/(w4+λ1+λ5+λ)

𝑝5,6= λ1/(w4+λ1+λ5+λ)

𝑝5,10= λ5/(w4+λ1+λ5+λ)

𝑝5,8= λ/(w4+λ1+λ5+λ)

𝑞6,1= 𝑤4𝑒− 𝑤4+𝜆5+𝑤1+𝜆2 𝑡

𝑞6,7= 𝜆5𝑒− 𝑤4+𝜆5+𝑤1+𝜆2 𝑡

𝑞6,5= 𝑤1𝑒− 𝑤4+𝜆5+𝑤1+𝜆2 𝑡

𝑞6,8= 𝜆2𝑒− 𝑤4+𝜆5+𝑤1+𝜆2 𝑡

𝑝6,1= w4/(w4+λ5+w1+λ2)

𝑝6,7= λ5/(w4+λ5+w1+λ2)

𝑝6,5= w1/(w4+λ5+w1+λ2)

𝑝6,8= λ2/(w4+λ5+w1+λ2)

𝑞7,6= 𝑤5𝑒−𝑤5𝑡 𝑝7,6= w5/w5 = 1

𝑞8,9= 𝑤3𝑒−𝑤3𝑡 𝑝8,9= w3/w3 = 1

𝑞9,3= 𝑤4𝑒− 𝑤4+𝜆5 𝑡

𝑞9,11= 𝜆5𝑒− 𝜆5+𝑤4 𝑡

𝑝9,3= w4/(w4+λ5+w1+λ2)

𝑝9,11= λ5/(w4+λ5)

𝑞10,5= 𝑤5𝑒−𝑤5𝑡 𝑝10,5= w5/w5 = 1

𝑞11,9= 𝑤5𝑒−𝑤5𝑡 𝑝11,9= w5/w5 = 1

Table 3

The path probabilities for the steady state are given w.r.t. Base State ‘0’ are as

V0,0 = 1(verified) , V0,1 = p0,1 = λ1/(λ1+λ4+λ)

V0,2 = (0,2)+(0,1,2)/(1-L3) = λ/(λ1+λ4+λ) = p0,1p1,2/(1-L3)

= (λ/λ1+λ4+λ)+{λ1λ2/(λ+λ1+λ4)(w1+λ2+λ4)}/{1-λ4w4/(w1+λ2+λ4)(w1+w4+λ2+λ5)}

= {λ1λ2(λn+λ4)2/(λ2+w1)λnλ4(λ1+λ+λ4)}+{λ/(λn+λ4)(λ1+λ+λ4)}

= {(λn+λ4)2(λ1λ2+λλ2+λw1)/λnλ4(λ1+λ+λ4)(λ2+w1)}

V0,3 = (0,1,2,3)/(1-L3)+(0,2,3)+(0,5,8,9,3)/(1-L4)(1-L6)+(0,1,6,8,9,3)/(1-L5)(1-L7)

Ri(t) µi=Ri*(0)

𝑅0(𝑡)= 𝑒− 𝜆1+𝜆4+𝜆 𝑡 µ0 = 1/(λ1+λ4+λ)

𝑅1(𝑡)= 𝑒− 𝑤1+𝜆2+𝜆4 𝑡 µ1 = 1/(w1+λ2+λ4)

𝑅2(𝑡)= 𝑒−𝑤2𝑡 µ2 = 1/w2

𝑅3(𝑡)= 𝑒−(𝜆𝑛+𝜆4)𝑡 µ3 = 1/(λn+λ4)

𝑅4(𝑡)= 𝑒−𝑤3𝑡𝑔 𝑡 µ4 = [1-g*[(w3)]/w3

𝑅5(𝑡)= 𝑒− 𝑤4+𝜆1+𝜆5+𝜆 𝑡 µ5 = 1/(w4+λ1+λ5+λ)

𝑅6 𝑡 =

𝑒− 𝑤4+𝜆5+𝑤1+𝜆2 𝑡

µ6 = 1/(w4+λ5+w1+λ2)

𝑅7 𝑡 = 𝑒−𝑤5𝑡 µ7 = 1/w5

𝑅8 𝑡 = 𝑒−𝑤3𝑡 µ8 = 1/w3

𝑅9 𝑡 = 𝑒−(𝑤4+𝜆5)𝑡 µ9 = 1/(w4+λ5)

