Interpolation approximations for queues in series - Nanyang ...

IIE Transactions (2013) 45, 273–290 C “IIE” Copyright ISSN: 0740-817X print / 1545-8830 online DOI: 10.1080/0740817X.2012.682699

Interpolation approximations for queues in series KAN WU1,* and LEON MCGINNIS2 1

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Nanyang Technological University, School of Mechanical & Aerospace Engineering, 50 Nanyang Avenue, Singapore, 639798, Singapore E-mail: [email protected] 2 Georgia Institute of Technology, Department of Industrial and Systems Engineering, 765 Ferst Drive, Atlanta, GA 30332, USA E-mail: [email protected] Received October 2010 and accepted March 2012

Tandem queues constitute a fundamental structure of queueing networks. However, exact queue times in tandem queues are notoriously difficult to compute except for some special cases. Several approximation schemes that are based on mathematical assumptions that enable approximate analyses of tandem queues have been reported in the literature. This article proposes an approximation approach that is based on observed properties of the behavior of tandem queues: the intrinsic gap and intrinsic ratio. The approach exploits the nearly linear and heavy-traffic properties of the intrinsic ratio, which appear to hold in realistic production situations. The proposed approach outperforms existing approximation methods across a broad range of examined cases. It is also demonstrated that the proposed approach has the potential when applied to historical data to achieve accurate mean queue time estimates in practical production environments. [Supplemental materials are available for this article. Go to the publisher’s online edition of IIE Transactions to view the supplemental file.] Keywords: Queueing systems, manufacturing systems modeling, process simulation

1. Introduction To manage production systems effectively, a reliable prediction of the trade-off between throughput and queue time is essential. In a single workstation setting, this trade-off is generally well understood. However, a practical production system usually deals with multiple process steps and is composed of many workstations. Describing this trade-off for such a production system is much harder. A critical aspect of production systems that contributes to the difficulty in describing the trade-off is dependence among workstations in series. Some well-studied approaches address this dependence by considering only the first two moments of the departure process (Whitt, 1983) or by assuming that all servers experience heavy traffic (Dai and Harrison, 1992). Although these approaches lead to tractable approximations, their use seems to be limited in practice (Reiner, 2009). In this article we present an approach based on two newly discovered properties of tandem queues with infinite buffers: the intrinsic gap and intrinsic ratio. These properties, as discussed in Section 4, can be estimated from

∗

Corresponding author

C 2013 “IIE” 0740-817X

production data drawn from manufacturing operations. Using the proposed approach, we show that the dependence in production lines can be modeled in a way that (i) the resulting performance approximations do not become more computationally complex with either an increase in the number of workstations or the system utilization; and (ii) significant gains in accuracy are achieved in most of the performed simulations, relative to existing approaches. Moreover, the approach we propose can be adapted to exploit historical operational data to achieve quite reliable queue time estimates. In order to convey the fundamental insight and basic computational processes associated with our approach, we focus our presentation on production systems that may be treated as multiple single servers in series; i.e., as tandem queues. Hence, all workstations (or stations) in this article are composed of a single server. We specifically examine tandem queues without rework or feedback, and the only external arrival process is at the first station of the tandem queue. To differentiate this external arrival process from the arrival processes at the subsequent stations, we also call it an initial arrival process. When a system is composed of two single servers in series, it is called a simple tandem queue. It is the simplest queueing system that exhibits dependence among servers in series.


274 There are many practical manufacturing systems that can be modeled as tandem queues. For example, any mixedmodel assembly process has the structure of a tandem queue. Tandem queues are the fundamental structure embedded in manufacturing systems. Studying its behavior will give us valuable insights to understand the behavior of a manufacturing system. Due to the non-renewal departure process, queueing networks can be analyzed exactly only for some special Markovian cases. Landmark papers in this regard are Jackson (1957) and Baskett et al. (1975). We refer to Serfozo (1999) and Chen and Yao (2001) for a state-of-the-art survey. Since tandem queueing models cannot be analyzed exactly under realistic assumptions, the only alternative to simulations is approximations. This has been realized by researchers during the past few decades. The most popular approximation scheme for tandem queues is the Queueing Network Analyzer (QNA) developed by Whitt (1983) in the 1980s. The approximation suggested in this article is distinctively different from QNA. It builds on the key idea that Jackson networks are not the only queueing networks that can be analyzed exactly. As we will indicate with an extensive set of experiments, this leads to a new family of approximations that are as explicit as QNA but incorporate dependence in a way that outperforms QNA. Virtually all existing tractable results in FirstCome–First-Serve (FCFS) queueing networks only hold exactly under memoryless assumptions, which can be restrictive. An important exception is formed by two early papers that build a cornerstone for our modeling approach. Specifically, Avi-Itzhak (1965) and Friedman (1965) investigated the behavior of tandem queues with constant service times. In particular, Friedman showed the following important properties. If customers arrive at the first stage and proceed through the stages in FCFS order with infinite buffers, then (i) for any sequence of customer arrival times, the time spent in the system by each customer is independent of the order of the stages; and (ii) under certain conditions, a tandem queueing system can be reduced to a corresponding system with fewer stages, possibly a single stage. This procedure is called a reduction method. Consequently, the total system queue time is determined solely by the bottleneck workstation; i.e., the system queue time of any job equals the time the same job would have been waiting in the queue of the bottleneck workstation, assuming the same arrival process in both systems. Therefore, we can analyze such a queueing system exactly even though there is dependence among workstations. For tandem queues, Jackson and Friedman’s results seem totally different at first sight, but indeed share a very important structure: in both, some servers in a tandem queue can see the external arrivals directly. In tandem queues with Poisson arrivals and exponential service times, all servers see Poisson arrivals in steady state, and in tandem queues with constant service times, the bottleneck sees the external arrival process. Based on this key insight,

Wu and McGinnis we define two characteristic attributes, which we call the intrinsic gap and the intrinsic ratio. New approximate models are developed based on those characteristic attributes without directly dealing with the non-renewal departure process. For example, when the external arrival process is Poisson and service times are independent and identically distributed (iid), the expected queue time (QT) of a simple tandem queue where the second queue is the bottleneck can be determined by interpolating between the two exact results (detailed explanation is given in Section 5): 2 1 + cs1 ρ1 1 ∼ QT = QT1 + QT2 = y2 2 1 − ρ1 μ1 2 1 + cs2 ρ2 1 + , 2 1 − ρ2 μ2 where y2 is the intrinsic ratio, ρ i is utilization, csi is the coefficient of variation of the service time, and μi is the service rate of station i. The accuracy of this approximation is determined by the value of y2 , where y2 is a function of service and inter-arrival time distributions. Two properties of the intrinsic ratio, which we call the nearly linear and heavy-traffic properties, enable quite good estimates of its value. Drawing from the results of extensive simulations, we argue in this article that the proposed approach outperforms earlier methods that are based on the parametric decomposition and diffusion approximation approaches. In this article, if not specified, we assume that the dispatching policy is FCFS, the buffer in front of each station is infinite, the service times of each workstation and the external inter-arrival times are iid, and the service time sequences of all workstations and the external inter-arrival time sequence are mutually independent. The rest of this article is organized as follows. A literature review is given in Section 2 and definitions of some critical terms are given in Section 3. For the case of simple tandem queues, Section 4 presents the intrinsic gap and intrinsic ratio along with some important properties and observations. In Section 5, the approximate model for simple tandem queues is introduced and its performance is validated under Poisson arrivals. In Section 6, the model is extended to many single-server tandem queues with general arrivals, and the adaptation to use historical data is discussed. In Section 7, we draw conclusions and discuss future work.

