The renewal function for an alternating renewal process, which has a Weibull failure distribution and a constant repair time

Reliability Engineering and System Safety 31 (1991) 321-343

The Renewal Function for an Alternating Renewal Process, Which Has A Weibull Failure Distribution and a Constant Repair Time

J. M. D i c k e y Department of Physics, Queens College, Flushing, New York 11367, USA (Received 28 October 1989; accepted 22 January 1990)

ABSTRACT A series expansion is developed for the nth order distribution functions of the Weibull distribution and the coef/i'cients are found by solving recurrence equations. The availability and the renewalfunction of a n alternating renewal process, which consists of a Weibull failure distribution and a constant repair time, are written as related double series. The distribution functions and the unavailability are calculated for several values of the parameters and the rate of convergence of the series is examined.

1 INTRODUCTION Many practical problems in the fields of reliability and risk assessment can be solved by applying the results of renewal theory (see, for example, Gnedenko et al. ~ and M a n n et al. 2). An important quantity which must often be calculated is the instantaneous unavailability of a component or a system. A simple renewal process can be used to describe a component that may be replaced when it fails. However, in order to describe a component for which a finite time elapses after failure, before it is ready for use again an alternating renewal process must be used. In this case there are two alternating sequences of independent random variables, which may be called the failure times and the repair times. A typical example, which occurs frequently in nuclear safety systems, is a component whose condition is 321 Reliability Engineering and System Safety 0951-8320/91/$03"50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

322

J. M. Dickey

continuously monitored. The pressure may be measured by a pressure gauge and recorded or a signal light in the control room may indicate the correct functioning of the component. When failure is detected, the component may then be repaired. In order to treat this problem mathematically, simple models must be chosen to describe the failure and repair processes. It is found empirically that the data on failure times of many different engineering components can be described in terms of a 'bath-tub' curve. ~'2 This curve shows, in general, three regimes, although some components may show only two of these regimes. In the initial 'burn-in' regime the hazard rate decreases as inherent weaknesses and defects are repaired. In the following 'normal operating' regime, failure is random so the hazard rate is constant. Finally in the 'wear-out' regime the cumulative effects of small irreversible changes result in an increasing probability of failure. The two-parameter Weibull distribution is a popular choice 3'4 as a mathematically simple approximation for the hazard rate. By suitable choice of the two parameters, it is possible to model a large portion of the bath-tub curve although it is not possible to model the complete curve. A component may have several different failure modes and the repair time may vary depending on the nature of the failure. There are several probability distributions that could be used for the repair time in the appropriate circumstances; for example, normal distribution, lognormal, delta function, etc. We will make the simplest assumption, namely that each repair takes exactly the same time. The two probability density functions of the alternating renewal process considered here are the two-parameter Weibull distribution and the delta function. The results reported in this paper have been incorporated in the code ' F R A N T I C I I - - A Computer Code for Time Independent Unavailability Analysis'. 5'6 This code has been used to calculate the unavailabilities of systems for use in the risk assessment of nuclear power plants and similar applications. Renewal theory leads to an integral equation for the renewal density. However, this equation is soluble for only a small number of distributions. For a general renewal process involving more complex distribution functions than those considered here, the renewal function can be calculated only approximately. There have been several different approaches to the numerical evaluation of the renewal function, e.g. direct numerical solution,7 Laplace transform techniques, 8 spline approximations 9 and some other procedures described in Cleroux and McConlalogue, 1° Baxter, 11 Kao,12 Whitt 13 and Tortorella. 14 We report in this paper an exact solution in the form of a series for the alternating renewal process described above. The series converges rapidly and may be evaluated easily on a computer. Thus the special alternating renewal process discussed in this paper may be used as a benchmark problem to check the approximations made and the

The renewalfunction for an alternative renewal process

323

programming techniques used in a general procedure for the numerical evaluation of an arbitrary renewal function. We show first how the renewal density can be written as a finite sum of the nth order distribution functions [eqn (17)]. We then show that each nth order distribution function of the Weibull distribution can be written as an infinite power series. The coefficients satisfy a recursion relation and so may be calculated. Finally, it is possible to calculate exactly the renewal function of the alternating renewal process and the corresponding unavailability. In the last section we report results of such calculations, using values of the parameters that are physically realistic and examine the rate of convergence of the series.

