Properties of the Geometric and Related Processes
W. John Braun,1 Wei Li,2 Yiqiang Q. Zhao3
1 Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Canada N6A 5B7
2 Department of Electrical and Computer Engineering, The University of Toledo, Mail Stop 308, Toledo, Ohio 43606-3390
3 School of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6
Received 4 April 2001; revised 1 April 2005; accepted 28 April 2005 DOI 10.1002/nav.20099 Published online 14 July 2005 in Wiley InterScience (www.interscience.wiley.com).
Abstract: Some properties of the geometric process are studied along with those of a related process which we propose to call the ␣-series process. It is shown that the expected number of counts at an arbitrary time does not exist for the decreasing geometric process. The decreasing version of the ␣-series process does have a finite expected number of counts, under certain conditions. This process also has the same advantages of tractability as the geometric process; it exhibits some properties which may make it a useful complement to the increasing geometric process. In addition, it may be fit to observed data as easily as the geometric process. Applications in reliability and stochastic scheduling are considered in order to demonstrate the versatility of the alternative model. © 2005 Wiley Periodicals, Inc. Naval Research Logistics 52: 607–616, 2005.
Keywords: geometric process; reliability; renewal theory; birth processes; stochastic scheduling
1. INTRODUCTION Lam [6] has investigated many important theoretical properties of the geometric process. The process gives trac- table solutions for some reliability problems (Lam [5] and Consider a machine which operates for a random amount Lam and Zheng [8]). Given a set of data, statistical infer- of time before being repaired and then operates again until ence can be easily carried out for this process (Lam [7]). the next repair is required, continuing to alternate between Thus, the model is an attractive candidate for reliability periods of operation and repair. Under many circumstances, problems involving nonstationary processes. Furthermore, the times between repairs (uptimes) would seem to most no specific distributional assumptions other than some mo- appropriately be modeled as a stochastically decreasing ment conditions are required for the uptimes or downtimes, sequence of random variables (e.g., Ross [9]), since the so the model should have wide applicability. machine may not always be repaired as good as new. For The model proposed in this paper shares many of the similar reasons, the sequence of periods during which the advantages of the geometric process, including simplicity. machine is being repaired (downtimes) may best be mod- In the succeeding sections of this paper, we shall examine eled as a stochastically increasing sequence of random some of its theoretical properties, as well as its applicability variables. to reliability and scheduling problems. We shall also show In this paper, we consider two simple stochastic processes that statistical inference for this new process is as straight- for modeling such deteriorating repairable machines. Both forward as for the geometric process. processes are generalizations of the renewal process. The The outline of the paper is as follows. In the next section, first is the geometric process, proposed by Lam [6]. The we describe the two models. Section 3 is concerned with the second is a related process which we propose as a counter- existence and nonexistence of the first moment of each of part to the geometric process. the process counts. Section 4 is devoted to the special case of exponential uptimes or downtimes, and is meant to con- vey information about the qualitative behavior of the ex- Correspondence to: W.J. Braun ([email protected]) pected counts (N(t)) in each case. Section 5 deals with the
