Reliability Analysis of Parallel System with Two Types of Failure and Preferable Repairman Policy

CHAPTER – 7

Reliability Analysis of Parallel System with Two Types of Failure and Preferable Repairman Policy

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Introduction

The old two chapters are devoted to analyse the two indistinguishable dependability theoretical accounts holding similar sort of failure. But in existent life systems, there is ever a possibility for the two non-identical systems for acquiring different sorts of failure. Keeping this in position there are many research workers who had analyzed many systems by utilizing uninterrupted every bit good as distinct distribution. Gupta et Al. [ 140 ] , Said [ 141 ] , Kadyan et Al. [ 142 ] and Prashar et Al. [ 143 ] had analyzed the dependability theoretical accounts dwelling of non-identical units with precedence and different manners of failure. Malik et Al. [ 144-145 ] had analyzed the two non-identical analogues units with precedence to mend capable to review. Kumar and Kadyan [ 146 ] discussed the non-identical system in which the one unit undergoes for fix upon falure while the 2nd unit is replaced by similar new 1. Vashney [ 147 ] had besides given his part by analysing non–identical analogue system possessing failure and fix clip as Geometric distributions. Haggag [ 148-149 ] had analyzed two dissimilar unit cold standby systems with three different failure and preventative care.

Now this chapter is devoted to analyse the two theoretical accounts of non-identical systems with the construct of review policy to observe the two types of failure utilizing regenerative point technique and discrete ( geometric ) distribution. Initially in Model-I, an automatic unit is in operative and manual unit is in cold standby but in instance of Model-II, both the units are in operative province. An review is being done foremost on the failed automatic unit, to observe the failure ( minor or major ) to avoid any confusion for maintenance man. But no such review policy has been discussed for the manual 1 because it has merely the minor sort of failure. In instance both the units were together in failed status so the penchant will be given to the manual 1 as compared to the automatic unit. The fix clip for child is less every bit compared to major.

Using regenerative point technique assorted of import steps of system effectiveness i.e. average clip to system failure, steady province handiness, busy period for maintenance man and inspector has been obtained. Graphs were besides been drawn to analyse the behaviour of average clip to system failure and net income map with regard to failure/repair rate.

7.2MODEL-I DESCRIPTION

The undermentioned premises have been associated with theoretical account:

  • System consists of two non-identical parallel units. Initially, automatic unit is operative and other manual is standby.
  • System is assumed to be in failed province when both units fall together in failed province.
  • Inspection strategy is being introduced to inspect the failed automatic unit. But the manual 1 is free from review policy.
  • A individual maintenance man is available to mend the fix both sort of failed unit with penchant to minor failure.
  • The repaired unit’s operate every bit good as new.

7.2.1 Terminology

:

Automatic unit is in operative manner

:

Automatic unit is in operative/standby manner.

:

Failed automatic unit is under review.

:

Unit of measurement is in major failure manner and is under repair/waiting for fix.

:

Unit of measurement is in minor failure manner and is under repair/waiting for fix.

Failed manual unit is under fix.

a/b

:

Probability of unit’s to acquire into failed province with minor/major failure.

/

:

Probability of an automatic unit to acquire into failed province or non.

:

Probability of the failed unit to be inspected satisfactory or non.

:

Probability of manual unit to acquire into failed province or non.

R

:

Repair rate of a failed automatic unit.

:

Probability and cumulative denseness map of first transition clip from regenerative provinceto.

:

Steady province passage chance from provinceto.

:

Mean visit clip in province.

©

:

Whirl of the two maps of non-negative variable.

Table 7.1: Terminology

Up States

,

,

Down State

,,

,

Figure 7.1: Passage Diagram

7.3TRANSITION PROBABILITIES AND SOJOURN TIMES

Since the system undergoes distinct failures so the figure of failures predating the first success follows the geometric distribution. The cumulative denseness map of first transition clip from regenerative provincetohas been defined by:

01=12=

13=14=

15=16=

20=30=24=35=

27=37=42=53=

61=62=

63=64=

65=70=

71=76=

( 7.1-7.18 )

The steady province passage chances from provincetocan be obtained from

It can be verified that

,,,

,.

