Markovian Modeling of Multiclass Deterministic Flow Lines with Random Arrivals: The Case of a Single-Channel

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1 2015 IEEE International Conference on Automation Science and Engineering (CASE) Aug 24-28, Gothenburg, Sweden Markovian Modeling of Multiclass Deterministic Flow Lines with Random Arrivals: The Case of a Single-Channel Sang-Yoon Bae and James R. Morrison, Member, IEEE Abstract Although there has been some success in the exact analysis of tandem queueing networks with finite intermediate buffers, equilibrium probabilities for the waiting time of customers remain elusive. Recently, for deterministic flow lines with random arrivals and a single customer class, exact channel decomposition has enabled Markovian modeling of the waiting time probabilities. Although exact channel decomposition results have been obtained for certain types of multi-class deterministic flow lines, stochastic analysis of customer delays remains unresolved. Here we demonstrate that certain types of single channel multi-class flow lines also possess a Markovian property for their customer delays. The explicit recursive relationship between the delays from one customer to the next is developed. Due to the complexity of the recursive relationship, we provide some guidance for constructing the state space and transition probabilities of a Markov chain modeling the delays. A computational example is provided. As flow lines can serve as good models for certain types of semiconductor manufacturing equipment, the results may ultimately lead to useful analytic models for such systems. I. INTRODUCTION Efforts to analyze flow lines and tandem queueing networks have been conducted for decades because of their importance for modeling manufacturing systems. Most existing studies on exact solutions of equilibrium probabilities for the customer delays are only applicable to systems with three stages or less. For systems with more than three stages, many results have focused on efficient numerical approximations; c.f., [1-6]. Some of these approximations can be very accurate. However, as observed in [7], exact solution approaches can be beneficial in several ways. Exact solutions may be more useful if they are significantly more accurate and the model is a good representation to the real system. Exact solutions can also serve to provide meaningful intuition about system behavior. Finally, exact solutions may prove helpful to develop improved approximations. Classic exact results for systems with three stages or less were obtained in [7-14]. More recently, there has been some progress in obtaining exact solutions of flow line models. In [15-17], (max,+)-algebra methods were exploited to obtain moments of customer waiting time for an arbitrary number of stages, a Poisson arrival process, and so-called non-overlapping service times. These (max,+) algebra methods were extended to obtain an expression that the equilibrium waiting times must satisfy in [18-19]. This result can be used to obtain the equilibrium probabilities in an approach similar to, but faster than, simulation. The authors are with the Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daeeon , Korea ( ames.morrison@kaist.edu; Exact analyses for the class of flow lines with deterministic service times, finite buffers, infinite arrival buffer and an arbitrary arrival process have been pursued. In [20], for a single class of customers, a recursion for the exit times from customer to customer was developed. An exact decomposition of such systems into segments of adacent stages called channels was pursued in [21]. Within each channel, there is well-structured delay experienced by the customers. Based on this delay structure, [22] obtained a Markovian property for the delays at each stage. However, these results are restricted to systems with a single class of customer. In [23], exact decomposition ideas were extended to deterministic flow lines with multiple classes of customers. A similar structure and behavior of delay was identified. Here we focus on a type of multiclass deterministic flow line (MDFL) systems with a single channel and satisfying three assumptions detailed in the sequel. We exploit the delay structure of [23] to develop a recursion from customer to customer for the delays experienced in each stage. The recursion proves that the delays possess a Markovian property. For any discrete-time renewal arrival process, the delays can thus be modeled as a Markov chain. Guidance is provided for how to construct the state space and transition probabilities for such a Markov chain. Numerical examples are provided. Since the resulting state space is infinite, truncation is used. The key results of this paper can be used for exact modeling of manufacturing flow lines with deterministic stages. Such systems can be useful for modeling automobile and semiconductor manufacturing. The results can provide intuition on the delay behavior in such systems. Though we restrict attention to single-channel MDFL systems, we hope the results may