𝑅10 𝑡 = 𝑒−𝑤5𝑡 µ10 = 1/w5

𝑅11 𝑡 = 𝑒−𝑤5𝑡 µ11 = 1/w5

Table 4

Here we use some notations for our convenience

L1 = (0,1,0) = p0,1 p1,0 =

λ1w1/(λ1+λ4+λ)(w1+λ2+λ4)

L2 = (0,5,0) = p0,5 p5,0 =

λ4w4/(λ1+λ4+λ)(w4+λ1+λ5+λ)

L3 = (1,6,1) = p1,6 p6,1 =

λ4w4/(w1+λ2+λ4)(w4+λ5+w1+λ2)

L4 = (5,6,5) = p5,6 p6,5 =

λ1w1/(λ1+λ5+w4+λ)(w4+λ5+w1+λ2)

L5 = (6,7,6) = p6,7 p7,6 = λ5/(w4+λ5+w1+λ2)

L6 = (5,10,5) = p5,10 p10,5 = λ5/(w4+λ1+λ5+w)

L7 = (9,11,9) = p9,11 p11,9 = λ5/(w4+λ5)

L8 = (9,3,9) = p3,9 p9,3 = λ4w4/(λ4+λn)(w4+λ5)

L9 = (3,4,3) = p3,4 p4,3 = λn/(λ4+λn)(1-g*wn)




+(0,1,6,5,8,9,3)/(1-L6)(1-L5)(1-L7)

= (p0,2p2,3+p0,1p1,2p2,3)/(1-L3)+p0,5p5,8p8,9p9,3/(1-L4)(1-L6)+p0,1p1,6p6,8p8,9p9,10/(1-L5)

(1-L7)+(p0,1p1,6p6,5p5,8p8,9p9,3+p0,5p5,6p6,8p8,9p9,3)/(1-L5)(1-L6)(1-L7)

= [λ1/(λ1+λ4+λ)+λ1λ2/(λ1+λ4+λ)(λ2+λ4+w1)]/[1-λ4w4/(w1+λ2+λ4)(w1+w4+λ2+λ4)]

+[λ4λw4/(λ+λ4+λ)(λ+λ1+λ5+w4)(w4+λ5)]/[1-λ1w1/(λ+λ1+λ5+w4)(w1+w4+λ2+λ5)

(1-λ5/w4+λ+λ1+λ5)]+[λ1λ4λ2w4/(λ1+λ4+λ)(w1+λ2+λ4)(w4+w1+λ2+λ5)(w4+λ5)]

/[1-λ5/(w1+w4+λ2+λ5)(w4/λ5+w4)]+[λ1λ4w1λw4/(λ1+λ4+λ)(w1+λ2+λ4)(w1+w4+λ2+λ5)

(λ1+λ+λ5+w4)(w4+λ5)+λ4λ1λ2λw4/(λ1+λ4+λ)(λ+λ1+λ5+w4)(w4+w1+λ2+λ5) (λ5+w4)]

/[(1-λ5/w1+w4+λ2+λ5)(1-λ5/λ+λ1+w4+w5) (w4/λ5+w4)]

V0,4 = (0,1,2,3,4)/(1-L3)+(0,2,3,4)+(0,5,8,9,3,4)/(1-L4)(1-L6)+(0,1,6,8,9,3,4)/(1-L5)(1-L7)

+(0,1,6,5,8,9,3,4)/(1-L6)(1-L5)(1-L7)

= (p0,2p2,3 p3,4+p0,1p1,2p2,3 p3,4)/(1-L3)+p0,5p5,8p8,9p9,3 p3,4/(1-L4)(1-p6)+p0,1p1,6p6,8p8,9p9,44

/(1-L5)(1-L7)+(p0,1p1,6p6,5p5,8p8,9p9,3 p3,4+p0,5p5,6p6,8p8,9p9,3 p3,4)/(1-L5)(1-L6)(1-L7)

= [λ1/(λ1+λ4+λ)+λ1λ2/(λ1+λ4+λ)(λ2+λ4+w1)]/[1-λ4w4/(w1+λ2+λ4)(w1+w4+λ2+λ4)]

