Transit vehicle reliability models

Mieroelectro,. Reliah., Vol. 22. No. 3, pp. 619-624, 1982.

0026.2714/82/030619-06S03.00/0

Printed in Great Britain,

© 1982 Pergamon Press Ltd.

TRANSIT

VEHICLE BALBIR

RELIABILITY

MODELS

S. D H I L L O N

Engineering M a n a g e m e n t Program, Department of Mechanical Engineering, Un{verslty of Ottawa, OTTAWA, Ontario, KIN 6N5, CANADA. (Received for publication 3rd D e c e m b e r 1981) ABSTRACT This paper presents two mathematical Markov models of repairable t r a n s i t systems. Laplace transforms of the state probability equations are developed. INTRODUCTION Similar models can be found in references [ I - 4 ] .

This paper presents

two repairable-system ( t r a n s i t ) models. Model I represents an operating automobile vehicle which may f a i l p a r t i a l l y or completely in three d i s t i n c t f a i l u r e modes. The p a r t i a l l y operating vehicle may also f a i l in three d i s t i n c t f a i l u r e modes. However, the p a r t i a l l y failed vehicle may be repaired to i t s f u l l y operational state.

The three d i s t i n c t states repre-

sent f a i l - s a f e , minor accident (vehicle repairable) and major accident (vehicle non-repairable) f a i l u r e modes of an automobile vehicle. The Model I I represents four-wheel automobile vehicle tires operating in alternating environments (normal to abnormal weather and vice-versa) with one standby t i r e .

Tire f a i l u r e ( f l a t ) rate and repair rate may change

due to change in weather from normal to abnormal. Therefore, this model can be used to predict the u n r e l i a b i l i t y of the vehicle due to f l a t tires in changing weather conditions.

Cars are a typical example of this model.

Assumptions The following assumptions are common to both models: I.

Failures are s t a t i s t i c a l l y independent.

2.

Failure rates are constant.

3.

Repaired system is as good as new.

620

BALBIR S. DHILLON

MODEL I The system t r a n s i t i o n diagram is shown in Figure I .

~fa(.~_x) ~7

g

~i

As

~fs(x) h

Figure 1.

System t r a n s i t i o n diagram.

The following assumptions are associated with Model I : I.

The system repair rate from the degraded state is constant. Failed system repair times are a r b i t r a r i l y d i s t r i b u t e d .

2.

The system can completely f a i l from i t s normal and degraded operating states.

System f a i l u r e rates include human and

hardware f a i l u r e s . 3.

The system may be degraded due to accident or other f a i l u r e s .

4.

System states are:

good (g); degraded (d); f a i l safe ( f s ) ;

f a i l e d and accident but system repairable ( f a ) ; f a i l e d , accident and the system is irreparable ( f i r ) .

The following notations are associated with Model I : Pi(t)

p r o b a b i l i t y that system is in state for

Pi(x,t)

t;

i = g, d, fa, f s , f i r .

p r o b a b i l i t y density (with respect to repair time) that the f a i l e d system is in state time of

u i ( x ) , qi (x)

x; for

i

and has an elapsed repair

i ~ fa, fs.

repair rate (a hazard rate) and pdf of repair time when system is in state time of x; for

~° 1

i , at time

i

and has an elapsed repair

i = fa, fs.

i t h constant system f a i l u r e rate; i = 1 (g to d), i = 2 (g to f s ) , i = 3 (g to f i r ) ,

i = 4 (g to f a ) ,

i = 5 (d to f s ) , i = 6 (d to f i r ) ,

i = 7 (d to f a ) .

Transit vehicle reliability

621

ud

constant degraded system repair rate (d to g).

S

Laplace transform variable. The associated system of differential equations [4] with Figure l are: 4 dPn(t) + ( z Xi ) Pg(t) = P d ( t ) dt i=l

+ i

Pfs(X't)

~d +

pfa(X,t)

~fa(X) dx

0

(i)

~fs (x) dx

0

dPd(t) dt + (nd dPfir(t) dt

+

7 i15ki ) Pd(t) = Pg(t) k 1

At

(3)

Pd(t) X6 + Pg(t)x3

aPi(x,t) aPi(x,t) --ax + --+at for

(2)

(4)

Pi (x't) ~i (x) = 0

i = fa, fs

pfa(O,t) = Pg(t) X4 + Pd(t) x7

(5)

pfs(O,t) = Pd(t) k s + Pg(t) k2

(6)

t = O, Pg(O) = I ; Pd(O) = P f i r ( O )

= pfa(X,O) = pfs(X,O)

=0.

The Laplace transforms of the solution are (7) - ( l l ) : 4

Pg(S) = Is + ~ Xi i=l

XI

7

{~d + k7 Gfa(S)+ ksGfs(S)}

S+nd+ i=:5 (7)

- {~4Gfa(S) + X2Gfs(S)}]-I where

Gi(s ) = ~ e-sx qi(x) dx; for 0

i = fa, fs

7 Pd(S) = Pg(S) kil(s + ~d + s xi) i=5

P f i r (s) =

Pn(s) s

{~3 +

pfa(S) = Pg(S){k4 +

(8)

XBX1

}

S+nd+ i -

1

k1~7 7 }{ S+Ud+ i_~5 ~i

(9)

l - Gfa(S)

(lO)

BALBIR S. DHILLON

622

X~X~ 7

Pfs(S) = Pg(S) {X2 +

l - Gfs(S)

}{

(11)

s

s+ Ud+ i---5 Xi MODEL I I The system transition diagram is shown in Figure 2. 4x

gn in

Normal weather

Abnormal Weather

~,__~ /

f

I

B

Cz

C(

B

4xs

I

:ds

gs

Figure 2.

