Transient Behavior of Two-Machine Geometric Production Lines

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1 Trasiet Behavior of Two-Machie Geometric Productio Lies Semyo M. Meerkov Nahum Shimki Liag Zhag Departmet of Electrical Egieerig ad Computer Sciece Uiversity of Michiga, A Arbor, MI , USA ( smm@eecs.umich.edu, liagzh@eecs.umich.edu) Departmet of Electrical Egieerig Techio Israel Istitute of Techology, Haifa 32, Israel ( shimki@ee.techio.ac.il) Abstract: Productio systems trasiets describe the process of reachig the steady state throughput. Reducig trasiets duratio is importat i a umber of applicatios. This paper is iteded to aalyze trasiets i systems with machies obeyig the geometric reliability model. The Markov chai approach is used, ad the secod largest eigevalue of the trasitio matrices is utilized to characterize the trasiets. Due to large dimesioality of the trasitio matrices, oly two-machie systems are addressed, ad the secod largest eigevalue is ivestigated as a fuctio of the breakdow ad repair rates. Coditios uder which shorter, rather tha loger, upad dowtimes lead to faster trasiets are provided. Keywords: Productio lies; Geometric reliability model; Productio rate; Trasiet behavior; Effects of upad dowtime. INTRODUCTION Productio systems ofte operate i trasiet regimes. Examples iclude pait shops of automotive assembly plats, some buffers are emptied at the ed of each shift due to techological costraits; this leads to productio losses i the subsequet shift (util the buffer occupacy reaches its steady state). Aother example are machiig departmets operatig with so-called floats, additioal work-i-process is built up by slow machies after the ed of a shift i order to prevet starvatios of fast machies i the subsequet shift, leadig to icreased productio durig the trasiets. Clearly, to quatify the performace of these systems, a method for aalysis of their trasiets is ecessary. Ufortuately, the literature offers very few publicatios i this regard. Specifically, Narahari ad Viswaadham (994) study trasiets i oe-machie productio systems, usig the idea of Markov process absorptio time. Mocau (25) develops a algorithm for a umerical solutio of the partial differetial equatio, which describes the evolutio of the probability desity fuctio of a buffer with Markov-modulated iput ad output flows. The closest to the curret study is the paper by Meerkov ad Zhag (28), which studies trasiets of serial productio lies with machies obeyig the Beroulli reliability model. Accordig to this model, each machie, beig either starved or blocked, produces a part durig a cycle time with probability p ad fails to do so with probability p, irrespective of what had happeed i the previous cycle time. Thus, Beroulli machies are memoryless, which simplifies the aalysis of the resultig systems. While the Beroulli model is applicable to some assembly operatios, it does ot describe well may others, icludig machiig, heat treatmets, washig, etc. Thus, a extesio of the results reported by Meerkov ad Zhag (28) is ecessary. This is carried out i the curret paper for machies obeyig the geometric reliability model, which is applicable to the maufacturig operatios metioed above. Due to the complexity of the resultig mathematical descriptio, oly the case of two-machie systems is addressed; loger lies will be aalyzed i the future work. The outlie of this paper is follows: Sectio 2 presets the model ad the problem formulatio. I Sectio 3, trasiets of idividual machies are aalyzed. Sectios 4 ad 5 are devoted to two-machie lies with short ad log buffers, respectively. The coclusios ad future work are give i Sectio 6. All proofs ad umerical justificatios are icluded i the Appedix. 2. MODEL AND PROBLEM FORMULATION 2. Model We cosider a two-machie productio lie (see Figure 2.) defied by the followig assumptios: N P, R P, R m m 2 b Fig. 2.. Two-machie geometric lie (i) Both machies have a idetical cycle time, τ. The time axis is slotted with the slot duratio τ. The state