+[λ4λw4/(λ+λ4+λ)(λ+λ1+λ5+w4)(w4+λ5)]/[1-λ1w1/(λ+λ1+λ5+w4)(w1+w4+λ2+λ5)

(1-λ5/w4+λ+λ1+λ5)]+[λ1λ4λ2w4/(λ1+λ4+λ)(w1+λ2+λ4)(w4+w1+λ2+λ5)(w4+λ5)]

/[1-λ5/(w1+w4+λ2+λ5)(w4/λ5+w4)]+[λ1λ4w1λw4/(λ1+λ4+λ)(w1+λ2+λ4)(w1+w4+λ2+λ5)

(λ1+λ+λ5+w4)(w4+λ5)+λ4λ1λ2λw4/(λ1+λ4+λ)(λ+λ1+λ5+w4)(w4+w1+λ2+λ5)(λ5+w4)]

/[(1-λ5/w1+w4+λ2+λ5)(1-λ5/λ+λ1+w4+w5) (w4/λ5+w4)]

V0,5 = (0,5)/(1-L4)(1-L6)+(0,1,6,5)/(1-L5)(1-L4)(1-L6)

= p0,5/(1-L4)(1-L6)+p0,1p1,6p6,5/(1-L5)(1-L4)(1-L6)

= (λ4/λ1+λ4+λ)/[1-λ1w1/(λ1+λ5+w4+λ)(w4+λ5+w1+λ2)(1-λ5/w4+λ+λ1+λ5)]+[λ1λ4w1

/(λ+λ4+λ)(w1+λ2+λ4)(w4+λ5+λ2+w1)]/[(1-λ1w1/λ1+λ5+w4+λ)(w4+λ5+w1+λ2)

(1-λ5/w4+λ+λ1+λ5)(1-λ5/w4+λ5+w1+λ2)]

V0,6 = (0,1,6)/(1-L5)+(0,5,6)/(1-L6)(1-L5) = (p0,1p1,6+p0,5p5,6/1-L6)/(1-L5)

= [λ1λ4/(λ+λ1+λ4)(w1+λ2+λ4)+{λ4λ1/(λ+λ1+λ4)(λ+λ1+λ5+w4)}(1-λ5/w4+λ1+λ5+λ)]

/[(1-λ5/λ4+λ5+w1+λ2)]

V0,7 = (0,1,6,7)+(0,5,6,7)/(1-L6) = (p0,1p1,6p6,7+p0,5p5,6p6,7/(1-L6)

= λ1λ4λ5/(λ+λ1+λ4)(w1+λ2+λ4)(w4+λ5+w1+λ2)+[λ4λ1λ5/(λ1+λ1+λ4)(λ+λ1+λ5+w4)]

/[(1-λ5/w4+λ1+λ5+λ)]

V0,8 = (0,5,8)/(1-L4)(1-L6)+(0,5,6,8)/(1-L5)(1-L6)+(0,1,6,8)/(1-L4)(1-L5)+(0,1,6,5,8)

/(1-L5)(1-L6)

= (p0,5p5,8)/(1-L4)(1-L6)+(p0,1p1,6p6,8)/(1-L4)(1-L5)+(p0,5p5,6p6,8p0,1p1,6p6,5p5,8)

/(1-L4)(1-L5)

= [λ4λ/(λ+λ4+λ1)(w4+λ1+λ5+λ)]/[(1-λ1w1/λ1+λ5+w4+λ)(w4+λ5+w1+λ2)

(1-λ5/w4+λ5+λ1+λ)]+[λ1λ4λ2/(λ+λ4+λ1)(w1+λ2+λ4)(w4+λ5+λ1+λ)]

/(1-λ1w1/λ1+λ5+w4+λ)(λ1+λ5+w4+λ)(w4+λ5+w1+λ2)(1-λ5/λ5+w4+w1+λ2)]

+[λ4λ1λ2/(λ1+λ4+λ)(w4+λ1+λ5+λ)(w4+λ5+λ1+λ2)+ λ1λ4w1λ/(λ+λ4+λ)(w2+λ2+λ4)

(w4+w1+λ2+λ5)(w4+λ+λ1+λ5)]/[(1-λ5/w4+w1+λ2+λ5)(1-λ5/w4+λ5+λ+λ1)]

V0,9 = (0,5,8,9)/(1-L4)(1-L6)+(0,5,6,8,9)/(1-L5)(1-L6)+(0,1,6,8,9)/(1-L4)(1-L5)

