Heuristics for allocation of reconfigurable resources in a serial line with reliability considerations

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1 IIE Transactions ISSN: X (Print) (Online) Journal homepage: Heuristics for allocation of reconfigurable resources in a serial line with reliability consierations Cheng-Hung Wu, Douglas G. Down & Mark E. Lewis To cite this article: Cheng-Hung Wu, Douglas G. Down & Mark E. Lewis (2008) Heuristics for allocation of reconfigurable resources in a serial line with reliability consierations, IIE Transactions, 40:6, , DOI: / To link to this article: Publishe online: 07 Apr Submit your article to this journal Article views: 117 View relate articles Citing articles: 12 View citing articles Full Terms & Conitions of access an use can be foun at Downloa by: [ ] Date: 10 July 2016, At: 04:06

2 IIE Transactions (2008) 40, Copyright C IIE ISSN: X print / online DOI: / Heuristics for allocation of reconfigurable resources in a serial line with reliability consierations CHENG-HUNG WU 1, DOUGLAS G. DOWN 2 an MARK E. LEWIS 3, 1 Institute of Inustrial Engineering, National Taiwan University, 1, Section 4, Roosevelt R, Taipei 106, Taiwan wuchn@ntu.eu.tw 2 Department of Computing an Software, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canaa L8S 4L7 own@mcmaster.ca 3 School of Operations Research an Information Engineering, Cornell University, 226 Rhoes Hall, Ithaca, NY 14853, USA mel47@cornell.eu Receive August 2005 an accepte August 2007 We consier the allocation of reconfigurable resources in a serial line with machine failures. Each station is equippe with non-iling eicate servers while the whole system is equippe with a finite number of reconfigurable servers. The reconfigurable servers are available to be assigne to any station an all servers are allowe to collaborate on a single job. We provie conitions for a policy to achieve throughput optimality. We also show in the two-station case that transition monotone optimal policies exist. We iscuss heuristics base on the two-server moel that reuce average holing costs significantly. These heuristics are compare to several heuristics from the literature via a etaile numerical stuy. Keywors: Markov ecision processes, reliability, stochastic scheuling theory 1. Introuction Due to rapily changing market emans an the popularity of customize proucts, many companies have investe in Reconfigurable Manufacturing Systems (RMSs) (DeGaspari, 2002; Narongwanich et al., 2003). That is, they have evelope systems in which some of the capacity can be configure in a relatively short perio of time to hanle several ifferent job types. To allow for this, the material hanling system in the RMS is usually capable of routing Work In Process (WIP). This makes ynamic resource allocation (or ynamic job re-routing) easily implementable without aitional costs. On the other han, while this may be the motivation for the evelopment of RMSs, there is another ae benefit; the alleviation of congestion ue to machine failures. A typical metho of ealing with machine failures is to buil enough reunancy into the system so as to have excess capacity at each station. Dynamic allocation of resources allows factory managers to reuce recovery costs by using the reconfigurable resources to cover for faile servers in the case of a machine breakown (see Freiheit et al. (2004), Anraóttir et al. (2007) an Wu et al. (2006)). Corresponing author Unfortunately, ue to the curse of imensionality most of the stuy of optimal allocation policies has been restricte to systems with only a few stations (usually two) an a few machines. In the particular case where reliability is consiere uner the average holing cost criterion the system is even less tractable. In orer to overcome the imensionality restriction, we prove that transition monotone policies are average cost optimal in a two-station tanem queueing system with reliability consierations. Base on the intuition obtaine from the two-station queueing network, heuristic resource allocation policies are evelope. This heuristic algorithm significantly reuces calculations an accoring to our simulation experiments performs very well uner both the long-run average holing costs an long-term average throughput criteria. Along the way we provie conitions for the stability of the network an optimality in terms of average throughput. By an large the literature on ynamic job re-routing or resource allocation in manufacturing systems focuses on reucing holing costs or increasing throughput without consiering machine reliability. One major approach to minimizing holing costs is clearing system analysis. The goal is, given a fixe number of jobs in the system (a prouction scheule), how might one empty the system at minimum cost. Uner this assumption, Farrar (1993) shows the existence of optimal policies that are transition monotone X C 2008 IIE

3 596 Wu et al. (monotone in the number of jobs at the secon station) in a two-station tanem queueing system with a flexible server. Ahn et al. (1999) proves that allocating both flexible servers to a single station is optimal in a two-station queueing system with two ientical flexible servers. Schiefermayr an Weichbol (2005) provie a complete solution for the optimal server scheuling policy for a moel very similar to that in Ahn et al. (1999). In two-station systems with reliability consierations an multiple eicate an reconfigurable servers, Wu et al. (2006) proves the existence of optimal transition monotone policies. With aitional assumptions, they also prove that the optimal switching curves that efine the optimal transition monotone policy shoul have slopes of at least 1. When external arrivals are allowe, Hajek (1984) shows the existence of transition monotone policies in a queueing network with two flexible servers. Ahn et al. (2002) shows that the result in Ahn et al. (1999) hols in a system with external arrivals uner an average cost criterion. Sennott et al. (2006) examine a moel with several workers eicate to stations an one floating worker an iscuss how the floating worker can be use to stabilize the system an to minimize holing costs. They allow for switching times, but o not consier failures. For a more etaile literature review of flexible server allocation problems reaers may wish to consult Hopp an Van Oyen (2004). In aition to minimizing holing costs, several papers have stuie the resource allocation problem to maximize throughput. Freiheit et al. (2004) use static allocation policies to examine server an buffer capacity allocation problems an try to improve system throughput. When all servers are reliable, Anraóttir et al. (2001) an Anraóttir an Ayhan (2005) examine tanem queueing systems with finite buffers. Anraóttir et al. (2007) stuy the ynamic resource allocation problem with both class an machine reliability consierations. They show that the maximum throughput is tightly boune by the solution of a linear program. They also prove that a time generalize roun-robin policy can approximate the maximum capacity. In the current work we stuy the problem with reliability