On the reliability of systems with switching lifetime distributions

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On the reliability of systems with

switching lifetime distributions

1. I nt roduct ion 1

2. Modeling a Single Component 4

3. Modeling a l-out-of-2:G System 15

4., Conclusion 25

5. References 26

E( .. ) mean value of a random variable, F(tl lifetime distribution,

fI(t;Tl modified ltd with mode switching at time T, according to strategy SI,

fI(tl ltd of a l-out-of-2:G system according to strategy SI,

fII(t;Tl rnodified ltd with mode switching at time T, according to strategy siz,

fI⁷ (t) ltd of a l-out-of-2:G system according to strategy S^{7 7,} G(tl

g(t) G' (tl g' ( t )

L, L' ltd M, M'

µ' µ' pdf r(t) r' ( t )

T

*

lifetime distribution in operational mode M, probability density function for G(t),

lifetime distribution in operational mode M', probability density function for G' (t),

random variables representing lifetimes, lifetime distribution,

two different operational modes,

failure rates of exponential distributions, probability density function,

the two considered repair strategies, failure rate in mode M,

failure rate in mode M',

time instant of mode switching, convolution operator.

ON THE RELIABILITY OF SYSTEMS WITH SWITCHING LIFETIME DISTRIBUTIONS

Abstract -

H. Bähring, J. Dunkel FernCniversität Hagen

FRG

In real systems components often work in essentially diffe- rent operational modes, characterized by changing load or environment conditions. Those modes result in different failure rates and lifetime distributions. In this paper we present a model for a single switch-over between two distinct lifetime distributions. We consider two different cases of sticking together the distributions. Naturally our

~odel can be generalized to multiple switching. As examples the single unit system and the l-out-of-2:G system are discussed at depth.

1. Introduction

Recent works on perf ormabi l i ty [Beau, Meye] show a strong dependence between load and failure rates of systems. Addi- tionally there is a strong correlation between failure rates and external influences. Usually slight modifications of load and environment conditions are taken into account by the time-dependence of the failure rate function. But in real systems components often work in essentially different

operational modes. In empirical studies i t is shown that there 1.s a very strong relationship between the failure rates and the different operational modes of the system.

Castillo and Siewiorek (CaSi] distinguish in their measure- 8ent-based analysis a user and a kernel mode. In kernel mode, the system will crash with greater probability.

To nodel such systems we have to consider changing failure rate functions and lifetime distributions. In the case of a constant failure rate, in each mode the classical ~arkovian approach allows to cope with state dependent failure rates [Rein]. In this work we show how to treat the case of gene- ral failure rate functions.

One example is a "warm" spare, spite of not being active

1..e. a component which - 1.n nevertheless suffers from external or internal stresses and therefore may be down at the moment when it has to replace a failed component. This characterizes the difference to a cold spare. General ly, the failure rate in a warm state is much smaller than that 1.n an active (hot) state. When the spare is switched into an active state the former deterioration must be taken into account when calculating the lifetime of the system. Think of a spare wheel 1.n a car, which looses pressure or gets damaged even in the trunk and therefore may be useless in the case of a tire puncture.

Another example is a component in a graceful ly degri:!-ding system, which has to process different loads, depending on the actual system configuration, i.e. an increased load, if a component has failed, and decreased load, if a repaired component is reintegrated. A lot of empirical studies of real systems [HsiT, IyRo] has shown that failure rates depend strongly on the workload and on the operational modes.

Two different strategies for the reconfiguration process

at

the time instant of a component failure are possible:

sr

The system is restartet without any maintenance of the not yet f a i l ed components, so tha t one has t.o regard their deterioration during the time prior to the actual reconfiguration, i.e. they are in a state not as good as new [BaPo].

srr

Before the restart of the reconfigured system all sur- viving components are tested, maintained, and, if necessary, repaired, so that they are in a state as good as new.

In all these cases switching of operational mode results in a different failure rate and lifetime distribution, respec- tively.

In this paper we present a model for a single switch-over between two distinct lifetime distributions. Of course our model can be generalized to multiple switching.

