AN INTERMITTENTLY USED SYSTEM WITH PREVENTIVE MAINTENANCE

Transcription

1 J. Operatins Research Sc. f Japan V!. 15, N. 2, June The Operatins Research Sciety f Japan AN INTERMITTENTLY USED SYSTEM WITH PREVENTIVE MAINTENANCE SHUNJI OSAKI University f Suthern Califrnia (Received August 2, 1971 and Revised Nvember 5, 1971) Abstract An intermittently used system with preventive maintenance is discussed. The Laplace-Stieltjes transfrm f the distributin and the mean time f the disappintment time are derived by using Markv renewal prcesses. The effect f the preventive maintenance plicy is als discussed. 1. Intrductin A system can be classified int ne f tw types accrding as whether it is always used r it is used intermittently. In this paper the reliability analysis f an intermittently used system is discussed. In particular, a mdel with preventive maintenance is discussed. Gaver [2] defined the cncept f reliability fr an intermittently used system, which is called the "disappintment time." That is, the disappintment time is the time f system failure during a usage perid, r f ccurrence f a demand during a system inperative perid, whichever ccurs first. We nte that a system is alternatively perative and inperative (r under repair). Gaver [2J btained the Laplace-Stieltjes (LS) transfrm f the distributin f the disappintment time and the mean f the disappintment time fr a ne-unit system. In this paper we shall discuss a ne-unit system with preventive maintenance. The 102

2 An Intermittently Used System 103 LS transfrm f the distributin f the disappintment time and the mean time f the disappintment time will be derived fr a ne-unit system with preventive maintenance. The effect f preventive maint"r:ance is discussed. In the subsequent discussins we shall cnsider tw mdels, Mdel I and Mdel II, in which the preventive mainter:ance plicies are different frm each ther. 2. Mdel I Cnsider a ne-unit system in wllich the,;ystem is alternatin~ly perative and under repair. The failure time f the unit is assumed t bey sme distributin F(t), and the repair time f the unit anther distributin G(t). Each switch ver is assumed t be instantaneolb. A unit functins perfectly upn rep3.ir. Cnsider the preventive maintenance plicy fr the unit. The time f the beginning f the preventive maintenance, measured frm :Le instant that the unit begins t be perative, is assumed t bey distributin A (t), i.e., we cnsider a randm preventive maintenance plicy. The hlding time f the preventive maintenance (r the preventin: repair time) is assumed t bey distributin B(t). We assume that the preventive maintenance time is stchastically shrter than the repair time, i.e., B(t»G(t). A unit functins perfectly upn preventive maintcl~al1ce. Finally cnsider the behavir f ccurrence f a need (r a use). The ccurrence time f a need is assumed t bey an expnential distributin a(t)=l-exp(-:\t), and the hlding time f a need a distril:utin P(t). Then we define the prbability P(t) that the system is nt used at time t, given that the system was nt used at t=o. "e have (1) P(t) = 1:: [a(t)*,b(t)]"* * ii(t) = [l-a(t)*,b(t)j(-l) * ii(t), n=o where we define that, in general, [y(t)j* = E(t) (i.e., a unit fuilctin), 00 [y(t)j"*= [y(t)j<"-ll* *y(t) (n;;:::i), [1- y(t)](-ll= 2: [y(tw*, and ),(t) = 11=0 l-y(t}. Further, the prbability P(t) that the system is used at time t, given that the system was nt llsed at t=o, is given by Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

3 104 Shunji Osaki (2) 1'(t) = :E [a(t) * {1(t)]"* * a(t) * p{t),,-0 = [1- a(t) * {1(t)] (-1) * a(t) * Mt). If we assume that {1 (t) = 1-exp( -,ut), we have (3) P(t) = _,u_ + _A_ e-(ah)/. A+,u A+,u Fr ur mdel, we define the fllwing fur states (which are the regeneratin pints except state S3): State s: The unit begins t be perative and the need des nt ccur. State S1: The unit begins t get repaired, and the need des nt ccur. State S2: The preventive ma'intenance begins, and the need des nt ccur. State S3: The disappintment time, the time f system failure during a usage perid, r f ccurrence f a demand during a system inperative perid, whichever ccurs first. The state transitin diagram (which crrespnds t the signal flw graph) fr ur mdel is shwn in Fig. 1. We shall btain each branch gain in Fig. 1. SO ':_-----'D$3 Fig The state transitin diagram. Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

