Availability analysis of transit systems

0026-2714/8453,00 + .00 © 1984 Pergamon Press Ltd.

Microelectron. Reliab., Vol. 24, No. 4, pp. 761-768, 1984. Printed in Great Britain.

AVAILABILITY

ANALYSIS

OF

TRANSIT

SYSTEMS

B A L B I R S. D H I L L O N Engineering Management P r o g r a m m e , Department of Mechanical Engineering, UNIVERSITY OF OTTAWA, Ottawa, Ontario K I N 6N5, CANADA.

(Received for publication Z4th April 1984) ABSTRACT This paper presents repairable

two mathematical

transit systems.

models of

State probability equations

for both models are developed. INTRODUCTION In an era, where time is too precious a reliable society.

transit system is a m u s t for modern day's Nowadays more attention

r e l i a b i l i t y and maintainability

is being given to

of the transit systems

during the design phase than ever before. result of benefits concepts

publications

references [~

fields.

on transit system reliability

[~ .

This may be the

gained from the application of reliability

in the aerospace and military

reference

to be wasted,

In addition,

[2-3 ] .

A list of

is given in

similar models are given in

This paper presents

two Markov models

of transit systems. Model

I is concerned with a vehicle being used in day

to day e n v i r o n m e n t

in the field.

any of the following i) ii) iii)

Operating

The vehicle can be in

three states:

normally in the field

Failed completely

in the field.

Repair workshop

When the vehicle fails completely

in the field,

attempt is made to repair the failed vehicle 761

an

in the field.

762

B.S. DHILLON

If this a t t e m p t into normal workshop.

is s u c c e s s f u l

service,

then the v e h i c l e goes b a c k

otherwise

it is towed to the r e p a i r

On the o t h e r h a n d w h e n the v e h i c l e

then it is d r i v e n the p a r t i a l l y

to the w o r k s h o p

for repair.

In o t h e r w o r d s ,

f a i l e d v e h i c l e can be d r i v e n to the w o r k s h o p .

The state space d i a g r a m of m o d e l Model

fails p a r t i a l l y

I is s h o w n in F i g u r e

i.

II is c o n c e r n e d w i t h two i d e n t i c a l v e h i c l e s

b e i n g used in day to day e n v i r o n m e n t in the field. two v e h i c l e

s y s t e m c a n be in any of the f o l l o w i n g

The four

states :

i)

Both vehicles

operating

One v e h i c l e o p e r a t i n g

ii)

one f a i l e d iii)

successfully

in the field.

and the o t h e r

in the field.

Both vehicles

iv)

successfully

failed

in the field.

Repair workshop.

W h e n one or b o t h v e h i c l e s

fail in the field an a t t e m p t

is m a d e to r e p a i r

the failed v e h i c l e or v e h i c l e s

fully operational

state

operating

successfully).

(i.e. w h e r e b o t h v e h i c l e s Furthermore,

v e h i c l e s c a n not be r e p a i r e d

b a c k to are

if b o t h failed

simultaneously

in the f i e l d

t h e n an a t t e m p t is m a d e to r e p a i r any one of the f a i l e d vehicle.

In b o t h c a s e s

if r e p a i r a t t e m p t s

in the field then the v e h i c l e s workshop. Figure

are u n s u c c e s s f u l

are towed to the r e p a i r

The state space d i a g r a m of m o d e l

II ~

shown in

2 .

ASSUMPTIONS The f o l l o w i n g a s s u m p t i o n s

are a s s o c i a t e d w i t h b o t h

models:

i) ii) iii) iv)

F a i l u r e s are s t a t i s t i c a l l y

independent.

R e p a i r e d v e h i c l e or v e h i c l e s Failure, Partially workshop.

are as g o o d as new.

r e p a i r and t o w i n g r a t e s are c o n s t a n t . failed vehicle

is d r i v e n to the r e p a i r

Transit systems

763

MODEL I

Ip

It

If

I Icompletely in the field

Figure i.

I i~

System transition diagram.

The following notations are associated with Model I: ith state of the system: i=0 nQrmally),

i=l

(means vehicle operating

(means vehicle failed

completely in the

field), i=w (means failed vehicle in the repair workshop). Pi(t)

probability that system is in state i, at time t, for i=0, 1 and w.

If

constant failure rate of the vehicle

(i.e. failed

completely in the field). Ip

constant failure rate of the vehicle failed partially in the field

(N.B. partially failed

vehicle is driven to the repair shop for repair). ~t

constant towing rate of the completely failed vehicle in the field.

~i

ith constant repair rate of the vehicle;

i=l,

[means the repair rate of the vehicle from state 1 to state 0), i=2( means the repair rate of the vehicle from state w to state 0). t

is time.