2 PRELIMINARY DEFINITIONS The general alternating renewal process1 consists of an alternating sequence of failure times (z'l, z~ .... ) and repair times (z~, z~, T'~.... ). All the failure times are statistically independent and have the same probability density,f (t), and distribution function, F(t). We will consider the special case when all the repair times are identical, that is z~ = ~. The probability density function of the repair times is thus a delta function, g(t) = 6(t - z), and the distribution function is a step function at T. There are two related simple renewal processes. The first process, which we will use here, consists of the sequence (t'n), where t'l = z'l !

t',=z+z,,

n>l

(1)

A failure occurs at each instant T', where n

T" = ~ ' , t"

(2)

m=l

This sequence forms a modified renewal process. The distribution function for t'l is F(t), and for the other times t'n is ~(t), where, from eqn (1),

dP(t)= { F ( t - z)

t>
(3)

, >

(4)

The corresponding density ~b(t) is 4,(t)

f(t - T)

t_<-c

324

J. M. Dickey

The renewal function for this process, H(t), is equal to the m e a n n u m b e r of failures to time t. The renewal density, h(t), is h(t) = H'(t)

(5)

The unavailability, q(t), is defined as the probability that the system is d o w n at time t. F o r a constant repair time z, q(t) = prob(failure occurred in the interval (t - z), t) = H ( t ) - H ( t - z)

(6)

=

(7)

h(t') dt' t--l*

3 T H E SERIES S O L U T I O N O F T H E I N T E G R A L E Q U A T I O N F O R THE RENEWAL DENSITY The integral equation for the modified renewal process described above is 1 h(t) = f ( t ) + ~ i h(t - x)(x) d x

(8)

The first term corresponds to the event that the first failure occurs at time t. The second term corresponds to the event that a failure occurs at an earlier time (t - x). This is followed by a repair and the next failure, which occurs at time t. Using eqn (4), h(t) = f ( t ) + .[j h(t - x ) f ( x - z) d x

=f(t) +

h ( y ) f ( t - z - y) d y

(9)

Since the integral only involves values of h for times y < t - r this equation can be solved iteratively. Alternatively, a series solution m a y be obtained by replacing h in the integrand of eqn (9). Write GO

h(t) = ~ , h.(t)

(10)

n=l

where

(11)

hx(t ) =f(t) h.(t) =

h. _ ~ ( y ) f ( t - z - y) d y

(12)

The renewal function for an alternative renewal process

325

Then h2(t) =

fo

- ' f ( y ) f ( t - z - y) d y

=f2(t - z) ha(t) =

(13)

f2(Y - z ) f ( t - z - y) dy

= f l - 2'f2(z)f(t -- 2z -- z) dz

= f a ( t . 2z)

(14)

h.(t) =f.(t - (n - 1)z)

(15)

or in general

where f~ is the n-fold convolution of the probability density,f Because of the upper limit in the integral, the series terminates for N = [t/r]. Therefore h(t) = fl(t) +f2(t - z) +f3(t - 2z) + . . . +fN+ t(t - Nz) N+I

(16)

= ~'~ f~(t -- (n -- 1)z) n=l

The nth order distribution function, F~, is related to f~: (17)

F~(t) = flf~(t' ) dt'

Thus the renewal function, H(t), can be found by combining eqns 5, 16 and 17: N+I

H(t)

=~

F~(t - (n -

1)z)

(18)

n=l

This formula could have been written down immediately from the definition of H(t), since the fight-hand side o f e q n (18) is the sum of the probabilities of the disjoint events that one failure or two failures and one repair, or three failures and two repairs, etc., have occurred prior to the time t.

J. M. Dickey

326

Substituting eqn (18) into eqn (6) we obtain an equation for the unavailability: N+I

q(t) = ~

N+I

F , ( t - (n -1)z) - ~

n=l

F,(t - nr)

n=l

N+I

= )'

[F~(t -- (n - 1)~) - F.(t - nr)]

(19)

n=l

Thus the renewal function and the unavailability for an alternating renewal process, in which the repair times are constant, m a y be calculated if the nth order distribution functions are known. However, explicit expressions for F. can be found only for a few probability distributions. In the next section we show that F. can be calculated for the Weibull distribution. 4 THE nTH ORDER DISTRIBUTION FUNCTIONS FOR THE WEIBULL DISTRIBUTION The probability density of the two-parameter Weibull distribution is

f(t) = 2fit t~- 1 exp ( - 2t p)

(20)

and the distribution function is

F(t) = 1 - exp ( - 2fl)