2. Literature review The parametric decomposition method (Kuehn, 1979) analyzes the nodes in the queueing networks separately by assuming that each node is stochastically independent. This dates back to Kleinrock’s independence assumption (Kleinrock, 1976). Whitt (1983) developed the QNA approach based on this assumption, Kingman’s heavy-traffic approximation (Kingman, 1965; Heyman, 1975), and Marshall’s

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Interpolation approximations for queues in series


equation (Marshall, 1968). The essential model for QNA is 2 ca + cs2 ρ 1 ∼ QT(G/G/1) = , (1) 2 1−ρ μ cd2 = ca2 + 2ρ 2 cs2 − (2ρ)(1 − ρ)μQT ∼ (2) = ρ 2 cs2 + (1 − ρ 2 )ca2 , where QT is the expected queue time (excluding service), ρ is the utilization, μ is the service rate (or capacity), σ a is the standard deviation of the inter-arrival time, σ s is the standard deviation of the service time, cd is the coefficient of variation of the departure intervals, ca is the coefficient of variation of the inter-arrival time, and cs is the coefficient of variation of the service time. Variability is the squared coefficient of variation. When the arrival process is Poisson, Equation (1) reduces to the Pollaczek–Khintchine formula (Khintchine, 1932; Pollaczek, 1932), which is exact. When the arrival process is independent and generally distributed, the percentage error of Equation (1) converges to zero in heavy traffic. Whitt (1985) pointed out that the performance of the parametric decomposition method deteriorates in the presence of high variability, especially in the arrival process, and the method tends to perform poorly when the service time is deterministic or nearly deterministic. Boxma (1979a, 1979b) investigated the simple tandem queue with identical (generally distributed but perfectly correlated) service times at both stations and showed analytically that Kleinrock’s independence assumption is violated. Reiser and Kobayashi (1974) used the diffusion process approximation to develop analytical models of computing systems by considering service time distributions of a general form. Through multidimensional-reflected Brownian motion, Harrison and Nguyen (1990) used QNET to approximate queueing networks. QNET is an analytical method for estimating the long-run average cycle time. Dai and Harrison (1992) developed QNET further to obtain numerical results. QNET is based on a Functional Central Limit Theorem (FCTL), involving a sequence of networks in which all nodes are assumed to be in heavy traffic. However, the convergence speed of FCLT can be very slow when service time variability is close to zero. In practical manufacturing systems, to ensure product quality, process time variability is required to be small. Furthermore, to reduce waste and maintain competitiveness, unnecessary WorkIn-Process (WIP) has to be eliminated, which eventually leads to small variability in service times. Due to its assumptions, the QNET algorithm tends to perform well in networks where all servers are heavily utilized with large service time variability. However, such a regime may not always be realistic in practice and there is no guarantee on the speed of convergence, especially when the service time variability is small. Furthermore, the computational complexity of the QNET algorithm grows with the size of the network. In order to overcome the computational complexity issues associated with QNET, Dai et al. (1994) de-

veloped the Sequential Bottleneck Decomposition (SBD) method to reduce the computational complexity by grouping workstations with similar utilizations and limiting these sub-networks to a reasonable size. In addition to analytical queueing models, discrete-event simulation models are sometimes used to predict system performance. Simulation can capture the dependence among workstations without making strong assumptions and potentially can give more reliable results. However, constructing and running a simulation model can be time consuming and the model only provides a snapshot of the system’s performance instead of a complete response profile. To overcome the snapshot issue, Yang et al. (2007) used simulation to construct a complete cycle time–throughput curve by a non-linear regression metamodel. However, to conduct the regression, many data points are needed from simulation.

3. Definitions Throughout this article, the term bottleneck refers to a throughput bottleneck, which is the workstation with the highest utilization. Thus, the utilization of a tandem queue is its bottleneck utilization. A variability factor (α) can be used to quantify the trade-off between queue time and throughput. Based on Kingman’s approximation, the variability factor for a single server is defined as QT(G/G/1) ρ 1 α= = QT(G/G/1) QT(M/M/1) 1−ρ μ 2 2 + c c a s ∼ , (3) = 2 where QT(G/G/1) is the expected queue time of a G/G/1 queue, and QT(M/M/1) is the expected queue time of an M/M/1 queue. This definition corresponds to what Hopp and Spearman (1996) call the variability factor in their VUT formulation. Figure 1 shows n single-server queues in series, where QTi is the mean queue time of the ith server. The external arrival process has rate λ with coefficient of variation ca1 . The mth server is the bottleneck server (BN in the figure). Embedded in the general tandem queueing system shown in Fig. 1 are two idealized tandem queues (see Fig. 2) that are the foundation for the proposed interpolation approach.

(λ, ca1)

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Fig. 1. n Single-server queues in series.

…

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Wu and McGinnis BSIA system queue times; that is,

QT1A

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Fig. 2. ASIA (left) and BSIA (right) systems.

Definition 1. In an ASIA system, QTi = QTiA , where QTiA is the mean queue time of the ith server when All See the Initial Arrivals (ASIA) directly. Since all queue times are mutually independent, it can be viewed as a decoupled system. The ASIA system is motivated by tandem queues with Poisson arrivals and exponential service times. In this situation, all workstations see Poisson arrivals (Burke, 1956), which is indeed the external arrival process at the first queue. For a given tandem queue with n servers in steady state, its ASIA system is composed of n independent systems, where each system contains a single server and sees the initial arrival process directly. If the initial arrival process is renewal, the ASIA system queue time can be estimated by Kingman’s approximation. While an ASIA system is inspired by tandem queues with exponential service times (i.e., the coefficient of variation is one), the following BSIA system is inspired by tandem queues with deterministic service times (i.e., the coefficient of variation is zero). Definition 2. In a BSIA system, in=1 QTi = QT1B , where QT1B is the mean queue time when the Bottleneck Sees the Initial Arrivals (BSIA) directly. Since all queue times are mutually dependent and simply determined by the bottleneck, the BSIA system is also called a Fully Coupled System (FCS). It is motivated by Friedman’s results: When service times are constant, the queue time of any job equals the time the same job would have been waiting in the queue of the bottleneck, assuming the same arrival process in both systems (Avi-Itzhak, 1965; Friedman, 1965). While Friedman’s result is a sample path argument, one can apply this result to stochastic models via an ergodic theorem and law of large numbers if the limiting average on the sample path exists (El-Taha and Stidham, 1999). Since Friedman’s results are applicable to tandem queues with any specified arrival process, it should be noted that a BSIA system may have either renewal or non-renewal arrival processes. Definition 3. For a server in tandem queues, its Intrinsic Gap (IG) is the difference between its expected ASIA and

(4)

where QT is the expected queue time. Two important ratios are defined based on the IG. Definition 4. For a server in tandem queues, its Intrinsic Ratio (IR) is defined as IR =

Actual QT − QT in BSIA system . IG

(5)

Definition 5. For a server in tandem queues, its Intrinsic Gap Ratio (IGR) is the ratio of its IG to its expected AISA system queue time, that is IGR =

IG . QT i n ASIA system

(6)

4. Property of simple tandem queues The fundamental insight of the proposed approximation comes from a careful analysis of simple tandem queues; i.e., carefully examining the behavior of the IG and IR. In a simple tandem queue, the first queue time can be approximated by Kingman’s equation if the arrival process is renewal. The challenge comes from the second queue time since its arrival process may not be renewal. There are two possible cases: (i) the second station has higher utilization and (ii) the first station utilization is at least as large as the second. We call the first situation a Simple Tandem Queue with Backend (STQB) bottleneck because the system queue time is dominated by the second queue time. Case (ii) is called a Simple Tandem Queue with Front-end (STQF) bottleneck. In this case, the queue time approximate errors at the second station are expected to make only a minor contribution to the system’s queue time, especially in heavy traffic. When the two stations have the same utilization, their queue times are about the same based on Kingman’s approximation; i.e., they both are ∼ ρ BN /(1 – ρ BN ), where ρ BN is the bottleneck utilization. Thus, it is treated as an STQF bottleneck since the second queue time approximate error is not as important as it is for the STQB bottleneck. Accurately predicting the bottleneck queue time for the STQB bottleneck is important but difficult, since the system’s queue time is dominated by the bottleneck queue time while the bottleneck faces a non-renewal departure process. On the other hand, since the external arrival process can be assumed to be renewal, for an STQF bottleneck, the bottleneck queue time, which dominates the system queue time in heavy traffic, can be approximated more reliably. Accurately approximating the second station queue time for an STQB bottleneck is more important than for an STQF bottleneck. Therefore, we will study the behavior of the second queue time for an STQB bottleneck in detail.



For the STQF bottleneck, we mainly focus on its performance when the service time Squared Coefficient of Variation (SCV) is smaller than one, since it is the practical range in manufacturing systems. We begin by analyzing the IG and IR for STQB and STQF bottlenecks. These results form the foundations to estimate the queue times for more than two stations in series. For simple tandem queues, the ASIA system expected queue time of the second station (QT2A) can be approximated by 2 2 ca1 + cs2 ρ2 1 , (7) QT2A ∼ = 2 1 − ρ2 μ2 2 2 where ca1 is the inter-arrival time SCV of the first station, cs2 is the service time SCV of the second station, μ2 is the mean service rate of the second station, and ρ 2 is the utilization of the second station. While the FCS queue time of the second station for a STQF bottleneck is zero, the FCS expected queue time of the second station (QT2C ) in STQB is given by 2 2 ca1 + cs2 ρ2 1 QT2C ∼ = 2 1 − ρ2 μ2 2 2 ca1 + cs1 1 ρ1 − , (8) 2 1 − ρ1 μ1 2 where cs1 is the service time SCV of the first station, μ1 is the mean service rate of the first station, and ρ 1 is the utilization of the first station, and other notation is the same as for Equation (7). The IG of the second station for an STQF bottleneck is 2 2 ca1 + cs2 1 ρ2 C A ∼ − QT = QT . IG = QT A = 2 2 2 2 1 − ρ2 μ2 (9)

The IG of the second station for an STQB bottleneck is 2 2 ca1 + cs1 ρ1 1 A C . IG = QT 2 − QT 2 = QT 1 = 2 1 − ρ1 μ1 (10) For an STQB bottleneck, the IG of the second station is exactly the same as the queue time of the first station. Therefore, this IG possesses the following nice property. Theorem 1. (Heavy-traffic property of IG for an STQB bottleneck.) In simple tandem queues with backend bottlenecks, the IGR of the second station goes to zero as its traffic intensity goes to one.

