Queue time and x-factor characteristics for semiconductor manufacturing with small lot sizes

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1 Proceedings of the 3rd Annual IEEE Conference on Automation Science and Engineering Scottsdale, AZ, USA, Sept 22-25, 27 TuRP-B3.2 Queue time and x-factor characteristics for semiconductor manufacturing with small lot sizes Kilian Schmidt (AMD), Oliver Rose (TUD) Abstract Small lot size is widely regarded as a promising means to achieve shorter cycle times in semiconductor manufacturing. The dominant contributor to cycle time is queue time. In this paper, we quantify how the queue time changes with lot size reductions by means of queuing theory and single-operation simulation. This includes an analysis of the factors shaping this queue time change and how their influence changes for different availability characteristics. Additionally, the x-factor changes resulting from the changes in queue time and raw process time are outlined. S I. INTRODUCTION MALL lot size has been identified by several IC manufacturers as a potential architectural shift to drive down cycle time (e.g. []). Limited simulation work on this change has been done by Wakabayashi et al [], but a solid analytical background on the expected queue time reduction is still missing. This paper aims at closing this gap and will outline which queue time and x-factor changes can be expected for smaller lot sizes. In the literature, there are a few papers that provide some background material. In [7], we analyzed the mandatory characteristics of models of raw processes for small lot sizes. Ref. [8] includes a high-level application of these models for a semiconductor fab. In [9], we analyzed changes of raw process time (RPT) and queue time (QT) changes for priority lots. As priority lots normally bypass the queue and are instantly processed on an available tool, their analysis is comparably easy because the corrupting influence of variability can be ignored. The opposite is the case for normal lots. Queue time is the dominant share of normal lot cycle time and its amount is considerably influenced by variability. Therefore, the change of variability that goes hand in hand with the lot size change is an important part of this paper s analysis. This paper is organized as follows. Section II illustrates the problem in detail. The theoretical discussion is given in Section III while Section IV discusses experimental approaches and results. Section V focuses on the Manuscript received July 3st, 27. Kilian Schmidt is with AMD Saxony LLC & Co. KG, Wilschdorfer Landstraße, MS I-IE, D-9 Dresden (phone: ; fax: ; kilian.schmidt@amd.com). Oliver Rose is with the Institute of Applied Computer Science, Dresden University of Technology. consequences that follow for x-factor changes. Section VI summarizes the paper. II. PROBLEM ILLUSTRATION Fig. illustrates the problem analyzed in this paper. The cycle time that lots spent at a particular operation is divided into queue time and raw process time. In Ref. [7] we showed how raw process time changes for smaller lot sizes, therefore we now turn to the next step, the analysis of queue time changes. In the illustration, we further divided the raw process time into processing t and a delay d because this simplifies the queue time analysis. The processing time t is given as t = n, () PR with n denoting the lot size and PR denoting the wafer process rate. The delay d is caused by the overlapping processing of consecutive lots and is often referred to as first wafer delay. Definition of first wafer delay is not uniform throughout the industry mainly as a consequence of different raw process time definitions. Slightly simplified we think of it as the time necessary to transport a wafer from the carrier to the actual process location and back into the carrier after processing. Universally valid it can be defined as d = RPT t, (2) such that all other formulas for different tool types or accuracy levels can be derived from the RPT formulas and definition generated in [7]. 25 wafer lot Queue Time Queue QT? Raw Process Time Processingt Delay d <25 wafer Queue QT Processingt Delay d lot Fig. : Illustration of the cycle time change under consideration /7/$ IEEE. 69

2 TuRP-B3.2 III. THEORETICAL DISCUSSION Queueing theory is a powerful tool for analyzing queueing behavior as its closed formulas permit exact insights into the relevant factors. Of course, this statement is limited to queueing systems for which these closed formulas exist but it is a very valuable starting point. Simulation experiments can pick up for other queueing systems (see Section IV). Queueing systems can be characterized by Kendall s notation as A/B/m with A denoting the distribution of arrival times, B describing the distribution of process times and m denoting the number of parallel machines (see [3]). Our theoretical and experimental analysis is independent of queuing discipline, therefore we don t specify queuing discipline in notation or discussion. In the semiconductor industry, interarrival times to a given workstation are usually highly variable and some research suggests abstracting them as exponential (see [6]). Therefore, we will use an exponential (Markovian - M) distribution for the arrival process. For the rare exceptions (e.g. workstations with preceding batch operations) that necessitate more general arrival distributions, [] gives helpful queueing time approximations. Process times are usually thought of as being fairly constant. This is a reasonable assumption for the pure process time t that we assume as being deterministic. However, if we look at process times from a logistical point of view we have to account for additional delays. These delays can be