The throughput rate of serial production lines with deterministic process times and random setups: Markovian models and applications to semiconductor manufacturing
Introduction
Serial production lines, also known as flow lines, with random behavior can serve as prototypical models for a host of manufacturing systems including production and assembly lines; cf., [1], [2], [3], [4] or, more recently [5], [6], [7], [8], [9], [10], [11]. As such, they have a rich history of research focus and have been applied in many contexts. However, excepting limited classes of systems, exact results do not exist and approximations and simulation methods are typically employed. We endeavor to obtain exact results for a class of such systems with practical application potential in semiconductor manufacturing.
Our performance metric of choice is the maximum production rate or just-in-time (JIT) throughput. The JIT throughput is the long term average rate at which customers exit the system when the input is never starved. That is, it is the throughput when customers arrive just-in-time to the flow line. This is equivalent to the maximum rate at which the system can produce customers. In [12], it was shown that the JIT throughput for two stage serial production lines can be expressed as , where E is the expected value and Si denotes the random service duration for stage i. Based on this result, in [12], [13], [14], explicit expressions for the JIT throughput under exponential, Erlang and uniform service distributions were obtained. For the three stage case, it was demonstrated in [15] that the JIT throughput is , where Si is the random service time for stage i. A variety of papers pursued exact performance results based on this fact under different service time distributions; cf., [15], [16], [17], [18], [19], [20]. Beyond these cases, exact analysis does not seem possible for stages with overlapping random service durations; cf., [1] or [4]. Refer also to [21].
Much recent work has focused on approximations based on aggregation [22], [23], [24], [25], [26] or decomposition methods [27], [28], [29], [30], [31]. These approximations can possess astonishing accuracy and have been used in many practical contexts; cf., [7]. However, like simulation (which can be computationally intensive relative to other methods), they do not as readily provide the same qualitative insights which may be possible via exact results. In addition, these approximations have not explicitly incorporated issues such as setups.
In clustered photolithography tools in semiconductor wafer manufacturing, various types of setups may be required. These setups have a large influence on the JIT throughput of system. While serial production lines have been used to study practical semiconductor manufacturing [32], [33], [34], [35], no closed form expressions considering setups have been obtained. Our efforts are motivated by this lack of results. We focus on obtaining exact results and closed form expressions where possible. To this end, we will restrict attention to serial production lines with deterministic service times. While the JIT throughput of such a deterministic system is trivially the inverse of the longest process time, allowing random setups dramatically complicates analysis. Yet, setups are of practical importance and essential for models of certain manufacturing environments.
For certain classes of randomly occurring setups inspired by clustered photolithography tools (CPTs), we will demonstrate that the maximum production rate can be exactly calculated. This is possible by exploiting the fact that, for JIT customer arrivals, the delay customers experience in the system can be modeled as a Markov chain. In certain cases, explicit expressions for the system throughput can be obtained. In others, the throughput can be calculated by solving for the equilibrium probabilities of the Markov chain. With these results in hand, we endeavor to apply them to models of CPTs. Such tools have a structure that is particularly amenable to our approach. CPTs are essential components of semiconductor wafer fabrication and can be modeled as a serial production line when in the process bound region; cf., [32], [33], [34], [35]. We incorporate practical features such as a wafer buffer before the scanner of the CPT and batch arrivals. Generally, the scanner has the longest process time in a CPT and there is a buffer to ensure its throughput. We compare the throughput obtained via our models with detailed simulations; they are acceptable. Despite limitations associated with the model assumptions, we hope that the results may prove useful in practical contexts where analytic throughput expressions are of value.
Though we describe the systems under consideration in detail in the next section, we require a few ideas now. A serial production line consists of M stages, labeled . Each stage consists of a single server or machine, which provides service. The process time for a customer at stage mi is deterministic with duration τi. There are no buffers between stages, excepting the arrival buffer before the first stage (it has infinite capacity). This restriction is easy to relax and we will do so in the next section. There is a bottleneck stage; it is the first stage mB whose is greater than or equal to all other process times.
Aside from any additional time that may be required for a setup, the service time for every customer in stage mi is τi. Recent theoretical results [36] suggest that it may be possible to relax this assumption and allow service times to depend on the customer. However, significant work will be required to allow this. We hope that the methods developed here will then be applicable in that context.
The classes of setups we consider are as follows. These classes are those for which exact analysis is possible and/or are relevant to modeling setups in CPTs.
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Type I setup: Bottleneck only. A customer may require the bottleneck stage alone to conduct a setup once it arrives to mB. The process time at the bottleneck for this customer will appear greater than .
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Type II setup: A sequence of stages . Before entering the system, a customer may require that all stages prior to and including some distinguished stage be vacant before a setup of those stages commences. Once the setup of those stages is complete, the customer may enter m1 and start its service. We will assume .
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Type III setup: Full-flush. In a full-flush setup, all stages should be vacant before the setup begins. After the setup is complete, the customer may enter the system.