Proof. By Definition 5, IGR = IG/QT A 2 . By Equations (7) and (10): 2 2 + cs1 /2 (ρ1 /(1 − ρ1 )) (1/μ1 ) QT 1 ∼ ca1 2 IGR = . A = 2 ca1 + cs2 /2 (ρ2 /(1 − ρ2 )) (1/μ2 ) QT 2 Since the second station is the bottleneck, ρ2 → 1 while ρ 1 is strictly smaller than one. Because the percentage error

277 of Kingman’s heavy-traffic approximation goes to zero as traffic intensity goes to one, IGR → 0 as ρ2 → 1. For an STQB bottleneck, when randomness exists, the second queue time as well as its ASIA system and FCS queue times approach infinity when the traffic intensity (ρ 2 ) approaches one. However, as the traffic intensity approaches one, the IG remains finite. Thus, the IGR of the second station goes to zero in heavy traffic. On the other hand, for an STQF bottleneck, the IGR for the second station is one across all traffic intensities. The heavy-traffic property does not hold for an STQF bottleneck. To gain further insight into the implications of Theorem 1, we conducted a number of simulation experiments for both STQB and STQF bottlenecks. In these experiments, we assumed Poisson arrivals (i.e., SCV = 1) and gammadistributed service times. In a series of experiments for “small” SCVs, values for the two stations were chosen from {0.1 (low), 0.5 (medium), 0.9 (high)}, resulting in nine experiment settings; i.e., (1, 0.1, 0.1), (1, 0.1, 0.5), (1, 0.1, 0.9), (1, 0.5, 0.1), (1, 0.5, 0.5), (1, 0.5, 0.9), (1, 0.9, 0.1), (1, 0.9, 0.5), and (1, 0.9, 0.9), where the first number in the bracket is the inter-arrival time SCV, the second one is the service time SCV of the first server, and the last one is the service SCV of the second server. In each experiment for the STQB bottleneck, the service time of the second station was always 30, and the first service time was chosen from {10, 20, 25, 29}. For the STQF bottleneck, the service time of the first station was always 30, and the second service time was chosen from {10, 20, 25, 30}. Hence, there were a total of 36 experiments for each STQB and STQF bottleneck. When the SCV of the service time is smaller than one, the simulation results for the STQB and STQF bottlenecks are shown in Appendices A and B in the online supplement. Each observation in the tables is the average of between 100 and 200 replications. Depending on the utilization and service time, each replication is the average of between 200 000 and 50 000 000 data points after a warm-up period of 50 years (i.e., 87 600 (at 10% utilization) to 832 200 (at 95%) data points) or longer. Among the 72 cases, their halfwidth 90% confidence intervals are all within 1.8% of their mean simulated queue times, and most are within 0.5% of the mean queue times. When the SCV of the service time is smaller than one, the IRs of the second station in the STQB bottleneck cases are shown in Fig. 3. Each curve represents one of the 36 experiments in Appendix A. All IRs are between zero and one and nearly constant as a function of utilization when utilization is less than 80%. When utilization increases, the IGR decreases based on Theorem 1, and the ratio of the confidence interval to the mean queue time also increases. If the simulation runs are not long enough, the IG can be smaller than the confidence intervals in heavy traffic. Hence, some of the IRs are irregular in heavy traffic due to the larger confidence intervals. When the SCV of the service time is smaller than one, the IRs of the second station in the STQF bottleneck cases

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Wu and McGinnis 100% 90% 80%

Intrinsic Ratio

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Fig. 3. IR versus utilization of the STQB bottleneck when the SCV of the service time is smaller than one (color figure provided online).

are shown in Fig. 4. Each curve represents one of the 36 experiments in Appendix B. Since the STQF bottleneck does not possess the heavy-traffic property of the IG, the IGR does not go to zero in heavy traffic. The IR behaves more regularly and is nearly constant as a function of utilization. It should be noted that for both the STQB and STQF bottlenecks, the IR is smaller than one when the SCV of the service time is smaller than one. The results from QNA and QNET were also computed along with the simulation results as shown in Appendices A and B in the online supplement. The results for QNA

were obtained from Equations (1) and (2). The approximation from the QNET was obtained by running the QNET code developed by Dai (1992). In STQB bottlenecks with small variability, the average errors from QNA and QNET are 11.0 and 13.1%, respectively. In STQF bottlenecks, the average errors from QNA and QNET are 261.9 and 80.1%. We additionally examined the case where the service time variability is greater than one for the STQB bottleneck. As before, we conducted 36 experiments with Poisson arrivals and gamma-distributed service times. The service times were as in the first series of experiments. However, in this

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Fig. 4. IR versus utilization of the STQF bottleneck when the SCV of the service time is smaller than one (color figure provided online).

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Fig. 5. IR versus utilization of the STQB bottleneck when the SCV of the service time is greater than one (color figure provided online).

series, the SCV of the service time was chosen from two, five, and eight. Each observation was taken as the average of 100 replications. Each replication was the average of between 200 000 and 1000 000 data points after a warm-up period of 50 years (87 600 to 832 200 data points) or longer. For the considered 36 cases, their half-width 90% confidence intervals are all within 3.0% of their mean simulated queue times, and most are within 1.0% of the mean queue times except for some heavy-traffic cases. The simulation results are shown in Fig. 5 and Appendix C in the online supplement. The average errors from QNA and QNET are 8.3 and 10.3%, respectively. When the SCV of the service time is greater than one, the IR is greater than one with a positive slope. Compared with queue time, which diverges to infinity as utilization goes to one, the second derivative of the IR is much smaller. In other words, the IR behaves much more linearly than its queue time. Based on Figs. 1, 2, and 3, we have the following observation. Observation 1. (Nearly linear relationship). The intrinsic ratio is approximately linear relative to traffic intensities. Since the IR is developed based on the ASIA and BSIA systems, it is possible to obtain the IR exactly when the arrival process is Poisson and the service times are exponential or constant. When the arrival process is Poisson, the IR is one if service times are exponential and is zero if the service times are constant at all utilizations. Hence, the IR slopes are zero in these two cases. From simulation results, the slope of the IR is also close to zero when the SCV of the service time is between zero and one. Although the slopes become positive when the SCV of the service

time is greater than one, their second derivatives are still much smaller than those of the queue time curves. More discussion on Figs. 3, 4, and 5 will be given in Section 4.1. 4.1. Observations and connections to prior research From Figs. 1 and 2, it can be observed that the queue time for the ASIA system acts like an upper bound when the arrival process is Poisson and the SCV of the service time is smaller than one. This has been proven rigorously under some special conditions; e.g., Tembe and Wolff (1974) gave an upper bound on the cycle time of the second station in an M/D/1 → M/1 system as ρ2 1 1 1 = + , (11) CT2 ≤ μ2 − λ 1 − ρ2 μ2 μ2 where CT is the expected cycle time (including service). This upper bound is simply the M/M/1 cycle time of the second station, and is indeed the same as the cycle time of the ASIA system. When the arrival process is Poisson and the first service time is exponential, the departure process of the first station is also Poisson (Burke, 1956). The second station can be modeled as an M/G/1 queue. The actual queue time of the second station is the same as the queue time of the ASIA system. The upper bound is tight in this case. When the arrival process is not Poisson but generally distributed, Niu (1980) claimed the following upper bound without proof: In a GI/D/1 → G/1 system, the stationary expected delay in front of the second station is smaller than it would be if there was no first station at all. Niu’s upper bound is the same as the queue time for the ASIA system.