( 7.19-7.24 )

7.3.1Mean Sojourn Times

Letbe the sojourn clip in provinceso “mean visit time” in provinceis obtained by:

so that

( 7.25-7.30 )

Mean visit clipof the system in provincewhen the system is to pass through intois given by:

=

,

,

,

,

( 7.31-7.36 )

The value of passage chances and intend visit clip are used to cipher the consequences for dependability, handiness of the system, and busy clip period of review and maintenance man.

7.4RELIABILITY AND MEAN TIME TO SYSTEM FAILURE

The clip taken by the system before its failure is known to be first transition clip to the failed province or life clip of the system. If random variabledenotes the life clip of the system so its expected value provides the average clip to system failure.

Letdenotes the chance of system that perform satisfactorily for atleasteras, when it get down ab initio from operative regenerative provinceThe dependability analysis of the system is obtained by analysing the average clip to system failure of system by the work outing the undermentioned equation:

( 7.37-7.41 )

By taking Geometric transmutation and work outing the above equations, we get

The MTSF ( average clip to system failure ) is:

where

1011213027712727

( 7.42 )

11213242776

( 7.43 )

7.5AVAILABILITY ANALYSIS

The per centum of clip that the equipment is under operation is called steady-state handiness. It characterizes the average behaviour of the unit. The handiness mapis defined as the chance that the unit is runing at clip T.

Letdenotes the chance of system to be in upstate at erawhen it started ab initio from regenerative province. Using simple probabilistic statement, the undermentioned return looks has been obtained:

00011

11122133144

155166

22200244277

33300355377

4422

5533

6611622633644655

77700711766

( 7.44-7.51 )

By taking Geometric transmutation and work outing the above equations, we get

and

The steady province handiness of system is obtained by:

After using ‘L’ Hospital Rule, we obtained:

where

201661202770124277661

216617271661

( 7.52 )

=10166120277023124277661

321661427761664651415

6116646524626314152462

6326162427761617271661

( 7.53 )

7.6Busy PERIOD ANALYSIS

7.6.1Busy period of inspector

Letbe the chance of the inspector that inspects the failure of a failed unit, before it acquire repaired by maintenance man. Using simple probabilistic statement, the undermentioned return looks has been obtained:

0011

1112213314

4155166

2200244277

3300355377

4422

5533

6661162263

3644655

7700711766

( 7.54-7.61 )

By taking Geometric transmutation and work outing the above equations, we get

The chance of review installation involved in inspecting the failed unit is obtained by:

=

After using ‘L’ Hospital Rule, we obtained:

where

312427766161624277616

( 7.62 )

andis the same as in handiness analysis.

7.6.2Busy period of maintenance man

Letbe the chance of fix installation being involved to mend the failed unit, when system starts ab initio from regenerative province. Using simple probabilistic statement, the undermentioned return looks has been obtained:

=01( tiˆ­1 ) ©

=1121314

1516

=22024

27

=33035

37

=442

=553

=6616263

6465

=7

( 7.63-7.70 )

By taking Geometric transmutation and work outing the above equation, we get

The chance of fix installation involved in fix of failed unit is obtained by:

=

After using ‘L’ Hospital Rule, we obtained:

= –

where

421661427761664651415611664652462631415

246263616242776167271661

( 7.71 )

andis same as in handiness analysis.

7.7GRAPHICAL INTERPRETATION AND PROFIT FUNCTION ANALYSIS

The expected net income for steady-state is obtained by:

( 7.72 )

where

: be per unit operative clip gross by the system.

& A ;:be per unit failed clip outgo on the system.

Harmonizing to the information analysis of this peculiar existent life state of affairs, the behaviour of the

net income map has been studied through graphs by the fixed values of certain parametric quantities a,,,,,andas

The dependability steps of system effectivity for above values are obtained as:

Average clip to system failure ( MTSF )= 15.583359

Availability ()= 0.75936

Busy period of Inspector ()= 0.076587

Busy period of maintenance man ()= 0.462811

Net income= 213.134

Figure: 7.2A graph reflects the behaviour of average clip to system failure with regard to failure ratefor different values of fix ratei.e. average clip to system failure lessening with addition in failure rate.

Figure: 7.3A graph reflects the behaviour of average clip to system failure with regard to mend ratefor different values of failure ratei.e. average clip to system failure increases with addition in fix rate.

Figure: 7.4A graph reflects the behaviour of Net income map with regard to failure ratefor different values of fix ratei.e. Net income decreases with addition in failure rate.

Figure: 7.5A graph reflects the behaviour of Net income map with regard to mend ratefor different values of failure rate

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