© 2005 Wiley Periodicals, Inc. 608 Naval Research Logistics, Vol. 52 (2005) problem of statistical inference—estimation of the process i Ն 1} is a stochastically decreasing sequence which can be ␣ Ͻ A Ն parameters from observed data. Section 6 describes how the used as a model for machine uptimes. If 0, {Xi , i expected values of N(t) can be numerically approximated 1} is stochastically increasing. In this case, this process can for each process. Section 7 describes two reliability and be used as a model for machine downtimes. scheduling applications. Section 8 summarizes our findings. It should also be noted that when the Yi’s are exponen- tially distributed and ␣ ϭ 1, NA(t) is a linear birth process and, when ␣ ϭ 0, NA(t) is a Poisson process. For other 2. THE TWO MODELS values of ␣, NA(t) is a birth process. ... Suppose Y1, Y2, is a sequence of independent and identically distributed random variables with distribution function F( y) and mean m Ͻϱ. 3. THE EXISTENCE OF E[NG(t)] AND E[NA(t)] The geometric process can be constructed as follows. Let The usefulness of the geometric process and its relatives Y in reliability and scheduling applications depends upon the G i X ϭ Ϫ , i ϭ 1, 2, 3, · · · first moment being finite. Many of the equations which i ai 1 result from such optimization problems contain terms in- volving this moment explicitly. Furthermore, numerical Ͼ for some a 0. If methods are usually required to calculate approximations for this moment; clearly, one should not attempt such a n calculation if the moment does not exist. Thus, it is impor- G ϭ G ϭ G S0 0 and Sn Xi , tant to establish easy-to-verify conditions for the existence iϭ1 or nonexistence of E[NG(t)] and E[NA(t)]. Lam [6] considered this problem for the geometric pro- then cess and derived the following analogue of the renewal equation, G͑ ͒ ϭ ͕ G Յ ͖͑Ն ͒ N t sup n : Sn t t 0
t is defined as a geometric process with parameter a. E͓NG͑t͔͒ ϭ F͑t͒ ϩ ͵ E͓NG͑a͑t Ϫ x͔͒͒ dF͑x͒, (1) When a Ͼ 1, Lam [5] observed that the XG’s form a i 0 stochastically decreasing sequence, and thus proposed NG(t) with a Ͼ 1 as a possible model for the number of Ͻ G and gave conditions under which a unique solution machine uptimes by time t. When a 1, the Xi ’s are G stochastically increasing. In this case, Lam proposed NG(t) E[N (t)] to this equation exists. Verifying these conditions as a model for downtimes. When a ϭ 1, NG(t) is a renewal amounts to checking whether a sequence of approximate process. solutions is uniformly bounded in any finite interval. Lam G A related process can be constructed as follows. Let concluded that E[N (t)] is finite when a Յ 1, and he gave an example in which these conditions are not satisfied and Ͼ Y the moment is not finite in the case where a 1. A i X ϭ ␣ , i ϭ 1, 2, 3, . . . , An additional necessary condition for the finiteness of i i E[N(t)] when a Յ 1isF(0) Ͻ 1. Furthermore, Lam’s example actually typifies the behavior of E[NG(t)] when ␣ where is a real-valued parameter. Set a Ͼ 1 as we now show.
n THEOREM 1: If a Ͼ 1 and F() Ͼ 0 for all Ͼ 0, then A ϭ A ϭ A G S0 0 and Sn Xi , E[N (t)] is infinite for all t Ͼ 0. iϭ1 PROOF: and define the point process E͓Y ͔ m A͑ ͒ ϭ ͕ A Յ ͖ G i N t sup n : Sn t . E͓X ͔ ϭ Ϫ ϭ Ϫ . i ai 1 ai 1 We shall call the process NA(t)an␣-series process. ␣ ϭ A ␣ Ͼ A When 0, N (t) is a renewal process. If 0, {Xi , Therefore, Braun, Li, and Zhao: Geometric and Related Processes 609
ϱ when 0 Ͻ x Ͻ . Therefore, ma G ϭ Ͻ ϱ Eͫ Xi ͬ , a Ϫ 1 iϭ1 E͓NG͑t͔͒ ϭ ϱ
Ͼ Ͼ and there exists 0 such that, for any n, for all t t1/a. Iterating this argument n times, we can see that
n ϱ am am E͓NG͑t͔͒ ϭ ϱ Pͩ XG Յ ͪ Ն Pͩ XG Յ ͪ Ͼ. i Ϫ i Ϫ ϭ a 1 ϭ a 1 i 1 i 1 Ͼ n Ͼ n for all t t1/a . Since a 1, we can make t1/a arbitrarily small. ᮀ Since REMARK 1: If F() ϭ 0 for some Ͼ 0, it is possible G n for E[N (t)] to be finite for small t, but the above proof can G Ͼ NG͑t͒ Ն n iff XG Յ t be modified to show that E[N (t)] is infinite for all t , i Ͼ iϭ1 where is such that F( ) 0.
