to be able to simulate these frequency deviations in our simulation, a common oscillator model is...

Comparison of Replacement Policies, and Renewal Theory ImplicationsAuthor(s): Richard E. Barlow and Frank ProschanSource: The Annals of Mathematical Statistics, Vol. 35, No. 2 (Jun., 1964), pp. 577-589Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2238511 .Accessed: 30/12/2014 11:02

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to TheAnnals of Mathematical Statistics.

http://www.jstor.org

This content downloaded from 80.66.191.254 on Tue, 30 Dec 2014 11:02:42 AMAll use subject to JSTOR Terms and Conditions

COMPARISON OF REPLACEMENT POLICIES, AND RENEWAL THEORY IMPLICATIONS

BY RICHARD E. BARLOW' AND FRANK PROSCHAN Boeing Scientific Research Laboratories

1. Introduction and preliminaries. Among the most useful replacement policies currently in popular use are the age replacement policy and the block replacement policy. Under an age replacement policy a unit is replaced upon failure or at age T, a specified positive constant, whichever comes first. Under a block replacement policy a unit is replaced upon failure and at times T, 2T, 3T, *2 . We shall assume for both policies that units fail permanently, independently, and that the time required to perform replacement is negligibly small. Block replacement is easier to administer since the planned replacements occur at regular intervals and so are readily scheduled. This type of policy is commonly used with digital computers and other complex electronic systems. On the other hand, age replacement seems more flexible since under this policy planned replacement takes into account the age of the unit. It is therefore of some interest to compare these two policies with respect to the number of failures, number of planned replacements, and number of removals. ("Removal" refers to both failure replacement and planned replacement.)

Block replacement policies have been investigated by E. L. Welker, 1959, R. F. Drenick, 1960, and B. J. Flehinger, 1962. Age replacement policies have been studied by G. Weiss, 1956, and Barlow and Proschan, 1962, among others.

The results of this paper depend heavily on the properties of distributions with monotone failure rate (Barlow, Marshall, and Proschan, 1963). If a unit failure distribution F has a density f, it can be verified by differentiating log F(x) that the failure rate r(x) = f(x)/F(x) is increasing (decreasing) if log F(x) is con- cave when finite (is convex on [0, oo ) ). We consistently use F for 1 - F. For mathematical convenience and added generality, we use this concavity (con- vexity) property as the definition of increasing (decreasing) failure rate whether a density exists or not. We shall refer to increasing failure rate by IFR and decreasing failure rate by DFR. It is also easy to show that F is IFR(DFR) if and only if

Fx(A) = [F(x + A) - F(x)]/F(x) is increasing (decreasing) for all x such that A > 0 and F(x) > 0. This implies F is IFR(DFR) if and only if (1.1) F(x - A)/F(x)

Received 22 July 1963; revised 18 November 1963. Now at the University of California, Berkeley.

577


578 RICHARD E. BARLOW AND FRANK PROSCHAN

is increasing (decreasing) in x for all A > 0. This property will be needed in Theorem 2.1.

The evaluation of the replacement policies considered also depends heavily on the theory of renewal processes (e.g., Smith, 1958, and Cox, 1962). A renewal process is a sequence { Xk} =Li of independent random variables with common distribution F. We also assume F(O0) 0 O. If the random variables are not identically distributed we call this a renewal sequence. Let us write N(t) for the largest value of n for which X1 + X2 + * - * + Xn _ t; in other words N(t) (the renewal random variable) is the number of renewals that will have occurred by time.t. The process {N(t); t > 01 is known as a renewal counting process. We will be primarily concerned with bounding the renewal function, M(t) = E[N(t)]. Previously, Feller, 1948, has given methods for bounding M(t) using bounds on F.