First we treat a single component. Then we consider the well known l-out-of-2:G system. Finally we show some remar- kable results for the exponential case.

2. Modeling a Single Component

Suppo~2 that component C works 1n two different operational modes M and M'. C has the mod2 dependent lifetime distribu- tions ( ltdl

G ( t ) , G' ( t l,

if i t starts in mode M, if i t starts in mode M' at time t=0.

( 1)

The corresponding probabi l ity densi ty functions ( pdf) of the life of C are

g(t) and g' (t), respectively. The corresponding failure rates are

r(tl = g(t)/[1-G(t)] r ' ( t ) = g ' ( t ) / [ 1-G ' ( t ) ] ^{( 2)} Furthermore we assume that for the second distribution

yE [ 0, 1> • This

·G' (tl the inverse G'-¹ (y)

holds true for most of reliability applications.

exists for the relevant

all

distributions in It is no restriction to regard only the case where

,

C starts working in mode M ( at t=0 l. We now consider a fixed but arbitrary time instant T>0. If C fails before T i t works in mode M during its whole lifetime. Otherwise if C survives T we suppose that i t changes into mode M' at T ( swi tching time).

Our f irst aim is to calculate for both strategies s:c and g:cr the modified lifetime distribution function Fr(t;T) and p:cr(t;T), which takes into account the mode changeat t=T.

Case s:r:

Our aim is to change at T from G(tl to some other monoto- nously increasing function F:r(t,T) which fulfills the con- ditions

FI(T;T) G(TI ( 3)

A s i•pl e way to find such a function i t to use part of another ltd, say, G' (t) from the moment T~ where it assumes the value G{Tl to ^{00 •} Figure 1 shows two examples for the determination of T'.

/

'{

^/

G„ tJ /

I / I

/

---

/

G~ (t)

-

0 ___Jllllfl;::C:::,..:=--__]l,__.J_ _ _ _ .L... _ _ _ _ _ _ ___

T~

T

T~

()

F ig. 1: On the determination of T'.

This yields

T' := G'-¹ {G(T)) , i.e, G' (T') = G(T)

( 4)

1& the time at which C would have the same deterioration as

if i t would have been started in mode M' at t=0, We define for brevity

6(T) .- T - T' ( 5 )

Then for all t~T the modified ltd fI(t;T) 1s given by

= G(T) ⁺

I:

^{g' (-r - Ö(T)) d,:}

= G(Tl +- G' (t-ö (Tl l - G' (T')

= G'(t-ö(T)l With (3) we get

{

^{G' G(t) (t-ö (T) l} , t<T, , t~T •

( 6)

( 7)

[For G(t)=G' (tl we get by (4) and (5) that ö(Tl=O and thus pr(t;Tl=G(t) for all t~O.]

For t=T we get:

fI(T;Tl = G' (T-ö(T)) = G' (G'-¹ (G(T))) = G(T) , which means that Fr(t;T) is continuous at t=T.

With (2) and (7) we get the failure rate r(t;T) as

r l t ; T l = {

r ( t )

r' (t-ö (T) l

t<T T~t •

By (]) we get the mean value of the ltd fI(t;T):

EI(T) =

J:

^{(1 -} fI(t;T)] dt

=

I: [

^{1 - G ( t) l dt +}

I: [

^{1 -} G' ( t- ( T-T' ) ) ] dt

( 8)

( 9)

= E(LIT) +

J:.

(1 - G' ( t l l dt

= E(L/Tl ⁺

1:

(1 - G' (t) l dt -

I:·

( 1 - G ' ( t l l d t

= E(L/Tl + E(L') - E(L' IT') ( 10) with

E(L,T)

1 pseudo mean value of G(tl, if component ^C is

"switched off" at time T even if it 1s still war- king at this moment, i.e. E(LJT) t·g(t) dt.

E(L' /T') the same value for G' (t) and T', assuming C is always working in mode M' since t=O,

E( L') the mean value of G' (t).

Example 1:

Assume that the lifetimes of C in both modes M and M' are exponentially distributed, i,e.