ultimately lead to models for multi-channel MDFL systems. The paper is organized as follows. In Section II, we describe the MDFL systems of interest and introduce preliminary helpful theorems from [23]. Theorems leading to recursions and a Markovian property for the delays between successive customers are provided in Section III. In Section IV, guidance for constructing a discrete-time Markov chain from the recursions in Section III and related computational methods are introduced. A numerical example is also included. Concluding remarks and future directions are provided in Section V. For brevity, proofs are omitted throughout. II. MODEL DESCRIPTION A. Multiclass Deterministic Flow Lines Flow lines, or tandem queueing networks are characterized by a series of n stages (n > 2) at which customers receive service in order. We assume throughout that each stage is /15/$ IEEE 649

2 attended by a single server. (Hybrid flow lines allow more than one server per stage.) Between each stage, a finite buffer may reside. We assume manufacturing blocking. That is, after receiving service from stage, a customer may only advance to stage if there is an available buffer slot (or the stage is available). If no slot is available, the customer languishes at stage and prevents further customers from receiving stage service. The customer will immediately advance when stage has available buffer or server space. There is an infinite buffer prior to the first stage. Customers arrive to the system as an arbitrary renewal process. Each customer w has a class c(w). The system is thus a multiclass flow line; see Figure 1. We assume that stages are reliable and the service time at each stage is deterministic and depends only on the customer class. We use i c(w) to denote the service time for customer w at stage i. Servers are non-idling; that is, they will serve a customer when one is available unless they are experiencing manufacturing blocking. According to [23], intermediate buffers can be equivalently modeled as a stage with 0 service time. Hereafter, we thus model every internal buffer slot as a stage. We assume that customers are served in first come first served (FCFS) manner. Table1 provides our notation. Notation K c(w) τ i C(w) Table 1. List of notation Definition Number of classes of customers Class index of customer w. Deterministic service time for a customer whose class is w at stage i. Τ C(w) i (w) m=i τ m B Index of the stage whose service time is maximum in the system. If several achieve the maximum, B is the least index among these.. τ B (w) Service time for customer w at stage B, τ B (w): = τ B C(w) + s(w); s(w) allows customer w a setup time. a w X w, Arrival time of the w th customer. Entry time of customer w to stage m i C w, Completion time of service of customer w at stage. C w,b : = X w,b + τ B (w), C w, : = X w, + τ C(w). B. d 0 (w) Delay of customer w to enter stage m 1, defined as d 0 (w): = X w,1 a w. d (w) Delay of customer w in stage m, defined as d (w): = Y(w) { } + S i (w) S 1 B (w) Figure 1. Flow line with n stages of production X w, (X w, + τ C(w) ), =1,, B-1. There is no delay in or after the bottleneck. Total delay of customer w in the system, Y(w): = d i (w) max{0, } i=1. Maximum possible total delay customer w can experience from stage i to stage. Defined as [τ B (w + m) τ C(w) m=i m ] (i<). Maximum total delay customer w can experience in the system. Defined as [τ B (w B + m) τ C(w) m ] m=1. B. Basic model assumption and preliminaries Our focus is on MDFL systems with single renewal arrival process (each arrival is randomly allocated a customer class in IID (independent identically distributed) fashion). The fundamental behavior of the system is described by the elementary evolution equations (EEEs): X w+1,1 = max {a w+1, X w,2 } X w+1, = max {X w+1, 1 + τ C(w+1) 1, X w, } X w+1,m = max {X w+1,m 1 + τ C(w+1) M 1, X w,m + τ C(w) M } As in [23], we start with two assumptions on our MDFL systems. Assumption 1 requires the process times for each class to be multiples of each other. Assumption A1: Process Times between Classes. The process times satisfy τ k+1 = η k τ k, for all = 1,,M and k=1,,k-1. K 1 Here, every η k (0,1) and we let η = i=1 η i. We refer to the stages whose service time (for a given class) is greater than all previous stages as dominating stages. Definition 2.1: Dominating stage. A stage i for which k i > k, for all < i and k=1,, K, is a dominating stage. The first stage is the first dominating stage. The last dominating stage is the bottleneck. Use to denote the number of dominating stages and = 1,, as the index of dominating stages. Let (1),, ( ) be the index of each of the dominating stages. We denote the index as the bottleneck stage as B, so that (1) = 1 and ( ) = B. Under Assumption A1, an MDFL can be decomposed into segments of stages referred to as channels. Each channel consists of the stages between and including any two adacent dominating stages. Definition 2.2: Channel. The series of stages ( ),, ( +1) is called channel. Assumption 2 enables analysis and restricts the process