REMARK 2: The condition F() Ͼ 0 holds for most we have frequently occurring distributions in reliability applications: for example, exponential, gamma, and Weibull. am G PͩN ͩ ͪ Ն nͪ Ͼ (2) A a Ϫ 1 For the ␣-series process, E[N (t)] Ͻϱin the increasing case, as well as in the decreasing case, provided that the random variable Y has a sufficient number of moments. for all n. Hence, THEOREM 2: Suppose F(0) Ͻ 1. am EͫNGͩ ͪͬ ϭ ϱ a Ϫ 1 1. If ␣ Ͻ 0, then E[NA(t)] is finite for all t Ͼ 0. Յ ␣ Յ ͉ ͉2ϩ␦ Ͻϱ ␦ Ͼ 2. If 0 1/2 and E[ Yi ] for some A Ͼ and 0, then E[N (t)] is finite for all t 0. Ͻ ␣ Ͻ (1ϩ␦)/(1Ϫ␣) Ͻϱ 3. If 1/2 1, and E[Yi ] for some ␦ Ͼ 0, then E[NA(t)] is finite for all t Ͼ 0. E͓NG͑t͔͒ ϭ ϱ ␣ ϭ 4. If 1, and all moments of Yi exist such that, for some fixed ␦ Ͼ 0, Ն ϭ Ϫ ϭ ϩ Ͻ for all t t1 am/(a 1). Set t2 t1/a , where 0 Յ n t1/a. Then, ͉ Ϫ ͉͑ ϩ␦͒ E͓e Yi m 1 /i͔ iϭ1
t2 E͓NG͑t ͔͒ ϭ ͵ E͓NG͑a͑t Ϫ x͔͒͒ dF͑x͒ ϩ F͑t ͒ A Ͼ 2 2 2 is bounded, then E[N (t)] is finite for all t 0. 0 5. If ␣ Ͼ 1, and F() Ͼ 0 for some Ͼ 0, then E[NA(t)] is infinite for all large enough t. t1/a Ն ͵ E͓NG͑t ϩ a Ϫ ax͔͒ dF͑x͒ ϩ F͑t ͒ 1 2 PROOF: 0
1. ␣ Ͻ 0: G A n ␣ A Ն ͵ E͓N ͑t ϩ a Ϫ ax͔͒ dF͑x͒ ϩ F͑t ͒ ϭ ϱ S ϭ ¥ ϭ Y /i Ͼ ¥ ϭ Y ,soN (t) Ͻ N(t), 1 2 n i 1 i i 1 i 0 where N(t) is an ordinary renewal process. Since E[N(t)] Ͻϱ(e.g., [4]), E[NA(t)] Ͻϱ. 2. 0 Յ ␣ Յ 1/2: since Take p ϭ 2 ϩ ␦, where ␦ Ͼ 0. Using an inequality of Burkholder (e.g., (3.3.14) of Ϫ ϩ Ͼ t1 ax a t1 Stout [11], p. 154), we have, for some constant C, 610 Naval Research Logistics, Vol. 52 (2005)
p Now, Doob’s inequality can be applied to yield the n m n ͉Y Ϫ m͉p ͯ A Ϫ ͯ Յ p/2Ϫ1 i new upper bound Eͫ Sn ␣ ͬ Cn Eͫ p␣ ͬ i i iϭ1 iϭ1 n 1 1 Ϫ ␣ ͑1ϩ␦͒/͑1Ϫ␣͒ ϭ O͑nϪp/2ϩ1Ϫ␣p͒. ͓͉ ͉͑1ϩ␦͒/͑1Ϫ␣͔͒ ͩ ͪ C sup E Mn 1ϩ␦ Յ Ͻϱ n m 1 n iϭ1 For large enough n, ¥n m/i␣ Ͼ t, since F(0) Ͻ 1 iϭ1 ϩ o͑n1Ϫ␣͒, ensures that m is positive. For such n, where C is a constant. This is finite (and hence, A n m n m E[N (t)] is also finite) if ͑ A Ͻ ͒ Յ ͯ A Ϫ ͯ Ͼ Ϫ P S t Pͩ S ␣ ␣ tͪ n n i i iϭ1 iϭ1 ͑1ϩ␦͒/͑1Ϫ␣͒ E͓͉Mn͉ ͔ Ͻ ϱ. (3) m p ͯ A Ϫ n ͯ ͫ ␣ ͬ To demonstrate this, we invoke Rosenthal’s in- E Sn iϭ1 i Յ ϭ O͑nϪp/2͒ equality m p n ͩ Ϫ ͪ ␣ t p/2 iϭ1 i n n E͓͉S ͉p͔ Յ C ͩ E͓X2͔ͪ ϩ C E͓͉X ͉p͔, n p i p i by Markov’s inequality. Finally, iϭ1 iϭ1
ϱ ϱ which holds for some constant Cp, when X1, E͓NA͑t͔͒ ϭ P͑NA͑t͒ Ն n͒ ϭ P͑SA Ͻ t͒ Ͻ ϱ. X2,..., Xn are independent random variables n ϭ Ն nϭ1 nϭ1 with E[Xi] 0, and p 1. Applying this to our ϭ ␣ Ϫ ␣ ϭ ϩ 3. 1/2 Ͻ ␣ Ͻ 1: problem, we take Xi Yi/i m/i , and p (1 ϭ A ␦)/(1 Ϫ ␣). Then we see that For this case, we make use of the fact that Mn Sn n ␣ Ϫ ¥ ϭ m/i is an example of a martingale. Spe- i 1 ͑1ϩ␦͒/͓2͑1Ϫ␣͔͒ cifically, as for the case when ␣ Յ 1/2, we have n 2 p E͓͉M ͉ ͔ Յ C ͩ ␣ͪ n p i2 ϱ ϱ iϭ1 1 A A E͓N ͑t͔͒ ϭ P͑S Յ t͒ Յ P͉ͩM ͉ Ն m ␣ Ϫ tͪ n n i n ͑1ϩ␦͒/͑1Ϫ␣͒ ϭ ϭ E͓͉Yi Ϫ m͉ ͔ n 1 i 1 ϩ C . p i͓␣͑1ϩ␦͔͒/͑1Ϫ␣͒ iϭ1 n mn1Ϫ␣ ϭ P͉ͩM ͉ Ն ͪ ϩ o͑n1Ϫ␣͒ n 1 Ϫ ␣ ␣ Ϫ ␣ Ͼ iϭ1 Since /(1 ) 1, both terms above involve convergent series as n 3 ϱ. Thus, (3) holds. ͑1ϩ␦͒/͑1Ϫ␣͒ n m 4. ␣ ϭ 1: ϭ P͉ͩM ͉͑1ϩ␦͒/͑1Ϫ␣͒ Ն n1ϩ␦ͩ ͪ ͪ n 1 Ϫ ␣ Using Mn as defined in the previous case, we have iϭ1 that for some ␦ Ͼ 0, ϩ o͑n1Ϫ␣͒ ϱ ͓ ͑ ͔͒ ϭ ͉͑ ͉ Ͼ ͑ ͒͒ for any fixed ␦ Ͼ 0. Markov’s inequality can be E N t P Mn log n applied to the last expression obtained, yielding the nϭ1 upper bound ϱ ϱ ͉ ͉͑ ϩ␦͒ ϩ␦ 1 ͉ ͉͑ ϩ␦͒ ϭ ͑ Mn 1 Ͼ 1 ͒ Յ ͓ Mn 1 ͔ P e n ␦ϩ1 E e . ͑ ϩ␦͒ ͑ Ϫ␣͒ n n 1 1 Ϫ ␣ 1 / 1 nϭ1 nϭ1 ͓͉ ͉͑1ϩ␦͒/͑1Ϫ␣͔͒ ͩ ͪ E M ϩ␦ n n1 m iϭ1 ͟n This last expression is finite whenever iϭ1 ϩ␦ ͉ Ϫ ͉ E[e(1 ) Yi m ] is bounded. n 1 ␣ Ͼ ϩ ͑ 1Ϫ␣͒ Յ ͓ ͉ ͉͑1ϩ␦͒/͑1Ϫ␣͔͒ 5. 