In this paper we show that, assuming an IFR(DFR) unit failure distribution, the number of failures in [0, t] is stochastically larger (smaller) under an age policy than under a block policy (Theorem 2.1). The number of planned replacements and the total number of removals is always stochastically smaller under an age policy than under a block policy (Theorem 2.2). By considering the number of failures and the number of removals per unit of time as the dura- tion of the replacement operation becomes indefinitely large, we obtain in Theo- rem 2.4 simple useful bounds on the renewal function for any failure distribution, and an improvement on these bounds by assuming IFR(DFR) failure distributions. In particular we show that the moments, binomial moments, and variance of a renewal random variable associated with an IFR(DFR) renewal process are dominated (subordinated) by the corresponding moments and variance of a related Poisson random variable. Inequalities for renewal sequence are also obtained.

2. Contrast between age and block replacement. Denote the number of renewals in [0, t] when replacement occurs only at failure by N(t) and let M(t) = E[N(t)]. Denote the number of failures in [0, t] under a block policy by N*(t) and under an age policy by N*(t), both having replacement interval T. The following theorem provides a stochastic comparison of the number of failures experienced under these policies. We assume F(O-) = 0 throughout.

THEOREM 2.1. If F is IFR(DFR), then

(2.1) P[N(t) > n] _(?) P[N*(t) ? n] ?(?) P[N* (t) ? n] for t ? 0, and n = 0, 1, 2, *. . Equality is attained for the exponential distribution F(x) = 1 - e xIl where 41 denotes the mean of F.

We defer the proof of Theorem 2.1 to Section 3. The following bounds on M(t) are an immediate consequence of Theorem 2.1 and will be useful later on. (See Theorem 2.4 and Theorem 4.2.)

COROLLARY 2.1. If F is IFR(DFR), then


REPLACEMENT POLICIES AND RENEWAL THEORY 579

(i) M~(t) > E [N* (t) I > ( < ) E[Ns (t)] (ii) M(t) ?( O

for all t : 0. PROOF. (i) This follows from Theorem 2.1 and the fact that

00

M(t) = E[N(t)] = , P[N(t) > n]. n-=1

(ii) Let T = t/lk and observe that for this replacement interval M(t) _ ( < ) E[N* (t)] = kM(T) = kM(t/k).

(iii) By (ii) M(kT)T/kT ? () T4A1, since limt>O M(t)/t = 1/u1 by the elementary renewal theorem (e.g., Smith, 1958).

(iv) Define b(t) = t - [Xi + X2 + * + XN(t)]. Then t h

M(t + h) - M(t) = f f [1 + M(h - u)] dFx(u) dx P[6(t) < x] t h

>()f [1 + M(h-u)] dF(u) d. P[a(t) < x] since Fx(u) is increasing (decreasing) in x. Therefore

M(t + h) - M(t) > ( n] ? P[NA(t) ? n] ? P[NB(t) ? n]

for all t> 0, n = 0, 1, 2, **.



We defer the proof to Section 3. COROLLARY 2.2.

(i M (t) _!! E[NA(t)] < E[NB(t)] (2.3) (ii) M(t) < kM(t/k) + k k = 1, 2, ...

(iii) M (t) > t/,u - 1 for all t ? 0.

PROOF. (i) This is an immediate consequence of Theorem 2.2. (ii) Let T = t/k and observe that for this replacement interval

M(t) ? E[NB(t)] = kM(T) + k = kM(t/k) + k. (iii) This follows from the elementary renewal theorem limetO,M(t)/t =

1/Mi. 11 The following theorem summarizes some well-known limit results from renewal

theory. THEOREM 2.3. ( i) limt,, oN (t )/t = 1imzt,.M( t )/t= 1y (ii) limt ooN*(t)/t = limet.E[N*(t)]/t = F(T)/f oFP(x) dx (iii) limt,coN*(t)/t = 1imt_,.oEW[N*(t)]/t -M1(T)IT. PROOF. (i) See e.g. Smith, 1958. (ii) The times between failures { Yi}j 71 for an age replacement policy con-

stitute a renewal process with distribution

(2.4) ftT(t) = P[Y, > t] = [F(T)]n F(t - nT) for nT < t < (n + 1) T. The expected value of Yi can be calculated from (2.4) to be E[YJ] = frOT F(x) dx/F(T). (Weiss, 1956, calculated higher moments.) Hence limt >N* (t)/t = F(T)/ f' F(x) dx by (i).

