Probability Analysis Of A Cold Standby Unit System With Slow Switching And Correlated Appearance And Disappearance Of Repairman

Probability Analysis Of A Cold Standby Unit System With Slow Switching And Correlated Appearance And Disappearance Of Repairman

Mohit Kakkar1, Ashok Chitkara2, Jasdev Bhatti3

1Department of Applied Sciences,Chitkara University,Punjab,India 2Chancellor, Chitkara University, Himachal Pradesh, India 3Department of Applied Sciences, Chitkara University, Punjab,India

Abstract

The aim of this paper is to present a reliability analysis of a cold standby unit system with the assumption that the switching is not instantaneous

.There is only one repair facility. Appearance and Disappearance of repairman are assumed to be correlated. Using regenerative point technique various reliability characteristics are obtained which are useful to system designers and industrial managers. Graphical behaviors of MTSF and profit function have also been studied.

Keywords: Transition Probabilities, MTSF, Availability, Busy Period, Profit Function.

1. INTRODUCTION

   Two identical unit cold standby systems have widely studied in literature of reliability theory, repair maintenance is one of the most important

   measures for increasing the reliability of the

   appearance and disappearance time of repairman taken as correlated random variables having their joint distribution as bivariate exponential .

2. SYSTEM DESCRIPTION

   System consists of two identical units , initially both units are not operative but the only one of them is sufficient for operating the system, other one is in cold standby. There is single repair facility. When one unit fails another unit takes the charge but switching is not instantaneous. The joint distribution of appearance and disappearance of repairman is taken to be bivariate exponential having density function. Each repaired unit works as good as new.

   Afr Aw

   a

   Afr Aw

   a

   S7

   S6 Afr

   Ao

   a

   Y

   system. Many authors have studied various system models under different repair policies [1-4] ,they have assumed that appearance and disappearance times of repairman are uncorrelated random variable. But in real situation rest period of the repairman depends on workload on the repairman. They have also assumed that the switching is instantaneous but in real life this is not so. Taking these facts into consideration in this paper we investigate a two unit cold standby system model assuming the possibility of slow switching and

   S2 So

   Ao X Ao

   As As

   r Y a

   Y

   S4 S5

   Aw Aw

   A

   A

   Asb

   o

   r r

   S1

   Afr,

   Asb

   a

   Afr,

   Asb

   a

   As Asb

   a

   As Asb

   a

   S3

   S8 Ao

   Afr

   a

   Y

   Fig.1 : Transition Diagram

   Fig.1 : Transition Diagram

3. NOTATIONS

   For defining the states of the system we assume the following symbols:

   A0 : Unit A is in operative mode

   AS: Unit A is in standby mode

   1. Transition Probability and Sojourn Times

      The steady state transition probability can be as follows

      P01= P18.67=

      Afr :

      Unit A is in failure mode

      (1 r )

      ( )( )

      Asb : Unit A(stand by) is being switched

      : Constant rate of repair of unit A

      : Constant rate of failure of unit A

      P02=

      P16=

      (1 r )

      (1 r )

      P10.6=

      P10.6=

      ( )( )

      ( )

      : Constant rate of switching

      (1 r )

      Aw: Unit A in failure mode but in waiting for repairman

      P13=

      P32.6= 2 2

      1 2 (1 r2 )

      X: Random variables representing the disappearance of repair man

      P20=

      (1 r )

      (1 r )

      P40=

      2

      2 1 (1 r1 )

      Y: Random variables representing the appearance of repair man

      P24=

      (1 r )

      P41.5=

      1 (1 r1 )

      2 1 (1 r1 )

      fi(x,y) Joint pdf of (xi,yi);i=1,2

      (1 r )ei xi y I (2 ( r xy ); X ,Y ,

      P01+P02=1 P18.67+P10.6+P10.3=1

      P16+P13=1 P20+P24=1

      i i i

      0 ri 1,

      0 i i i

      P41+P48.5=1 P80+P88.7=1

      ( r xy) j

      (01-20)

      whereI0 (2

      i i ri xy )

      j 0

      i i i

      ( j!)2

      Mean sojourn times:

      ki (Y/X): Conditional pdf of Yi given Xi=x is given by

      r x y

      0

      1

      (1 r)

      1

      = ie

      i i i

      I0 (2 (i i ri xy )

      2 (1 r)

      i i

      i i

      gi(.): Marginal pdf of Xi= (1 r )ei (1ri ) x

      hi(.): Marginal pdf of

      1

      1

      1

      qij (.),

      i :

      Yi= (1 r )ei (1ri ) y

      i i

      i i

      pdf &cdf of transition time from regenerative states pdf &cdf of transition time from regenerative state Si to Sj.

      Mean sojourn time in stateSi.