2 of each machie (up or dow) is determied at the begiig of each time slot. (ii) Both machies obey the geometric reliability model, i.e., if s() { = dow, = up} deotes the state of a machie at time slot, the trasitio probabilities are give by P [s( + ) = s() = ] = P, P [s( + ) = s() = ] = P, P [s( + ) = s() = ] = R, P [s( + ) = s() = ] = R, P ad R are referred to as the breakdow ad repair probabilities, respectively. (iii) The buffer is characterized by its capacity N <. The state of the buffer is determied at the ed of each time slot. (iv) Machie m is ever starved; it is blocked durig a time slot if it is up ad the buffer is full. (v) Machie m 2 is ever blocked; it is starved durig a time slot if it is up ad the buffer is empty. Note that these assumptios imply, i particular, that time depedet failures are addressed ad the blocked before service covetio is used; that is why N. Note also that the average upad dowtime of the machies are T up = /P ad T dow = /R ad the machie efficiecy is e = T up /(T up + T dow ). 2.2 Problems Give the above model, the productio system at had is described by a ergodic Markov chai. As it is well kow (Meerkov ad Zhag (28)), the trasiets of such a system are characterized by the secod largest eigevalue (SLE) of its trasitio matrix. With this i mid, the problems addressed i this paper are as follows: Aalyze the secod largest eigevalue of a idividual geometric machie as a fuctio of P ad R. I particular, ivestigate the effect of T up ad T dow o SLE, uder the assumptio that the machie efficiecy e is fixed. Carry out similar aalyses for two-machie lies. I additio, ivestigate explicitly the trasiets of the productio rate, P R(), i.e., the probability that m 2 is up ad the buffer is ot empty at time slot =, 2,.... Note that the steady state productio rate, P R( ) =: P R ss, of a productio lie defied by assumptios (i)- (v) ca be evaluated usig the method developed i Li ad Meerkov (23). Here we are iterested i how P R() approaches the steady state value P R ss. The iterest i the effect of T up ad T dow o the trasiets stems from the followig: It is well kow (see Li ad Meerkov (29)) that for a fixed e, shorter T up ad T dow lead to a larger P R ss tha loger oes; decreasig T dow by a give factor leads to a larger P R ss tha icreasig T up by the same factor. Do similar effects exist i the case of trasiets as well? I other words, do shorter T up ad T dow lead to faster trasiets tha loger oes? These ad other similar questios are aswered i this paper. 3. TRANSIENTS OF INDIVIDUAL MACHINES Let x i (), i {, }, be the probability that the machie is i state i durig time slot. The, the evolutio of the vector x() = [x () x ()] T ca be described by x( + ) = Ax(), x () + x () =, (3.) The eigevalues of A are [ R P A = R P λ =, λ = P R, ]. (3.2) ad, therefore, the dyamics of the machie states ca be expressed as x () = ( e) + [x () ( e)]( P R) ( = ( e) ) e λ, (3.3) x () = e + [x () e] ( P R) = e ( + e ) λ, (3.4) = x () e = ( e) x (). (3.5) To ivestigate the effects of upad dowtime o the trasiets, cosider λ as a fuctio of R for a fixed e, i.e., ( ) λ (R) = e R R = R e. The behavior of λ as a fuctio of R is illustrated i Figure 3.. From this figure, we coclude: For < R < e, loger upad dowtimes lead loger trasiets. For R = e, the machie has o trasiets. Such a machie ca be viewed as a Beroulli machie. For e < R <, The evolutio of the machie states is oscillatory (sice λ < ) ad, more importatly, shorter upad dowtimes lead to loger trasiets. λ e Fig. 3.. Behavior of λ as a fuctio of R R

3 Next, we address the issue of separate effects of uptime ad of dowtime o the trasiets. Recall that, as metioed i Sectio 2, icreasig the uptime by a factor +α, α >, or decreasig the dowtime by the same factor lead to the same steady state performace for a idividual machie sice e = + T. (3.6) dow (+α)t up However, the trasiet properties resultig from both cases are differet. Ideed, cosider a geometric machie with breakdow ad repair probabilities P ad R, respectively. Let λ u deote the SLE of the machie with the uptime icreased by (+α), α > ad λ d deote the SLE for the same machie with the dowtime decreased by the same factor. The, Theorem 3.. For a idividual geometric machie, if e >.5, λ u > λ d, (3.7) T dow + α > 2. (3.8) This theorem implies that if the machie efficiecy is larger tha.5 ad the decreased dowtime is larger tha two cycle times, decreasig the dowtime leads to faster trasiets tha icreasig the uptime, preservig the steady state productio rate i both cases the same. 4. TRANSIENTS OF 2-MACHINE LINES WITH N = For a serial lie with two geometric machies, the state of the system ca be deoted by a triple (h, s, s 2 ), h {,,..., N} is the state of the buffer ad s i {, }, i =, 2, are the states of the first ad the secod machie, respectively. The behavior of the