G(t) = 1 - exp(-µ·t) G' (t) = 1 - exp(-µ' ·t) , · ( 11) with the corresponding failure rates r(t)=µ and r'(t)=µ'.

Then for 6(T) we obtain

6(T) = T - [ 1 / µ' · 1 n ( 1-G (Tl ) l =

For F:r(t;T) we get

- exp(-µ·t) F:r(t;T) =

µ'-µ

µ' • T •.

exp( [µ'-µ] •T) ·exp(-µ' ·t)

t<T , t~T

( 12)

( 13)

with the corresponding failure rate

r ' ( t ; T l ^{r { :}

t<T , t~T .

Figure 2 shows the graphs for rI(t;T) ^{i n} the cases

a) µ'=µ , b) µ'=2•µ , c) µ'=0.5·µ .

/J

r(t;TJ b)

-~

-+--- - - - - -- -- - - - a)

0

1 1

j c)

r·-·-·-·---·-·-

1 1

1

1

T

l

'

l 1

( 14 >

t

F ig. 2: The graphs f or rI ( t; ^T) for different re lations between µ and µ'.

Notice the jump at time instant T, i. e. the failure rate function is not continuous, which is not very realistic in many appl ications. ( See example 2 for a continuous change of the failure rate function.)

Figure 3 shows the ltds p:i:(t;T), G(t) and G' (t).

1

0,5

0 T t

Fig. 3: The ltds for different relations between µ and µ'.

One can see that after T the graph converges

{ :::::: ^}

towards 1 for the case

{

^b) c) than in the case a)

place.

µ' =µ , where no mode change takes

The corresponding mean value is given by:

E:x:(T) = 1/µ·[1 - exp(-µ•T)] + 1/µ'·exp(-µ'•T') - • E ( L ) • G (. T ) + E ( L ' ) • [ 1 - G ' ( T ' ) J

= E(L) ·G(T) + E(L' l · [ l - G(T) l (uslng (4) l , with

E(L) the mean value of the lifetime, assuming that C always works in mode M since t=O,

E(L' > the mean value of the lifetime, assuming that C always works in rnode M' since t=O,

(15)

By (15) we see that Er(T) is just the average value of the two mean values E(L) and E(L' ), weighted by the unreliabi-

lity G(Tl and the reliability 1-G(T) of the component C at time T, respectively.

As should be expected, the mean lifetime of C

• remains unchanged with µ'=µ

• increases with µ'<µ ,

• decreases with ^{µ 1}

>

µ •

Example 2:

Figure 4 shows the well-known bathtub curve [DhSi l with three distinct periods of failure behaviour, viz.

• the first period which corresponds to the infant mor- tality ( burn in), characterized by a decreasing fai- ,lure rate,

• the second one of the normal operating time with a constant failure rate,

• and the third one being the wear-out period with increasing failure rate,

r(t J

O burn

,n

useful life ~ r out ^t

Fig. 4: The bathtub curve of failure rate.

To demonstrate the power of our approach we now consider the last two operating periods. We approximate the failure rate in these periods by an exponential distribution

G(t) = 1 - exp(-µ•t) , µ>O, and a Weibul distribution

G' (t) = 1 - exp(-<l·t"') , <l,ß>O , ( 16 l with the failure rate r' (t) = <l•ß•ta-i , respectively.

The resulting failure rate rr(t;T) is shown in figure 4. To obtain a continuous failure rate in t=T, a is chosen tobe

a := µ·T· (ß·T)-a ( 17)

The inclination of the wear-out failure rate can be changed

by modifying the parameter ß.

r(f)=µ

~---~~---

0

- -- ---

- - r'(t-~(TJ

T

Fig. 5: Approximation of the bathtub curve.

Obviously this is a good approximation with the restriction that there exists no derivative at the switching point T.

With (7) we get the fat lined graph of figure 6. For clarity this figure also shows the exponential and the Weibul distribution (dashed lines).

We can see that after the switching point T the modified ltd fI(t;T) assumes higher values than the exponential distr ibution, which ref l ects the increasing fai l ure rate.