1: o n E sup Mn 1ϩ␦ 1ՅnϽϱ n iϭ1 ϱ ϱ E͓Sn͔ ͑1ϩ␦͒/͑1Ϫ␣͒ ͓ A͑ ͔͒ ϭ ͑ Ͻ ͒ Ͼ ͩ Ϫ ͪ 1 Ϫ ␣ E N t P Sn t 1 (4) 1Ϫ␣ t ϫ ͩ ͪ ϩ o͑n ͒. nϭ1 nϭ1 m Braun, Li, and Zhao: Geometric and Related Processes 611
ϭ ¥n by Markov’s inequality. Since E[Sn] m iϭ1 1 ␣ E͓aNG͑t͔͒ Յ 1/i , we can ensure that the series in (4) diverges ͑1 Ϫ a͒t if we take t Ͼ m ¥ϱ 1/i␣. ᮀ ϩ 1 iϭ1 m REMARK 3: The moment condition in part 4 holds, for which, together with (6), gives example, when Yi is exponentially distributed. d 1 E͓NG͑t͔͒ Յ . 4. QUALITATIVE BEHAVIOR OF THE dt ͑1 Ϫ a͒t ϩ m PROCESSES—EXPONENTIAL CASE Ͻ Again, this can be solved, with E[NG(0)] ϭ 0, to give the THEOREM 3: If a 1 and Y1 is exponentially distrib- uted with mean m, then right-hand inequality of (5). Similar arguing gives the left-hand side of (5): ͑1 Ϫ a͒t logͩ ͪ 1 a ͑1 Ϫ a͒t m ϩ 1 NG͑t͒ E͓a ͔ Ն G logͩ ϩ 1ͪ Յ E͓NG͑t͔͒ Յ . ͓ ϪN ͑t͔͒ ͑1 Ϫ a͒ ma 1 Ϫ a E a
(5) which, together with (6) and (8), gives
PROOF: For j ϭ 0, 1, 2, . . . , define p (t) ϭ P(NG(t) ϭ j d 1 j). Then, a standard argument leads to E͓NG͑t͔͒ Ն . dt ͑1 Ϫ a͒t ϩ m 1 a pЈ͑t͒ ϭ Ϫ p ͑t͒ 0 m 0 This, together with E[NG(0)] ϭ 0 gives the result. We can also obtain upper and lower bounds for E[NA(t)]. and ϭ A ϭ ϭ Define pj(t) P(N (t) j) for j 0, 1, 2, . . . . The usual arguments for continuous-time Markov chains lead to aj ajϪ1 Ј͑ ͒ ϭ Ϫ ͑ ͒ ϩ ͑ ͒͑Ն ͒ p j t pj t pjϪ1 t j 1 , m m ͑j ϩ 1͒␣ j␣ pЈ͑t͒ ϭ Ϫ p ͑t͒ ϩ p Ϫ ͑t͒, j ϭ 0,1,2,.... j m j m j 1 from which it easily follows that (9) d 1 E͓NG͑t͔͒ ϭ E͓aNG͑t͔͒, (6) Multiplying through by j, and summing gives dt m d 1 d ͑1 Ϫ a͒ E͓NA͑t͔͒ ϭ E͓͑NA͑t͒ ϩ 1͒␣͔. (10) E͓aNG͑t͔͒ ϭ Ϫ E͓a2NG͑t͔͒, (7) dt m dt m For 0 Յ ␣ Յ 1, we have and
d 1 ␣ d ͑1 Ϫ a͒ E͓NA͑t͔͒ Յ ͑E͓NA͑t͔͒ ϩ 1͒ . E͓aϪNG͑t͔͒ ϭ . (8) dt m dt ma Solving this differential inequality with E[NA(0)] ϭ 0 2NG(t) 2 NG(t) Since E[a ] Ն E [a ], (7) gives gives
͑ Ϫ ͒ 1/͑1Ϫ␣͒ d G͑ ͒ 1 a G͑ ͒ ͑mt͑1 Ϫ ␣͒ ϩ 1͒ Ϫ 1, if ␣ Ͻ 1, E͓aN t ͔ Յ Ϫ E2͓aN t ͔. E͓NA͑t͔͒ Յ ͭ dt m emt Ϫ 1, if ␣ ϭ 1.
Solving this differential inequality, together with the initial To obtain a lower bound, we multiply (9) through by j␣ G condition E[aN (0)] ϭ 1, we have and sum to obtain 612 Naval Research Logistics, Vol. 52 (2005) d ␣͑␣ ϩ 1͒ E͓NA͑t͒␣͔ ϭ E͓͑NA͑t͒ ϩ 1͒␣NA͑t͒␣Ϫ1͔ dt 2m ␣͑␣ ϩ 1͒ Ն E͓NA͑t͒2␣Ϫ1͔. 2m
If ␣ Յ 1/2, then we have
d ␣͑␣ ϩ 1͒ E͓NA␣͑t͔͒ Ն ͑E͓NA͑t͒␣͔͒2Ϫ1/␣. dt 2m
This can be solved together with the condition E[NA␣(0)] ϭ 0 to give
t͑1 Ϫ ␣2͒ ␣/͑1Ϫ␣͒ E͓NA͑t͒␣͔ Ն ͩ ͪ . 2m
From (10), we have
d 1 E͓NA͑t͔͒ Ն E͓NA͑t͒␣͔, dt m which, with E[NA(0)] ϭ 0, gives
Figure 1. Least-squares fits of geometric and ␣-series models to 1 Ϫ ␣ 1 Ϫ ␣2 ␣/͑1Ϫ␣͒ E͓NA͑t͔͒ Ն ͩ ͪ t1/͑1Ϫ␣͒. software failure data. m 2m