764

B.S. DHILLON

The following differential equations are associated with Figure 1: dP 0 (t) dt +(If+ip) P0(t) =~iPl(t)+~2Pw(t)

(1)

dP 1 (t) - -

dt

+(~l+It ) "Pl(t)

(2)

=If.P0(t)

dP w (t) dt

+~2"Pw(t)

(3)

=I t.Pl(t)+l P "P0(t)

At t=0, P0(0)=I, P11 (0) =Pw (0) =0 Solving equations we

(i) -(3) , by using Laplace transforms,

get [5]: A --

-

L

-

Cl "c2

+I 1 (~i~2+1t'~2) ClC 2

Cl t

÷

• e

c I (Cl-C 2 )

A I eC2t Cl(Cl-C 2 )

(4)

Where 2 A H ~ICI+~2CI+CI!t+CI+~I~2 + ItS2 -B~ /B2-4(~l.~2+It~+~21f+If.lt+~llp+Ip.lt) Cl,C2 =

B ~

P1 (t) =

If-~ 2 ClC 2

(6)

(7)

(~l+~2+If+Ip+It)

+-Cllf+If'~2

(5)

LClC2 +

(Cllf+If.~2) ] c2t Cl (CI_C2) Je

D E clt [ D E 1 e c2t P (t) + .e + ! w ClC 2 c I (Cl-C 2) " c?c 2 c I (ci-c2)

D ~ If. It+ ~llp+Iplt

(8)

(9)

(10)

Transit systems

765

(i1) E ~ Ip.Cl+lf-lt+~l

kp+lp'lt

As t becomes very large, the equation

(4) reduces to

UI~ 2+It.U 2 Avss ClC 2

(12)

Where

Avss

is the vehicle steady state availability.

(13)

ClC 2 -- ~i.~2+It.~2+~2 If+If.lt+~l Ip +i P .i t

MODEL II

Both vehicleS opQrating normally ~w

2~

21f

Dne vehicle failed =ompletely in the field, other opera~ ing normally f

/ Failed vehicles in the repairshop

v

=IW+I t

~ff

us

~f

Both vehicles failed completel~ in the field

Figure 2.

System transition diagram.

The following notations are associated with Model II: i

ith state of the system

: i=0

(means both vehicles

766

B.S. DHILLON

operating

normally),

completely

i=f

in the field,

Pi (t)

i=w

for i=0,

normally), in the

in the repair

shop).

that system is in state i, at time t,

f, ff and w.

probability

Pi

failed completely

(means failed vehicles

probability

failed

other one operating

i=ff( means both vehicles field),

(means one vehicle

that system is in state i; for i=0,

f,

ff and w. constant

~f

failure rate of the vehicle

completely constant

~w

vehicle constant

At

in the field). failure rate of the vehicle

partially

(i.e. failed

in the field

is driven

failed

(N.B. partially

to the repair

failed

shop for repair).

towing rate of the completely

failed

vehicle. ith constant i=s

repair

rate of the failed vehicle;

(means the vehicle

to state f), i=f

repair rate from state ff

(means the vehicle repair rate

from state f to state

0), i= ff

( means

the vehicle

repair rate from state ff to state 0), i=w the vehicle t

repair

(means

rate from state w to state 0).

is time. The following

with Figure

differential

equations

are associated

2:

dP 0 (t) dt +(21f+2lw)P0(t)

= Pw (t) "Ww+Pf (t).~f+Pff (t).~ff

(14)

dP w (t) dt

+ ~w.Pw(t)

(15)

= P0(t).2lw+Pf(t).~+Pff(h).2~ t

e~l+l w

dPf (t) dt +(lf+e+~f)Pf(t)

t

= Pff(t).~s+P0(t).21f

(16)

Transit systems

767

dPff (t) dt +(21t+~s+~f f) .Pff(t) = Pf(t).If

(17)

At t=0, P0(0)=I, Pw(0)=Pf(0)=Pff(0)=0

Setting the derivatives of equations

(14)-(17) equal

to zero and utilizing the relationship P0 + Pw + Pf+Pff=l, we get A.~ w

(18)

p = 0

2

A(21f+21W+~w)+(~w-~f) (21t+~s+~f f) 2lf+21f(~w-~ff)

(19)

A E (If+~+~f) (21t+~s+~ff)-~s.I f 2 Pw = i-[ A+21f(21t+~s+~ff)+21fA 1

P0

(20)

21f.P0(21t+~s+~ff) Pf

(21)

=

(If+e+~f) (21t+~s+~ff)-~s.l f 2

21f "P0 Pff =

(22) (If+~+~f) (21t+~s+~ff)-~s.lf

Where

P0' Pw' Pf are steady state probabilities.

Plots of equations (18), (20), (21) and (22) are shown in Figure 3 for the varying value of the failure rate,lf, of the vehicle failed completely in the field.

Figure 3 shows

that for the increasing value of , If, the corresponding values of P

and P decrease. w 0 for n number of vehicles.

This model can be generalized

768

DHILLON

B.S.

1.00lW=0.2,

It=O.l,

~f=0.6,

0.80 ~s =0.5, ~w=0.7,

~ff=0.8,

=lw+l t =0.3

0.60

H H

Pff 0.40 -

o

/Pw 0.20

Pf PO I

0.00

I

0.50

I

1.00

1.50

i

2.00

i

i

2 50

3.00

If Figure 3.

State probability plots.

REFERENCES i.

B.S. Dhillon,

Bibliography of literature on transit

system reliability,

Microelectronics

and Reliability,

Vol. 22, 1982. pp. 641-651. 2.

B.S. Dhillon,

Transit Vehicle reliability models,

Microelectronics

and Reliability, Vol. 22, 1982, p. 619-

624. 3.

B.S. Dhillon, weather,

RAM analysis of vehicles in changing

Proceedings of the Annual Reliability and

Maintainability Symposium, 4.

B.S. Dhillon, Management,

1984, pp. 48-53.

Systems Reliability,

Petrocelli Books,

Maintainability and

Inc., New York, 1983

(Books distributed by Van Nostrand Reinhold Company

,

New York). 5.

B.S. Dhillon,

The Analysis of the reliability of multi-

state device networks,

Ph.D. Dissertation, Available from

the National Library of Canada, Ottawa, 1975.