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Type IV setup: A sequence of stages . Before entering some distinguished stage , a customer may require that all stages conduct a setup. They must be vacant of customers prior to the commencement of the setup. Once complete, the customer may enter and proceed with service.
Multiple classes of setups may be required for a single customer. For example, some customer may demand a Type II setup from the stages , , followed by a Type I setup at the bottleneck. Note that a Type III setup includes both Type II and Type IV setups. As such, we need not consider them together. We consider
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Type V setup: Both Type I and Type II.
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Type VI setup: Both Type I and Type III.
We study Type I, II and III setups separately. A Type IV setup that is not subsumed by another class of setup cannot be modeled by our methods. Detailed simulation will be required. We address Type V and VI setups separately. As mentioned, our focus is on the JIT throughput, the maximum production rate, of such a system; call it α. The contributions of the work are as follows.
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For Type I and III setups and Type II setups (with or ), we determine α explicitly (Proposition 1, Proposition 2, Proposition 3, Proposition 4).
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For Type II setups, with certain conditions on SII to be detailed later, we show that the system can be modeled as a Markov chain (Proposition 6), identify its recurrent states (Lemma 4) and determine when the equilibrium probabilities, and thus α, can be obtained explicitly (Proposition 7, Proposition 8). Otherwise, we give the balance equations that will allow the calculation of α (Proposition 9).
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For Type V and Type VI (with or ) setups, we obtain α explicitly (Proposition 10, Proposition 11, Proposition 12).
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Incorporate practical features relevant to CPTs (Section 6).
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Compare the results against those from detailed simulation of a process bound CPT with wafer handling robots (Section 7).
The remainder of the paper is organized as follows. In Section 2, we describe our systems of interest. We also introduce relevant known results. In Section 3, we derive the JIT throughput under Type I and Type II setups. In Section 4, we focus on Type II setups with a certain condition on SII. We identify cases where the throughput can be explicitly obtained. In Section 5, we consider combination setup Types V and VI. We apply the results to serial production line models of CPTs in Section 6. In Section 7, a detailed simulation of a process bound CPT is compared with our models. The simulation includes wafer transport robots and is based on industrial CPT data from the literature. Concluding remarks are provided in Section 8.
Hereafter, we use the term flow line. It is synonymous with serial production line. An early abbreviated version of a subset of the work reported here appeared in conference form in [37].
Section snippets
Preliminaries
We first describe deterministic flow lines and relevant known results. Type I, II and III setups are also detailed.
Maximum production rate
In this section, we derive expressions for the maximum production rate, or JIT throughput for Type I, II and III setups. The throughput can be derived with a little effort for Type I and III setups. For Type II setups, the throughput can be derived explicitly when . Otherwise, the limiting probabilities for internal delays are required to calculate throughput. When the distinguished setup stage resides in the last channel and (), the internal delays can be
Delay in the last channel
For a Type II setup with distinguished stage in the last channel, we require . We next show that obeys a Markovian property and, as such, a Markov chain model can be constructed. The equilibrium probabilities, and thus the JIT throughput, can be extracted from this model.
Multiple setup types: Types V and VI
We now consider Type V and Type VI setups. Type VI setups are simpler; we consider these first.
Type VI setups allow Type I and Type III setups to occur. For each customer, let , , and pN denote the probability the customer prompts a Type I, Type III, Type I and III, or no setup, respectively, independently for each customer. As such, . Proposition 10 For a deterministic flow line under Type VI setups, the JIT throughput
Features relevant for practical CPTs
We now turn our attention to the study of features that are relevant for CPTs in semiconductor wafer manufacturing. In this context, the “customer” is a wafer on which the semiconductor devices are created. We will hereafter use the term wafer in place of customer. Identical wafers are typically processed in a group called a wafer lot, or simply lot. Similarly, we use the term process module (or simply module) in place of stage. A reticle is a patterned piece of glass through which light is
Numerical example and simulation results
In this section, we compare the throughput of our models under Type I and II setups with simulations of full CPTs (including wafer transport robots, wafer lots and a pre-scan buffer) based on actual data for a deep UV CPT [44]. Type III setups are similar to the Type I case. The robot movements are dictated by the longest waiting pair (LWP) method [45]. CPTs consist of process modules which provide different services: cool plate (CP), hot plate (HP), post-scan bake hot plate (PEB), edge
Concluding remarks
In this paper, we studied the production rate, or JIT throughput, of deterministic serial production lines with a variety of random setup types. In all cases considered, Markov chain models were constructed. Closed form analytic expressions for the production rate could be obtained for some setup types and particular parameter values. For others, the Markov chain was studied to develop equations from which the JIT throughput could be obtained numerically. Practical features such as batches of
Acknowledgments
This work was partially supported by National Research Foundation of Korea (NRF) Grant 20110005696. The authors are grateful for the guidance of the reviewers and editors. The paper is much improved for their guidance.
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