280 From Figs. 1 and 2, it also can be observed that the FCS queue time behaves like a lower bound when the arrival process is Poisson and the SCV of the service time is smaller than one. Based on Friedman (1965), this lower bound is tight when service times are constant. When the arrival process is Poisson and the SCV of the service time is greater than one, it can be verified with Appendix C (in the online supplement) that the ASIA system queue times are always smaller than the simulated queue times. An interesting observation is that all of the above observed upper or lower bounds are consistent with the predictions of Kingman (1965) and Marshall (1968) even though the departure process is not renewal. For example, when the arrival process is Poisson and the SCV of the service time is greater than one, the equations presented in Kingman (1965) and Marshall (1968) predict that the ASIA system queue time is a lower bound at the second station. Based on the above observations, we propose the following conjecture. Conjecture 1. (Upper and lower bounds for the second queue time with Poisson arrivals.) In a simple tandem queue with Poisson arrivals, if the SCV of the service time of the first station is smaller than one, the mean queue time of the ASIA system is an upper bound and the mean queue time of the BSIA system is a lower bound of the second mean queue time. If the SCV of the service time of the first station is greater than one, the mean queue time of the ASIA system is a lower bound of the second station’s mean queue time. Because of Conjecture 1 and Theorem 1, the second mean queue time for an STQB bottleneck with Poisson arrivals is expected to be bounded within the intrinsic gap, and the IGR of the second station goes to zero in heavy traffic. These nice properties give us a reliable way to estimate the second mean queue time for an STQB bottleneck.

5. Approximate models of simple tandem queues In approximating the queue time of simple tandem queues, the STQB bottleneck is the crucial case, since the system’s queue time is dominated by the second queue. In general, the input process for the second queue is non-renewal and thus is not as well understood as the renewal case, for which we have Kingman’s approximation. We begin by developing the approximate models for an STQB bottleneck in Section 5.1 and then for an STQF bottleneck in Section 5.2. 5.1. Approximate models for an STQB bottleneck Based on the heavy-traffic property and nearly linear relationship of the IRs, we propose to approximate the bottleneck queue time in an STQB system by 2 2 ca1 + cs2 ρ2 1 A ∼ QT 2 = QT 2 − (1 − y2 ) × IG = 2 1 − ρ2 μ2

Wu and McGinnis −(1 − y2 )

2 2 ca1 + cs1 2

ρ1 1 − ρ1

1 , μ1

(12)

where y2 is the IR between the first and second stations. When y2 has a value of one, Equation (12) is the second queue time in the ASIA system. When y2 has a value of zero, Equation (12) is the second queue time in the FCS. The heavy-traffic property of the IG (Theorem 1) plays an important role in approximating the second queue time of the STQB system. When ρ 2 goes to one, IGR goes to zero. Hence, the queue time approximate error caused by any imprecise estimate of the IR also becomes small. The value of the IR is determined by four factors: external arrival process, service time ratio (first service time/second service time), and the SCVs of the service times of the first and second stations. If the IR can be approximated by the first and second moments of the interarrival and service times, it can be presented as a function of those parameters: (13) IR : y2 ∼ = f λ, c2 , ST1 , ST2 , c2 , c2 , a1

s1

s2

where λ is the arrival rate, and STi is service time of the ith station. The challenge is to determine a good estimate of y2 . Two heuristics are proposed for estimating y2 . 1. Make it equal to the coefficient of variation of the first station service time. 2. Determine its value from the QNA queue time approximation at a utilization of 80%. The first heuristic is a direct result of the previous observation: When y2 is either zero or one, Equation (12) is either the second queue time in the FCS or the ASIA system, respectively. Using cs1 to approximate y2 seems to be a reasonable choice, since cs1 is zero in the FCS and is one in the ASIA system. We call the first heuristic the IR method with the first service time coefficient of variation (or IRCS1) and the second heuristic the IR method with QNA (or IR-QNA), since they are based on the properties of the IR. As shown in Appendix A in the online supplement, when the arrival process is Poisson and service time variability is smaller than one, the average error of the first heuristic is 3.3% (cf. 11.0% for QNA and 13.1% for QNET). Since the first queue time can be computed exactly in this case (and approximated by Kingman’s approximation for general arrivals), it is important to see the impact of the errors on the system’s queue time. If we give a weight to the approximate errors by “Second QT/System QT,” the weighted average errors are 1.8% for the first heuristic, 6.3% for QNA, and 10.3% for QNET. When the SCV of the service time is greater than one, the first heuristic does not perform well as shown in Appendix C in the online supplement. In this situation, we may simply use QNA or the second heuristic. The second heuristic is driven by consideration of the approximate errors of QNA in Appendices A and C in the online supplement. For the STQB cases examined, when

281



the arrival process is Poisson and the service times are gamma distributed, QNA gives minimum errors for the second station when the bottleneck utilization is around 80%. By making Equation (1) identical to Equation (12), y2 can be obtained as follows: 2 1 + cs2 ρ2 1 ∼ QT2 = 2 1 − ρ2 μ2 2 1 + cs1 ρ1 1 − (1 − y2 ) 2 1 − ρ1 μ1 2 2 ca2 + cs2 1 ρ2 ∼ , (14) = 2 1 − ρ2 μ2 2 ∼ 2 2 2 and thus: where ca2 = ρ1 cs1 + (1 − ρ12 )ca1 2 1 + cs2 ρ2 1 ∼ y2 = 1 − 2 1 − ρ2 μ2 2 2 ca2 + cs2 1 ρ2 − 2 1 − ρ2 μ2 2 1 + cs1 ρ1 1 . × 2 1 − ρ1 μ1

(15)

where ρ 2 is 0.8. By substituting for y2 in Equation (12), we can obtain the approximate queue time at other utilizations. Because ρ 2 is pretty high, together with Theorem 1, we can expect to obtain good approximations at the other points in heavy traffic. When the arrival process is Poisson and service time variability is smaller than one, the average error of IR-QNA is 5.6% (cf. 11.0% for QNA and 13.1% for QNET) as shown in Appendix A in the online supplement. The weighted average errors are 3.7% for the second heuristic, 6.3% for QNA, and 10.3% for QNET. When the SCV of the service time is greater than one, the average error of the second heuristic is 14.6% (cf. 8.3% for QNA and 10.3% for QNET) as shown in Appendix C in the online supplement. The weighted average errors are 9.4% for IR-QNA, 5.6% for QNA, and 6.9% for QNET. Although the second heuristic has a larger average error, it performs the best in heavy traffic. The error is only 1.8% at 95% utilization (cf. 20.1% for QNA and 2.7% for QNET). This is consistent with our observations about Fig. 5: since the slope of IR is positive when the SCV of the service time is greater than one, having only a good estimate of the IR at 80% utilization cannot guarantee a good estimate of the queue time in light traffic. Therefore, a potential approach when the SCV of the service time is greater than one is to use QNA when the utilization is less than 80% and use IR-QNA in heavy traffic. In practical manufacturing systems, small service time variability is preferred to maintain competitiveness. The IR method performs well in this region for an STQB bottleneck. Furthermore, for an STQB bottleneck, when the service time variability is smaller than one, the IR method is the only one that possesses the heavy-traffic property among the three (see Appendix A in the online supple-

Table 1. Performance of different approximate methods for STQB

SCV < 1 SCV > 1 SCV < 1 SCV > 1

QNA

QNET

IR-CS1

IR-QNA

6.3% 5.6% — M/L

10.3% 6.9% — —

1.8% 23.5% H/M/L —

3.7% 9.4% H H

ment). QNA has its minimum errors at between 70 and 80% utilization. QNET performs poorly when the SCV of the service time is small. The weighted average errors of different methods are summarized in the upper part of Table 1. The suggested methods for different SCV ranges are noted in the lower part of Table 1, where H is heavy traffic, M is medium traffic, and L is light traffic. Note that IR-QNA performs consistently well in heavy-traffic conditions. From the results for IR-CS1, one may conclude that the SCV of the first station has a greater impact on the IR than does the SCV of the second station. When the SCV of the second station falls into a different range than that of the SCV of the first station (e.g., the SCV of the first station is smaller than one, whereas the SCV of the second station is greater than one), this observation is still supported by the other two simulation experiments in Appendix F in the online supplement. Hence, the SCV ranges in Table 1 refer to the SCV of the first station. 5.2. Approximate models for simple tandem queues with front-end bottlenecks For the STQF bottleneck, the IG is determined by Equation (9) instead of Equation (10). Since the BSIA system queue time is zero for the second station in an STQF bottleneck, the second queue time for an STQF bottleneck can be approximated as 2 2 ca1 + cs2 1 ρ2 A ∼ , (16) QT2 = x2 × QT2 = x2 2 1 − ρ2 μ2 2 2 2 where the IR is x2 ∼ , ST1 , ST2 , cs1 , cs2 ). = f (λ, ca1 In Appendix B (in the online supplement), when the arrival process is Poisson, QNA gives the smallest error when the traffic intensity is greater than 80%. Specifically, it works well at 80% when the two stations have the same service time (i.e., the same parameter for the STQB bottleneck), and it works well in heavy-traffic conditions when there is a distinct bottleneck (i.e., the second service times are 10, 20, or 25). An important cause of dependence between two consecutive stations is the WIP in front of the first station. When WIP exists, the inter-departure times between consecutive jobs become correlated and thus the departure process is non-renewal in nature. Based on the extensive simulation results for the STQF bottleneck, when the external arrival process is Poisson and the service times are gamma distributed, the absolute errors decrease in heavy-traffic