consierations uner the minimum average holing cost criterion. Moreover, we provie conitions that guarantee throughput optimality in the infinite-buffer case. It turns out that the heuristic we propose achieves a throughput close to maximum an can be slightly moifie (as iscusse) to achieve this maximum. The remainer of the paper is organize as follows. A mathematical escription of the queueing moel an stability conitions are provie in Section 2. In Section 3, we give conitions for policies to guarantee throughput optimality. We use Markov ecision processes to prove the optimality of a transition monotone policy in two-station queueing networks with reliability consierations in Section 4. Using this result, a heuristic for larger systems (more than two stations) is provie in Section 5. Robustness of the heuristic iscusse in Section 5 is evaluate by a iscrete event simulation in Section 6. The heuristic is also compare to several other policies from the literature. We conclue the paper in Section Preliminaries We assume that N operations (at N stations) must be performe in a fixe orer on every job or customer that enters the system. Job inter-arrival times are assume to be exponentially istribute with rate λ. The service requirement for each job is exponentially istribute with a mean of one at each station. For the kth station of the system, there are M k eicate non-iling servers that can only serve customers at that station. There are M r aitional generalize (Anraóttir et al., 2007) reconfigurable servers that have constant service rates at all stations. These reconfigurable servers can be assigne to any station with zero setup time an costs (see Fig. 1). Throughout the rest of the paper, when consiering an N-station system, we label the station that receives arrivals from outsie of the system as station 1 an the others sequentially with the last one before exiting the system being labele station N. We say station n is upstream (ownstream) from station n if n < n (n > n ). Let (r,l),l {1, 2,...,M r } be the lth reconfigurable server an (k,l) inicate the lth server at station k, where k {1, 2,...,N} an l {1, 2,...,M k }. We assume that the service rate of eicate server (k,l)isμ k,l an the reconfigurable server (r,l) has service rate μ r,k,l at station k. The failure an repair times are assume to be exponentially istribute with rates α k,l (α r,l ) an β k,l (β r,l )for eicate server (k,l) (reconfigurable server (r,l)). When a server failure occurs, the service rate of the corresponing server is reuce to zero. If the number of servers is more than the number of jobs at any station, multiple servers can collaborate on a single job; the service rates are aitive. This assumption is usual when large components are being prouce an more than one machine can be assigne to a single prouct. Since the only ifference between this assumption an the case when servers are not allowe to collaborate is when the system is lightly loae, we believe that this serves as a reasonable approximation in either case. A brief numerical stuy to this effect is provie in Section 6.1. The state space is S ={q 1, q 2,...,q N, m 1,1, m 1,2,...,m 1,M1, m 2,1,..., m N 1,MN 1, m N,1,...,m N,MN, m r,1,...,m r,mr }, where q k is the queue length at station k, incluing the one currently in service, m k,l {0 : faile, 1 : operational} is the status of server (k,l), an m r,l is the status of reconfigurable server l. Let be the set of all non-iling, non-anticipating (measurable) policies. Each policy escribes where the reconfigurable resources shoul be allocate for all states, for all time (own servers are not assigne). We next provie conitions uner which we can guarantee the existence of a

4 Allocation of reconfigurable resources 597 N Server (1,1) faile Server (2,1) N Server (N,1) Server (1,2) Server (2,2) faile Server (N,2) Dynamic Reconfigurable Resources Allocation Server (r,1) Server (r,2) Server (r,3) Fig. 1. An example of an N-station serial line with reconfigurable servers. policy that achieves stability. This will be useful in both the throughput an average cost analysis to follow. In essence, it is an application of the results presente in Anraóttir et al. (2007). Note that the availability of each server can be moele as a two-state (on-off) continuous-time Markov chain. Let a k,l be the long-run proportion of time that server (k,l)is operational an δr,n,l π be the proportion of time that reconfigurable server (r,l) is assigne to station n for a policy π. That is α k,l a k,l = α k,l + β k,l δ π r,n,l = lim sup T 1 T T 0 1 {π assigns (r,l) to station n} (t)t, where 1 A (t) is the inicator of event A at time t. Consier the following Linear Program (LP), which we call LP(1): max λ M k s.t. a k,l μ k,l + δ r,k,l μ r,k,l λ for all k {1, 2,...,N}, (1) N δ r,k,l a r,l for all l {1, 2,...,M r } (2) k=1 δ r,k,l 0 for all k {0, 1,...,N} an l {1, 2,...,M r }. The above LP fins the maximal arrival rate, λ, such that the average service rate is greater than or equal to λ (the first constraint) an guarantees that the total proportion of time each reconfigurable server is allocate oes not excee its availability (the secon constraint). Define the following optimality criteria. Definition 1. Let D π (t) enote the number of epartures from the system by time t uner policy π. The throughput of policy π is efine by lim t D π (t)/t. Definition 2. Suppose for each job at station k, there is a linear holing cost accrue at a fixe rate h k. The expecte average cost of policy π is efine by lim t t N 0 n=1 h neq π n (s)s, t where Q π n (s) represents the (ranom) queue length of station n at time s uner policy π. The remainer of this paper is evote to searching for optimal policies uner these two criteria. In the latter case, we pose heuristics that perform very well uner the throughput criterion while approximating the average cost case for any finite number of stations. 3. Throughput optimality of general scheuling policies In this section we show several general results on throughput optimality. In particular, we show that it is not possible to achieve a throughput larger than λ uner very general conitions. Moreover, if the reconfigurable servers are such that μ r,k,l = γ r,l μ r,k for some strictly positive γ r,l an μ r,k, we show that if a scheuling policy satisfies a conition on the allocation when the queues are large it is throughput optimal. This service rate structure inclues the important