(21)

2 is called the scale parameter since the value of this parameter fixes the scale of the time axis. In this section, for simplicity, we will use units for time such that 2 = 1.0. /3 is the shape parameter; values o f / / < 1.0 correspond to a c o m p o n e n t that shows 'burn-in' behaviour and values o f / / > 1.0 correspond to a c o m p o n e n t that shows 'wear-out' behaviour. The special c a s e , / / = 1.0, corresponds to a time independent hazard rate and the probability distribution reduces to the exponential distribution. Smith and Leadbetter I s have proved that the renewal function of a simple process based on the Weibull distribution can be written as a uniformly convergent infinite series. We will show that the nth order distribution functions and the renewal function o f the alternating renewal process m a y also be written as an infinite series. Assume that each F. can be expressed in terms of a power series that is uniformly convergent: oo

r.(t) = E k=l

(_)k- 1An,ktk# l"(kfl + 1)

(22)

The renewalfunctionfor an alternativerenewalprocess

327

where F(x) is the gamma function. However, from eqns (20) and (21), oo

Fl(t) =

(_)k- 1

(23)

kt

k=l oo

fa(t) = 2

(_)k- 1kfl ~'e- 1 k!

(24)

k=l SO

AI'k =

r(kp + 1) k!

(25)

From the definition of the nth order distribution function, eqn

(17),

F.(t) = ~i F._ l(x)f(t -- x) dx

(26)

Use eqns (22) and (24) to substitute in eqn (26):

(_)k- ~A..k r(kp + 1) k=l

. - l,q F(qfl + q=l

=

~2

1)

JE2

( - ) p - Ipfl (t - P!

p=l

A"-x'~(Pfl) flxqtJ(t-x)PtJ-ldx

( _ ) p + , - 2 r(qp + 1)p!

(27)

q=Ip=l

The integral in eqn (27) is a standard form and yields x6

tp+qfl r(qfl + 1)F(pfl) (28) F(qfl + pfl + 1) Substituting in eqn (27), cancelling F(qfl + 1) and using F(pfl + 1)=pflF(pfl), (__)k_ IA k= t

/kO

F(pfl + 1) 1) tip+q)~ (--)p + ~ - 2 A " - X'~p!F((p + q)fl +

.,k F(kfl + 1) -q= 1 p = l

(29)

For brevity, let us define ~.:

r(n# + 1) ~Zn =

n!

(30)

J. M. Dickey

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The renewal function for an alternative renewal process

331

O n equating the coefficients o f the same powers o f t in eqn (29), we can obtain the following recurrence equations for the coefficients A.,k: A l,k = ~tk

for all k

A..k = 0

for k < n

A.,. = ( - ) " - 1~]

for all n

(31)

k-1

A.,k = -- ~ ' , A._ l,~ak_i

for k > n

j=n-1 These equations can be solved iteratively for A., k. Table 1 shows the coefficients f o u n d for several values o f ft. C o m b i n i n g eqn (22) with eqns (18) a n d (19) the renewal function a n d the unavailability can be written as double series: ~

H(t) = 1 - exp ( - 2t p) +

(

_

- 1

2kEt -- (n - - 1 ) Z ] r(k/~ + 1)

k#

(32)

n=2k=n q(t) = -- exp ( -- 2t p) -- exp [ -- 2(t -- z)~] N

o¢

__ )k-1 An,k~'k{[t--(n--1) } F "f]k#-(t-nz)kIJ ( k+ / 1)~

_~~(

(33)

n=2k=n T h e Weibull distribution reduces to the exponential distribution for the special case f l = 1.0. T h e solution for an alternating renewal process consisting o f a constant repair and an exponential distribution is well known, I so we will include it as a check on the general results. F o r fl-- 1.0, f r o m eqn (25) 0c,--1.0 identically. T h u s the recurrence relations, eqn (31), reduce to A.,. = ( - 1)"- 1

An,k= An.k-

l --

(34) A._ X,k- 1

for k > n

Substituting in eqns (22) a n d (33) we easily recover the s t a n d a r d results:

F.(t) = 1 - e._ ~(t) exp ( - t)

(35)

where

~ e .( t ) = m=O

tm m--~.

(36)

332

J. M. Dickey

and N

q(t)

= 1 -- ~ , (t -- nz)" exp [--(t -- nz)] n!