282

Wu and McGinnis


Table 2. Mean queue times and the 90% confidence intervals Util (%)

QT 1

90% CI

QT 2

90% CI

QT 3

90% CI

QT 4

90% CI

QT 5

90% CI

10 20 30 40 50 60 70 80 90 95

1.07 2.31 3.75 5.46 7.51 10.01 13.14 17.17 22.50 25.86

0.22 0.18 0.14 0.13 0.14 0.17 0.15 0.20 0.20 0.20

1.49 3.27 5.45 8.16 11.66 16.37 22.96 33.06 49.89 63.91

0.21 0.19 0.16 0.16 0.17 0.18 0.17 0.25 0.34 0.30

2.11 4.74 8.11 12.59 18.92 28.57 44.90 78.57 183.81 398.97

0.17 0.15 0.15 0.17 0.16 0.22 0.27 0.36 0.85 1.43

1.27 2.76 4.55 6.75 9.54 13.23 18.30 25.72 37.60 46.59

0.19 0.17 0.16 0.16 0.18 0.17 0.21 0.22 0.24 0.31

0.73 1.56 2.48 3.57 4.84 6.37 8.25 10.62 13.70 15.61

0.23 0.18 0.17 0.17 0.17 0.16 0.16 0.18 0.18 0.21

conditions but will not go to zero (except for the case with equal service times). Similar to the STQB bottleneck, the IR (x2 ) can be approximated using QNA as follows: 2 2 ca2 + cs2 ρ2 1 QT2 ∼ = 2 1 − ρ2 μ2 2 2 ca1 + cs2 1 ρ2 = x2 , (17) 2 1 − ρ2 μ2 2 ∼ 2 2 2 and thus where ca2 = ρ1 cs1 + (1 − ρ12 )ca1 2 2 2 2 / ca1 . x2 = ca2 + cs2 + cs2

(18)

where ρ 2 is 0.8 when μ1 = μ2 , and ρ 2 is 0.999 when μ1 < μ2 . By substituting for x2 in Equation (16), the approximate queue time at other utilizations can be obtained. As shown in Appendix B in the online supplement, the average error for this approach is 74.3% (cf. 261.9% for QNA and 80.1% for QNET) when the arrival process is Poisson and the service time variability is smaller than one. As we have explained, the approximate errors of the second queue time for an STQF bottleneck have a lower impact on the system queue time approximation. If we give a weight to the approximate errors by Second QT/System QT, the weighted average error becomes 1.6% for the IR method, 6.4% for QNA, and 1.2% for QNET. The performances of the QNET and IR methods are about the same. However, since the system queue time is dominated by the first station in an STQF bottleneck, the weighted average errors at the second station are small, and the approximate errors of the STQF bottleneck have little impact on the gross system queue time.

6. Approximation for multiple single-server queues in series The properties of the IG and IR play a key role in the proposed approximate model of simple tandem queues. It is important to find out if these properties carry over to tandem queues with more than two stations.

We examined several cases of five stations in series via simulation. The first case had mean service times of 20, 25, 30, 25, and 20 with Erlang(2) distribution (i.e., the SCV is 0.5). The arrival process was taken to be Poisson. The traffic intensity at the bottleneck (i.e., the third station) varied from 10 to 95%. One hundred replications were performed at each specific input rate. Each of the 100 replications was composed of 200 000 data points (i.e., 20 000 000 data points in total) after a 50-year warm-up period; i.e., 87 600 (at 10% utilization) to 832 200 (at 95%) data points were discarded. For different utilizations, the mean queue times of each station and their half-width 90% confidence intervals are shown in Table 2. The ASIA system and BSIA system queue times of the five stations are as follows: 2 1 + cs2 ρ2 1 , 2 1 − ρ2 μ2 2 1 + cs2 ρ2 1 = 2 1 − ρ2 μ2 2 1 + cs1 ρ1 1 − , 2 1 − ρ1 μ1 2 1 + cs3 ρ3 1 = , 2 1 − ρ3 μ3 2 1 + cs3 ρ3 1 = 2 1 − ρ3 μ3 2 1 + cs2 ρ2 1 − 2 1 − ρ2 μ2 2 1 + cs1 ρ1 1 − , 2 1 − ρ1 μ1 2 1 + cs4 ρ4 1 = , QT4C = 0, 2 1 − ρ4 μ4 2 1 + cs5 ρ5 1 = , QT5C = 0. 2 1 − ρ5 μ5

QT2A = QT2C

QT3A QT3C

QT4A QT5A

Because the third station is the bottleneck, the FCS queue times of the fourth and fifth station are zero. The ASIA

283

Interpolation approximations for queues in series Table 3. ASIA and BSIA system queue times, intrinsic gaps and intrinsic ratios 2nd Server

10 20 30 40 50 60 70 80 90 95

LB

UB

Gap

IR (%)

0.63 1.44 2.50 3.92 5.89 8.74 13.11 20.33 33.75 45.39

1.70 3.75 6.25 9.38 13.39 18.75 26.25 37.50 56.25 71.25

1.07 2.31 3.75 5.46 7.51 10.01 13.14 17.17 22.50 25.86

79.9 79.3 78.6 77.8 76.9 76.2 74.9 74.1 71.7 71.6

LB

UB

Gap

−0.06 2.50 2.56 0.05 5.63 5.58 0.45 9.64 9.20 1.38 15.00 13.62 3.33 22.50 19.17 7.38 33.75 26.37 16.41 52.50 36.09 39.78 90.00 50.22 130.11 202.50 72.39 337.73 427.50 89.77

system queue time, FCS queue time, IGs, and IRs of each station are listed in Table 3. Figure 6 shows the IR values of stations 2 to 5. The nearly linear relationship holds in the five stations in series. The relationship exists regardless of whether or not the arrival process of the previous station is renewal in nature. It is reasonable to assume that Observation 1 is applicable to multiple stations in series. This hypothesis is tested in Sections 6.2 and 6.3. 6.1. Approximate models for multiple single-server queues in series In an FCS, all stations behind (or after) the system bottleneck are classified as non-bottlenecks and have zero queue times. If a station is identified as the next bottleneck before (or in front of) the system bottleneck, all stations

4th Server

5th Server

IR (%)

LB

UB

Gap

IR (%)

LB

UB

Gap

IR

84.8 84.1 83.3 82.3 81.3 80.4 79.0 77.2 74.2 68.2

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.70 3.75 6.25 9.38 13.39 18.75 26.25 37.50 56.25 71.25

1.70 3.75 6.25 9.38 13.39 18.75 26.25 37.50 56.25 71.25

74.5 73.7 72.7 72.0 71.2 70.6 69.7 68.6 66.8 65.4

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.07 2.31 3.75 5.45 7.50 10.00 13.13 17.14 22.50 25.91

1.07 2.31 3.75 5.45 7.50 10.00 13.13 17.14 22.50 25.91

68.2 67.4 66.2 65.4 64.5 63.7 62.9 62.0 60.9 60.3

between the main and the second bottleneck are classified as non-bottlenecks and have zero queue times. Based on this insight, we develop a procedure to approximate the queue times of n stations in series; see Fig. 7. Our procedure uses the IR for each station except for the first station. If the station is identified as a bottleneck, the IR is yi . If it is a non-bottleneck the IR is xi . After presenting the method, we will discuss how to approximate the IRs. Procedure 1. (Queue time for each station in a tandem queue.) Stage I: Decomposition by bottlenecks 1. Identify the index of the system bottleneck (BN 1 ), where μ BN1 = min μi , for i = 1 to n. Let k = 1. If more than one station has the minimum service rate, BN 1 = min i, where μi = μ BN1 .

100% 90% 80% 70%

Intrinsic Ratio


Util (%)

3rd Server

60% 50% 40%

2nd Server

30%

3rd Server

20%

4th Server

10%

5th Server

0% 0%

10%

20%

30%

40%

50%

Utilization

Fig. 6. The IRs of five single queues in series.