4 An Intermittently Used System 105 Frm state S, three transitins can be cnsidered: The transitin t state SI is an event that the system fails befre the preventive maintenance, and at that time the system is nt used, the transitin t state S2 is an event that the preventive maintenance begins befre the system failure, and at that time the system is nt used, and the transitin t state S3 is an event that the system fails r the preventive maintenance begins, and at that time the system is used. Thus we have the fllwing three branch gains: (-! ) qol(s) = r e- st P(t) A(t) df(t), (5) Q02(S) = re-si P(t) F(t) da(t), r (6) Q03(S) = e- st P(t) A(tl df(t) + re-si P(t) F(t) da (t). 0 Frm state S1> tw transitins can be cnsidered: The transitin t state S is an event that the repair is cmpleted befre the ccurrence f a need. The transitin t state S3 is an event that a need ccurs befre the cmpletin f repair. r Thus we have (7) Ql0(S) = e- st e-i<t dg(t) = O(s+)..), (8) 00 _ ).. QI3(S) = J e- sl G(t)).. e- A1 dt = --[1-0(s+)..)], s+).. where O(s) dentes the LS stransfrm f G(t). Frm state S2' tw transitins can be cnsidered. techniques III (7) and (8), we have Using the similar (9) Q20(S) = re-si e-i<1 db(t) = 11(s+)..), Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

5 106 Shunji Osaki (10) 00 A q23(s) = J e- s / B(t) A e.j../ dt = -- [I-~(s+A)]. S+A Defining that state S is a surce and state S3 is a sink in Fig. 1, and applying Masn's gain frmula [3J, we pave the fllwing system gain: (11) IPO(s) = q03(s) + qol(s) Q13(S) + q02(s) Q2J(S), 1- qol(s) Ql0(S)- Q02(S) Q20(S) which is the LS transfrm f the distributin f the disappintment time. The mean time f the disappintment time is given by using Masn's gain frmula [3J: (12) T = ~+ Q01(0) ~1 + Q02(0) ~2 1- Q01(O)Ql0(0)- Q02(O)Q20(0), where each branch gain %(s) is replaced by Q;j() except that Q;3(S) (i= 0, 1, 2) is replaced by ~;, the uncnditinal mean in state Si, i.e., (13) 3 dq;j(s) I ~;=- L: --. j-i ds 5-0. In ur mdel, each ~; (i=0, 1, 2) is given by (14) ~ = J t A(t) df(t) + J t F (t) da (t), 0 (15) ~1 = [1-G(A)]/A., (16) ~2 = [I-~(~)]/A.. In particular, if we assume that (17) A (t) = { ~ fr t < t fr t?:. t, i.e., if we assume the cnstant time preventive maintenance plicy, we have Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

6 '0 (18) ~ = J F(t) dt, An InteTmittenUy Used System 107 and the mean time T in (12) is a functin f t. In the remaining part f this sectin, we shall discuss the effect f the preventive maintenance plicy. T simplify the discussin we cnsider the equilibrium alternating renewal prcess instead f the alternating renewal prcess. COX [IJ, p. 85 discussed three pssibilities t btain the equilibrium alternating renewal prcess frm the alternating renewal prcess. If the behavir f the needs beys the equilibrium alternating renewal prcess, we have (19) P(t) = _fl_ A,+fl (20) - A, P(t) =-- A,+fl which are independent f time t. We shall discuss the effect f the preventive maintenance plicy assuming the equilibrium alternating renewal prcess fr the behavir f the needs. First cnsider the mean time f the disappintment time given in (12). Using (19) and (20), we have (21) where P=fl!(A,+fl). Next cnsider the mean time f the disappintment time withut preventive maintenance. As t----> in (21), we have (22) E(X) + P 1-~(A,) ;\, T = --I--p--;:"a-(A,-) where Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