5 598 Wu et al. special cases where the reconfigurable servers have service rates that are either inepenent of the server or inepenent of the station. Since for most of this section we consier a fixe policy, we suppress the superscript π. We first introuce some ieas from Dai (1999). Let Q k (t) be the queue length at station k at time t. Also, efine T k,l (t) to be the amount of time that eicate server l at station k has been busy up until time t an T r,k,l (t) to be the amount of time that reconfigurable server l has been busy at station k up until time t. Suppose {x n, n 0} is a sequence of nonnegative real numbers such that lim n x n =.Ifwelet a flui limit be efine by Q k (t) = lim x 1 n n Q k(x n t), T k,l (t) = lim x 1 n n T k,l(x n t), T r,k,l (t) = lim T r,k,l(x n t), x 1 n n then as in Section 3.2 of Anraóttir et al. (2007), we have that any flui limit satisfies the following set of conitions, calle the flui moel: M 1 Q 1 (t) = Q 1 (0) + λt μ 1,l T 1,l (t) μ r,1,l T r,1,l (t), M k 1 M k Q k (t) = Q k (0) + μ k 1,l T k 1,l (t) μ k,l T k,l (t) + μ r,k 1,l T r,k 1,l (t) (3) μ r,k,l T r,k,l (t), 2 k N, (4) such that Q k (t) 0, 1 k N, T k,l (0) = 0, 1 l M k, 1 k N, T r,k,l (0) = 0, 1 l M r, 1 k N, 0 t T k,l (t) a k,l, 1 l M k, 1 k N, t T k,l (t) = a k,l whenever Q k (t)>0, 1 l M k,1 k N, N 0 t T r,k,l (t) a r,l, 1 l M r, k=1 an T k,l ( ), T r,k,lr ( ) are non-ecreasing for 1 l M k, 1 k N, 1 l r M r. The erivatives above exist almost everywhere, as T k,l (t) an T r,k,l (t) are Lipschitz (note that T k,l (t) T k,l (s) t s ). From this point on, erivatives will be unerstoo to be taken on the conition that they exist. Note that the equations above o not completely specify T r,k,l (t). Any aitional conitions woul refine the flui moel. We introuce the following notions of stability for the flui moel (see Dai (1999)). The flui moel is sai to be weakly stable if for each solution to the flui moel with Q k (0) = 0, Q k (t) = 0fort 0. Conversely, the flui moel is sai to be weakly unstable if there exists an s > 0 such that for every solution to the flui moel with Q k (0) = 0, N Q k=1 k (s) 0. The first result shows that any arrival rate greater than λ results in an unstable system, no matter what the scheuling policy. Proposition 1. If λ>λ, with probability one, N k=1 Q k(t) as t. Consequently, the long-run average cost is infinite. Proof. By contraiction. Suppose λ>λ, an that Q k (s) = 0 for all 1 k N an all s > 0. To prove the result we construct a solution (using λ) to LP(1); this contraicts the efinition of λ. Since the eicate servers can only be busy while they are available, set T k,l (s) = a k,l s. Suppose we set T r,k,l (s) = δ r,k,l s, then Equations (3) an (4) imply (after iviing through by s): M 1 0 = λ μ 1,l a 1,l μ r,1,l δ r,1,l, (5) 0 = M k 1 M k μ k 1,l a k 1,l μ k,l a k,l + μ r,k 1,l δ r,k 1,l μ r,k,l δ r,k,l, (6) with the constraint that: N 0 δ r,k,l a r,l. (7) k=1 The fact that Equations (5) to (7) have a solution for λ> λ contraicts the efinition of λ, as Equations (5) to (7) provie a feasible solution for LP(1). Thus, the flui moel is weakly unstable an the result follows from Theorem of Dai (1999). Now, let all of the reconfigurable servers be such that μ r,k,l = γ r,l μ r,k for all l = 1, 2,...,M r an k = 1,...,N. Consier the following assumption. Assumption (LQ): t T r,k,l (t) = a r,l for 1 l M r. k: Q k (t)>0 Assumption (LQ) implies that when some stations have long queues the reconfigurable servers are assigne to serve at one (or some) of the highly loae stations. It may seem that this assumption is ifficult to check, however, we will provie conitions that can be impose on any policy that ensure that (LQ) hols. Before oing so, we show that a policy satisfying (LQ) achieves throughput optimality. To o this, we efine a policy π to be rate stable (see Dai (1999)) if

6 Allocation of reconfigurable resources 599 lim t D π (t)/t = λ for λ<λ. The notion of rate stability is weaker than stability in the usual sense since no result about the existence of a stationary istribution is obtaine. However, since λ is the highest throughput rate that can be hope for, policies satisfying (LQ) are calle throughput optimal. Proposition 2. If there exists strictly positive γ r,l an μ r,k such that μ r,k,l = γ r,l μ r,k for all l = 1, 2,...,M r an k = 1,...,N, an the (reconfigurable) server scheuling policy π satisfies (LQ), then π is rate stable. Proof. If we efine x r,k,l = δ r,k,l γ r,l, LP(1) may be rewritten as max λ, M k s.t. a k,l μ k,l /μ r,k + x r,k,l λ/μ r,k for all k {1, 2,...,N}, N x r,n,l a r,l /γ r,l for all l {1, 2,...,M r }, n=1 x r,k,l 0 for all k {0, 1,...,N} an l {1, 2,...,M r }. Call this LP(x). Now we turn to the flui moel. Differentiating Equations (3) an (4) an evaluating at time 0 yiels: t Q 1 (t) t=0 = λ M 1 μ 1,l t T 1,l (t) t=0 γ r,l μ r,1 t T r,1,l (t) t=0, (8) t Q M k 1 k (t) t=0 = μ k 1,l t T k 1,l (t) t=0 M k μ k,l t T k,l (t) t=0 + γ r,l μ r,k 1 t T r,k 1,l (t) t=0 γ r,l μ r,k t T r,k,l (t) t=0. (9) Let U be the set of stations such that: t Q k (t) t=0 > 0fork U. We first show by contraiction that U must be empty. Assume U is non-empty. We have by (LQ) that: t T r,k,l (t) t=0 = a r,l. k U Also, for each fixe eicate server l: t T k,l (t) t=0 = a k,l for any k U. By the efinition of U, for all k U we have that the instantaneous arrival rate to station k must be greater than the instantaneous service rate (at time 0). Also, λ is an upper boun on the instantaneous arrival rate to station k (at time 0). In other wors, we have for k U: M k μ k,l t T k,l (t) t=0 + γ r,l μ r,k t T r,k,l (t) t=0 M k = μ k,l a k,l + γ r,l μ r,k t T r,k,l (t) t=0 <λ λ. (10) By the efinition of λ, this combination is not possible as the inequality (10) cannot hol for all k U. Inee if this were the case, letting γ r,l T r,k,l (t) t=0 /t play the role of x r,k,l in LP(x) woul imply that the solution to LP(x) woul be less than λ. As a result, our assumption on U must be incorrect an U is empty. This means that for k = 1,...,N, /t Q k (t) t=0 = 0. Since there is nothing special about time 0, we can use the above argument for any time t. It follows that Q k (t) = 0 for all t, an in turn the flui moel is weakly stable. The result now follows from Corollary of Dai (1999). Note that (LQ) is not sufficient to guarantee throughput optimality for a more general rate structure, as seen by the following example. There are two stations, two reconfigurable servers, no fixe servers an the reconfigurable servers are not subject to failure. Let μ r,1,1 = μ r,2,2 = 1 an μ r,1,2 = μ r,2,1 = 2. Suppose that server i gives priority to queue i as much as possible while ensuring (LQ) hols. Then if for some ε>0, the arrival rate is 1 + ε, then server i will at some point always remain at queue i (as uner the propose policy, both queues are unstable an as a result (LQ) is always satisfie). Thus, the maximum throughput in this case can be no more than one. However, it is easy to see by simply assigning server 1 permanently to station 2 an server 2 permanently to station 1, the maximal throughput is two. It remains to comment on more natural conitions that guarantee (LQ). This will help in constructing heuristics that are near throughput optimal while attempting to minimize holing costs. Consier the following two simple moifications. Assumption (LQ1): There exists a threshol L < such that if Q k (t) L for at least one k, then at time t the reconfigurable servers are all serving the station(s) with the longest queue. Assumption (LQ2): There exists a threshol L < such that if Q k (t) L for at least one k, then at time t the reconfigurable servers are all serving stations with queue length at least γ Q max (t)where0<γ 1 an Q max (t) is the largest queue length at time t. (Note that (LQ1) is (LQ2) with γ = 1.)