Furthermore, in this period i t is obtained by parallel shifting the Weibul distribution with a distance given by ö ( T).

t

1

0,5

0

T

- - - - - ^- - - - - - -/

I '

/ /

----

^,.,-,,-/

^{/ /}

6/T)

- - - ' l o l

r'

/

/ I

/G'(t}

- ~

/ /

Fig. 6: The ltd px(t;T) of the approximating failure rate.

Case s:r:r:

In this case the modified lifetime distribution pxx(t) is given by (for the proof see [BaHe])

[

G(t)

G(T) ·+ [1 - G(T)] ·G' (t-T)

t(T , t2:T •

( 18)

(Hint: the first term G(T) is the probability that C fails before T and the second term ist the probabi 1 i ty that C survives T, but fails before t, supposing i t works in mode M' since T, As Cis assumed tobe as good as new in T, the

latter probability depends only on the difference t-T.)

Formula (2) yields the corresponding failure rate

{

_{r' (} ^r(t) _{t l t~T .} ^{t(T , ( 19 l}

The mean value of prr(t) is given by

=

I:

(1 - G(t) ]dt + (1-G(T)

l·J:

(1-G' (t-T) J dt

= E ( L / T l + [ 1-G ( T > l • E ( L ' l , ( 20 l

with E(LjTI and ECL' l as defined in (10).

Example 3:

L'nder the same assumptions as in example 1 formula ( 18 l yields

prr(t;T) = pr(t;T) for all t~0 • ( 21 l

This obviously results from the memoryless property of the exponential distribution. Therefore the corresponding mean valu~ is given by (15).

3. Modeling a l-out-of-2:G System

Figure 7 shows the reliability block diagram of a l-out-of- 2:G system S composed of the two components Ci and C₂ with the corresponding ltds G1 (tl and G2 (tl, respectively.

s

Fig. 7: Reliability block diagram of a l-out-of-2:G system Assuming stochastical ly independent components - as usual in classical reliability theory - we get the well known simple results for the ltd F(t) of S, viz. for

• hot redundancy:

1 22 l

• cold redundancy:

.( 23)

(with '*' for the convolution and gi(t) for the pdf of Gi ( ^{t ) ) .}

Now we treat the more realistical case of stochastical dependence but only in the sense that, caused by the fai-

lure of one component, the failure rate of component changes. Notice that there is a l lowed between the cornponents as long as define this state tobe mode M.

the survi ving no dependence both work. We

The prev ious l y regarded time .T may be interpreted as the instant at which the f irst component failure occurs. ^l'\s this instant is non-deterministic, T is a random variable ( wi th va 1 ues , ) . Mode M' corresponds to the sta te where only one component is alive.

Because we now distinguish all

index i, i=l,2.

cons ider two components quantities calculated

Ci, C2 we in section

have 2 by

to an

For the ltd of S we get with the modified ltd F(t;,l according to (7) and (18), respectively:

F(tl

( 24 l

This is true, because for i,j=l,2

• gi(,) dt approximates the probability that Ci fails in the interval [,,,+d,) for d,

•

O ,

• [Fj(t;t)-Fj(,;-r)J 1s the probability that component Cj fails in [t,t1, if i t is working in mode M' after the failure of Ci (iTj) at t,

• the two terms of (24) reflect the cases that component Ci or C2 fails first, respectively.

With (24) we obtain the mean system lifetime Es:

[l - F(t)) d t . ( 25 l

Again we consider the different cases

sx

and

sxx.

Case s:r:

With (6), (8) and (24) we get the ltd Fr(t) of the system s as

( 26)

( 27)

Equation (27) is easily interpreted as follows:

itj, i,j=l,2 is the probability that Ci fails in the time interval [0,t] but

cj

survives t in mode M' [compare (7)]. Therefore the sum of those integrals for i=l,2 is just the probability that exactly one component fails before t.

• G1 (t)+G2(t)-G1·G2(t) is the probability that at least one component fails before t.

• The unreliability Fr(t) is the probability that at least one component fails minus the probability that exactly one component fails before t.