Ͻ ␣ Յ If 1/2 1, then similar arguing, together with so that the least-squares estimate of the line relating log(Xi) Jensen’s inequality, leads to to i would lead to estimates of log(a) and hence a. The estimation procedure for the ␣-series process is very d ␣͑␣ ϩ 1͒ similar to that taken for the geometric process. The basic E͓NA͑t͒␣͔ Ն ͑E͓NA͑t͔͒͒2␣Ϫ1, dt 2m idea is as follows. If observed data is X1, X2,...,Xn, then
ϭ Ϫ ␣ ϭ  Ϫ ␣ ϩ which, together with (10), gives log Xn log X0 log n log n ,
d2 ␣͑␣ ϩ 1͒ where X is a random variable with distribution F( x), and ͓ A͑ ͔͒ Ն ͑ ͓ A͑ ͔͒͒2␣Ϫ1 0 2 E N t 2 E N t . ϭ  dt 2m E[log X0] . Linear least-squares can again be used to estimate  and ␣. The solution to this is For an example of data where a stochastically increasing process might be appropriate, we consider the software ͑ Ϫ␣͒ failure data given in Crowder et al. [2]. In fitting the geo- 1 Ϫ ␣ 1 ϩ ␣ 1/ 1 E͓NA͑t͔͒ Ն ͫ ͱ ͬ t1/͑1Ϫ␣͒. ᮀ metric process, we find that the estimated slope and inter- m 2 cept are .0650 (standard error ϭ .02, p-value ϭ .005) and 1.00, respectively. The data and fitted line are exhibited in 5. PARAMETER ESTIMATION the top half of Figure 1. An estimate of a can be obtained by exponentiating We next consider the problem of estimating model pa- Ϫ.065 which gives us aˆ ϭ .937. Thus, we are modeling the rameters as did Lam [7] for the geometric process model. data with a stochastically increasing geometric process. Lam noted that If the ␣-series model is fit, the intercept is found to be not significant ( p-value ϭ .22). The slope is .761 (s.e. .08, Ϫ10 log͑Xi͒ ϭ log͑Yi͒ Ϫ ͑i Ϫ 1͒log͑a͒ p-value ϭ 8 ϫ 10 ) after dropping the intercept from the Braun, Li, and Zhao: Geometric and Related Processes 613
6. NUMERICAL CALCULATION OF E[NG(t)] AND E[NA(t)]
When a Յ 1, there are few cases in which E[NG(t)] can be computed analytically. Therefore, an approximation technique is needed. For the renewal case (a ϭ 1), this problem has been well-studied, and a stable solution method is given by Xie [12]. Chaudhry [1] has also given a useful method for solving (1) when a ϭ 1. These methods will not work directly when a Ͻ 1. However, an adaptation of the method of collocation (e.g., de Boor [3]), which is used for solving differential equations is one possible method of solution. In contrast to the geometric process where a renewal equation analogue is available, it does not seem to be possible to obtain an integral equation whose solution is exactly E[NA(t)]. The following inequalities can be ob-
tained by conditioning on the value of Y1.
THEOREM 4: If E[NA(t)] exists, then
t E͓NA͑t͔͒ Ն F͑t͒ ϩ ͵ E͓NA͑t Ϫ x͔͒ dF͑x͒ dx 0 Figure 2. Least-squares fits of geometric and ␣-series models to air-conditioning failure data. and model. The data and fitted line are exhibited in the bottom half of Figure 1. The estimate of ␣ is ␣ˆ ϭϪ.761 which t A A ␣ corresponds to a stochastically increasing ␣-series process. E͓N ͑t͔͒ Յ F͑t͒ ϩ ͵ E͓N ͑2 ͑t Ϫ x͔͒͒ dF͑x͒ dx. Figure 2 relates to an example where a stochastically 0 decreasing process might be appropriate: the air condition- ing failure data given in Crowder et al. [2]. When fitting the The solutions of these inequalities provide upper and geometric model, we obtain slope and intercept estimates of lower bounds on E[NA(t)]. Again, a collocation method Ϫ.0566 and 4.58, respectively, leading to the estimate aˆ ϭ may be the most appropriate approach. 