times in a channel. Assumption A2: Process Times within a Channel. The service times of customers of class k satisfy the relation τ k k, for β(α) < < β(α + 1) for all α = 1,, σ 1, ητ 1 and for all > B+1. With these two assumptions, [23] demonstrated that the delay in each stage in a channel is well structured. This structure allowed for the development of a recursion from customer to customer for the total delay in each channel. In addition, delays cannot occur in stages after the bottleneck stage. As such, we hereafter omit the stages and delays after the bottleneck stage. Throughout, we restrict attention to MDFL systems with a single channel. Finally, this paper adopts Assumption A3. Assumption A3: Service Times between First and Bottleneck Stage. The least bottleneck service time is greater or equal to the greatest service time of the first stage. min k {τ B k } max k {τ 1 k } 650

3 Table 2. MDFL service times under A1, A2 and A3 Table 2 provides service times for an MDFL system which satisfies Assumptions A1, A2 and A3, and possesses a single channel. III. MARKOVIAN PROPERTY OF DELAYS IN SINGLE-CHANNEL MDFL SYSTEMS This section provides a recursion for the delays in a single-channel MDFL system from customer to customer. The proofs are omitted for brevity. Note however that seven lemmas were helpful to structure the proof they are not provided here. Theorem 3.1 provides a recursion for customer delays when two successive customers arrive simultaneously. The general non-zero interarrival case is addressed subsequently. Throughout, {.} + = max{., 0}. Theorem 3.1: Recursion between Delays of Two Successive Customers with Simultaneous Arrival: Let d 0 (w + 1) denote the delay of customer w + 1 at stage when a w+1 = a w. Let be the index of the first stage in which customer w faces delay (if there is one). The following recursions hold. d 0 0 (w + 1) = d 0 (w) + τ 1 C(w) + d 1 (w). If Y(w) = 0 (in which case customer w faces no delay in any stage), d 0 (w + 1) = 0 for = 1,, B-2. 0 d (w + 1) = {τ B (w) τ C(w) 1 + i=1 ( τ C(w) i τ C(w+1) i )} +. If = 1, for = 1,, B-1. If Y(w) 0 and 1, In the case c(w + 1) > c(w), so that τ C(w) > τ C(w+1) for every stage - When τ C(w) 1 (τ C(w) + d (w)) τ i C(w+1) ) d 0 (w + 1) = 0 for = 1,, -1. d 0 (w + 1) = d (w) + d +1(w) 1 i=1 (τ i C(w) + (τ C(w) i τ C(w+1) i=1 i ) + τ C(w) C(w) +1 τ 1 for = +1,, B-1 - When τ C(w) 1 (τ C(w) + d (w)) τ i C(w+1) ) d 0 (w + 1) = 0 for = 1,, d 1(w + 1) = d (w) 1 i=1 (τ i C(w) + 1 (τ C(w) i τ C(w+1) i=1 i ) + τ C(w) C(w) τ 1 for =,, B-1 In the case that c(w + 1) = c(w) - When τ C(w) 1 (τ C(w) + d (w)) 0 d 0 (w + 1) = 0 for = 1,, -1. d 0 (w + 1) = d (w) + d +1(w) + τ C(w) C(w) +1 τ 1 for = +1,, B-1 - When τ C(w) 1 (τ C(w) + d (w)) < 0 d 0 (w + 1) = 0 for = 1,, d 1(w + 1) = d (w) + τ C(w) C(w) τ 1 for =,, B-1 - In the case that c(w + 1) < c(w), so that τ C(w) < τ C(w+1) for every stage, and using d 0 (w + 1) to denote the delay at stage of customer w + 1 when c(w + 1) = c(w), d 0 (w + 1) = [d 0 (w + 1) (τ C(w+1) i τ C(w) i ) max{0, (τ C(w+1) i τ C(w) i=1 i ) 1 0 (w + 1) }] + for = 1,, B-1. i=1 d i In general, however, customers do not arrive simultaneously. As stated in Theorem 3.2, a recursion for the delays with non-simultaneous arrival can be obtained from the simultaneous case delays via a much simpler expression. Theorem 3.2: Recursion between Delays of Two Successive Customers with Non-Simultaneous Arrival: Recall that d 0 (w + 1) is the delay customer w + 1 would face at stage if a w+1 = a w. For general inter arrival time a w+1 > a w, the following holds: d (w + 1) = [d 0 (w + 1) max{0, (a w+1 a w ) 1 i=0 d 0 i (w + 1) }] + for = 0,, B-1 Together, these enable a Markovian property. Theorem 3.3: Markovian property. The customer class and delays {c(w), d 0 (w), d 1 (w), d (w)}, w = 1, 2,, in a single channel MDFL under Assumptions A1, A2 and A3 with a continuous-time renewal arrival process can be 651

4 modeled as a discrete-time continuous-state Markov process. If the arrival process is discrete-time (and service durations are rational), the resulting process is a Markov chain. In this result, the Markov process or chain evolves from customer to customer. As such, using the more standard terminology, we refer to it as discrete-time (even though the time index is really the customer index). For single class deterministic flow lines, analogous results can be obtained for systems with an arbitrary number of channels; see [22]. However, for our MDFL systems, the structure that is present in the single class case becomes more complicated. As such, results for an MDFL with more than one channel remain open questions. IV. MARKOV CHAIN MODELING AND ANALYSIS The recursive relationships in Theorems 3.1 and 3.2 allow one to obtain a customer s delays with only information on the prior customer s delays. For a discrete-time system, Theorem 3.3 states that these delays can be modeled as a Markov chain. However, it is not immediately clear what are the recurrent classes (if one or more exist) or the transition probabilities between the states. In this