Case g:r:r:

With (18) and (24) we get the ltd F:r:r(t) of the system S by ( see [BaHe] l :

Equation i,j=l,2:

( 28) is easy to understand,

(28)

since for

• gi(t) ·[l - Gj(t)] dt approximates the probability that Ci fails in the interval [t,t+dt] for dt• O, but Cj survives the instant t ,

• Gj' (t-t) is the probability that component Cj fails 1.n the interval [t,t], after i t has been renewed at t, the failure instant of Ci, i.e. its lifetime is not langer than t-t •

• The two terms of ( 28) again consider the cases that component C1 and C2 fails first, respectively.

Example 4:

Suppose that all lifetimes of the components in both modes M and M' be exponentially distributed, i.e. for i=l,2:

Gi' (t) = 1 - exp(-µi't) ( 29)

With (21) and (24) we know that under these assurnptions the ltds of case

sr

^{and case}

srx

are the same, i.e. we can also neglect the exponents I and II.

For each component Ci li=l,2) we get the function 6i(t) and the ltd Fir(t) frorn 112) and (13) by adding the index i.

Calculating the ltd Fit) of the system S according to (271 yields:

( 30 l

We have to distinguish the following cases a) and b) with certain sub-cases:

Here

Fit) = 1 - exp(-[µ1+µ2l•t) ¹ 31)

_ _ µ_2---·[exp(-µ 1 '·t) - exp(-[µ1+µ2J·t) l µ1+µ2-µ1'

i)

For the corresponding mean value we obtain by (25):

· Es = 1 ( 32)

As an important special case we consider µ,_ ¹ = 2·µ;__ ^, i=l,2, µ ^1. =t=J.J 2 , which means doubling of failure rates e.g. by increasing load after the failure of one component.

The function F(t) is given by:

( 33)

and the mean lifetime by:

= 1

2 · [

^{E ( L}1 ) + E ( L2 ) ] , ( 34)

with E(L;__) the mean lifetime of Ci in mode M.

Figure 8 shows F(t) for µ2 =2·µ1 in comparison with the original ltds G1 (t) and G2 (t) according to (29).

Additionally the ltd Fsi(t) is given for the case of stochastical independent components.

1

0,5

0 T t

Fig. 8: The ltds in the exponential case.

iil A second special case deals with identical components Ci and C2 , i.e. µ=µ1=µ2, µ'=µ1'=µ2', µ't2•µ •

In this case we get:

µ' 2•µ

F(t) = 1 + - - - exp(-2·µ•t) - ---•exp(-µ'•t) (35)

2•µ-µ' 2•µ-µ'

= 1 - (1/(2•µ-µ')] · [2•µ•exp(-µ't)-µ' ·exp(-2µt)], and

Es= 2•µ 1 1

+ -µ'

= 1

2

^{-E(L) + E(L') , ( 36 l}

with E(L) and E(L') as in (15). (34) is just the mean lifetime of a 2-out-of-2:G system of two identical

components which after failure is totally replaced by one cold spare.

If we define ß:=µ'/µ, we get with 1361:

2·µ ⁺ µ' Es= 2 . µ • µ '

1 2 ⁺ ß 2 ⁺ ß

= - - - =

~µ 3ß 3ß ( 371

with E.i=3/(2µ1 the mean value of system lifetime in the case of stochastical independent components. The last equation shows that already for ß=l.2, i.e. for a about 20 per cent increased failure rate e.g. caused by load balancing, the relative deviation between our approach and the normally implied case of stochastical

independent components is:

r( ßl 1 - ß

= 2 .

2 ⁺ ß ⁼ -12.5 % ^{( 38)} which means that the mean l ifetime is about 12. 5 % shorter than i t is if computed under the unrealistic assumption of stochastical independence.

Figure 9 shows the graph of the relative deviation r ( ^ß).