system is described by a ergodic Markov chai. For N =, the trasitio probability matrix is: [ ] A A A = 2, (4.) A 3 A 4 ( R) 2 ( R) P ( R) R ( R) ( P ) A = R ( R) R P, R 2 R ( P ) ( R) P ( R) ( P ) A 2 = R P R ( P ) ( R) P P 2 ( R) 2 R P P ( P ) ( R) R A 3 = ( P ) ( R) ( P ) P R ( R), ( P ) R ( P ) 2 R 2 ( R) P P 2 R P P ( P ) A 4 = ( P ) ( R) ( P ) P ( P ) R ( P ) 2 ad s are zero-matrices of appropriate dimesioalities. The eight eigevalues of A are: [, P R, P R, ( P R) 2, ( R) 2,,, ]. (4.2) Clearly, the two eigevalues P R represet, as it follows from Sectio 3, the dyamics of the idividual machies; the eigevalue ( P R) 2 represets the trasiets of a pair of idividual machies (ote that the states of the machies i model (i)-(v) are determied idepedetly); therefore, the remaiig o-zero eigevalue ( R) 2 ca be viewed as describig the trasiets of the buffer. The last statemet is supported by the followig two argumets: First, usig the otatios λ m = P R, λ b = ( R) 2, the trasiets of the states, i.e., x h,i,j () = P [h() = h, s () = i, s 2 () = j], =,,..., ca be represeted as x h,i,j () = x h,i,j ( + Bλ b + Cλ m + D(λ 2 m) ), (4.3) h {, }, i, j {, }, =,, 2,..., x h,i,j = lim x h,i,j() ad B, C ad D are costats defied by iitial coditios. Theorem 4.. Cosider a serial lie with two idetical geometric machies ad N =. Assume that iitially the machies are i the steady states, i.e., P [s () = ] = P [s 2 () = ] = e. (4.4) The, i expressio (4.3), C = D =, i, j, h {, }. Thus, if the machies are i the steady states, the eigevalue ( R) 2 ideed characterizes the trasiets of the buffer. The secod argumet is as follows: Recall that if R = e, the machies ca be viewed as obeyig the Beroulli reliability model. I this case, the machies have o trasiets, ad the trasiets of the system are defied by λ b = ( e) 2, which, as it follows from Meerkov ad Zhag (28), is equivalet to the Beroulli case with p = e. From (4.2), it is ot immediately clear which of the eigevalues is the SLE. Obviously, the SLE ca be either P R or ( R) 2, i.e., either λ m or λ b. Which oe is, i fact, the SLE depeds o the relatioship betwee P

4 ad R. To ivestigate whe λ m or λ b is SLE, cosider the simplex < P < R < i the (P, R)-plae (see Figure 4.). Each poit (P, R) implies e >.5 ad each lie, P = kr, k <, represets a set of poits (P, R) with idetical efficiecy e = +k. Let λ deote the SLE, i.e., The, it ca be show that λ = max{ λ m, λ b }. { λm, if < P < R( R), λ = λ b, if R( R) < P < ( R)(2 R), (4.5) λ m, if (2 R)( R) < P <. This leads to the partitioig of the simplex accordig to SLE as show i Figure 4.. Thus, i area I, the trasiets of the system are defied by a idividual machie; i area II, the trasiets are defied by the buffer; i area III, the trasiets are agai defied by the machie, however, sice the eigevalue i this area is egative, the trasiets i area III are oscillatory. P P = R( R) P = ( R)(2 R) II. λ = ( R) 2 III. λ = P R e =.75.2 I. λ = P R R Fig. 4.. Partitioig of the simplex < P < R < accordig to SLE Next, we characterize the effects of shorter ad loger upad dowtimes o the duratio of trasiets. Theorem 4.2. Cosider a geometric lie with two idetical machies ad N =. The, for ay fixed e >.5, the SLE is a mootoically decreasig fuctio of R for R (,.5). Thus, for T dow > 2, shorter upad dowtimes lead to faster trasiets tha loger oes, eve if machie efficiecy e >.5 remais the same. This pheomeo is illustrated i Figure 4.2. PR()/PRss R = Fig Trasiets of P R for e =.9 I additio, the followig ca be obtaied regardig the effects of icreasig uptime or decreasig dowtime o system trasiets: Theorem 4.3. Cosider a geometric lie with two idetical machies ad N =. Let λ u ad λ d deote the SLEs resultig from icreasig the uptime by (+α), α >, ad decreasig its dowtime by the same factor, respectively. The, uder assumptio (3.8), λ u > λ d. (4.6) Thus, the qualitative effect of the uptime ad the dowtime o the trasiets i two-machie lies with N = remais the same as that for idividual machies: uder (3.8), it is better to reduce the dowtime tha icrease the uptime i order to shorte the trasiets. This pheomeo is illustrated i Figure 4.3. PR()/PRss Decreased dowtime Icreased uptime Fig Trasiets of P R with icreased uptime or decreased dowtime for e =.7, R =. ad e =.9 5. TRANSIENTS OF 2-MACHINE LINES WITH N 2 A direct aalytical ivestigatio of trasiets i twomachie geometric lies with N 2 is all but impossible due to high dimesioality