+(0,1,6,5,8,9)/(1-L5)(1-L6)+(0,1,2,3,9)/(1-L3)(1-L9)






V0,10 = (0,5,10)/(1-L4)+(0,1,6,5,10)/(1-L5) = (p0,5p5,10)/(1-L4)+(p0,1p1,6p6,5p5,10)/(1-L5)




= [λ4λ5/(λ+λ1+λ4)(λ+λ1+λ5+w4)]/[{(1-λ1w1)/(λ+λ5+λ4+w4)(w4+w1+λ5+λ2)}

+{(λ1λ4w1λ5)/(λ1+λ2+λ4)(λ4+λ2+w1)(w4+w1+λ2+λ5)(w4+λ1+λ5+λ1)}]

/(1-λ5/w4+w1+λ5+λ2)

V0,11 = (0,5,8,9,11)/(1-L4)(1-L6)+(0,5,6,8,9,11)/(1-L5)(1-L6)+(0,1,6,8,9,11)/(1-L4)(1-L5)

+(0,1,6,5,8,9,11)/(1-L5)(1-L6)+(0,2,3,9,11)/(1-L3)(1-L9)+(0,1,2,3,9,11)/(1-L3)(1-L9)






Mean Time to System Failure (MTSF) (T0): The regenerative un-failed states to which the system

can transit(initial state ‘0’), before entering any failed state are: ‘i’ = 0,1,,5,6 taking ‘ ’ = ‘0’.

𝑝𝑟 𝜉

𝑠𝑟 (𝑠𝑓𝑓 ) 𝑖 𝜇𝑖

Π𝑚1≠𝜉 1−𝑉𝑚1𝑚1

𝑖 ,𝑠𝑟 ÷ 1− 𝑝𝑟 𝜉

𝑠𝑟 (𝑠𝑓𝑓 ) 𝜉


𝑠𝑟

T0 = [(0,0)µ0+(0,1)µ1+(0,5)µ5+(0,6)µ6]/[1-(0,1,0)/(1-L3)-(0,5,0)/(1-L4)(1-L6)

-(0,5,6,1,0)/(1-L5)(1-L6)-(0,1,6,5,0)/(1-L5)(1-L6)]

= [(µ0+p0,1µ1+p0,5µ5)+( (p0,5p5,6+p0,1p1,6)µ6]/[(1-p0,1p1,0)/(1-L3)-p0,5p5,0/(1-L4)

(1-L6)-p0,5p5,6p6,1p1,0/(1-L5)(1-L6)-p0,1p1,6p6,5p5,0/(1-L5)(1-L6)

= (λ2+λ4+w1+λ1)/[(λ1+λ+λ4)(w1+λ2+λ4)-λ1w1]

Availability of the System: The regenerative states at which the system is available are ‘j’ = 0,1,3

and the regenerative states are ‘i’ = 0 to 4 taking ‘ξ’ = ‘0’ the total fraction of time for which

the system is available is given by

A0= 𝑝𝑟 𝜉𝑠𝑟 𝑗 𝑓𝑗 ,𝜇𝑗

Π𝑚1≠𝜉 1−𝑉𝑚1𝑚 1

𝑗 ,𝑠𝑟 ÷ 𝑝𝑟 𝜉𝑠𝑟 𝑖 𝜇 𝑖

1


𝑖 ,𝑠𝑟

A0 = 𝑉𝜉 ,𝑗𝑗 , 𝑓𝑗 , 𝜇𝑗 ÷ 𝑉𝜉 ,𝑖𝑖 ,𝑓𝑗 , 𝜇𝑖1 Taking f0 = f1 = f5 = f6 = f3 = f9 = 1, rest one are zero

= V0,0µ0+V0,1µ1+V0,3µ3+V0,5µ5+V0,6µ6+V0,9µ9/(V0,0µ0+V0,1µ1+V0,2µ2+V0,3µ3

+V0,4µ4+V0,5µ5+V0,6µ6+V0,7µ7+V0,8µ8+V0,9µ9+V0,10µ10+V0,11µ11)

Busy Period of the Server: The regenerative states where server ‘j’ = 1,2,3,4 taking base state ξ =