You can see that for

• ß=0, i.e. µ'=0: r(ßl=l, since with (371 E9 =+^{00 ,}

• ß<l, i.e. µ'<µ: r(ßl>0, mean lifetime increases,

• ß=l, i.e. µ'=µ: r(ß)=0, no change of mean lifetime,

• ß>l, i.e. µ'>µ: r(ßl<0, mean lifetime decreases,

• ß• ^{00 ,} i.e. µ' • ^{00 :} r(ß)• -2, since

E₉ =(1/3)·E.i = ½·1/µ with (37) which is the mean lifetime of a 2-out-of-2 :G system. (This is obvious

as µ ' ~⁰⁰ means that the system fails with the first failing component.)

1

0

- 1

-2

1

:-deterioration

1

1 1

1mprove-.

ment ,

1

i 0

ß

---+---~---~t=-=-=---4

Fig. 9: The graph of the relative deviation r(ß).

b) µ.:i..'=µ1+µ 2 , i=l or i=2. Here the integral terms in (30) are given by (ifj):

µ :1 • ex p ( - µ .:i.. ' • t ) ·

f:

e xp ( - [ µ 1 + µ 2 .-µ .:i.. ' 1 • 't ) d 't

F(t)= 1 - (1 + µ'•t)·exp(-µ't) •

This yields the mean lifetime:

1 ⁺ µ' µ'2

( 39 l

( 40 l

( 41)

iil For the case of identical cornponents with doubled failure rates in mode Yl', i.e. µ=µ 1 =µ2 and µ' c:2•µ we get:

f9 (t)= 1 - (1 + 2•µ·t)·exp(-2µt) ,

and

1 ~ 2•µ 4·µ2

( 42 >

( 43)

4. Conclusion

In ·this paper we have given an approach to consider the fa i 1 ure behaviour of components which work 1.n different operational modes. It 1.s more realistic than assum1.ng stochastical independence, s1.nce changing load or environment conditions certainly leads to different failure rates and lifetime distributions. We have distinguished two cases of sticking together two distributions at the time instant of mode changing:

1. the component keeps on working in a state not as good as new, corresponding to the already suffered deterioration,

2. the component keeps on working in a state ^as good a.s new, covering the case of maintenance, repair and eventual replacement.

(The latter was already investigated in [BaHe],)

In this paper we have investigated only the case of a single switch over. Our further efforts will mainly concern multiple (e.g. periodical) mode changes. Thereafter the investigation of multi component systems will be in order, typical ly those, where changes of mode happen on certain component failures.

We thank Prof. Dr. Schneeweiß for fruitful discussions.

[BaPr]

[BaHel

[Beau]

[CaSi]

[DeSi]

[HsIT]

[IyRo]

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Baehring, H., Heidtmann, K.: Reliability analysis of systems with periodical ly changing operat ion modes (in German).

Informatik-Fachberichte, vol. ^61, Springer, Stuttgart, 1981, pp. 277-286

Beaudry, M.D.: Performance-related reliability measures for computing systems.

IEEE, Trans. on Computers, vol. C-27, June 1978, pp. 5 4 0-5 4 7 .

Castillo, X., Siewiorek, D.P.: A workload depen- dent software reliability prediction model.

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Hsueh, M. C., Iyer, R. K. , Trivedi, measurement-based performability model multiprocesso system.

K. S .• : for

A a

2nd Intern. Workshop on Appl ied Mathematics and Performance/Reliability Models of Computer/Commu- nication Systems, Rome, 1987, pp. 337-352.

Iyer, R.K., Rosetti, D.P.: A measurement-based model for workload dependence of CPU errors.

IEEE, Trans. on Computers, vol. C-35, June 1986, PP• 511-519.

---

[MCSTJ

[ :'1eye l

[Rein]

McConnel, S.R., mesurement and

Siewiorek, D.P.: Tsao, M,M., The analysis of transient errors 1.n digital computer systems.

Proc. 9th Int. Symp. Fault-Tolerant Computers, 1979, pp.67-70.

Meyer, J.F.: Closed-form bility. IEEE, Trans. on July 1982, pp. 648-657.

solutions of performa- Computers, vol. C-31,

Reinschke, K,: Reliability of systerns German). Berlin, VEB-Verlag Technik, 1973

(in