(iii) Let N* (T) denote the number of failures in (i - 1) T, iT] following a block replacement policy. Then

limt 0N*B(t)/t = limn- ooE N*i (T)/nT = E[N*j(T)]/T = M(T)/T. 11 i=l4

For a generalization of (iii), see, e.g., Flehinger, 1962. Using these limit results we can improve (iii) of (2.2) and (iii) of (2.3). THEOREM 2.4.

(2.5) () M(t) _ t/ f tP(x) dx-1 t/,ul- 1. (ii) If F is IFR(DFR), then

(2.6) M(t) ) tF(t)/f P(x) dx


PROOF. (i) By Corollary 2.2 (i), E[NA(t)] < E[NB(t)]. By Theorem 2.3 (ii) and

(iii)

limt>,E[NA(t)]/t = 1/f F(x) dx ? limt T/ F(x) dx-1 forall T > 0.

Obviously jT F(x) dx < i,u implies T/IfT F(x) dx -1 ? TI, - 1. (ii) By Corollary 2.1 (i), E[N*(t)] _(_) E[N*(t)]. By Theorem 2.3 (ii) and (iii) M(T)/T = limt+O:E[N*(t)]/t ?(>) limt,o E[N*(t)]/t

= F(T)/f F(x) dx ) 14A1.

The last inequality follows by noting that HT(t) [see (2.4)] is decreasing (increasing) in T if F is IFR(DFR). 11

Smith, 1961, has given the inequality M(t) < 1/F(t) which is true for all F. Since

tF(t) P(x) dx < t / (x) dx ? I/F(t).

(ii) of Theorem 2.4 is an improvement on Smith's result when F is IFR. (This observation is due to the referee.)

Equality is approximately attained in (2.5) for intervals (t - E, t) where t = k,ul(k = 1, 2, * * * ) by the distribution degenerate at the mean Al, and of course M(t) > 0 is sharp for t < Al . Equality is attained in (2.6) for the Poisson process. Note that inequality (2.6) is an improvement on (2.5) in the DFR case since

t ~ ~~t IIt It tF(t)/ F(x) dx > f F(x) dx/ F(x) dx = t/ ] (x) dx - 1.

From (2.5) and (2.6) we see that

1/i - 1/T < M(T)/T < F(T)/f P(x) dx ? 1/#l

when F is IFR. Hence when F is IFR the expected numbers of failures per unit of time under the two replacement policies do not differ by more then 1/T in the limit as t -> oo.

3. Proofs of the Theorems of Section 2. Proof of Theorem 2.1. Assume F is IFR. First let us suppose 0 < t < T, where

T is the replacement interval. Let P[N(t) _ n I x] denote the probability that



N(t) > n, given that the age of the unit in operation at time 0 is x. Then we shall show that

(3.1) P[N(t) > n I x] _ P[N*(t) _ n I x] _ P[N*(t) _ n]. For n = 0, (3.1) is trivially true. For n > 0, we can rewrite (3.1) as

t t 3(-1))(t- u) dFx(u) _ F0-1) (t - u) dF.j(u)

> f F-) (t - u) dF(u)

where F(n) (t) denotes the n-fold convolution of F with itself and F.(u) = [F(x + u) - F(x)]/F(x).