of the resultig Markov trasitio matrices. Therefore, we resort to approximatios. Clearly, the dyamic behavior of the productio rate is give by P R() = P [buffer is ot empty at ]P [m 2 is up at ]. (5.) The secod term i the right had side of this expressio, as it follows from Sectio 3, is give by + e λ m, (5.2) is defied i (3.5). We approximate the first term by reducig the geometric lie to a Beroulli oe with the machies defied by ad the buffer capacity p Ber = R P + R (5.3) N Ber = [NR + ], (5.4) [x] deotes the earest iteger to x. For such a lie, P R Ber (), =,,..., ca be easily calculated (see Meerkov ad Zhag (28)). We use P R Ber () to approximate the first term i (5.) takig ito accout that oe time slot i the Beroulli lie is cosidered as oe dowtime i the origial geometric lie. I additio, sice i the Beroulli lie, the flows i ad out of the

5 buffer are statioary, we assume that the first machie of the geometric lie also reaches its steady state. This leads to the approximatio P R() = P R Ber ( T dow ) ( + e λ m) 2, (5.5) the additioal multiplier (+ e λ m) accouts for the trasiets of the first machie. The accuracy of (5.5) has bee ivestigated umerically usig 5, lies costructed by selectig the parameters radomly ad equiprobably from the followig sets: e [.6,.95], (5.6) R [.5,.5], (5.7) N {2, 3,..., 4}. (5.8) A typical example is show i Figure 5., the accuracy ɛ() is defied by ɛ() = P R() P R( ) P R() P R( ). (5.9) As oe ca see, the accuracy is sufficietly high. N = 5 N = N = 2 PR()/PRss PR()/PRss PR()/PRss P R()/P R ss.2 Geometric lie Beroulli lie Geometric lie Beroulli lie Geometric lie Beroulli lie 5 ɛ() ɛ() ɛ().5.5. ɛ() Fig. 5.. Illustratio of the accuracy of expressio (5.5) for e =.9 ad R =. Usig approximatio (5.5), the effects of upad dowtime o the trasiets ca be evaluated. Sice this is carried out umerically, we formulated the results as umerical facts. Numerical Fact 5.. Cosider a geometric lie with two idetical machies havig e >.5 ad N 2. The, for ay T dow > 2, shorter upad dowtimes lead, practically always, to faster trasiets tha loger oes. Numerical Fact 5.2. Uder coditio (3.8), reducig dowtime leads, practically always, to shorter trasiets tha icreasig uptime. As it is show i the justificatio of these umerical facts, the term practically always is quatified as 99% for Numerical Fact 5. ad 96% for Numerical Fact CONCLUSIONS AND FUTURE WORK This paper provides a characterizatio of trasiets i two-machie geometric productio lies. It is show that, i some cases, the system s trasiets ca be aalyzed by separatig the trasiets of the machies ad the trasiets of the buffer. Whe the buffer is of capacity, this separatio is exact; for loger buffers the separatio is approximate. I either case, it is show that if the machies efficiecy is greater tha.5 ad the average dowtime is larger tha two cycle times, shorter upad dowtimes lead to faster trasiets tha loger oes. Uder the same coditio, it is show that a reductio i dowtime leads to faster trasiets tha a similar icrease of the uptime. Future work will address trasiets i geometric lies with more tha two machies ad productio lies with other machie reliability models, e.g., expoetial, Weibull, logormal, etc. For o-markovia machies, the effect of the coefficiets of variatio of upad dowtime o the duratio of trasiets will be ivestigated. REFERENCES J. Li ad S. M. Meerkov. Due-time performace i productio systems with markovia machies. I S. B. Gershwi, Y. Dallery, C. T. Papadopolous, ad J.M. Smith, editors, Aalysis ad Modelig of Maufacturig Systems, chapter, pages Kluwer Academic, Bosto, MA, 23. J. Li ad S. M. Meerkov. Productio Systems Egieerig. Spriger, 29. S. M. Meerkov ad L. Zhag. Trasiet behavior of serial productio lies with beroulli machies. IIE Trasactios, 4(3):297 32, 28. S. Mocau. Numerical algorithms for trasiet aalysis of fluid queues. I Proceedigs of 5th Iteratioal Coferece o the Aalysis of Maufacturig Systems, pages 5 2, Zakymthos, Greece, 25. Y. Narahari ad N. Viswaadham. Trasiet aalysis of maufacturig systems performace. IEEE Trasactios o Robotics ad Automatio, (2):23 244, 994. Appedix A. PROOFS AND JUSTIFICATIONS Proof of Theorem 3.: It follows from (3.3) that λ u = P R, + α (A.) λ d = P ( + α)r. (A.2) Solvig iequalities λ u λ d > ad λ u λ d > results i λ u λ d >, if ( + α 2 )R < e, λ u λ d <, if ( + α 2 )R > e. It follows immediately from (3.8) that ( + α 2 )R < ( + α)r <.5 < e < e.