setups or preventive and corrective maintenance and inflate the process time to a higher value called effective process time t e. Because these delays only occur for some lots, the effective process time includes variability quantified in its coefficient of variation c e. The resulting effective process time is a random variable following a general distribution (G) shaped by t e and c e. Following the above notation, there are many M/G/m queueing systems in a semiconductor fab. We start the analysis with the special case of a single tool, the M/G/ queueing system. The queueing time QT for these systems is given by + ce² u QT( M / G /) te 2 u = (3) where u denotes the utilization of the tool. For queueing systems featuring parallel tools, there is, in general, no closed formula that provides exact solutions. However, the approximation 2( m+ ) + ce² u QT( M / G / m) = 2 m ( u) gives excellent results although some improvements can t e (4) be made for low utilizations (see [2]). In our case the exactness of the above approximation is sufficient though. We generally think of utilization as a function of the number of wafers to be processed that is independent of the lot size. This is obviously true for the number of parallel tools m as well, leaving the same lot size dependant factors in both Equations (3) and (4). Two parameters are left that define the queue time changes for lot size reductions, the effective process time t e and its coefficient of variation c e. Under the assumption of full time availability and thus deterministic process times, the formula can be further simplified to u QT( M / D /) t 2 u = (5) because in this case t e does not differ from t and c e equals to zero. In this queueing system, the queue time decreases proportionally to the process time t and considering () also proportionally to lot size. However, full time availability is uncommon in semiconductor manufacturing. As a consequence, this first result can only serve as a reference point. Different types of downtimes have different impacts on variability. The most important distinction is between preemptive and non-preemptive downtimes. Preemptive outages occur right in the middle of a process. Typically, these are outages for which there is no control as to when they happen (e.g. failures). In contrast, non-preemptive outages require the tool to be idle before they can happen. This means that we have some control as to exactly when they occur. This is usually the case for planned maintenance activities or setup times. Setup times elude easy analysis. Intelligent setup and dispatching policies seek to avoid setup occurrence and it is therefore difficult to generally estimate how setup frequency would change for lot size reductions. Because of this lack of clarity and the very limited number of tools that is significantly impacted by setups (most notably implant tools), we don t further analyze setups within this paper. However for completeness, we have to state that the previous assumption of utilization being independent of lot size does not necessarily hold for tools with significant setup frequency. A. Preemptive Outages For preemptive downtimes, [5] provides formulas for the parameters of interest, for the effective process time t e and t t e = (6) A 7

3 TuRP-B3.2 2 ( + c ) A( A) 2 2 m c c r e = + r (7) t for its coefficient of variation c e. In these formulas A denotes the equipment availability, c the coefficient of variation of the processing time t (which equals zero for constant t ), c r the coefficient of variation of repair times and m r the mean repair time. It is important to note that the availability A refers only to preemptive outages. It is defined as Variability degrades the queueing time reduction. Moderate availability reductions already lead to significantly smaller queueing time reductions than for the deterministic case. There is always some reduction in queueing time although the reduction approaches zero for very high variability In Fig. 3, we vary the full lot processing time t and the availability A for constant mean times to repair m r of 4 hrs. m f A= (8) m + m f by the parameters mean time to failure m f and mean time to repair m r. We assumed t to be deterministic, therefore c equals zero and we can further minimize the number of variables by assuming repair times to be exponentially distributed which means that c r equals one. This leaves the three variables t, m r and A defining the queueing time change for smaller lot sizes. We illustrate the shape of this change in an example. In Fig. 2, we show the relative queuing time for half lots compared to full lots for varying mean time to repair m r and availability A. The process time t is a constant.5 hrs at full lot size and half that value for the smaller lot size Mean time to repair m r r % 84% 88% 92% 96% % Availability A Fig. 2: Relative queueing time for halving the lot size depending on mean time to repair m r and availability A Fig. 2 illustrates several findings: For half the lot size the queueing time is between half the queueing time of full lots and the full lot size value. At % availability (which is the deterministic process time case) the highest relative queueing time reduction is achieved Process time t % 84% 88% 92% 96% Availability % Fig. 3: Relative queueing time for halving the lot size depending on process time t and availability A Fig. 3 further illustrates the influence of t on the queueing time reduction. Short process times degrade the queueing time reduction. With respect to the denominator of (7), the shape of the t -curve is concave enabling a wider range of significant queueing time reductions. Yet, the key insight from Fig. 3 remains that the queueing time reduction is higher at