1.06. This corresponds to the decreasing geometric process. Alternatively, to obtain an approximation to E[NA(t)], We note that the standard error of the slope estimate is .033, one could appeal to the following idea. Define a new family corresponding to a p-value of .11. Thus, there is only one of processes, indexed by k, using weak evidence that the process is truly decreasing. How- ever, as pointed out in Section 3, if the process is actually decreasing, the estimated expected number of failures in any n Y Y time interval is infinite, which does not seem reasonable. i ϭ k ϩ A Ն ␣ ␣ S ϩ , n k, ␣ SA ϭ i k k 1,n Fitting the -series process, we obtain slope and intercept k,n Ά iϭk estimates of Ϫ.0404 and 4.87, respectively, leading to ␣ˆ ϭ 0, n Ͻ k, .0404. Again, this corresponds to a stochastically decreasing process. This time, though, the expected number of failures in any finite time interval is finite as long as it is believable and that the 2 ϩ ␦ moment of the interfailure time random ␦ Ͼ variable exists, for some 0. Again, the p-value for the NA͑t͒ ϭ sup͕n : SA Յ t͖. slope is too large (standard error ϭ .28, p-value ϭ .16) to k k,n state conclusively that the process is truly a decreasing process. Conditioning on Yk leads to: 614 Naval Research Logistics, Vol. 52 (2005)
A ϭ THEOREM 5: If E[Nk (t)] exists, then for k 1, 2, . . . , Let Xi denote the amount of time that the machine oper- ates before its ith failure, and let NG(t) denote the corre- tk␣ y sponding geometric process; since the machine is deterio- ͓ A͑ ͔͒ ϭ ͫ A ͩ Ϫ ͪͬ ͑ ͒ ϩ ͑ ␣͒ rating, we assume that the process is decreasing with pa- E N t ͵ E N ϩ t ␣ dF y F tk . (11) k k 1 k ϭ 0 rameter a 1/b. The expected working age of the machine at time T is then given by A Ϫ ␣ If the integrand above is replaced by E[Nk (t y/k )], then A NG͑T͒ NG͑T͒ an approximation to E[Nk (t)] can be obtained. The quality 1 Ϫ E͓b ͔ m E X ϭ mͩ ͪ Յ . ͫ iͬ of this approximation depends upon the condition 1 Ϫ b 1 Ϫ b iϭ1
A A ͉E͓N ͑s͔͒ Ϫ E͓N ϩ ͑s͔͉͒ ϭ lim sup k k 1 0. (12) Ͼ k3ϱ sՅt If the repair times Ri have mean mR 0 and are indepen- dent and identically distributed and independent of NG(t), When this condition holds, the following algorithm for then it follows from Theorem 1 and the subsequent remark approximating E[NA(t)] is certain to converge. (The ap- that the expected length of a replacement cycle is infinite. proximation to E[NA(t)] is denoted by Eˆ [NA(t)].) The expected cost of a cycle is given by
1. Solve NG͑T͒ NG͑T͒ Ϫ ϩ ␥ ϩ ␥ ϭ ␥ Ϫ ϩ ␥ ␣ Eͫ T Riͬ R R T Eͫ Riͬ, tk y ͓ A͑ ͔͒ ϭ ˆ ͫ Aͩ Ϫ ͪͬ ͑ ͒ ϩ ͑ ␣͒ iϭ1 iϭ1 E N t ͵ E N t ␣ dF y F tk k k k 0 ␥ ␥ where is the repair cost rate, and R is the replacement ˆ A ʦ ␣ cost. By Theorem 3, this quantity must also be infinite. for E[Nk (s)], for all s [0, tk ]. 