FXf(u) is the distribution of the time to the first failure when the age of the unit in operation at time 0 is x and planned replacement is scheduled for T - x if no failure intervenes. We need specify the distribution FT (u) only on [0, t]: Fx(u) = [F(x + u) - F(x)]/F(x) if u ? T - x

= [F(T) -F(x) + F(T)F(u-T + x)]/F(x) if T-x < u ? t. To prove (3.1') we need only show (3.2) F.(u) > FXT(u) _ F(u) forO < u < t since F(n-1) (t - u) is decreasing in u. For u < T - x

F.(u) = FxT(u) = [F(x + u) - F(x)]/F(x) ? F(u) since F is IFR. For t - x ? u < t

[F(x + u-T + T) -F(T)]/F(T) ? F(x + u- T) implies F(x + u) ? F(T) + F(T) F(u - T + x) and so

F.(u) = F(x + u) - F(x) > P(x) - P(T) + P(T)F(u - T + x) = F.T(u) P(x) P (x)

proves the first inequality in (3.2). Also for T - x < u _ t, F(u - T + x)/F(u) is increasing in u by (1.1) since we may assume x < T. Therefore

F(u - T + x)/F(u) < F(x)/F(T) since 0 _ u < T. Rearrangement yields

F(u) > F(T) F(u - T + x)/F(x), so that

F,T(u) = [F(x) - F(T) + F(T)F(u - T + x)]/F(x) > F(u)



which completes the proof of (3.2). From (3.2) we deduce that for x > 0 and O < t < T

P[N(t) > n lx] > P[N*(t) > n I x] > P[N*(t) _ n]. Now suppose kJT < t < (k + 1)T, where k > 1. The proof proceeds by in-

duction on k. Assume (3.1) is true for 0 < t < kT. For n = 0, (3.1) is trivially true. For n > 0, write

n T

P[N(t) _ n] = zf {P[N(T) = |6(T) = x]P[N(t -T) > n-r r I(T) = x] } dx P[S(T) < x]

n T

P[N* (t) _ n] = E f {P[N*(T) r I 6(T) = x]P[N*(t - T) r=O

> n-rI6(T) = x]} dxP[S(T) < x] and

n T

P[Ns(t) _ n] - E f {P[N*(T) r=OO

= rI6 a (T) = x]P[N*(t-T) _ n-r] } dx P[S(T) < x] where 6(T) is a random variable denoting the age of the unit in use at time T. By inductive hypothesis

P[N(t- T) _ n-r r 6(T) = x] _ P[N*(t- T) _ n-r r 6(T) = x] ? P[N*(t - T) _ n - r].

Also

P[N(T) = r I J(T) = x] = P[N*(T) = r 6(T) = x] =P[N*(T) = rI6(T) =x]

since all three policies coincide on [0, T]. Hence (3.1) follows for kT ? t < (k + 1) T for all k > 1 by the axiom of mathematical induction.

For F DFR the proof is similar with the inequalities reversed. 11 Proof of Theorem 2.2 (Due to Albert W. Marshall). Let {Xk} l denote a

realization of the lives of successive components. We shall compute what would have occurred under an age and under a block replacement policy. Let T'( TB) denote the time of the nth removal under an age (block) replacement policy. Then

T = min(TP1 + T, TP-1 + Xn), T = min(TB1 + a, TB1 + Xn) where a (O < a _ T) is the remaining life to a scheduled replacement. Since initially T' = TB, we have by induction T' > TB . Thus for any realization



{Xk} k= NA(t) is smaller than NB(t). By a similar argument N(t) is smaller than NA(t) for any realization. 11

4. Renewal theory consequences. A renewal process is an IFR(DFR) renewal process if the underlying distribution F is IFR(DFR). This does not imply that N(t), the renewal random variable associated with an IFR(DFR) renewal process, is IFR(DFR). (See Barlow, Marshall, Proschan, 1963.) However, just as the geometric (exponential) distribution is a natural comparison distribution for IFR and DFR discrete (continuous) random variables, the Poisson process serves as a natural comparison process for IFR and DFR renewal processes. In Corollary 2.1 we saw that the mean of an IFR(DFR) renewal random variable is dominated (subordinated) by the mean of an associated Poisson random variable. This is also true of the binomial moments and, indeed, even the variance.

We define the mth binomial moment, Bm(t), as

Bm(t) = (')P[N(t) = j].

The following result is well known and we omit the proof. LEMMA 4.1. For any renewal counting process {N(t); t > O}

Bm(t) = M(m) (t) where M(m) (t) denotes the m-fold convolution of M(t) E[N(t)].