tools running at smaller rates. B. Non-preemptive outages There is no simple queueing formula for non-preemptive outages that is applicable for our analysis. It is required that the downtimes can be attributed completely to a lot, which does not match our definition of preventive maintenance. For setup times the applicability is given and [5] provides formulas for this case. We analyze non-preemptive downtimes within the experimental analysis in Section IV. Some helpful assumptions on non-preemptive outages can be gained from intuition though: At lower utilizations the variability impact of nonpreemptive outages is lower because longer downtime intervals happen with no lots in queue. At higher utilizations the difference between preemptive and non-preemptive downtimes will disappear because the one lot that can finish processing represents only a small part of the queue. The part of effective process time t e in addition to t is utilization dependant. For low utilizations it is 7

4 TuRP-B3.2 smaller for non-preemptive than for preemptive outages. IV. EXPERIMENTAL DISCUSSION The experimental analysis for non-preemptive outages is not only important for preventive maintenance analysis that is not possible with queueing theory but also because outages that in reality are preemptive are often modeled as nonpreemptive ones in fab-simulations. This is due to fact that usually fab simulation programs are not able to simulate preemptive outages. Therefore, we need to be aware of the impact of this difference when judging fab simulation results. We start the experimental analysis with a station consisting of one tool and outages caused by failures. The mean times to failure and to repair again follow an exponential distribution as do interarrival times. We assume an availability A of 94%, a mean time to repair m r of 4 hrs and a process time t of the full lot of.5 hrs. As we expect utilization dependencies for these experiments we run the experiments for utilizations from % to 99%. Fig. 4 shows the results for this experiment. We depict the different changes. The green dots represent the queueing time ratio of half lot size to full lot size, the orange dots represent the ratio of quarter lot size to full lot size and the blue dots the ratio of quarter lot size to half lot size. The straight lines are the ratios calculated with (3), (6), and (7) for preemptive downtimes preemptive preemptive preemptive % 2% 3% 4% 5% 6% 7% 8% 9% % Fig. 4: Relative queueing time for reducing the lot size depending on utilization for a single tool (downtime cause: failures) At high utilizations, the values for preemptive downtimes are approached as expected based on the discussion in Section III.B. For lower utilization, downtimes seem to cause similar queueing times independent of the lot size. This can be explained by the following consideration. At low utilizations more queueing time at single tool stations occurs while waiting for the downtime to finish and not while waiting for the previous lot to finish. Therefore, lots cannot make use of the higher lot service rate of small lot sizes to a significant extent. Another expectation from Section III.A is shown here. Because the queueing time benefit from the lot size reduction decreases for smaller t, the blue dots generally lie above the green dots although both ratios represent a lot size reduction by 5%. The second experiment is run with the same conditions as the first one except that the station now has 4 parallel tools. Fig. 5 displays the results for this experiment preemptive preemptive preemptive 2% 3% 4% 5% 6% 7% 8% 9% % Fig. 5: Relative queueing time for reducing the lot size depending on utilization for 4 parallel tools (downtime cause: failures) Again, for high utilizations the values for preemptive downtimes are approached. At lower utilizations the behavior is contrary to the station consisting of a single tool. With four tools at hand downtimes of one tool do not cause a queue at low utilizations as the lots can make use of the other tools. Therefore, at lower utilizations the deterministic case is approached. High utilizations are generally expected in the semiconductor industry because tools are very capitalintensive. For this prevalent case both preemptive and nonpreemptive outages lead to similar results, therefore the differentiation is not a must and the queueing time approximation works very well. We continue the analysis with experiments considering downtimes that we have more control of like in the case of planned maintenance activities. These downtimes occur in fixed intervals and their length is characterized by low variability. In the experiments, we assume one planned maintenance activity that happens every 24 hours and lasts for one hour on average. Additionally, we assume another maintenance activity that happens every 4 weeks and lasts for 2.32 hours on average. This leads to a downtime impact of exactly 6% which matches the average downtime of our previous experiments. The downtimes itself follow a triangular distribution with a range of +/-2%. Under these conditions we perform experiments for a single tool station and a station with 4 parallel tools. Fig. 6 shows the results for the single tool station. 72