7 600 Wu et al. Uner (LQ1) an by the fact that Q k (t) is Lipschitz, if Q k (t) > 0 for at least one k, then there exists a subsequence {xn 1Q k(x n s), n 1}, such that for some M 0 an h > 0, Q k (x m s) L for all m M an s [t, t + h]. This means that for large m, the longest queue is serve on [t, t + h]. Taking limits implies that all of the reconfigurable servers are serving a queue where Q k (s) > 0fors [t, t + h]. That is, the rate at which the busy time increases is a rl an (LQ1) implies (LQ). A similar argument hols for (LQ2). 4. Minimizing average holing costs Anraóttir et al. (2007) use a flui moel to show that a time generalize roun-robin policy is stable (in the sense of yieling a stationary istribution) when λ<λ. Moreover, they show that any throughput rate less than λ can be achieve. In systems with only one server at each station, Theorem 4.1 of Dai an Meyn (1995) shows that all moments of the queue lengths are finite when the flui moel is stable. Since all service times are exponentially istribute, accoring to Kaufman et al. (2005), their proof carries over to systems with a finite number of servers at each station. We conclue that if λ<λ : lim t N n=1 h n E π x Q n(t) <, (11) where π is a time generalize roun-robin policy as escribe in Anraóttir et al. (2007). That is, there exists a time generalize roun-robin policy with finite average cost. Moreover, applying Theorem of Puterman (1994) yiels that there exists a Markovian ranomize policy, say π, with the same average cost. Assume now that uniformization (Serfozo, 1978) has been applie so that we consier the iscrete-time equivalent to the continuous-time problem efine. The cost function for each state x S an reconfigurable allocation a (the action) is c(x, a) := N n=1 h nq n. Let x n S be the state of the system after the nth ecision epoch an y n be the action chosen uner policy π. Define the n-stage expecte total iscounte cost for initial state x as follows: [ ] n 1 V n,α (π, x) := E π x α k c(x k, y k ), (12) k=0 where α (0, 1] is the iscount factor an V 0,α = 0. Let V α (π, x) := lim n V n,α (π, x) enote the infinite-horizon expecte total iscounte cost of policy π. The long-run average cost uner π is now efine: g(π, x) = lim sup n V n,1 (π, x). (13) n A policy π is optimal uner the respective criterion if G(π, x) = inf π G(π, x) for all x X, where G = V n,α, V α or g. It shoul be clear that any non-iling policy generates a Markov chain such that all states communicate. The following result is now simple but important. Proposition 3. Suppose λ<λ an consier the iscrete-time equivalent to the continuous-time problem originally pose. There exists a Markov, ranomize policy, π such that g( π) <. Proof. The fact that the Markov chain generate uner any non-iling policy is uniformizable is trivial since the transition rate out of each state is boune. Thus, the average cost uner a fixe policy π in the continuous-time problem is the same as that in the iscrete-time equivalent moulo, a multiplicative constant. Applying Equation (11) to Definition 2 yiels the result. The goal is to fin a policy π that will minimize the long-run average holing cost per unit time over an infinite planning horizon The two-station moel Let N = 2 an let i an j represent the number of customers at stations 1 an 2, respectively. From this point, we assume that the service rate for each reconfigurable server is inepenent of the station, in other wors μ r,k,l μ r,l for all r an l. Denote the server status (0 = faile, 1 = operational) of the lth eicate server at stations 1 an 2 an the reconfigurable server by ζ l, η l an θ l, respectively. The (now simplifie) state space is X ={(i, j, s = (ζ,η,θ)) i, j Z +,ζ = (ζ 1,ζ 2,...,ζ M1 ), η = (η 1,η 2,...,η M2 ), θ = (θ 1,θ 2,...,θ Mr ),ζ l {0, 1},η l {0, 1},θ l {0, 1}}. Without loss of generality, assume the uniformization rate := λ + k,l {μ k,l + α k,l + β k,l }+ l {μ r,l + α r,l + β r,l }=1. Suppose h 1 (h 2 ) represents the holing cost rate per customer per unit time for station 1 (2). The cost function when in state x = (i, j, s) an choosing the allocation action a is c(x, a) = ih 1 + jh 2. In orer to ease notation, we efine two operators F an R to encoe the failure an repair transitions. Thus, for failure transitions F l (ζ ):= (ζ 1,ζ 2,...,ζ l = 0,...,ζ M1 ) an repair transitions R l (ζ ):= (ζ 1,ζ 2,...,ζ l = 1,...,ζ M1 ). Furthermore, let x, y = x i y i an 1 be a vector of all ones. For any real number a R, efine a + = max{0, a}; the positive part of a. Then, min {a, b} =a [a b] +. Define the mapping Hv(i, j,ζ,η,θ)fromx to R as follows: Hv(i, j,ζ,η,θ) = θ l μ r,l min{v(i 1, j + 1,ζ,η,θ), v(i, j 1,ζ,η,θ)}+u(i, j,ζ,η,θ) + w(i, j,ζ,η,θ) = θ,μ r (v(i 1, j + 1,ζ,η,θ) [v(i 1, j + 1,ζ,η,θ) v(i, j 1,ζ,η,θ)] + ) + u(i, j,ζ,η,θ) + w(i, j,ζ,η,θ), (14)

8 Allocation of reconfigurable resources 601 where v(i, 1,ζ,η,θ) = v(i, 0,ζ,η,θ) an v( 1, j,ζ,η,θ) = v(0, j 1,ζ,η,θ), u(i, j,ζ,η,θ) = λv(i + 1, j,ζ,η,θ) + μ 1,l ζ l v(i 1, j + 1,ζ,η,θ) l + μ 2,l η l v(i, j 1,ζ,η,θ) l + ( 1 ζ,μ η, μ θ,μ r )v(i, j,ζ,η,θ), an w(i, j,ζ,η,θ) = α 1,l v(i, j, F l (ζ ),η,θ) l + α 2,l v(i, j,ζ,f l (η),θ) l + α r,l v(i, j,ζ,η,f l (θ)) + β 1,l v(i, j, R l (ζ ),η,θ) l l + β 2,l v(i, j,ζ,r l (η),θ) + β r,l v(i, j,ζ,η,r l (θ)). l l Note that u encoes external arrivals, service completions of eicate servers an ummy transitions while w encoes the server failures an repair transitions. The following are calle the finite horizon optimality equations, Discounte Cost Optimality Equations (DCOE) an the Average Cost Optimality Equations (ACOE), respectively. v n,α (i, j,ζ,η,θ) = ih 1 + jh 2 + αhv n 1,α (i, j,ζ,η,θ) (15) v α (i, j,ζ,η,θ) = ih 1 + jh 2 + αhv α (i, j,ζ,η,θ) (16) g + h(i, j,ζ,η,θ) = ih 1 + jh 2 + Hh(i, j,ζ,η,θ). (17) One might note that the optimality assumes that all of the reconfigurable servers are to be allocate to one station or the other; never