A stationary renewal process {Xk}k'=1 is one for which X1 has distribution t

F (t) = [ (x) dx/41

and Xk(k = 2, 3, ... ) are independently distributed according to F. Denote a stationary renewal counting process by {N(t); t > O}. It is known (Cox, 1962, page 46) that E[N(t)] = t/1,1 for this process. A useful comparison can be made between renewal counting processes and their associated stationary processes.

THEOREM 4.1. Let {N(t): t ? O} denote a renewal counting process governed by F with mean,ui . Let {I (t); t > O} denote the associated stationary process. If

(4.1) f F(x) dx/P(t) ? (?) > for all t > O then

(i) P[ST(t) ? n] >(?)P[N(t) > n]

(ii) BMn(t) E (N(t)) E jn(t 00

forn, m-0,1,2, *yandO ? t < ct2.



PROOF. By hypothesis

F(t) (x) dx/Mul >(?) F(t). Therefore

t t

P[X(t) ? n] Fj Fn-'(t - x) dF(x) _ F(n-') (t - x) dF(x) = P[N(t) > n]

proving (i). By (i), E[N(t)] = M(t) _ E[l(t)]. Since E[N(t)] = t/1u (Cox, 1962, p. 46).

(ii) follows from Lemma 4.1. To obtain (iii) note that x' = E>=lx(x - 1) * (x - m + 1)SZ where

Sn are Stirling numbers of the second kind (Jordan, 1950, p. 168). Therefore E[Nn(t)] = Ej n=1m! Bm(t)Sn. Since Sn' are positive (Jordan, 1950, p. 169), (iii) follows from (ii). I

The importance of Theorem 4.1 stems from the interpretation of (4.1). Note that (4.1) implies that the mean residual life of a unit aged t is less (greater) than the mean life of a new component. If F is a IFR(DFR) with mean /., then (4.1) is true. Of course, the converse is not true and (4.1) is a significant weakening of the IFR(DFR) assumption.

THEOREM 4.2. If {N(t); t > 0} is an IFR(DFR) renewal counting process, then

Var[N(t)] ?(>) E[N(t)] = M(t). The inequality is sharp.

PROOF. Assume {N(t); t > 0} is an IFR renewal counting process. Since t

Var[N(t)] = [2M(t - x) + 1 - M(t)] dM(x) we need only show

t f [2M(t - x) - M(t)] dM(x) < 0.

But M(x) < M(t) - M(t - x) by (iv) of Corollary 2.1 implies that we need only show fo [M(t - x) - M(x)] dM(x) ? 0. Clearly

t t/2

f [M(t - x) - M(x)] dM(x) = f [M(t - x) - M(x)] dM(x)

+ f [M(t - x) - M(x)] dM(x). t/2

Let y = t - x, then t t/2

t/2



Hence we need only show t12

J [M(t-x) -M(x)] dM(x) tl2

< f [M(t - x) - M(x)] d[M(t) - M(t - x)].

This follows immediately, since M(t - x) - M(x) is non-increasing in x, [M(t - x) - M(x)] ? 0 for 0 < x < t/2 and M(x) ? M(t) - M(t - x).

All inequalities are reversed if F is DFR. Equality is attained by the Poisson process. II

It is well known (Feller, 1948) that limt_O Var[N(t)]/E[N(t)I = 2/,2 ? 1

when F is IFR. Therefore, Theorem 4.2 is in a sense an extension of this result to all t.

5. Generalizations. Next we obtain a generalization of the inequality M(t) < ( >) t/M, which holds when successive replacements have different failure distributions but a common mean. The method of proof is quite different from that used in Theorem 2.1 or Theorem 4.1. We will need to define the generalized renewal function

(5.1) Mo(t) = F1(t) + F1 * F2(t) + F1 * F2 * F3(t) + Note that Mo(t) is the expected number of renewals in a stochastic process in which the first unit has distribution F1, its replacement has distribution F2, etc.