(37)

/,,,,,,,d n=O

To compare with the results of Smith and Leadbetter, ~5 it should be noted that for a simple renewal process, i.e. with instantaneous repair, the renewal function, Ho(t), can be written as

Ho(t) = ~ , F.(t)

(38)

n=l

This is also the limit of expression (18) as ~ tends to 0. Thus the expansion coefficients of Ho(t ), given by Smith and Leadbetter, ~5 A k

tk~ ( _ ) k - 1Ak F(kfl + 1)

Ho(t ) =

(39)

k=l

are given in terms of the coefficients for the F,, defined in this paper by k

Ak = ) ' , A,,k

(40)

n-1

It can easily be shown that the recurrence relations ofeqn (31) can be reduced to the recurrence relations for Ak, which are given in Smith and Leadbetter. 1 5

5 R E S U L T S OF C A L C U L A T I O N S A computer code has been written to solve the recurrence equations for the coefficients A,,,k and to calculate the nth order distribution functions and the unavailability. We report some typical results using realistic values for the parameters. 5 In practice the mean failure time will be much larger than a repair time. The mean failure time of the component is taken to be 3.3333( + 5) hours. Several values of the shape parameter, fl, are used to show the consequences of different degrees of burn-in or wear-out behaviour. The parameter ), can be found from the mean failure time, TF, by the standard expression for the Weibull distribution: 2-

F(I + 1/fl) tF

(41)

The renewal function for an alternative renewal process

333

13 0.5 0.7 0.8 0.9

f--

1.0 1.1 1.2 1.3

/f__ 58

1.5

/

2.0

8

9" 8

% Fig. I.

Unavailability for times less than about 1 week, T F = 3-3333(5) h, z = 20 h and values o f fl shown.

Results corresponding to other failure times may be found by appropriately scaling the time axis. The repair time is taken to be 20 hours for the first three figures. The unavailability is plotted versus time--in Fig. I for times up to about a week, in Fig. 2 for times up to about a year and in Fig. 3 for times up to about 40 years. The unavailability always tends to an asymptotic value, qo, as t--, oo 1 and qo = - -

~+TR

(42)

The rate at which the asymptote is approached depends on the value of ft. The asymptote is approached most rapidly for fl = 1"0 and q is within a few per cent of the asymptote at time t = TF for the values chosen of fl > 1.0. However, the asymptote is approached much more slowly for values of fl < 1-0. Table 2 summarises the times required for q(t) to be within a given percentage of qo for several ft. Figure 4 shows the effect of the repair time on the behaviour of q(t) for

J. M. Dickey

334

0.5

8 1.0 1.1 -.i

--©

g

t

Y 'o~o

zo.o0

20.00

~0.00

4~.00

50.00

60.00

70,0o

TIME [MRS]

Fig. 2.

~0 ,,,00

90.00

[XlO-- J

100.0o

U n a v a i l a b i l i t y for times less t h a n a b o u t 1 year, T F = 3.3333(5) h, ~ = 20 h a n d values

of fl shown.

1.0

~o! ~'ig. 3.

~.®

,~=

,~.-~

~,.~

T IHE [ H R $ ]

:

~'00

~'00

Ix ~001o"3

J~tX~.00

'

Unavailability for times up to about 40 years, Tr = 3.3333(5)h, z = 2 0 h and values o f fl shown.

The renewal function for an alternative renewal process

335

TABLE 2 Asymptotic Behaviour of q(t): The Time, t/Tr, for which [q(t)-qo]/qo Equals the Stated Percentage, TF= 3"3333(5) h, T = 20h

[3

Time (t/L)

0"5 0-7 1.3 1"5

20%

10%

5%

1'95 0.64 0'33 0"47

3'63 1"20 0"60 0"71

6"10 2'05 0'87 0"92

2~13=0.

5 ~=1.0

'I(hr$)

~.. M~

.

--

~=1.5

~,.

%Too Fig. 4.

I0.00

I ZO,O0

,~.Oe

40O . OTM I {HRS)~' E O0

~.00

"I~.00 {X~iotPO)

~,00

t~.O0

Unavailability for several values of the repair time, z, TF = 3"3333(5) h and fl = 0-5, 1"0 and 1'5.

336

J. M. Dickey

~ %

%.oo Fig. 5.