ivie between stations. The next result states that the optimality equations are sufficient when iviing the servers among stations is allowe. Proposition 4. The minimums in Equations (15), (16) an (17) are sufficient to solve the more general problem where servers can be ivie among the two stations. That is, there exists an optimal policy where reconfigurable servers are never ivie among the two stations. Proof. Consier the infinite-horizon iscounte cost moel an allow servers to be ivie between stations. The DCOE become: v α (i, j,ζ,η,θ) = ( Mr + α min k l {0,1},,...M r { ih 1 + jh 2 θ l μ r,l k l v α (i 1, j + 1,ζ,η,θ) + θ l μ r,l (1 k l )v α (i, j 1,ζ,η,θ) + u(i, j,ζ,η,θ) )} + w(i, j,ζ,η,θ), where w an u are as efine above an k l is the ecision variable enoting where the l th server is assigne (to station 1 (1) or 2 (0)). A little algebra yiels: v α (i, j,ζ,η,θ) = min k l {0,1},,...M { r α θ l μ r,l k l v α (i 1, j + 1,ζ,η,θ) } + α θ l μ r,l (1 k l )v α (i, j 1,ζ,η,θ) + ih 1 + jh 2 + αu(i, j,ζ,η,θ) + αw(i, j,ζ,η,θ) { = min α θ l μ r,l k l [v α (i 1, j + 1,ζ,η,θ) k l {0,1},,...M r } v α (i, j 1,ζ,η,θ)] + α θ l μ r,l v α (i, j 1,ζ,η,θ) + ih 1 + jh 2 + αu(i, j,ζ,η,θ) + αw(i, j,ζ,η,θ). Now notice that if v(i 1, j + 1,ζ,η,θ) v(i, j 1,ζ,η,θ) ( ) 0 inicates whether all of the k l shoul be set to zero or one (θ l an μ r,l are non-negative for all l). In other wors, all servers shoul be assigne to one station or the other as esire. The other cases are similar. It is well known that a solution to Equation (15) exists an satisfies v n,α = V n,α. In Equation (16), if a solution v α exists it is such that v α = V α. In Equation (17), it is also well known (at least when all states communicate) that if a solution, (g, h), to the ACOE exists then g = g (x) for all x X an h is calle the relative value function (see for example Puterman (1994)). We next show the existence of solutions to the DCOE an ACOE uner certain assumptions. We also iscuss the convergence of the value iterates uner both the iscounte an average cost criteria. Note that this implies a metho for approximating solutions. We then use value iteration to prove the existence of optimal transition monotone policies. Since the holing costs are non-negative an linear an there are only two actions available for each state, Proposition 5 follows irectly from Proposition 9.17 of Bertsekas an Shreve (1996). Proposition 5. For each α (0, 1), the iscounte cost optimality Equations (16) have a unique solution v α an v α = lim n v n,α. Proposition 6. Suppose that λ<λ an x X, the following hol: 1. The optimal average cost may be compute by v m,1 (x) g = lim(1 α)v α (x) or g = lim α 1 m m. 2. Let z X be a istinguishe state an V α (z) < for α (0, 1). There exists a limit function h(x) of h αn (x) :=

9 602 Wu et al. V αn (x) V αn (z), whereα n 1 (see Definition of Sennott (1999)). 3. The average cost optimality Equations (17) have the solution (g, h) where g an h are efine in parts 1 an 2. Proof. Recall from Proposition 3 that there exists a Markov ranomize policy that has a finite average cost, say J π <. Since the cost function is linear in the queue lengths, note that {(i, j) c((i, j), a) = ih 1 + jh 2 J π } is a finite set. Accoring to Proposition 4.3 of Sennott (1996) the assumptions of Theorem (incluing the postulate finiteness of V α (z)), Proposition an Theorem of Sennott (1999) hol. Applying these theorems irectly an noting that the irreucibility of the state space implies that all states are recurrent when λ<λ yiels the esire results Optimality of transition monotone policies in two-station networks In this section, we exten the results on transition monotone policies of Wu et al. (2006) to systems with eicate servers at station 2 an external arrivals uner the iscounte an average cost criteria. Throughout the section we assume M 1 = 0. Note from Equation (15) that it is optimal to allocate all available reconfigurable servers to station 1 (2) if v m,α (i 1, j + 1,ζ,η,θ) ( )v m,α (i, j 1,ζ,η,θ). Definition 3. We say that a policy is transition monotone (see Hajek (1984) an Farrar (1993)) if for any fixe state (i, j,ζ,η,θ), v m,α (i, j 1,ζ,η,θ) v m,α (i 1, j + 1,ζ,η,θ) implies v m,α (i, j + k 1,ζ,η,θ) v m,α (i 1, j + k + 1,ζ,η,θ) for all k 0. In other wors, if it is optimal in state (i, j, ζ, η, θ) to allocate the reconfigurable servers to station 2, it is optimal to allocate the reconfigurable servers to station 2 for all states (i, j,ζ,η,θ), where j j. In the next theorem we show the existence of optimal transition monotone policies for each m-stage problem. We then apply Propositions 5 an 6 to get that the same property hols uner the infinite-horizon iscounte or average cost criteria by taking the appropriate limits. Theorem 1. Suppose λ<λ an M 1 = 0 (no eicate server at station 1). For all (i, j,ζ,η,θ) X, α (0, 1] an m Z +, the following hol: 1. v m,α (i, j,ζ,η,θ) is non-ecreasing in j; 2. v m,α (i 1, j + 2,ζ,η,θ) v m,α (i, j,ζ,η,θ) v m,α (i 1, j + 1,ζ,η,θ) v m,α (i, j 1,ζ,η,θ). Proof. By inuction. The results hol trivially for m = 0. Assume that they hol for m an consier m + 1. To show that the first assertion hols, let (i, j,ζ,η,θ) X. A little algebra yiels: v m+1,α (i, j + 1,ζ,η,θ) v m+1,α (i, j,ζ,η,θ) = h 2 + α<θ,μ r > [min{v m,α (i 1, j + 2,ζ,η,θ), v m,α (i, j,ζ,η,θ)} min{v m,α (i 1, j + 1,ζ,η,θ),v m,α (i, j 1,ζ,η,θ)}] + α[u m,α (i, j + 1,ζ,η,θ) u m,α (i, j,ζ,η,θ)] + α[w m,α (i, j + 1,ζ,η,θ) w m,α (i, j,ζ,η,θ)]. (18) Since u m,α an w m,α are positive linear combinations of v m,α, the inuction hypothesis yiels that they are non-ecreasing in j. Moreover, min{v m,α (i 1, j + 2,ζ,η,θ),v m,α (i, j,ζ,η,θ)} min{v m,α (i 1, j + 1,ζ,η,θ),v m,α (i, j 1,ζ,η,θ)}, since the inuction hypothesis yiels v m,α (i 1, j + 2,ζ,η,θ) v m,α (i 1, j + 1,ζ,η,θ) an v m,α (i, j,ζ,η,θ) v m,α (i, j 1,ζ,η,θ). This completes the proof of the first assertion. To prove