60%

70%

80%

90%

100%

284

Wu and McGinnis (μ1, cs1) (λ, ca1)

1

(λ, cd1)

(μl, csl) …

(λ, cdl)

l

BNp

…

BN2

(μm, csm) (λ, cdm) … m

(μn, csn) n

BN1


Fig. 7. n single queues in series.

2. Identify the index of the next bottleneck (BNk+1 ) in front of the previous one (BNk ), where μ BNk+1 = min μi , for i = 1 to BNk – 1. If more than one station has the minimum service rate, BNk+1 = min i, where μi = μ BNk+1 . 3. If BNk+1 = 1, stop. Otherwise, let k = k + 1, go to Step 2. Stage II: Determining the model for each station 4. Let:

ρ1 1 QT 1 = α1 , 1 − ρ1 μ1 2 2 ca1 + cs1 . α1 = 2

and i = 2,

where

5. If station i is marked as a bottleneck, then: QT i = QTi − (1 − yi )IG = αi A

− (1 − yi )

i −1

ρi 1 − ρi

1 μi

QTk,

k=1

otherwise:

QTi = xi αi

ρi 1 − ρi

1 , μi

where αi =

2 + csi2 ca1 2

.

station to the newest identified bottleneck (not included). At first when no bottleneck has been identified, the subsystem is the same as the original system. The subsystem then gradually becomes smaller until it is composed of a single station, which is the first station of the tandem queue. The first equation in Step 5 applies the heavy-traffic property in Theorem 1 to the bottleneck stations in tandem queues with more than two stations. As we have shown previously, the IG of the second station in an STQB bottleneck is the mean queue time of the first station. When a tandem queue has more than two stations, we first need to identify all bottlenecks. The IG of each bottleneck is the total mean queue time of all stations from the first station to the station just before that bottleneck. Since all those upstream stations have lower utilizations than the bottleneck utilization, the IGR goes to zero in heavy-traffic conditions due to the heavy-traffic property. In a subsystem, for the non-bottleneck stations after the bottleneck, their utilizations are less than or equal to the bottleneck utilization. The queue time approximation errors at the non-bottleneck stations make only a minor contribution to the system’s queue time (similar to the non-bottleneck in an STQF bottleneck system). From Procedure 1, we can also see the importance of the ASIA systems. The queue time of each station is determined by the IR and its variability in the ASIA systems, not by the true variability in the original systems.

6. If i = n, stop. Otherwise, let i = i + 1, go to Step 5. Due to the property of FCSs, the above procedure identifies the next bottlenecks within each subsystem, and each subsystem is composed of the stations from the first

6.2. Implementation without historical queue times Procedure 1 gives us a way to approximate the queue time of each station in tandem queues, but it does not specify

Table 4. Mean and SCV of the inter-arrival times and service times in Cases D-1 to D-6

D-1 D-2 D-3 D-4 D-5 D-6

Mean SCV Mean SCV Mean SCV Mean SCV Mean SCV Mean SCV

Arrival

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

1 0 1 8 1 8 1 8 1 0 1 0

0.6 1 0.6 1 0.6 0 0.9 0 0.6 8 0.9 8

0.6 1 0.6 1 0.6 1 0.6 1 0.6 1 0.6 1

0.6 1 0.6 1 0.6 1 0.6 1 0.6 1 0.6 1

0.6 1 0.6 1 0.6 1 0.6 1 0.6 1 0.6 1

0.6 1 0.6 1 0.6 1 0.6 1 0.6 1 0.6 1

0.6 1 0.6 1 0.6 1 0.6 1 0.6 1 0.6 1

0.6 1 0.6 1 0.6 1 0.6 1 0.6 1 0.6 1

0.6 1 0.6 1 0.6 1 0.6 1 0.6 1 0.6 1

0.9 1 0.9 1 0.6 1 0.6 1 0.6 1 0.6 1

— — — — 0.9 1 0.9 1 0.9 1 0.9 1

285

Interpolation approximations for queues in series Table 5. Queue time approximations of nine stations in series in Case D-1 Simulation


Station number 1 2 3 4 5 6 7 8 9 Total Total absolute error Total absolute % error

QNA

QNET

SBD

IR method

QT

90% CI

QT

Error %

QT

Error %

QT

Error %

QT

Error %

0.290 0.491 0.607 0.666 0.706 0.731 0.748 0.775 5.031 10.045

2.41 1.43 1.32 1.20 1.42 1.78 1.34 1.68 4.31

0.45 0.61 0.72 0.78 0.82 0.85 0.87 0.88 7.99 13.97 3.93 39.09

55.17 24.64 17.90 17.42 16.79 16.51 16.19 13.58 58.74 39.09

0.45 0.66 0.74 0.79 0.82 0.84 0.85 0.86 6.97 12.98 2.94 29.22

55.17 34.42 21.91 18.62 16.15 14.91 13.64 10.97 38.54 29.22

0.45 0.66 0.74 0.79 0.82 0.84 0.85 0.86 4.05 10.06 1.98 19.68

55.17 34.42 21.91 18.62 16.15 14.91 13.64 10.97 −19.50 0.15

0.45 0.58 0.67 0.74 0.78 0.82 0.84 0.86 6.25 11.98 1.93 19.23

55.17 17.72 10.31 10.38 10.77 11.57 12.25 10.55 24.23 19.23

how to approximate the parameters (xi or yi ). Similar to reasoning given in Section 5, the following heuristic uses QNA to estimate the IR. Procedure 2. (Intrinsic ratio method based on QNA.) 1. Use Procedure 1 to determine which queue time model to use for each station. Let k = 2. 2. If the unknown variable in the queueing model is xk , go to a. If it is yk , go to b. a. If STBNk = STk, compute xk using QNA at 80% system utilization, where ST is the mean service time and BNk is the immediate bottleneck in front of station k. Otherwise, compute xk using QNA at 99.9% system utilization. b. Compute yk using QNA at 80% system utilization. If k = n, go to 3. Otherwise, let k = k + 1, then go to Step 2. 3. Use xi (or yi ) to approximate the queue times of the ith station at other traffic intensities. The selection of the percentages in Step 2 derives from observing the results in Appendices A, B, and C in the online supplement. The performance of Procedure 2 was tested using six test problems that were first presented in

Suresh and Whitt (1990) and Dai et al. (1994). There are 9 stations in Cases D-1 and D-2 and 10 stations in Cases D-3 to D-6 (see Appendix D in the Online Supplement). The service time distribution is deterministic if the SCV is zero, exponential if the SCV has a value of one, and hyperexponential if the SCV has a value of eight. The parameters used in each case are listed in Table 4. Each expected queue time was obtained from 10 replications of 30 000 arrivals after discarding the first 2000 data points. The performance of the IR method was compared with QNA, QNET, and SBD. Except for the last two columns in Table 5, all of the data in the tables are directly cited from Dai et al. (1994). To examine the performance of each approximate model objectively, we did not compare the total queue time percentage errors since small net percentage errors could result from large negative and positive errors compensating for each other. Instead, we compared the total absolute percentage error, which is the ratio of the total absolute error to the total system queue time from simulation, where the total absolute error is the summation of the absolute differences between the simulated and the approximated queue times. The ratio of the half-width 90% confidence interval to the mean queue time is shown in the third column next to the simulated queue times. For each method, the

Table 6. Performance comparison among the four models for Cases D-1 to D-6 QNA

D-1 D-2 D-3 D-4 D-5 D-6 TTL

QNET

SBD

IR method

Sys. QT

Abs. error

% Error

Abs. error

% Error

Abs. error

% Error

Abs. error

% Error

10.05 45.27 45.50 53.25 22.10 60.12 236.27

3.93 23.95 25.51 7.73 1.08 12.31 74.52

39.09 52.91 56.08 14.52 4.90 20.48 31.54

2.94 28.19 30.57 6.55 2.39 14.66 85.30

29.22 62.27 67.20 12.31 10.79 24.39 36.10

1.98 10.27 12.04 9.44 6.42 13.81 53.95

19.68 22.68 26.47 17.73 29.03 22.97 22.84

1.93 12.05 13.06 6.09 2.96 13.05 49.14

19.23 26.62 28.71 11.44 13.41 21.70 20.80

286 (λ, ca1)

Wu and McGinnis (μ1, cs1)

(μ2, cs2)

(μ3, cs3)

(μ4, cs4)

(μ5, cs5)

1

2

3

4

5

heavy-traffic conditions. Hence, it would be nice to develop an approach that has not only the advantages of analytical models (i.e., generates complete response profiles quickly) but also those from simulation (i.e., considers the dependence among stations) and only uses one or two data points from simulation or historical data since historical queue times are often available (e.g., from the performance of the previous month). Because the approach combines the analytical models of queueing theory and empirical data from simulation or history, it is an important class of hybrid simulation/analytic models (Shanthikumar and Sargent, 1983). Since simulation can be replaced by historical data in our approach, and the analytic model specifically refers to queueing models, we call our approach a hybrid queueing approach. To ensure the applicability, a hybrid queueing approach should require as few data points as possible while solving the analytically intractable problems. In Procedure 2, to estimate the IR, we used QNA to obtain the queue times for a specific traffic intensity. If historical data are available, the following procedure that may be applied with a single historical utilization or with two distinct utilizations can be proposed.