•~o

.~o (

.~o TIME /TAU ) .~o

:lO0000hrs 100,1000hrs

,*.oo

~'.2o

~'.,o

:.~0

?.~0

2.®

Unavailability for the exponential distribution,/~ = 1, for several repair times.

three values of r: 1"5, 1-0 and 0.5. When the unavailability is normalised by dividing by q0, the resulting curves are very close for times greater than the repair time. Figure 5 shows for comparison the familiar results for the special case fl = 1.0 or exponential distribution. A new time scale is used since the asymptotic value is approached for t ~ z for the values of the parameters suitable for practical applications in risk assessment: The nth order distribution functions, F,, are plotted in Fig. 6(a)-(c) for values of fl = 1.5, 1.0 and 0.5. In each case TF has been taken to be 1.0. For larger values of fl the higher order distribution functions do not become significant until longer times, as would be expected for components showing 'wear-out' effects. In contrast, for r = 0.5 the higher order distribution functions become relatively significant at much shorter times, reflecting the fact that this value offl represents a component that is prone to fail relatively early, or in other words is showing 'burn-in' behaviour. Thus in the double series expressions for H ( t ) and q(t), it is necessary to sum to larger values o f n as fl is decreased. For completeness Fig. 7 shows the renewal function Ho(t) calculated from the above distribution functions. The rate of convergence of the double series for q(t) is examined in Table 3. Let us define q, as the contribution to q from the nth order distribution function: q,( O = F~(t -- (n - 1)3) - F~(t -- nz)

(43)

In the calculation of q~ a sufficient number of terms in the k series must be

337

The renewal fanction for an alternative renewal process TABLE 3 The Percenratage Contribution of Each Term, q., to q(t) at the Stated Time, Tr = 3"333 h, T = 2 0 h (a)/~ = 1.5

Time = N

0.25T F (%)

0.5T r (%)

I'OTF (%)

2"OTv (%)

1

94 6

83 16 1

57 35 8 1

16 40 30 11 3

2 3 4 5 6 7 8

4"OTF (%) --

6 19 29 24 14 6 2

(b)/~ = 1'3

Time = N

0"25 TF (%)

0"S TF (%)

I'O TF (%)

2"OTF (%)

1

90 10

76 21 3

50 37 11 2

16 35 30 14 5 1 1

2 3 4 5 6 7 8 9

4"OTF (%) 1

7 18 25 23 15 7 3 1

(c)/~ = 1.0

Time = N

0"25T F (%)

0"ST F (%)

I'OT F (%)

2"OTF (%)

4"OTF (%)

1

78 2

61 30 8 1

37 37 18 6 2

13 27 27 18 9 4 1

2

2 3 4 5 6 7 8 9 10

7 14 20 20 16 10 6 3 1

338

J. M. Dickey

T A B L E 3---contd. (d) fl = 0.7 Time= N

O,~T v (%)

0.5~ (%)

1.OTv (%)

2.0~ (%)

4.0T v (%)

1 2 3 4 5 6 7 8 9 10 11 12 13

56 31 10 2

40 34 17 7 2 1

22 29 23 14 7 3 1

9 17 21 19 14 9 5 3 1

2 6 10 13 15 14 12 9 7 5 3 2 1

te)/~ =

0.5

Time = N

0"25 T r (%)

0.5 T r (%)

1"0 T r (%)

2"0 T r (%)

4"0 T F (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

34 32 19 9 4 !

22 27 22 14 8 4 2 1

12 19 20 17 12 8 5 3 2 1

6 11 14 15 14 12 9 7 5 3 2 1 1

2 4 6 8 10 10 10 10 9 7 6 5 4 3 2 1 1 1

The renewal function for an alternative renewal process

339

~. = 1 . 4 1 4 2 =0.5

TF = 1.0

J

Fa

co

F"5

c:]

•20

,40

.60

.80

1.00

1,20

1.40

1.60

1.80

2.00

TIME

(a) Fig. 6. D i s t r i b u t i o n functions for the Weibull distribution, F I - F 6 for (a) / 3 = 0 . 5 ; ~. = 1.4142, T r = 1-0.

used, since alternate terms in this series may partially cancel. Table 3 shows the proportional contributions of q, at different times for several values of ft. For any given value offl, it is necessary to include more terms in the series as the time increases. For practical calculations, 5 however, one may use the asymptotic expression, eqn (42), for long times. The time required for a given accuracy is illustrated in Table 2. For/3 > 1.0, as/3 increases the Weibull probability density, eqn (20), becomes more sharply peaked at t = Tv. Correspondingly the contribution to q from the term with n ,~ t / T r becomes increasingly dominant. For example, from Table 3(a), for/3 = 1.5, at t = 2T~, q2 contributes 40% of the total value of q. For small values of/3 < 1.0, more terms in the n series must be used [see Table 3(d) and (e)]. This is consistent

J. M. Dickey

340

X=I.0 ~. !