the secon assertion, note that: v m+1,α (i 1, j + 2,ζ,η,θ) v m,α (i, j,ζ,η,θ) = 2h 2 h 1 + α<θ,μ r > (min{v m,α (i 2, j + 3,ζ,η,θ),v m,α (i 1, j + 1,ζ,η,θ)} min{v m,α (i 1, j + 1,ζ,η,θ),v m,α (i, j 1,ζ,η,θ)}) + α[u m,α (i 1, j + 2,ζ,η,θ) u m,α (i, j,ζ,η,θ)] + α[w m,α (i 1, j + 2,ζ,η,θ) w m,α (i, j,ζ,η,θ)], (19) an v m+1,α (i 1, j + 1,ζ,η,θ) v m+1,α (i, j 1,ζ,η,θ) = 2h 2 h 1 + α<θ,μ r > (min{v m,α (i 2, j + 2,ζ,η,θ),v m,α (i 1, j,ζ,η,θ)} min{v m,α (i 1, j,ζ,η,θ),v m,α (i, j 2,ζ,η,θ)}) + α[u m,α (i 1, j + 1,ζ,η,θ) u m,α (i, j 1,ζ,η,θ)] + α[w m,α (i 1, j + 1,ζ,η,θ) w m,α (i, j 1,ζ,η,θ)]. (20) By the inuction hypothesis, it is straightforwar that u m,α (i 1, j + 2,ζ,η,θ) u m,α (i, j,ζ,η,θ) u m,α (i 1, j + 1,ζ,η,θ) u m,α (i, j 1,ζ,η,θ) an w m,α (i 1, j + 2,ζ,η,θ) w m,α (i, j,ζ,η,θ) w m,α (i 1, j + 1,ζ,η,θ) w m,α (i, j 1,ζ,η,θ) for all j > 1 because u m,α an w m,α are positive linear combinations of vα m. If j = 1 an there is a service completion at station 2, we nee to show v m,α (i 1, 2,ζ,η,θ) v m,α (i, 0,ζ,η,θ) v m,α (i 1, 1,ζ,η,θ) v m,α (i, 0,ζ,η,θ) since they are part of the expansion of u m,α (i 1, 3,ζ,η,θ) u m,α (i, 1,ζ,η,θ) an u m,α (i 1, 2,ζ,η,θ) u m,α (i, 0,ζ,η,θ), respectively. However, this follows from the first assertion of the theorem (alreay proven). Consier now the terms involving minimums: min{v m,α (i 2, j + 3,ζ,η,θ),v m,α (i 1, j + 1,ζ,η,θ)} min{v m,α (i 1, j + 1,ζ,η,θ),v m,α (i, j 1,ζ,η,θ)} = v m,α (i 2, j + 3,ζ,η,θ) v m,α (i 1, j + 1,ζ,η,θ) [v m,α (i 2, j + 3,ζ,η,θ) v m,α (i 1, j + 1,ζ,η,θ)] + + [v m,α (i 1, j + 1,ζ,η,θ) v m,α (i, j 1,ζ,η,θ)] +, (21) an min{v m,α (i 2, j + 2,ζ,η,θ),v m,α (i 1, j,ζ,η,θ)} min{v m,α (i 1, j,ζ,η,θ),v m,α (i, j 2,ζ,η,θ)}

10 Allocation of reconfigurable resources 603 = v m,α (i 2, j + 2,ζ,η,θ) v m,α (i 1, j,ζ,η,θ) [v m,α (i 2, j + 2,ζ,η,θ) v m,α (i 1, j,ζ,η,θ)] + + [v m,α (i 1, j,ζ,η,θ) v m,α (i, j 2,ζ,η,θ)] +. (22) Define x = min{0, x} for all x R (note that this is not the negative part). Note that for A, B, C, D R, A C an B D, implies A A + + B + C C + + D +. Inee A A + + B + C C + + B + C C + + D +. Let A = v m,α (i 2, j + 3,ζ,η,θ) v m,α (i 1, j + 1,ζ, η,θ), B = v m,α (i 1, j + 1,ζ,η,θ) v m,α (i, j 1,ζ,η,θ), C = v m,α (i 2, j + 2,ζ,η,θ) v m,α (i 1, j,ζ,η,θ), D = v m,α (i 1, j,ζ,η,θ) v m,α (i, j 2,ζ,η,θ). The fact that A C an B D follows from the inuction hypothesis. The previous argument implies A A + + B + C C + + D +. Thus, the secon assertion hols for stage m + 1 an the proof of the theorem is complete. This leas to the following structural result; the main theorem of this section. Theorem 2. Suppose λ<λ an M 1 = 0 (no eicate server at station 1), the following hol: 1. There exists a transition monotone policy that is infinite horizon iscounte cost optimal. 2. There exists a transition monotone policy that is average cost optimal. Proof. First note that by taking the limit as m in statement 2 of Theorem 1 we have: v α (i 1, j + 2,ζ,η,θ) v α (i, j,ζ,η,θ) v α (i 1, j + 1,ζ,η,θ) v α (i, j 1,ζ,η,θ). (23) If we next take the limit along a subsequence α n 1 in Equation (23), statement 2 of Proposition 6 an Equation (23) yiel: h(i 1, j + 2,ζ,η,θ) h(i, j,ζ,η,θ) h(i 1, j + 1,ζ,η,θ) h(i, j 1,ζ,η,θ), (24) where h satisfies the ACOE on recurrent states of the optimal policy. Recall that it is infinite-horizon iscounte cost optimal to allocate all available reconfigurable servers to station 1 (2) in state (i, j,ζ,η,θ)ifv α (i 1, j + 1,ζ,η,θ) ( ) v α (i, j 1,ζ,η,θ). Thus, if it is optimal to allocate the reconfigurable servers to station 2 in state (i, j, ζ, η, θ), then Equation (23) implies it is optimal to allocate all of the resources to station 2 for all (i, j + k,ζ,η,θ) where k 0. Similarly for the average cost case. Theorem 2 implies the existence of a threshol function L(i,ζ,η,θ), where for any state (i, j,ζ,η,θ) X, allocating all reconfigurable servers to the secon station is optimal if an only if j L(i,ζ,η,θ). This simplifies consierably the search for an optimal allocation policy (see Section of Puterman (1994)). Moreover, only the curve L nees to be store in the lookup table as oppose to actions for every state. We conclue this section with a remark on the system with eicate servers at the upstream station. To complete the proof of Theorem 1 in systems with eicate servers at station 1, the aitional bounary conition woul require vα m(0, j + 2,ζ,η,θ) vm α (0, j + 1,ζ,η,θ) vα m(0, j + 1,ζ,η,θ) vm α (0, j,ζ,η,θ) for all j an m. That is to say vα m (0, j,ζ,η,θ) is convex in j. However, sufficient conitions for the convexity of vα m (0, j,ζ,η,θ)inj remain unclear. Consier the following example. Example 1. Suppose there are two stations with two reconfigurable servers an two eicate servers at each station. As an approximation, assume a fixe buffer capacity of 30 before each station an an external arrival rate λ<0.5λ. Suppose h 1 = 1; h 2 = 3; μ 1,1 = μ 1,2 = 7; μ 2,1 = μ 2,2 = 10; α 1 = 0.4; α 2 = 0.1; β 1 = 6; β 2 = 0.3; μ r = 5; α r = 0.1; β r = 0.3 an λ = 7. Let i = 9, j = 10, ζ ={1, 1}, η ={0, 0} an θ ={0, 0}. Then h(i, j + 1,ζ,η,θ) 2h(i, j,ζ,η,θ) + h(i, j 1,ζ,η,θ) That is to say that h is not convex in j at (i, j,ζ,η,θ). We note the non-convexity exhibite in example 1 is not pathological. Other examples in the literature can be foun in Hajek (1984) an Weber an Stiham (1987). Even within the example there were several other states that exhibite non-convexity. Of course this oes not imply that optimal policies are not transition monotone. Quite the contrary, our numerical work suggests that there exists an optimal transition monotone policy when M 1 > Heuristic allocation policies We have shown the existence of optimal transition monotone resource allocation policies in a two-station system. This simplifies the search for an optimal policy an the implementation of this policy for small problems. However, fining an average-cost optimal policy (or