section, we provide guidance for developing a discrete-time Markov chain for the customer delays in a discrete-time MDFL system. A. Markov Chain Model for the Delays In single class deterministic flow line (DFL) systems with arbitrary arrival process, it was shown in [20] that an exit time recursion from customer to customer exists. As a consequence of the structure of this recursion, the exit times are independent of the order of the stages and the total delays are the same as in a similar system with the bottleneck stage placed at the front of the DFL system. Thus, the total delay a customer experiences is the same as they would face in a G/D/1 queueing system with service time τ B. Exploiting this property for stable systems, [22] demonstrated that for all except a finite number of states, the equilibrium delays in a DFL system can be obtained from G/D/1 queuing results. To obtain the equilibrium probabilities for the remaining finite number of states, a finite number of balance equations can be solved. However, there is no known exit recursion for MDFL systems. As such, no method allows one to immediately identify all but a finite number of equilibrium probabilities. Thus, for a discrete-time MDFL, the Markov chain for the customer delays has an infinite number of states and a corresponding infinite number of balance equations. B. Truncation and State Space The infinite number of states in a discrete-time MDFL system arises due to the possibility of unbounded delays in the infinite buffer serving stage 1. To numerically resolve an unbounded state space, it is common to truncate to a finite number of states. For this purpose, here, in our Markov chain model, if a customer s queuing time in the infinite buffer prior to stage 1 would exceed a constant value, the customer arrival is ignored and the state remains unchanged. There is a tradeoff in choosing the maximal delay constant. Larger values improve the fidelity of the resulting model. Smaller values reduce the size of the state space and enable computation. C. Constructing States and Transition Probabilities Our focus continues to be an MDFL system with discrete-time renewal arrival process, so that there is a Markov chain model for the delays (see Theorem 3.3). For convenience, we specify our interarrival time probability mass function and class selection probability function. The approach can readily be employed in general with slight notational adustment. We use (c(w), d 0 (w), d 1 (w), d (w)) as our state vector. Let K, B, and b denote the number of customer classes, index of the bottleneck stage, and maximum allowable delay in the infinite buffer prior to stage 1 (after which the arrival is reected due to our truncation), respectively. Further, let m(k, ): = τ 1 B τ k. It is the maximum amount of delay that a customer of class k in stage can experience. Let C = {1, 2,, K}, D 0 = {0, 1, 2,, b} and D k = {0, 1,, m(k,)}. The state space S of all communicating states is contained in the following set: K S [{k} D 0 D k k 1 D ] Φ k=1 Some elements of the set may not be reachable from an initially empty system (they are transient). For convenience, we label the state s = (c(w), d 0 (w), d 1 (w), d (w)) S with the index ind(c(w), d 0 (w), d 1 (w), d (w)) = K d 0 (w)(1 + i=1 m(c(w), i) ) + i=1 d i (w) k k=1 N S + wheren S k b m(k, 1) m(k, B 1), is the number of elements in with class k. We now turn our attention to the one-step transition probability matrix and use p and q to denote the parameter for our assumed interarrival time distribution and parameter for class determination, respectively. Let pc and nc denote the class of the previous customer and class of the next customer, respectively. (Here previous is synonymous with current we take the vantage of a time instant between two customers and call them in this manner.) Consider a state s with index i, let zero(i, nc) be the next state for the simultaneous arrival case when the previous and next customer classes are pc and nc, respectively (pc is deduced from the index i). This can be calculated from Theorem 3.1 Similarly, let next( i, nc, ) be the next state when the interarrival time is, previous (that is, the current) 652

5 state index is i, and the next class is nc. This can be obtained by Theorem 3.2. Let return(nc) be state (nc, 0, 0, 0). It is convenient to create the one-step transition probability matrix via Algorithm 1. Once the algorithm is complete, the matrix PL 1 [ ] PL Φ addresses transition probabilities from all elements in to all elements in. Many of these elements are transient states. Removing those columns with all zero entries, as well as the corresponding rows, gives the transition probability matrix for those states in S. Algorithm 1: Constructing the Transition Probability Matrix for i = 1, 2,, do 0. PC i class of the state with index i 1. PL i (0,, 0) of length, (PL i () is the th element) for nc = 1, 2,, K do 2. Get zero(i, nc) using Theorem Snc sum of all delays in state zero(i, nc) 4. Dz delay to enter stage 1 in state zero(i, nc) for = 1, 2,, Snc-1 do if Dz > b then 5. PL i (i) PL i (i) + P(PC i, nc, ) else 6. n1 ind(next( i, nc, )) 7. PL i (n) PL i (n) + P(PC i, nc, ) end if end for 8. n2 ind(return(nc)) 9. PL i (n2) PL i (n2) + 1 r=snc P(PC i, nc, r) end for end for D. Example In this subsection, an example MDFL system satisfying Assumptions A1, A2, and A3 is considered. We focus on the delay behavior in response to the interarrival time distribution parameter p. Suppose there are 2 classes and 3 stages with τ 1 1 = 10, τ 2 1 = 5, τ 3 1 = 15, τ 1 2 = 8, τ 2 2 = 4, τ 3 2 = 12. The maximum delay in the first stage we allow for truncation is b = 30. Set q = 0.9. We vary p over the range 0.01 to There are 1085 state candidates 113 among them are not reachable. There are 972 states in the resulting communicating class. The state candidates removed are (1, i, 0, 0) for i = 1,, 21 and (2,, 0, k) for = 1,, 23 and k = 0, 1, 2, 3. The transition probability matrix can be obtained via Algorithm 1; we omit the details. Since there is a single communicating class, the system is positive recurrent (we have truncated the state space and the original system has loading less than 1 for all p values considered), and the period of every state is 1, one can solve Figure 2. Equilibrium probabilities in our example the standard balance equations for the equilibrium probabilities. Figure 2 depicts the equilibrium probabilities as the interarrival time distribution parameter p is varied. In the figure, each row represents a different value of p as indicated. Across each row, every column of width 1 pixel represents one state. The value of the equilibrium probability for each such state is converted to a color. The maximum equilibrium probability for all states with fixed p value is set to the color black. The minimum nonzero value is set to white (zero values are also set to white). In between, a log 10 scale is used to transition from white to black. (A linear scale does not show the structure clearly.) This is done separately for each p value, so the colors from one p value to the next are not comparable they only should be compared with a fixed p value. The left half of each row corresponds to states with class k = 1. The right half to states with class k = 2. Within each class, states with smaller total delay appear on the left (they are indexed according to the index function above). As p increases, the interarrival times decrease on average (the loading increases). As p increases, the probability mass shifts to the right within each class as evidenced by the fading of the black on the left in each class. It is interesting to note that, as the probability shifts to the right, it does not do so evenly. The mass becomes clustered into narrow bands to the right in each class. These bands correspond to the full internal delay cases for each external delay value. For each state s = (c(w), d 0 (w), d 1 (w), d (w)), the total delay is d(s) = d 0 (w)+ d 1 (w)+ + d (w). There may be numerous states that achieve this same total delay. Let (d) be the sum of the probabilities for all states with total delay d. Figure 3 depicts (d) for the various values of d as Figure 3. Sum of the probabilities for total delay with various p 653

6 Figure 4. Expected delays of each modules with various p a function of p. As p increases, the probability mass shifts from small values of d to larger values of d. Figure 4 depicts the expected delays at each stage as a function of p. As p increases (and so too does the loading) these delays increase toward their maximum values. V. CONCLUDING REMARKS In this paper, we investigated the delay in multiclass deterministic flow line (MDFL) systems with random arrivals. Our focus was on MDFL systems possessing channel decomposition properties (enforced via Assumptions A1 and A2). For single channel systems in which the service times at the first stage never exceed the service times at the bottleneck stage (Assumption A3), two theorems providing recursions for the delays from one customer to the next were discussed. (The proofs including seven lemmas leading to the results were omitted for brevity.) Together, these theorems can be used to show that the delays in single channel MDFL systems under Assumptions A1, A2 and A3 possess a Markovian property for any renewal arrival process. Focusing on discrete-time systems, the state consisting of the customer class and vector of delays is a discrete-time time-homogeneous Markov chain. We explored how to construct the state space and one-step transition probabilities. An example was provided exploring the manner in which the system behavior was influenced by the customer arrival rate. There are several directions for future work. Can the Markovian property be extended to systems with more than one channel? Can Assumption A3 be removed? The resulting Markov chain has an infinite number of states which we truncated for computational convenience. Can an exit recursion be developed that allows for the immediate determination of all but a finite number of the equilibrium probabilities as in [22]? 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