THEOREM 5.1. Let F1 , F2 , F3, X be non-degenerate IFR(DFR) distributions with common mean g, and assume

Fi(t) 9 G(t) = 1 - e-IJl for t > 0 and i = 1,2, .. Then

(5.2) Mo(t) < ( > ) tVl for t > 0. PROOF. Assume Fi (i- 1, 2, *) are IFR. First suppose F1 = F2=

Then 00 00

MoW(t) = F(k)(t) < E G(k)(t) = tl 1 = k=1 k=lI

for 0 < t ? . (See Barlow, Marshall, 1963.) Suppose there exists r > Aul such that Mo( r) = T/i1. Then, since F has mass in [0, r]

rl, = MoQr) = f [1 + Mo(r - x)] dFi(x) < [1 + T dF1(x)



or

r/Al < Fi(r) + f Fi(x) dx/Mi which implies

PlO (x) dxlFt(7) ]< Al

But this contradicts (2.6) of Theorem 2.4. Hence Mo( r) < r/y,, 0 < r < , for this special case.

The argument proceeds by induction. Suppose the theorem is true for all sequences of distributions of the form H1 , H2, H.. , Ik = Hk+l = Hk+2 = where the Hi satisfy the IFR assumption.

M1(t) = F2(t)+F2*F3(t)+ + F2*F3 ****Fk+1(t)

+ F2 *F3** *Fk+l *Fk+l(t) +

and

Mo(t) - [1 + Mj(t - x)] dF,(x).

As before, Mo(t) < t/,ul for 0 < t < Al . Suppose there exists r > Al such that Mo(r) = r/li . This implies

7

Iri f [1 + Ml(r - x)] dFl(x) < F1(r) +] Fi(x) dx and

J 1(x)/Fl(Tr) < u.

This is a contradiction. Theorem 5.1 follows by the axiom of mathematical induction.

All inequalities are reversed for DFR distributions. We note that Smith (1960) and Chow and Robbins (1963) have considered

sequences of non-identically distributed random variables and have given conditions under which limt-..M(t)/t = 1/Mi.

The method of proof used in Theorem 5.1 can be used to generalize the bound on M(t) in yet another direction.

THEOREM 5.2. Assume F has density f, fatlure rate r(x) = f(x)/[F(x)], and mean i .

(i) If r(x) > a for all x, then M(t) < tilM + I/iam - 1. (ii) Suppose there exists 8 > 0 such that f(x) > O for 0 < x t/M, + /I3Ml - 1. Equality is attained in (i) and (ii) for the Poisson process.



PROOF. If F(x) = 1 - e 1" the bounds are attained. Hence suppose F(x) # 1 - e 1. Then (5.3) inf., r(x) < ll,ul < SUp$, r(x) (see Barlow Marshall, Proschan, 1963). (5.3) implies a < 1/ti < g, and 1/ayl - 1 >0, l/f3il - 1 < 0. Since M(O) = 0,

t/1.t + 1/,uB - 1 < M(t) 0 and suppose there exists 0 < r < oo such that

M(r) = r/l, + i/ayl - 1

and M(t) < t/M,1 + 1/oaMl-1 for t aoF(x) and so

P(r)= ff(x) dx > af (x)dx

which implies

FF(x) dx r 1

a contradiction. Hence, actually

M(t) < ti/l + 1/aMi - 1 when F(t) # 1 - e7t1'.

(ii) If ,B = oo, the inequality follows from Corollary 2.2 (iii). Hence suppose f3 < oo. There exists 0 < r < oo such that

M (r) = r/MA + 1//3Mi - 1


REPLACEMENT POLICIES AND RENEWAL THEORY, 589

and M(t) > t/M,I + 1/l3M-1 for t < r.