5 TuRP-B % 2% 3% 4% 5% 6% 7% 8% 9% % Fig. 6: Relative queueing time for reducing the lot size depending on utilization for a single tool (downtime cause: planned maintenance) The curves do not look fundamentally different to the ones for outages caused by failures. At low utilizations lot size changes do not make much of a difference for the same reasons. And the higher the utilization the more significant the difference becomes. However, higher reductions are achieved for downtimes caused by planned maintenance activities which again highlights the negative influence that variability has on the benefit of lot size reduction. At very high utilizations beyond 9% the curves show a swing downwards that is surprising at first. The queueing time reduction approaches the deterministic case values there. This in turn means that the main source of queueing time must be the same as in the deterministic case. And this makes sense. At high utilization the main cause for waiting time are other lots in the queue and at the tool, whereas the planned maintenance is a relatively small contributor. Fig. 7 displays the results for the station with 4 parallel tools % 3% 4% 5% 6% 7% 8% 9% % Fig. 7: Relative queueing time for reducing the lot size depending on utilization for 4 parallel tools (downtime cause: planned maintenance) Again, the curves in this figure very much resemble the ones for outages caused by failures. There are two major differences. Because of the lower variability the values nearly match the deterministic case over a wider utilization range. And at high utilizations there is again the swing downwards for the same reasons as for the single tool. We summarize the key conclusions of the experimental analysis:. For outages caused by failures and reasonable utilizations, (3) and (4) give excellent means for analyzing and benchmarking simulated or actual lot size reductions. It is less important whether the outages are preemptive or non-preemptive. 2. Because of its lower impact on variability, planned maintenance downtimes have a lower negative impact than failures on the effectiveness of lot size reduction with regard to queue time reductions. V. X-FACTOR CHANGES Based on raw process time and queueing time changes, we derive the resulting x-factor changes, where the x-factor is defined as the ratio of cycle time to raw process time which can be expressed through the parameters used above as QT + t + d x =. (9) t + d As an example, we will consider the case of halving the lot size. For this change t is halved, d stays constant and the new QT is somewhere between 5% and % of the original value depending on the degree of variability and the value of t. This leads to the non-intuitive conclusion that the x-factor can actually improve for the change to smaller lot sizes, especially for long delays d and low variability. Leaving the dependence on t aside, Table gives qualitative x-factor changes that happen for this change. In this table the plus sign(s) stand for a positive x-factor change which means a reduction of the x-factor value. The minus sign(s) stand for a negative x-factor change, which is an increase in its value. Table : Comparison of the likely x-factor changes for the switch to smaller lot sizes High variability Medium variability Low variability Short d -- - o Medium d - o + Long d o + ++ VI. SUMMARY In this paper, we discussed how queueing time changes for the shift to smaller lot size. The dependencies of this shift on type, frequency and length of outages have been shown and quantified. The queueing time reductions for smaller lot size have been shown to be very significant. For the switch from 73

6 TuRP-B to 2 wafer lot size currently discussed in the industry the reduction can be up to >5%. However, current availability characteristics will mostly lead to a smaller reduction. We also showed that variability not only drives queueing time in general, it also corrupts the effectiveness of the lot size reduction. The less variability in a system, the higher the relative queueing time improvement is for lot size reductions. Therefore variability reductions remain a key contributor to cycle time reduction efforts in the semiconductor industry that can be leveraged with lot size reduction. Furthermore we pointed out that for lot size reductions x- factor changes are possible in both directions depending on variability of the equipment and the size of the delay d. Further research will analyze the queueing time reduction based on the switch from batch to mini-batch or single wafer tools. Additionally, we will analyze how the results presented within this paper hold within a fab simulation model. ACKNOWLEDGMENT The authors thank AMD s Industrial Engineering Department for their helpful contributions. Special thanks go to Andreas Kirchberg for his valuable suggestions. REFERENCES [] O. Bonnin, D. Mercier, D. Levy, M. Henry, I. Pouilloux, E. Mastromatteo. Single-Wafer/Mini-Batch Approach for Fast Cycle Time in Advanced 3-mm Fab. IEEE Transactions on Semiconductor Manufacturing, Vol. 6, No. 2, pp. -2, 2. [2] O. J. Boxma, J. W. Cohen and N. Haffels. Approximations of the mean waiting time in an M/G/s Queueing System. Operations Research 27, pp. 5-27, 979. [3] D. Gross and C. M. Harris. The fundamentals of queueing theory. John Wiley & Sons, Inc. 3 rd edition, 998. [4] F. Hillier and G. Lieberman. Introduction to Operations Research, McGraw Hill Inc., 24. [5] W.J. Hopp and M.L. Spearman. Factory Physics. McGraw Hill Inc., 2. [6] J. K. Robinson. The P-K Formula at [7] K. Schmidt, O. Rose and J. Weigang. Modeling Semiconductor Tools for small lot size fab simulations, Wintersim Conference, 26. [8] K. Schmidt and O. Rose. Development and simulation assessment of semiconductor fab architectures for fast cycle times, Ph.D Colloquium of the Simulation and Visualization Conference, Magdeburg, 27. [9] K. Schmidt. Improving priority lot cycle times, ASMC 27 [] T. Wakabayashi, S. Watanabe, Y. Kobayashi, T. Okabe, A. Koike. High-speed AMHS and its operation method for 3mm QTAT fab, IEEE Transactions on Semiconductor Manufacturing, Vol. 7, No. 3, pp , 24. [] W. Whitt. Approximating the GI/G/m Queue. Production and Operations Management 2, pp. 4 6, Advanced Micro Devices, Inc. 74