4X

s

System transition diagram.

The associated assumptions with Model I I are: I.

System operates in alternating environments. For example, normal weather to stormy weather and vice-versa.

2.

The system (automobile vehicle) has four wheels ( t i r e s ) and one standby t i r e .

The standby t i r e cannot f a i l ( f l a t ) in the

standby mode. 3.

As soon as the t i r e f a i l u r e occurs, the failed t i r e is immediately replaced with the standby.

4.

System with more than one f l a t t i r e is assumed failed.

5.

The alternating weather rates are constant.

6.

Tire f a i l u r e ( f l a t ) rate and repair rate may (or may not) change due to change in weather from normal to abnormal.

7.

All vehicle tires (including standby) are identical.

8.

The t i r e repair rate is constant.

9.

System states are:

normal weather good - four tires operating

with one standby (gn), normal weather degraded - four tires operating with no standby (dn), normal weather failed ( f n ) , abnormal weather good - four tires operating with one standby

Transit vehicle reliability

623

(gs), abnormal weather degraded - four" tires operating with no standby (ds), abnormal weather failed (fs). The following notations are associated with Model I I : Pi(t)

probability that system is in unfailed state for

~i

i , at time

t;

i = gn, dn, fn, gs, ds, fs.

constant rate of having f l a t t i r e ; i = n (normal weather), i = s (abnormal weather).

ui

constant rate of repairing a f l a t t i r e ; i = n (normal weather), i = s (abnormal weather). constant weather transition rate from normal to abnormal

B

constant weather transition rate from abnormal to normal

s

Laplace transform variable. The associated system of d i f f e r e n t i a l equations with Figure 2 are:

At

dPgn(t) dt + (m + 4Xn) Pgn(t) = Pdn(t) Un + Pgs(t) B

(12)

dPdn(t) dt + (4~n + ~ + Un) Pdn(t) = Pds(t) B + Pgn(t) 4~n

(13)

dPfn(t) dt + ~ Pfn (t) = Pdn(t) 4~n + Pfs (t) B

(14)

dpgs(t) dt + (4Xs + B) Pgs(t) = Pds(t)

(15)

us + Pgn(t) ~

dPds(t) d ~ + (B + 4Xs + us) Pds(t) = Pdn(t) ~ + Pgs(t) 4Xs

(16)

dPfs(t) ~ dt+

(17)

B Pfs(t) = Pds(t) 4Xs + Pfn (t)

t = O, Pgn(O) = l ; Pdn(O) = Pfn(O) = Pgs(O) = Pds(O) = Pfs (0) = 0 The Laplace transforms of the solution are (18) - (30):

4Us~n ~ ~ 4~n~n 4Xn~ ~nB 4~s~nB Pgn(S) = [s+~+4~n - { ~ + { AIAIA3A2A4 + A~4 } + T I }

4B~s~n~ ~B } ] - I { AIA2A~A4 + ~ Al - s + 4~n + ~ + ~n A 2 -= s + 13 + 4X s + }Js A 3 =- [I - ~B

]

(18)

(Ig) (20) (21)

BALBIR S. DHILLON

624

A4 - s + B + 4~, s

4UsXs

(22)

A2A 3

A s - (s + ~)

(23)

A6 - (s + B)

(24)

A7 =- (I -

(25)

B~

4~n~ 4~ s Pdn(S) = [B { AIAIA2A3 + ~

4~n~

4~ s

Pds(S) = Pgn(S) [ AIA-~3 + ~

4~s~n~ { AIAIA2A3A4

4Us~n~

+

cL

c~

} ]

- { AIAgA3A 4 + ~

4~

n

}} + A T ]

Pgn(s) (26) (27)

4UsXn~ Pgs(S) = [ A1A2A3A4 + ~-4 ] Pgn(s)

(28)

4x n Pfn(S) = [ Pdn(S) A T +

(29)

4BXs A ~ 6 • Pds(S)]/A7

4~s 4B~s~ 4~ pfs(S) = [ A T + AsA6A6A7 ] Pds(S) + asA6A7n Pdn(S)

(30)

REFERENCES I.

B.S. Dhillon, C. Singh, Engineering R e l i a b i l i t y :

New Techniques and

Applications, John Wiley and Sons, New York (1981). 2.

B.S. Dhillon, R e l i a b i l i t y Engineering in Systems Design and Operation, Van Nostrand Reinhold Co., New York (In Press).

3.

B.S. Dhillon, Bibliography of Literature on Transit System R e l i a b i l i t y , Microelectronics and R e l i a b i l i t y ,

4.

(In Press).

D.R. Cox, The Analysis of Non-Markovian Stochastic Process by Supplementary Variables, Proc. Camb. Phil. Soc., 51, pp. 433-441 (1955).