13 : 1.0

TF= 1"0

F2

i' g

v

-o.oo

.~o

,io

,eo

Fig. 6.--contd.

.so TIME

1.oo

i.~o

i'.4o

(b) fl = 1"0, 2 = 1"0, TF = 1'0.

a,so

~.so

2,o0

The renewal function for an alternative renewal process

341

X = 0.8577 I~ = 1.5 TF=I.0

P

• 40

.110

,IN)

I .IX)

I ,~

1,40

TIME

F i g . 6.--contd.

(c)/~ = 1"5, 2 = 0"8577, T r = 1.0.

I.E0

I.W

J. M. Dickey

342

0.5

8

~.0

5

•~0

Fig. 7.

.,tO

.60

.CO

TINE

I ,CO

1.20

1.40

1.60

I .SO

2,00

Renewal function, Ho(t), for several values of fl, T F = 1.

with Fig. 7 which shows the early importance o f the high order distribution functions for a c o m p o n e n t with burn-in behaviour. Several terms give a significant contribution, e.g. from Table 3(e), fl = 0-5, at time t = 4TF, qs, q6, qT, qs, each contribute 10% to the total value o f q. In general the contributions to q are more broadly distributed a m o n g the q, as /3 is decreased. We have shown that for the parameters o f practical interest, z << TF, the series expansion for q(t) rapidly converges and the asymptote is approached for times less than or comparable to TF. The expression deduced for q(t) is easily implemented on a computer and is thus o f practical value in unavailability calculations.

The renewalfunction for an alternative renewalprocess

343

ACKNOWLEDGEMENT Part of this work was done while a guest scientist at Brookhaven National Laboratory, Upton, New York, USA.

REFERENCES 1. Gnedenko, B. V., Belyaev, Y. K. & Solovyev, A. D., Mathematical Methods of Reliability Theory. Academic Press, 1969. 2. Mann, N. R., Schafer, R. E. & Singpurwalla, N. D., Methods for Statistical Analysis of Reliability and Life Data. John Wiley & Sons, New York, 1974. 3. Weibull, W., A statistical theory of the strength of materials. Ing. Vetenskaps Akad Handbuch, 151 (1951) 1. 4. Kao, J. H. K., A graphical estimation of mixed Weibull parameters in lifetesting of electron tubes. Technometrics, 1 (1959) 389. 5. Vesely, W. E., Goldberg, F. F., Powers, J. T., Dickey, J. M., Smith, J. M. & Hall, R. E., FRANTIC II--a computer code for time dependent unavailability analysis. NUREG/CR-1924, BNL-NUREG-51355, 1981. 6. Dickey, J. M., Ginzburg, T. & Hall, R. E., Time dependent unavailability of a monitored component. BNL-NUREG-51457, NUREG/CR-2332, 1981. 7. Deligonu, Z. S., An approximate solution of the integral equation of renewal theory. J. Appl. Prob., 22 (1984) 926. 8. Jaquette, D. L., Approximations to the renewal function m(t). Opns. Res., 20 (1972) 722. 9. Deligonu, Z. S. & Bilgen, S., Solution of the Volterra equation of renewal theory with Galerkin techniques using cubic splines. J. Stat. Comput. Simul., 37 (1984) 45. 10. Cieroux, R. & McConlalogue, D. J., A numerical algorithm for recursivedefined convolution integrals involving distribution functions. Mgt. Sci., 22 (1976) 1138. 11. Baxter, L. A., On the tabulation of the renewal function. Technometrics, 24 (1982) 151. 12. Kao, E. P. C., Computing the phase type renewal and related functions. Technometrics, 30 (1988) 87. 13. Whitt, W., Approximating a point process by a renewal process, I: two basic methods. Opns. Res., 30 (1982) 125. 14. Tortorella, M., Comparing renewal processes with application to reliability modelling of highly reliable systems. Technometrics, 31 (1989) 357. 15. Smith, W. L. & Leadbetter, M. R., On the renewal function for the Weibull distribution. Technometrics, 5 (1963) 393. 16. Abramowitz, M. & Segun, I. A., Handbook of Mathematical Functions. National Bureau of Standards, AMS 55, 1970.