Fig. 8. Five stations in series.

percentage error (compared with simulation) is in the column to the right of the approximate value. The result of Case D-1 is shown in Table 5. The performance of the four models is summarized in Table 6. The system’s queue time comes from the simulations. The absolute error is the difference between the simulation results and the approximations. Detailed data for Cases D2 to D-6 are available in the online supplement. For the six cases considered, the IR method gives the smallest average error of 20.8% (cf. 31.5% for QNA, 36.1% for QNET, and 22.8% for SBD). A potential issue with the considered test cases is that the bottleneck utilization is fixed at 90%, which cannot give us a complete picture of the performance. Furthermore, most of the service times are exponential and the SCVs of arrivals are either zero or eight, which are not representative of practical manufacturing systems. To explore the model performance in general situations, we provide additional simulation experiments in the next section. An implicit assumption of almost every queueing model is the existence of the mean and SCV of the service time. However, in practice, even estimating the mean service time may not be a trivial operation (Wu and Hui, 2008; Wu, et al., 2011). A common way to estimate the SCV of the service time is to analyze a fair amount of historical data. However, if we have such historical data, we can also analyze the historical queue time at some specific utilizations. Hence, it is reasonable to explore the performance of IR methods when historical queue times are used to estimate the IR.

Procedure 3. (Single-point IR method with historical data.) 1. Find the historical queue time of each station at a specific utilization. 2. Use the historical queue times in the equations in Procedure 1 to solve for xi (or yi ) for the ith station, where i = 1, . . ., n. 3. Use xi (or yi ) to approximate the queue times at other traffic intensities. Procedure 3a. (Two-point IR method with historical data.) 1. Find the historical queue times of each station at two specific utilizations. 2. Based on the equations in Procedure 1, calculate xi (or yi ) for the ith station at the two utilizations, where i = 1, . . ., n at the two utilizations. 3. Based on the xi (or yi ) at the two utilizations, extrapolate (or interpolate) for xi (or yi ) at other utilizations and

6.3. Implementation with empirical queue times: The hybrid queueing approach Simulation is a viable option to predict production system performance because it can capture the dependence among stations. However, running simulation models for a complete response profile is time consuming, especially under

Table 7. Mean and SCV of the inter-arrival times and service times in Cases E-1 to E-10 E-1

E-2

E-3

E-4

E-5

E-6

E-7

E-8

E-9

E-10

Mean SCV Mean SCV Mean SCV Mean SCV Mean SCV Mean SCV Mean SCV Mean SCV Mean SCV Mean SCV Arrival 1st 2nd 3rd 4th 5th

— 20 25 30 25 20

1 0.5 0.5 0.5 0.5 0.5

— 25 28 30 20 25

1 0.25 0.25 0.25 0.25 0.25

— 20 23 25 28 30

1 0.25 0.25 0.25 0.25 0.25

— 30 28 25 23 20

1 0.25 0.25 0.25 0.25 0.25

— 30 25 28 20 30

1 0.25 0.25 0.25 0.25 0.25

— 23 25 28 20 30

1 8 0.25 0.8 0.5 1

— 23 25 28 20 30

1 8 2 5 4 1

— 23 25 28 20 30

1 0.36 0.25 0.8 0.5 0.64

— 23 25 28 20 30

0.1 0.36 0.25 0.8 0.5 0.64

— 23 25 28 20 30

5 0.36 0.25 0.8 0.5 0.64

287

Interpolation approximations for queues in series Table 8. Performance comparison (in percentages) among the five models for Cases E-1 to E-10


QNA

QNET

IR method with QNA

Single-point (80%) IR method w/ historical data

Two-point (70/80%) IR method w/ historical data

Utilization (%)

E-1

E-2

E-3

E-4

E-5

E-6

E-7

E-8

E-9

E-10

MAPE (%)

10 20 30 40 50 60 70 80 90 95 Ttl. Abs. (%) 10 20 30 40 50 60 70 80 90 95 Ttl. Abs. (%) 10 20 30 40 50 60 70 80 90 95 Ttl. Abs. (%)

20.2 19.2 17.4 14.8 11.4 7.4 4.7 6.0 10.6 15.9 12.0 3.9 3.7 3.8 3.9 4.1 4.3 4.6 5.5 7.5 8.9 7.1 10.1 9.3 8.8 8.4 8.0 7.5 6.7 5.5 3.5 1.5 3.9

51.0 48.1 43.0 35.9 27.3 17.8 8.0 8.3 18.6 29.9 22.9 7.0 6.6 6.1 5.9 5.9 6.3 7.6 11.4 23.9 40.2 25.5 14.4 13.8 12.9 12.1 11.1 10.5 9.6 8.1 5.3 2.8 5.8

48.2 45.2 39.9 32.7 23.6 13.3 10.4 16.3 26.9 36.7 28.4 12.2 12.1 12.1 12.3 12.9 14.2 16.8 22.1 34.4 47.3 33.9 28.2 27.2 26.1 24.7 23.2 21.6 19.5 16.3 10.3 5.1 11.4

52.7 49.7 44.7 38.0 30.2 22.1 14.5 7.8 3.0 1.5 8.0 1.8 2.4 3.2 3.8 4.2 4.6 4.8 4.5 3.7 2.4 3.3 12.4 11.5 10.7 10.1 9.3 8.4 7.4 6.4 4.6 2.6 4.9

49.2 45.9 40.5 32.9 24.2 15.1 12.7 10.6 11.6 11.9 13.6 7.7 7.6 7.4 7.3 7.2 7.2 7.3 7.6 7.8 8.8 8.1 16.0 15.5 15.0 14.3 13.7 13.0 12.2 11.4 10.0 9.9 10.8

8.8 10.0 10.1 10.5 11.1 11.5 11.7 11.5 10.9 11.3 11.2 22.1 21.2 20.6 20.0 19.7 19.5 19.9 20.9 23.0 24.8 22.6 24.0 19.4 15.7 12.9 10.6 9.6 10.6 11.2 10.7 11.6 11.3

18.2 18.8 17.8 15.4 12.1 8.1 6.2 7.9 12.2 21.8 14.6 22.0 17.1 12.8 9.5 7.5 6.0 5.4 5.0 5.5 7.8 6.8 36.0 29.9 24.3 19.2 14.9 11.7 9.9 7.3 4.4 8.1 8.8

19.7 18.2 15.7 12.1 7.9 6.7 5.8 8.8 13.0 16.0 12.8 5.5 5.3 5.2 5.2 5.3 5.5 6.1 7.2 9.7 11.0 9.0 13.1 12.8 12.5 12.1 11.7 11.0 10.2 8.9 7.1 4.9 7.3

4916.7 440.3 153.3 78.9 49.5 35.1 28.3 24.8 25.1 29.0 31.0 6465.9 591.7 213.3 111.9 68.4 44.5 30.0 19.7 12.2 8.4 20.1 7518.3 698.4 259.3 141.1 89.8 60.6 42.0 27.2 13.8 5.5 23.4

44.9 26.3 30.4 30.7 25.4 23.5 24.5 31.2 42.8 53.6 41.8 78.9 68.6 61.1 55.3 51.2 48.6 47.7 49.9 58.4 69.2 60.6 78.7 67.6 58.8 51.2 46.0 42.2 37.4 31.0 20.9 12.8 25.6

10 20 30 40 50 60 70 80 90 95 Ttl. Abs. (%)