‘0’, the total fraction of time for which the server remains busy is

B0= 𝑝𝑟 𝜉𝑠𝑟 𝑗 ,𝑛𝑗

Π𝑚1≠𝜉 1−𝑉𝑚1𝑚 1

𝑗 ,𝑠𝑟 ÷ 𝑝𝑟 𝜉𝑠𝑟 𝑖 𝜇 𝑖

1


𝑖 ,𝑠𝑟

B0 = 𝑉𝜉 ,𝑗𝑗 ,𝑛𝑗 ÷ 𝑉𝜉 ,𝑖𝑖 , 𝜇𝑖1 Taking ni = µi

= (V0,1µ1+V0,2µ2+V0,5µ5+V0,6µ6+V0,7µ7+V0,8µ8+V0,9µ9+V0,10µ10+V0,11µ11)

/(V0,0µ0 +V0,1µ1+V0,2µ2+V0,3µ3+V0,4 µ4+V0,5 µ5+V0,6µ6+V0,7µ7+V0,8µ8+V0,9µ9

+V0,10µ10+V0,11µ11)

Expected Number of Server’s Visits (V0): The regenerative states where the repair man do this job

j = 1 Taking base state ‘ξ’ = ‘0’, the number of visit by the repair man is given by

V0= 𝑝𝑟 𝜉𝑠𝑟 𝑗

Π𝑘1≠𝜉 1−𝑉𝑘1𝑘1

𝑗 ,𝑠𝑟 ÷

𝑝𝑟 𝜉𝑠𝑟 𝑖 𝜇 𝑖1

Π𝑘2≠𝜉 1−𝑉𝑘2𝑘2

𝑖 ,𝑠𝑟

V0 = 𝑉𝜉 ,𝑗𝑗 ÷ 𝑉𝜉 ,𝑖𝑖 , 𝜇𝑖1 = (V0,1+V0,2+V0,4+V0,5+V0,6+V0,7+V0,8+V0,9+V0,10+V0,11)

MTSF (T0) =




Taking ni = μi ,i = 0, using value we get

Now we discuss special cases D when λi = λ for i = 1, 2 ,4 and wi = w, i = 1,2,4 and repair rate after

degradation is given by recurrence relation.

λn = (1+α)λn-1,where α→ degradation parameter depending upon following conditions.

1. The working environment of system, in which unit A is working.

2. The mechanism of repair or preventive maintenance adopted by the server.

3. The number of intervals for which the system goes under preventive maintenance.

4. The material or strength of material used for manufacturing the unit:

Here we take λn-1 = 0.2 (for first time) then α = 0.5, λn = (1+0.5)0.2, λn = 0.3

MTSF

T0 = N / D, where D = λ-{3w(2w+λ)}/5(λ+w)2

N = (1/3λ)+{1/4(2λ+w)}+(1/3λ+w)[(2λ2+2w

2+wλ)/3(λ+w)

2+(w/4)(2λ

2+2w

2+λw)]

We get

T0 w = 0.7 w = 0.8 w = 0.9

λ = 0.2 0.8120 0.8392 0.8516

λ = 0.25 0.7655 0.7890 0.8005

λ = 0.3 0.7132 0.7415 0.7780

Table 5

MTSF table for different failure rate and repair rate

Mean Time To System Failure Graph

Figure 2

Availability of the System

A0 = N/D

N = (3λ2+5λ+w)/(2λ+w)+λ

2(2w+λ)(2λ+w)/3(λ+w)

4w+2λ(λ+w)/w(2λ+3w)+4(λ)

2

/(λ+w)

D = 8λ2/3(2λ+w)(λ+w)+(2λ+w)/3(w+λ)

2[λ+λw/(λ+w)

2+(1/3)]

A0 w = 0.7 w = 0.8 w = 0.9

λ = 0.2 0.6128 0.6521 0.6910

λ = 0.3 0.4215 0.4615 0.4985

Table 6

0.6

0.65

0.7

0.75

0.8

0.85

0.9

λ = 0.2 λ = 0.25 λ = 0.3

w = 0.7

w = 0.8

w = 0.9




Availability of the System Graph

Figure 3

PROFIT FUNCTION OF THE SYSTEM

Profit analysis of the system can be done by using the profit function

P0 = C1A0 – C2B0 – C3V0

Where C1 = Revenue per unit time the system is available

C2 = Cost per unit time the server remain busy for the repair.