Then

- + -1 =VTM(r) = [1 + M(r - x)] dF(x)

> IAI ~+-]t dF(x) since F has mass in [0, r]. Therefore

T+1 >F(r) + F(x) dx > 1 Mi #A1 i +1 MI Mi Ml

since r(x) ? ,B. This is a contradiction and therefore M(t) > tiMi + l/3AM - 1

for O < t < oo when F(t) 4 1-- tlPI. 11 Acknowledgment. Igor Bazovsky, 1962, conjectured (2.5) and (2.6).

We are indebted to Albert W. Marshall and Ronald Pyke for help and advice. REFERENCES

BARLOW, R. E., and MARSHALL, A. W. (1963). Bounds on distributions with monotone hazard rates. Boeing Scientific Research Laboratories Document D1-82-0247.

BARLOW, R. E., MARSHALL, A. W., and PROSCHAN, F. (1962). Properties of probability distributions with monotone hazard rate. Ann. Math. Statist. 34 375-389.

BARLOW, R. E., and PROSCRAN, F. (1962). Planned replacement. Studies in Applied Proba- bility and Management Science, edited by K. J. Arrow, S. Karlin, and H. Scarf, Stanford Univ. Press, 63-87.

BAZOVSEY, I. (1962). Study of maintenance cost optimization and reliability of shipboard machinery. ONR Contract No. Nonr-37400 (00) (FBM), United Control Corp., Seattle, Washington.

CHOW, Y. S., and ROBBINS, H. A. (1963). A renewal theorem for random variables which are dependent or non-identically distributed. Ann. Math. Statist. 34 390-395.

Cox, D. R. (1962). Renewal Theory. Methuen, London: Wiley, New York. DRENIC}, R. F. (1960). Mathematical aspects of the reliability problem. J. Soc. Indust.

A ppl. Math. 8 125-149. FELLER, W. (1948). On Probability Problems in the Theory of Counters. Courant Anni-

versary Volume. Interscience, New York. FLEHINGER, B. J. (1962). A general model for the reliability analysis of systems under

various preventive maintenance policies. Ann. Math. Statist. 33 137-156. JORDAN, C. (1950). Calculus of Finite Differences. Chelsea, New York. SMITH, W. L. (1958). Renewal theory and its ramifications. J. Roy. Statist. Soc. Ser. B 20

243-302. SMITH, W. L. (1960). On some general renewal theorems for non-identically distributed

variables. Proc. Fourth Berkeley Symp. Math. Statist. Prob. II 467-514. SMITH, W. L. (1961). On necessary and sufficient conditions for the convergence of the re-

newal density. Trans. Amer. Math. Soc. 104 79-100. WEISS, G. (1956). On the theory of replacement of machinery with a random failure time.

Naval Res. Log. Quart. 3 279-293. WELKER, E. L. (1959). Relationship between equipment reliability, preventive maintenance

policy, and operating costs. Proc. Fifth Nat. Symp. on Rel. and Qual. Control. 270.


Article Contentsp.577p.578p.579p.580p.581p.582p.583p.584p.585p.586p.587p.588p.589