6 Thus, uder coditio (3.8), λ u > λ d. Proof of Theorem 4.: For matrix A give i (4.3), there exists a osigular matrix Q such that Thus, A = Q ÃQ, Ã = diag[ λ b λ m λ m λ 2 m ]. x( + ) = Ax() = Q ÃQx() = Q Ã Qx(), Ã = diag[ λ b λ m λ m (λ 2 m) ]. Hece, the evolutio of the states ca be expressed as (A.3) x 3 () = P 2 [R 2 ( e) RP e] ( R + P + R 2 ) (R + P ) 2 =, x 4 () = P 2 [R( e) P e] ( R + P + R 2 ) (R + P ) 2 =, x 5 () = R[2RP e( e) R2 ( e) 2 P 2 e 2 ] ( R + P + R 2 ) (R + P ) 2 =. Therefore, due to (A.6) ad (A.7), C = D =. Proof of Theorem 4.2: Sice P R ad ( R) 2 are both mootoically decreasig fuctios of R o (,.5) for a fixed e, the SLE of the system is a mootoically decreasig fuctio of R o (,.5). Proof of Theorem 4.3: It follows from Theorem 3. that x h,i,j () = x h,i,j [ + B x 2 ()λ b + ( C x 3 () + C 2 x 4 ())λ m + D x 5 ()(λ 2 m) ], I additio, λ u m > λ d m. (A.8) h {, }, i, j {, }, =, 2,..., (A.4) B, C, C2 ad D are costats, Thus, λ u b = ( R) 2 > [ ( + α)r] 2 = λ d b. x i () = q i x() ad q i is the i-th row of Q. The, it follows from (4.3) that C = C x 3 () + C 2 x 4 (), D = D x 5 (). For matrix Q, it ca be obtaied that [ q3 q 4 q 5 ] P 2 (A.5) (A.6) (A.7) = ( R + P + R 2 ) (R + P ) 2 R 2 RP R 2 RP R 2 RP R 2 RP R R P P R R P P R3 R 2 R 2 P 2 R R3 R 2 R 2 P P P 2 R P P Moreover, iitial coditio (4.4) implies that x h,,j () = x h,i, () = e, h,j h,i x h,,j () = x h,i, () = e. h,j h,i I additio, sice m ad m 2 are idepedet, x h,i,j () = 2e( e), h,i j x h,, () = ( e) 2, h x h,, () = e 2. h. λ u = max( λ u m, λ u b ) > max( λ d m, λ d b) = λ d. Justificatio of Numerical Fact 5.: This justificatio was carried out by evaluatig the settlig time of productio rate, t sp R, which is the time ecessary for P R to reach ad remai withi ±5% of its steady state value, provided that the buffer is iitially empty. A total of, lies were geerated with e ad N radomly ad equiprobably selected from the sets (5.6) ad (5.8), respectively. For each lie, thus costructed, t sp R is evaluated usig approximatio (5.5) as a fuctio of R. As a result, we obtaied that t sp R is a mootoically decreasig fuctio of R o R (,.5) i 99% of all cases studied. Thus, we coclude that shorter upad dowtimes lead, practically always, to faster trasiets, i.e., Numerical Fact 5. holds. Justificatio of Numerical Fact 5.2: To justify this umerical fact, the 5, lies geerated as metioed i Sectio 5 were used to ivestigate the effects of icreasig uptime or decreasig dowtime o t sp R. To accomplish this, we selected α radomly ad equiprobably from the set α {.5,.,..., } ad evaluated the settlig times t u sp R ad td sp R, resultig from icreasig uptime by (+α) ad decreasig dowtime by (+α), respectively. It tured out that t u sp R was loger tha t d sp R i 96.2% of all cases studied. For the remaiig 3.88% of cases, t u sp R was shorter tha td sp R by at most cycle time. Therefore, we coclude that Numerical Fact 5.2 takes place. Thus, uder (4.4),