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.