2. For i ϭ k, k Ϫ 1,..., 2,set Thus, the decreasing geometric model is not suitable if we desire an optimal T policy. This has already been pointed out by Stadje and Zuckermann [10], but for different rea- ti␣ y ˆ ͓ A ͑ ͔͒ ϭ ˆ ͫ Aͩ Ϫ ͪͬ ͑ ͒ ϩ ͑ ␣͒ sons. E N Ϫ t ͵ E N t ␣ dF y F ti . i 1 i i We next consider this optimal T policy using the ␣-series 0 model. We assume that the repair periods are i.i.d., and we want to choose T to minimize A ˆ A The estimate for E[N (t)] is given by E[N1 (t)]. A simple bounding argument shows that A E͓cost͔ ϪT ϩ ␥E͓N ͑T͔͒E͓Ri͔ ϩ ␥R ϭ , E͓cycle length͔ T ϩ E͓NA͑T͔͒E͓R ͔ ͉ ˆ ͓ A͑ ͔͒ Ϫ ͓ A͑ ͔͉͒ Յ ͉ ͓ A͑ ͔͒ i sup E N1 t E N t sup E Nk s sՅt sՅt where Ri is the length of the ith repair period. Under the Ϫ ͓ A ͑ ͉͔͒ ͑ ␣ ͒ ͑͑ Ϫ ͒␣ ͒ ͑ ͒ E Nkϩ1 s F k t F k 1 t ···F t . conditions of Theorem 3.2, this has a nontrivial solution. This solution can be obtained numerically, using the meth- This converges to 0 as k 3 ϱ under condition (12). ods of Section 5. The second problem we wish to discuss concerns the scheduling of a set of M independent jobs (with mean 7. RELIABILITY AND SCHEDULING processing times m ) on a deteriorating machine so as to APPLICATIONS i minimize the expected flow time (see Ross [9]). If the
This section contains a brief discussion of some reliabil- machine uptimes Yi follow a decreasing geometric process, ity and scheduling applications. An optimal T policy and a and downtimes are independent and identically distributed
flow-time minimization problem are described in which the with mean mD, then the expected flow time is decreasing geometric and ␣-series processes are used to model machine uptimes. M M M i First, we consider the problem of operating a deteriorat- E͓C ͔ ϭ ͑M Ϫ j ϩ 1͒m ϩ m E NAͩ Y ͪ , i j D ͫ j ͬ ing machine which is to be replaced when its total operating iϭ1 jϭ1 iϭ1 jϭ1 time reaches a prespecified level. This is the kind of prob- lem (policy T) addressed by Lam [5]. We will show that the where Ci is the completion time of job i. By Theorem 1, the geometric model may not be appropriate, but the ␣-series expected flow time is infinite. Thus, the geometric model is, model can be used here. again, not suitable. Braun, Li, and Zhao: Geometric and Related Processes 615
We next consider this scheduling problem using the ␣-se- process only allows for logarithmic growth or explosive ries model: Minimize the expected flow time in case of M growth, but nothing in between. independent jobs, Y1, Y2,...,YM with means mi. Machine Statistical inference can be done for both processes uptimes follow the decreasing ␣-series process, and down- equally easily, using a least-squares approach. We have times are i.i.d. with mean mD. demonstrated how both the increasing and decreasing mod- The expected flow time is els (for both types of processes) fit actual data. As a further demonstration of the tractability of the pro-
M M M i cesses, we have suggested algorithms for computing numer- A G E͓C ͔ ϭ ͑M Ϫ j ϩ 1͒m ϩ m E NAͩ Y ͪ ical approximations for E[N (t)] and E[N (t)] where these i j D ͫ j ͬ iϭ1 jϭ1 iϭ1 jϭ1 moments exist. Specifically, it is possible to obtain numer- ical approximations as solutions of integral equations or systems of integral equations. In both cases, the computa- and tions are not prohibitive. These approximations find appli- cation in formulas arising in reliability and scheduling ap- 1/͑1Ϫ␣͒ plications. ͩi ͪ i Yj As for the case of renewal processes, asymptotic approx- jϭ1 NAͩ Y ͪЏ͑1 Ϫ ␣͒ . imations to NA(t) and