6.1 5.4 4.5 3.7 2.9 2.0 1.0 0.0 1.3 2.1 1.7

10.0 9.0 7.7 6.6 5.3 3.9 2.2 0.1 2.6 4.2 3.4

12.4 11.2 9.9 8.4 6.7 4.8 2.6 0.2 3.6 5.2 4.3

8.8 7.8 6.7 5.7 4.4 3.2 1.7 0.0 2.2 3.0 2.6

7.1 6.2 5.2 4.2 3.2 2.1 1.2 0.1 1.9 3.2 2.5

32.5 27.4 22.8 17.9 13.8 9.1 4.8 0.2 5.0 7.8 7.0

33.1 27.1 21.6 16.6 12.1 8.1 4.4 0.1 5.4 9.7 7.6

3.7 3.2 2.7 2.3 1.8 1.1 0.5 0.3 0.5 0.9 0.8

5686.6 508.4 175.0 85.4 46.9 25.5 12.4 0.1 7.9 9.5 13.4

70.2 54.8 42.9 32.7 23.9 17.3 11.6 0.2 10.7 9.5 13.0

5.6

10 20 30 40 50 60 70 80 90 95 Ttl. Abs. (%)

1.2 0.9 0.7 0.5 0.3 0.3 0.2 0.0 0.6 1.4 0.8

4.8 3.7 2.8 1.9 1.0 0.4 0.2 0.1 1.1 2.4 1.5

10.8 8.2 6.0 4.1 2.3 1.0 0.2 0.2 1.5 3.0 2.1

4.5 3.5 2.7 1.9 1.1 0.5 0.1 0.0 0.9 1.6 1.1

4.7 3.7 2.9 1.9 1.2 0.5 0.0 0.1 1.1 2.3 1.5

8.1 6.3 4.6 3.1 1.8 0.9 0.1 0.2 1.4 3.5 2.1

8.7 6.7 4.9 3.3 1.9 0.7 0.1 0.1 2.3 5.7 3.0

2.6 2.1 1.5 1.1 0.8 0.3 0.1 0.3 0.2 0.5 0.4

4935.6 403.3 123.5 52.7 25.1 12.6 0.1 0.1 3.9 5.5 7.4

45.6 28.5 25.1 23.8 20.6 16.7 0.2 0.2 5.8 4.2 7.2

2.7

19.6

19.7

11.3

288


use it to approximate the queue times at other traffic intensities. Procedures 2, 3, and 3a were validated by considering 10 cases composed of five stations in series. The first station is the only station that is fed by the exogenous arrival process as shown in Fig. 8. In each case, the SCV of the service time is different. The bottleneck can be the first, third, or fifth station. The mean bottleneck service time was always 30. The service times were taken to be iid and followed a gamma distribution. The external arrival process was assumed to be Poisson in the first eight cases. The parameters used in each case are listed in Table 7. The traffic intensity of the bottleneck was varied from 10 to 95%. In total, 100 replications were conducted for each expected queue time. Each replication was composed of between 200 000 and 400 000 data points after a 50-year warm-up period (i.e., 87 600 (at 10%) to 832 200 (at 95%) data points were discarded). The performance of Procedure 2 (IR method with QNA) and Procedures 3 and 3a (IR method with historical data) were compared with QNA and QNET and the results are listed in Table 8. The absolute percentage errors at all utilizations and the total absolute percentage error of each case were computed along with the simulation results. Detailed data for Cases E-1 to E-10 are available in the online supplement. In the IR method with historical data, it was assumed that the historical data were available at 80% utilization for the singlepoint approach and at both 70 and 80% utilizations for the two-point approach. In addition to the simulation results at 10 utilization levels, which represent the performances in the future, we ran the simulation at 70 and 80% utilizations with different random seeds to generate comparable “historical” data. Since there is no obvious way to feed historical queue times into QNA and QNET, the five methods in Table 8 can be viewed as two parts: the first part compares the QNA, QNET, and IR methods with QNA, where no historical queue time data are used. In the second part, which is composed of the IR method with historical data, the historical queue time is used to improve the results further. In the 10 examined cases, the two-point IR method with historical data (Procedure 3a) always performs the best, and the one-point IR method with historical data (Procedure 3) is the second best in nine cases. From the smallest to the largest, the Mean Absolute Percentage Error (MAPE) of all 10 cases are 2.7, 5.6, 11.3, 19.6, and 19.7% for the two-point IR method, one-point IR method, IR method with QNA, QNA, and QNET, respectively. In the first part, where no historical data are used, the IR method with QNA has already outperformed QNA and QNET. When the historical data are used, the improvement becomes even larger. Considering the confidence intervals of the simulation, the 2.7% error from the two-point IR method is an impressive result. It should be noted that all IR approaches perform well in heavy-traffic conditions, but QNA and QNET perform

Wu and McGinnis poorly under those conditions. QNA performs the best at between 60 and 80% utilization (similar to what was observed in Section 5 for an STQB bottleneck). The average errors from QNA and QNET are about the same. However, in terms of average errors, QNA ranges from 8.0 to 41.8%, whereas QNET ranges from 3.3 to 60.6%. In addition to heavy-traffic conditions, the IR method with historical data performs very well when the utilizations are close to the historical utilizations since it has a better prediction of the IR. In practice, we care more about the performance in this region, since utilization usually changes gradually rather than abruptly. Since the singlepoint and two-point IR methods with historical data have smaller errors and are easy to implement, they are attractive alternatives to QNA and QNET.

7. Conclusions We have presented a new approximation approach for the estimation of the mean queue times of serial production lines based on observed properties. Due to the heavy-traffic property, the resulting model performs very well for STQB bottlenecks. While the IR-based approximate models perform well in most of multiple single-server tandem queue experiments, the performances of QNA and QNET are less reliable. When compared with each other, neither QNA nor QNET clearly dominates in our test problems. An interesting observation is that, when the SCV of the service time is small, QNET does not converge in heavy-traffic conditions. Our approach to approximating the cycle time does not depend on the assumptions required in mathematical models. Instead, we have identified a fundamental property of tandem queues, namely, the IR, and exploited this property to deal directly with dependence among workstations. We have achieved notable improvement in the approximation errors relative to prior approaches. An improved understanding of the IR is needed in order to compute it more accurately or perhaps even exactly. Furthermore, the behavior of the IR in other situations, such as multi-server workstations, multiple products, process batches, or limited buffers has not yet been investigated. Since a general queueing network is analytically intractable, approximations are expected. Due to the significant improvement in computation speed and the discovered underlying structure of tandem queues, the hybrid queueing approach offers an appealing alternative to the pure analytical or simulation approaches. In practice, a common way to estimate the SCV of the service time is to analyze historical data. If we have these data, we may also analyze the historical queue time. When such data are available, the hybrid queueing approach could be applied to reduce the complexity of problems that are analytically intractable. Concurring with Shanthikumar and Sargent (1983), we hope that more research can be done in this promising direction. Of course, not all production systems are tandem queues, and an interesting question is whether the IG and IR

289

Interpolation approximations for queues in series concepts can provide a basis for estimating system mean queue times in more complex manufacturing networks. This is a topic of ongoing research, with promising initial results (Wu and McGinnis, 2012).


Acknowledgements The first author would like to thank Dr. Hayriye Ayhan and Bert Zwart for their guidance and Dr. Craig Tovey for his unreserved encouragement on this endeavor. The authors are grateful to the area editor, Dr. Jeffrey Kharoufeh, and the anonymous referees for their insightful comments, which made this article more rigorous. The first author is grateful to Yawen Cheng. Her careful proofreading considerably improved the quality of this article.

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Biographies Kan Wu is an Assistant Professor at Nanyang Technological University. He received an M.S. degree in Industrial Engineering and Operations Research, an M.E. degree in Nuclear Engineering from the University of California at Berkeley, and a Ph.D. degree in Industrial and Systems Engineering from Georgia Institute of Technology. He has held senior engineer positions with Tefen Ltd., and Taiwan Semiconductor Manufacturing Company and been an IE manager at Inotera Memories Inc., and the CTO and founding team member at Sensor Analytics Inc. His Ph.D. dissertation was awarded the third place for the IIE Pritsker

290 Doctoral Dissertation Award in 2010. His research interests are primarily in the areas of queueing theory, with applications in the performance evaluation of supply chains and manufacturing systems.


Leon McGinnis is the Gwaltney Professor of Manufacturing Systems at Georgia Tech. He is internationally known for his leadership in the material handling research community and his research in the area of discrete-event logistics systems. He has received several awards for his innovative research, including the David F. Baker Award from IIE, the Reed-Apple Award from the Material Handling Education Founda-

Wu and McGinnis tion, and the Material Handling Innovation Pioneer Award from Material Handling Management Magazine. He is author or editor of seven books and more than 110 technical publications. At Georgia Tech, he has held leadership positions in a number of industry-focused centers and programs, including the Material Handling Research Center, the Computer Integrated Manufacturing Systems Program, the Manufacturing Research Center, and the newly formed Product/Systems Lifecycle Management Center. His current research explores the application of PLM technologies to the design and management of highly capitalized factories.