Issue Table of ContentsThe Annals of Mathematical Statistics, Vol. 35, No. 2 (Jun., 1964), pp. 475-948Front MatterDistributions of Matrix Variates and Latent Roots Derived from Normal Samples [pp.475-501]Limit Theorems for the Maximum Term in Stationary Sequences [pp.502-516]Randon-Nikodym Derivatives of Stationary Gaussian Measures [pp.517-531]On the Coefficient of Coherence for Weakly Stationary Stochastic Processes [pp.532-549]Some Structure Theorems for Stationary Probability Measures on Finite State Sequences [pp.550-556]Limit Distributions of a Branching Stochastic Process [pp.557-565]Some Theorems Concerning the Strong Law of Large Numbers for Non- Homogeneous Markov Chains [pp.566-576]Comparison of Replacement Policies, and Renewal Theory Implications [pp.577-589]A Continuous Kiefer-Wolfowitz Procedure for Random Processes [pp.590-599]On Pre-Emptive Resume Priority Queues [pp.600-612]Graphical Methods for Internal Comparisons in Multiresponse Experiments [pp.613-631]Interactions in Multidimensional Contingency Tables [pp.632-646]Maximum Likelihood Estimation with Incomplete Multivariate Data [pp.647-657]Applications of the Calculus for Factorial Arrangements II: Two Way Elimination of Heterogeneity [pp.658-672]Singular Weighing Designs [pp.673-680]Extended Group Divisible Partially Balanced Incomplete Block Designs [pp.681-695]Generalized Least Squares Estimators for Randomized Fractional Replication Designs [pp.696-704]Orthogonality in Analysis of Variance [pp.705-710]Estimates of Effects for Fractional Replicates [pp.711-715]Simultaneous Confidence Intervals for Contrasts Among Multinomial Populations [pp.716-725]Asymptotically Nonparametric Inference in some Linear Models with One Observation per Cell [pp.726-734]A Characterization of Multisample Distribution-Free Statistics [pp.735-738]Rankings from Paired Comparisons [pp.739-747]Asymptotic Distribution of Distances Between Order Statistics from Bivariate Populations [pp.748-754]A Property of Some Symmetric Two-Stage Sequential Procedures [pp.755-761]On Two-Sided Tolerance Intervals for a Normal Distribution [pp.762-772]The Linear Hypothesis and Large Sample Theory [pp.773-779]Confidence Region for a Linear Relation [pp.780-788]On the Admissibility of Some Tests of Manova [pp.789-794]Sufficiency in Sampling Theory [pp.795-808]Two Estimates of the Binomial Distribution [pp.809-816]A New Proof of the Pearson-Fisher Theorem [pp.817-824]A Bayesian Approach to Some Best Population Problems [pp.825-835]Invariant Prior Distributions [pp.836-845]Asymptotic Behavior of Bayes' Estimates [pp.846-856]Max-Min Probabilities in the Voting Paradox [pp.857-862]NotesMemoryless Strategies in Finite-Stage Dynamic Programming [pp.863-865]A Limit Theorem for Random Interval Sampling of a Stochastic Process [pp.866-868]Non-Singular Recurrent Markov Processes have Stationary Measures [pp.869-871]On Adding Independent Stochastic Processes [pp.872-873]The Dependence of Delays in Tandem Queues [pp.874-875]A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices [pp.876-879]A Note on Idempotent Matrices [pp.880-882]On the Line Graph of the Complete Bipartite Graph [pp.883-885]Bayesian Bio-Assay [pp.886-890]Significance Probability Bounds for Rank Orderings [pp.891-894]Pseudo-Inverses in the Analysis of Variance [pp.895-896]Balanced Designs with Unequal Numbers of Replicates [pp.897-899]On Double Sampling for PPS Estimation [pp.900-902]

Book Reviewsuntitled [pp.903-906]untitled [pp.907-910]untitled [pp.911-912]untitled [pp.913-914]untitled [pp.915-916]untitled [pp.917-919]untitled [pp.920-922]

Correction Notes and Acknowledgment of Priority: Corrections to "Some Extensions of the Wishart Distribution" [p.923]Correction Notes and Acknowledgment of Priority: Corrections to "The Non-Central Wishart Distribution and Certain Problems of Multivariate Statistics" [pp.923-924]Correction Notes and Acknowledgment of Priority: Corrections to: "Some Results on the Distribution of Two Random Matrices used in Classification Procedures" [p.924]Correction Notes and Acknowledgment of Priority: Correction to "Combinatorial Results in Fluctuation" [p.924]Correction Notes and Acknowledgment of Priority: Correction to "Estimating the Parameters of Negative Exponential Population from One or Two Order Statistics" [p.925]Correction Notes and Acknowledgment of Priority: Acknowledgment of Priority: Distribution of Sum of Identically Distributed Exponentially Correlated Gamma-Variables [p.925]Abstracts of Papers [pp.926-940]News and Notices [pp.941-948]Back Matter

to be able to simulate these frequency deviations in our simulation, a common oscillator model is...

Documents