7 108 Shunji Osaki (23) E(X) = r tdf(t) = r F(t) dt. 0 Cmparing tw equatins (21) and (22), we have Therem. If (i) B(t»G(t), (ii) the failure rate r(t) f the failure time distributin F(t) is strictly increasing, and (iii) (24) 1-pG(),.) 1'= ( ) >, p[e(),.)-g(),.)] ~E(X)-~-J L ",+p, then we can adpt a suitable interval length t and the mean time (21) fr the mdel with preventive maintenance is greater than that f (22) fr the mdel withut preventive maintenance. Prf. We shall shw that (21) is greater than (22) under the abve assumptins. It is clear that the denminatrs f (21) and (22) are psitive. Then the numeratr f subtracting (22) frm (21) is given by t [J I-G(),.) l-e(),.) ] (25) N(t) = F(t) dt+ PF(t) X + PF(t) X We have x [1- pg(a.)] - [1-PF(t) G(A.)- PF(t) E(A.)] X [E(X)+P 1-~(X) J. (26) N(O)=- P(l;:P) [E(A.)-G()")]-[I-PE(A.}]E(X) <0, (27) N(=) = 0. Differentiating N(t) with respect t t, we have (28) dnd (t) = F (t) [1- pc(x)] t + pj(t) [E(A.)- G(X)] [_1_ - E(X)], )"+p, Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

8 An Intermittently Used "System 109 where we assume that F(t) has a density!(t). Nting the failure rate r(t)=j(t)jf(t) and setting zer in (28), we have a t* such that (29) r(t*) = 1--.:-p=-G--'(i\-=-) p [~(i\) - G(;\,)] [E(X) - i\!,u ] Using the assumptins that B(t»G(t), r(t) is an increasing functin f t, the inequality (24), and N(O)<w and N(w) =0, we can shw that N(t) is a unimdal functin and there exists a i such that N(i) =. Thus, if we chse a t>i, we have N(t»O, which cmpletes the prf. 3. Mdel 11 In the previus mdel we assumed that the preventive maintenance is made thugh the system is during a usage perid, which causes the disappintment time. In this sectin, t avid the disappintment time caused by the preventive maintenance, we cnsider a new mdel that the preventive maintenance is nt made while the system is during a usage perid. That is, we assume that the preventive maintenance is nly made while the system is nt in a usage perid. The states fr this new mdel are the same in the preceding sectin, and the signal flw graph fr this new mdel is the same as Fig. 1. We shall, hwever, btain each branch gain. Fr states S1 and S2' the branch gains are the same as given by (7) (10). Cnsider the branch gains frm state S t states S1' S2' and sa. The transitin frm state S t state S1 is an event that (i) the system fails befre the preventive maintenance and at that ~ime the system is nt in a usage perid, r (ii) the preventive maintenance is nt made because the system is in a usage perid and that the system fails while the system is nt in a usage perid. Thus, we have (30) Q01(S} = re-si P(t) A(t} df(t} 00 I + I e- s ' P(t} U P(t} da (t) ] df(t).. 0 Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

9 110 Shunji Osaki If we cnsider a similar event just mentined abve except that the system fails while the system is in a usage perid, we have the branch gain t state S3: (31) Q03(S) = re-si P(t) A(t) df(t) 00 t + J e- sl P(t) [J P(t) da (t) ] df(t). 0 The branch gain t state S2 is the same ne in (5). Thus we have all branch gains fr this new mdel. The LS transfrm f the distributin t the disappintment time is given in (11) and the mean time t the disappintment time is als given in (12), where branch gains q;(s) (i= 1,3) are given in (30) and (31), respectively, and the uncnditinal mean ~ in state S is given in (13). In particular, if we cnsider the cnstant time preventive maintenance plicy in (17), we have (32) ~= J F(t) dt+p(t) J F(t) dt. t. \Ve can btain a similar therem that the preventive mair,tenance plicy is effective in the sense f the mean time. \Ve, hwever, mit the frm f therem because the result is cmplicated. 4. Cnclusin We have discussed an intermittently used system with preventive maintenance, and derived the LS transfrm f the distributin f the disappintment time and the mean time f the disappintment time. We have further shwn the effect f the preventive maintenance plicy. We believe that the preventive maintenance plicy is imprtant in actual situatins. We have described ur mdel in a reliability cntext. We can, hwever, describe the same mdel in traffic, scheduling, and ther cn- Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.

10 An Intermittently Used System 111 texts. S, ur mdel is applicable in sme fields. The wrk reprted in this article was supprted by the Natinal Institutes f Health under Grant K. GM References [ 1] Cx, D.R., Renewal Thery, Methuen, Lndn, [2] Gaver, D.P., "A Prbability Prblem Arising in Reliability and Traffic Studies," OPeratins Research, 11 (1964) [3] Osaki, S., "System Reliability Analysis by Markv Renewal Prcesses," J. OPns. Res. Sc. Japan, 12 (1970), Cpyright by ORSJ. Unauthrized reprductin f this article is prhibited.