      8 6

      (21-24)

      : Symbol of ordinary Convolution

      A(t)

      t

      B(t) A(t u)B(u)du

      0

      : symbol of stieltjes convolution

      t

      0

      0

      A(t) B(t) A(t u)dB(u)

4. ANALYSIS OF CHARACTERISTICS

   1. MTSF (Mean Time to System Failure)

      To determine the MTSF of the system, we regard the failed state of the system as absorbing state, by probabilistic arguments, we get

      0 Q01 Q02 2 (t)

      2 Q24

      Q20

      0

      (t)

      0

      0

      Taking Laplace Stieltjes transforms of these relations and solving for ** (s) ,

      ** (s) N (s)

      Let Bi(t) be the probability that the repairman is busy at instant t, given that the system entered regenerative state I at t=0.By probabilistic arguments we have the following recursive relations for Bi(t)

      0 D(s)

      B0 q01 B1(t) q02 B2 (t)

      (25-27)

      B W (t) q

      B (t) q

      B (t) q

      B (t)

      Where

      1 1 10.6 0

      18.67 8

      10.3 0

      N 1 2

      P02

      B2 q20 B0 (t) q24 B4 (t)

      B4 q41 B1(t) q48.5 B8 (t)

      D 1 P10 P01 P20 P02 P30 P03 P40 P04

      (28-29)

      B8 W8 (t) q88.7 B8 (t) q80 B0 (t)

      (39-43)

      0

      0

   2. Availability Analysis

      Taking Laplace transform of the equations of busy

      period analysis and solving them for

      B* (s) ,we get

      Let Ai (t) be the probability that the system is in up-state at instant t given that the system entered regenerative state i at t=0.using the arguments of

      the theory of a regenerative process the point wise

      B* (s) N2 (s)

      0

      0

      D1 (s)

      availability Ai (t) is seen to satisfy the following

      In the steady state

      (44)

      recursive relations

      B (sB* (s)) N2

      A M (t) q A (t) q

      A (t)

      0 lim 0 D

      0 0 01 1

      02 2

      s0 1

      A1 q10.6 A0 (t) q18.67 A8 (t) q10.3 A0 (t)

      A2 M2 (t) q20 A0 (t) q24 A4 (t)

      Where N2 1(P01P80 P02P24P80 )

      (45)

      A q A (t) q A (t)

      8 (P01P18.67 P02P24P48.5 P02P24P41P18.67 )

      4 41 1

      48.5 8

      (46)

      A8 q88.7 A8 (t) q80 A0 (t)

      (30-34)

      D1 is already specified.

      4.4 Expected Number of Visits by the

      Now taking Laplace transform of these equations

      Repairman

      and solving them for

      A* (s), We get

      We defined as the expected numer of visits by

      * N1 (s)

      0

      0

      the repairman in (0,t],given that the system initially

      1

      1

      A0 (s) D (s)

      (35)

      starts from regenerative state Si

      By probabilistic arguments we have the following recursive relations for Vi (t)

      The steady state availability is

      V (t) q

      (1V (t)) q

      V (t)

      N 0 01

      1 02 2

      A

      A

      (sA* (s))

      (sA* (s))

      1

      1

      0 lim 0 D

      V1 (t) q10.6 V0 (t) q18.67 V8 (t) q10.3 V0 (t)

      s0 1

      Where

      V2 (t) q20 V0 (t) q24 V4 (t)

      V4 (t) q41 (1 V1(t)) q48.5 (1 V8 (t))

      N1 P80(0 2 P02)]

      D1 P80[0 1 (P01 P02P41P24 ) 2 P02 4 P02P24 ]

      8 (P01P18.67 P02P24P48.5 P02P24P41P18.67 )

      (36-38)

      (47-50)

      0

      0

      Taking Laplace stieltjes transform of the equations of expected number of visits

      4.3 Busy Period Analysis of The Repairman

      And solving them forV ** (s) , we get

      V

      V

      0

      0

      ** (s)

      N3 (s)

      D1 (s)

      (51)

      increases. Also for the fixed value of failure rate, the profit is higher for high correlation (r).From the fig.-4 it is clear that profit decreases linearly as disappearance rate of repairman increases. Also for the fixed value of disappearance rate, the profit is

      In steady state higher for high correlation (r).

      V (sV * (s)) N3

      0 lim 0 D

      Where

      s0 1

      (52)

      N3 0 (1 P24)P80 2 (P02 P41P88.7 P02P48.5 P88.7 )

      (4 )P02P24P80 8 (1 P24P02)

      (53)

      D1 is already specified

5. PROFIT ANALYSIS

   The expected total profit incurred to the system in steady state is given by

   P C0 A0 C1B0 C2V0

   Fig. 2:MTSF vs Failure Rate

   Fig. 2:MTSF vs Failure Rate

   (54)

   Where

   C0 =Revenue/unit uptime of the system

   C1 =Cost/unit time for which repairman is busy

   C2 =Cost/visit for the repairman

6. CONCLUSION

   For a more clear view of the system characteristics w.r.t. the various parameters involved, we plot curves for MTSF and profit function in figure-2 and figure-3 w.r.t the failure parameter ( ) of unit A for three different values of correlation coefficient (r1 =0.25, r2 =0.50, r3

   =0.75), between X and Y , while the other

   parameters are kept fixed as

   .005, .02, 0.001,C0 400,

   C1 200,C2 40, .004

   Fig. 3:Profit vs Failure Rate

   Fig. 3:Profit vs Failure Rate

   From the fig.-2 it is observed that MTSF decreases as failure rate increases irrespective of other parameters. the curves also indicates that for the same value of failure rate,MTSF is higher for higher values of correlation coefficient(r),so here we conclude that the high value of r between appearance and disappearance tends to increase the expected life time of the system.From the fig.-3 it is clear that profit decreases linearly as failure rate

   Fig. 4:Profit vs Disappear Rate

   Fig. 4:Profit vs Disappear Rate

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