C3 = Cost of visit of the server.

Conclusion: - From the tables and analytical discussions, we see that increase in the repair rate (w)

increases the availability of the system and the mean time to system failure, whereas increase in failure

rate decrease the availability and mean time to system which should be so practically. We also see

from the expression of busy period of server that when we increase the repair rate busy period of

server decrease and when we increase the failure rate then busy period of server also increase. Also

from expression for expected number of inspections by the repair man we see that when we increase

the repair rate the expected number of inspections by repairman increase and when we increase the

failure rate then the expected number of inspections by repairman is also increase. In future,

Researchers can evaluate the parameters, when repair rate sand failure rate are variable and also

discuss the cost and profit benefit analysis. Further results can also be applied to find the waiting time

of units and number of server visits, as if the states where the server is on prime visit or on a secondary

visit are determined separately using the formula. Since the cost of secondary visit is usually less than

primary visit of server, therefore the system can be run with low maintenance cost. Various system

parameters can also be evaluated taking any state as base state. As failure rates are beyond control, so

determine the repair rates and reduce the fixing the target of availability management can cost of

maintenance. The Regenerative Point Graphical Technique is useful to evaluate the key parameters of

the system in a simple way, without writing any state equations and without doing any lengthy and

cumbersome calculations.

References: - 1. Kumar, J. Kadyan, M. S., Malik, S. C. & Jindal, C. (2014): Reliability Measures of a Single-Unit System Under

Preventive Maintenance and Degradation With Arbitrary Distributions of Random Variables, Journals of

Reliability and Statistical Studies: ISSN 0974-8024, Vol. 7, p 77 – 88.

2. Liu., R. and Liu, Z. (2011): Reliability Analysis of a One-Unit System With Finite Vacations, Management

Science Industrial Engineering (MSIE) International Conference, p 248-252.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

λ = 0.2 λ = 0.3

w = 0.7

w = 0.8

w = 0.9




3. Malik, S. C. (2008): Reliability Modeling and Profit Analysis of a Single-Unit System with Inspection by a Server

who Appears and Disappears randomly, Journal of Pure and Applied Mathematika Sciences, LXVII (1-2), p 135-

146.

4. Nakagawa, T. and Osaki, S. (1976): Reliability Analysis of a One-Unit System with Un-repairable Spare Units

and its Optimization Application, Quarterly Operations Research, 27(1), p 101-110.

5. Goel, P. & Singh, J. (1997): Availability Analysis of A Thermal Power Plant Having Two Imperfect Switches,

Proc. (Reviewed) of 2nd

Annual Conference of ISITA.

6. Gupta, P., Singh, J. & Singh , I. P. (2004): Availability Analysis of Soap Cakes production System – A Case

Study, Proc. National Conference on Emerging Trends in Manufacturing System, SLIET, Longowal (Punjab) Pb

283-295.

7. Kumar, S. and Goel, P. (2014): Availability Analysis of Two Different Units System with a Standby Having

Imperfect Switch Over Device in Banking Industry, Aryabhatta Journal of Mathematics & Informatics, ISSN:

0975-7139, Vol. 6, No. 2, p 299-304.

8. Gupta V. K. (2011): Analysis of a single unit system using a base state.: Aryabhatta J. of Maths & Info. Vol. 3 (1)

pp 59-66.

9. Chaudhary Nidhi, Goel P., Kumar Surender (2013): Developing the reliability model for availability and behavior

analysis of a distillery using Regenerative Point Graphical Technique.: ISSN (Online): 2347-1697, Vol. 1 (iv),

Dec, 2013, pp 26-40.

10. Sharma Sandeep, P., (2015): Behavioral Analysis of Whole Grain Flour Mill Using RPGT. ISBN 978-93-325-

4896-1, ICETESMA-15, pp 194-201.

11. Ritikesh, Goel (2015): Availability Modeling of Single Unit System Subject to Degradation Post Repair After

Complete Failure Using RPGT. ISSN 2347-8527, September, 2015.

12. Goyal, Goel (2015): Behavioral Analysis of Two Unit System with Preventive Maintenance and Degradation in

One Unit after Complete Failure Using RPGT. ISSN 2347-8527, Vol. 4, pp 190-197.

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