even structural results) for an N-station system is still intractable. In this section, we use the two-station solution to evelop an easily implementable heuristic. As we will see, it performs well when compare to several heuristics currently in practice. Instea of solving the resource allocation problem for the whole system, our two-pairing heuristics reuce the computation effort by looking at only two stations at a time. For simplicity, suppose we start with the first two stations an the optimal (two-station) policy is to allocate all of the resources to the first station. The heuristic is to follow this avice, allocate the resources to station 1 an stop. On the other han, if the optimal policy allocates the resources to the secon station, the heuristic consiers an optimal policy for a two-station moel that consists of the secon an thir station. This process continues until we reach the last station. A formal escription of these two-pairing algorithms follows:

11 604 Wu et al. Two-Pairing Upstream ( TPU) [or Two-Pairing Downstream (TPD)]: In a system with N stations: Step 1. Initialization. 1.1 For each positive integer N > n 1, consier the two-station subsystem that has only the reconfigurable resources an stations n an n + 1 of the original system. Assume that the subsystem an the original system have the same service, failure, repair an external arrival rates. Fin the optimal resource allocation policy of each subsystem an store the optimal policy in a lookup table π n. 1.2 Set n = 1[for TPD: let n = N 1]. Step 2. Station evaluation. Suppose the current state s = (q 1, q 2,...,q N, (m 1,1,...,m 1,M1 ),..., (m N,1,...,m N,MN ), (m r,1,..., m r,mr )) S. 2.1 Let s = (q n, q n+1, (m n,1,...,m n,mn ), (m n+1,1,..., m n+1,mn+1 ), (m r,1,..., m r,mr )). Fin the optimal two-station action for state s in the lookup table π n (s ). 2.2 If the optimal two-station action in Step 2.1 is to allocate the reconfigurable server at the upstream [for TPD: ownstream] station, then go to Step 3. Otherwise, go to Step If n = N 1[for TPD: If n = 1], then let n = N [ for TPD: let n = 0] an go to Step 3. Otherwise, let n = n + 1 [for TPD: let n = n 1] an go to Step 2.1. Step 3. Allocation. Allocate the reconfigurable server to station n [ for TPD: station n + 1]. One might observe that TPU an TPD choose the allocation of reconfigurable resources without consiering nonajacent stations. We next efine a family of two-pairing heuristics that alleviate this concern. General two-pairing heuristics: In a system with N stations: Step 1. Initialization. 1.1 For each positive integer n {1, 2,...,N} an n {n + 1,...,N} consier a two-station system that has only the reconfigurable resources an stations n an n of the original system. Assume that the two-station system an the original system have the same service, failure, repair an external arrival rates. Fin the optimal resource allocation policies of each twostation system an store the optimal policy in a lookup table π n,n. 1.2 Let L be an N N matrix an L i,j be the (i, j)th element of L. Initialize L = Let n = 1. Step 2. Station evaluation. Suppose the current state s = (q 1, q 2,...,q N, (m 1,1,...,m 1,M1 ),..., (m N,1,...,m N,MN ), (m r,1,..., m r,mr )) S. 2.1 For each n {n + 1,...,N}, let s = (q n, q n, (m n,1,...,m n,mn ), (m n,1,...,m n,m n ), (m r,1,...,m r,mr )). Fin the optimal twostation action for state s in the lookup table π n,n. 2.2 If the suggeste action in Step 2.1 is to allocate the reconfigurable servers at the ownstream station, let L n,n = 0 an L n,n = 1. Otherwise, L n,n = 1 an L n,n = If n = N 1, continue. Otherwise, n = n + 1 an go to Step 2.1. Step 3. Station weights. Step 2 yiels pairwise comparisons between any two stations n an n. Define f n,n (h n, h n, n n) to be a set of weight functions calculate from holings costs h an the istance between n an n. Let Y n = N n =1 L n,n f n,n (h n, h n, n n). Y n is calle the weight associate with station n. Step 4. Allocation. Allocate the reconfigurable server to a station with the heaviest weight. The general two-pairing heuristics eserve more comment. The pairwise comparisons of the secon step escribe which stations ominate when consiere as two-station moels. This information is store in the matrix L. Each station is then weighte base on the holing cost rates an the istance between two stations by the (ecision-maker specifie) function f. Of course using ifferent weight functions causes the heuristic family to generate ifferent policies. Moreover, it shoul be clear that TPU an TPD are special cases of general two-pairing heuristics. In Section 6.3 we examine several choices of the function f.inthe next section we provie the etails of our numerical stuy comparing each of the heuristics along with several from the literature. 6. Simulation escription an results In this section, we analyze the performance of the heuristic policies in a four-stage serial line with the buffer capacity before each station fixe at 30. Each station is equippe with two ientical eicate servers. There are two aitional reconfigurable servers that can be allocate to any station without setup time or costs. In such systems, 20 ifferent parameters are require to escribe the moel. If for each of the 20 parameters require to escribe the system we consiere two values, a complete esign woul require 2 20 > 10 6 combinations. For each combination, we are require to solve several Markov ecision processes with computational complexity of 2 S 2 in each iteration (Puterman (1994), section 4.5, p. 93), where S enotes the number of ifferent states. In each two-station subsystem of this moel, S equals to = This esign is intractable. Thus, in orer to efficiently conuct a simulation stuy, we use ranomly generate system parameters. We also restrict the range of each parameter to ensure that reasonable systems are chosen for simulation evaluation.