E[NA(t)] can also be obtained. These j m jϭ1 will be the subject of another paper. We have demonstrated the applicability of the decreasing ␣-series process in situations where the geometric process Therefore, minimizing expected flowtime involves the Ϫ ␣ may not be as useful because of the nonexistence of 1/(1 ) moments of the job processing times. In cases G 1/(1Ϫ␣) E[N (t)]. We have attempted to give some scope to the where E[Y ] increases with mj, the expected flow Ͻ Ͻ ...Ͻ variety of applications in reliability and scheduling for this time is minimized when m1 m2 mM. alternative model, but there are many potential applications that we have not addressed in this paper. 8. CONCLUDING REMARKS
The geometric process, NG(t), is an interesting and po- ACKNOWLEDGMENTS tentially useful model for deteriorating machines. It is trac- table and easy to use. In this paper, we have introduced an This work was supported by research grants from the alternative model, the ␣-series process NA(t), which shares Natural Sciences and Engineering Research Council of Can- these characteristics. The two processes may be viewed as ada (NSERC). Wei Li would also like to acknowledge that complementary to one another. For example, the increasing his work was partially supported by the National Natural geometric process may be appropriate for modeling ma- Science Foundation of China (NNSFC) with Contract No. chine downtimes, while the ␣-series process can be used for 60474067. The authors are grateful to Hao Yu for assistance modeling machine uptimes. with the proof of Theorem 3.2, and they acknowledge the Both the increasing geometric and the increasing ␣-series very useful comments of an associate editor as well as two processes have a finite first moment under fairly general referees, which led to a substantial improvement in the conditions. However, the decreasing geometric process usu- paper. ally has an infinite first moment. The decreasing ␣-series process has a finite first moment under fairly general con- ditions, if ␣ Յ 1. Thus, for modeling successive uptimes, REFERENCES the decreasing ␣-series process will have wider applicability than the decreasing geometric process. [1] M.L. Chaudhry, On computations of the mean and variance To gain some appreciation for the qualitative behavior of of the number of renewals: A unified approach, J Oper Res both processes, we studied the exponential case in some Soc 46 (1995), 1342–1364. detail and showed that the increasing geometric process [2] M.J. Crowder, A.C. Kimber, R.L. Smith, and T.J. Sweeting, exhibits logarithmic growth, while the decreasing ␣-series Statistical analysis of reliability data, Chapman and Hall, London, 1991. process exhibits power law and exponential growth, de- ␣ Ͻ ␣ ϭ [3] C. de Boor, A practical guide to splines, Springer, Berlin, pending upon whether 1or 1. This is a further 1978. indication as to how the ␣-series process provides a comple- [4] W. Feller, An introduction to probability theory and its mentarity aspect: In the exponential case, the geometric applications, Wiley, New York, 1971, Vol. II. 616 Naval Research Logistics, Vol. 52 (2005)
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