12 Allocation of reconfigurable resources 605 Let U(0, 1) be a uniform (0,1) ranom number. The algorithm that is use to generate system parameters works as follows. 1. h 1, h 2, h 3, h 4 : ranomly choose from {1, 2, 3, 4}. 2. μ 1,μ 2,μ 3,μ 4,μ r : ranomly choose from {5, 6, 7, 8, 9, 10}. This ensures that the system is not extremely asymmetric, which is reasonable for most manufacturing systems. Meanwhile, sufficiently large arrival an service rates also help to ensure that the simulation collects enough ata within a fixe time perio. 3. α 1,α 2,α 3,α 4,α r : choose from 0.5 U(0, 1). This assumption makes service rates failure rates; a common characteristic in most systems. 4. β 1,β 2,β 3,β 4,β r : ranomly choose γ 1,γ 2,γ 3,γ 4 an γ r from {3, 7, 11, 15, 19}, an let β i = α i γ i. This forces a server s expecte failure time to be equal to 1/4, 1/8, 1/12, 1/16, or 1/20 of the total expecte time from repair to repair. In other wors, the expecte longterm average server reliability is in (0.75, 0.95). 5. λ: use the previously generate parameters to obtain the optimal solution λ of LP(1). Then, let λ = ( U(0, 1))λ. The generate arrival rate allows us to focus on systems with meium to high utilization; which is realistic in many manufacturing systems. In each simulation run, the first 500 units of time are eeme the warm-up perio; no statistics are collecte. After the warm-up perio, system statistics are collecte for 5000 units of time. Accoring to the simulation run length an the system parameter generation algorithm there are on average between an external arrivals. (However, in 91% of the generate systems, there were between an external arrivals.) 6.1. Comparison with several heuristics in practice Before introucing the simulation results, we first list the policies that we compare to TPU an TPD: 1. Upstream First Heuristic (UPF): allocate the reconfigurable resources to the furthest upstream station which is non-empty. This heuristic is similar in spirit to the bucket brigae policies in Bartholi an Eisenstein (1996) an Bartholi et al. (2001), except we inclue the presence of eicate servers. Bartholi an Eisenstein (1996) show that this heuristic has higher throughput than other current inustry practices. Of course they o not consier eicate servers or server reliability. 2. Downstream First Heuristic (DTF): allocate the reconfigurable resources to the furthest ownstream station that is non-empty. This heuristic is similar to the expeite heuristic in Van Oyen et al. (2001), except for the presence of eicate servers. 3. Trouble Shooting Heuristic (TRS): allocate the reconfigurable servers to the furthest ownstream station when there is no machine failure in the system an that station is non-empty. If there is a machine failure in a station an the station is not empty, use the reconfigurable servers at that station to compensate for the capacity loss. If there are several servers faile in the system, use the reconfigurable servers at the ownstream stations when the ownstream station with the faile server is not empty (Freiheit et al., 2004). 4. Time Generalize Roun-Robin Policy 1 (RR1): reconfigurable servers stay at station n for n time units an then move to the next station. When the buffer is empty at station n before the n time units are finishe, the reconfigurable servers move to the next station to prevent iling. In the time generalize roun-robin policy 1, we assume that N n=1 n = 10/μ r. 1 As previously note, in systems with reliability consierations, optimal throughput rates can be achieve by time generalize roun-robin policies when buffers have infinite capacity Anraóttir et al. (2007). 5. Time Generalize Roun-Robin Policy 2 (RR2): this policy is the same as the Time Generalize Roun- Robin Policy 1, except that we assume N n=1 n = 20/μ r. 6. Two-Pairing Downstream (TPD): since TPD is similar to TPU an can be simulate without aitional calculation, we inclue TPD in our stuy. In each simulation, we use common ranom numbers to reuce the variance of simulation results between policies. For each policy, we use the same ranom number sequences to generate the next server failure an repair times. We also use common ranom number sequences to generate the service time of each job. Using common ranom number sequences shoul provie better accuracy in comparing ifferent policies. For each interval of utilizations ρ, [0.6, 0.7), [0.7, 0.8), [0.8, 0.9) an [0.9, 1.0), we simulate 100 ifferent sets of system parameters. For each set of system parameters, we run sufficient replications such that 95% confience intervals have with less than 5% of the corresponing sample mean. This requires 30 replications for 389 systems an 50 replications for the remaining 11. In orer to facilitate unerstaning of the simulation results, we list a complete set of simulation outputs for one example system. Example 2. Input Parameters: holing cost rates at each station: (h 1, h 2, h 3, h 4 ) = (2, 3, 4, 2); 1 In Anraóttir et al. (2003), a small value of N n=1 n is suggeste to reuce the average holing costs. However, an extremely small N n=1 n is not practical in many manufacturing systems. We mention here that N n=1 n = 5/μ r oes not improve the average holing costs much over N n=1 n = 10/μ r. On average the ifference was less than 1%.