Abstract In manufacturing systems, batch building processes are very common, as goodsare often transported or processed in batches and must therefore be collected before thesetransport or processing steps can occur. In this paper, we present a method for the per-formance analysis of general batch building processes in material flow systems under thetimeout and capacity rules. The proposed model allows for stochastic collecting times andincorporates no restrictions with respect to the number of arriving units and their interarrivaltimes. The accuracy of the discrete-time approach is demonstrated by comparing this ap-proach with a discrete-event simulation model in continuous-time. Subsequently, the modelis applied to two cases: a transportation case from the health care industry and the processof building a batch for a batch processor.
Keywords Batch building · Discrete-time modeling · Queueing theory · Stochastic finiteelements
1 Introduction
In industrial practice, the analysis of material and information flows in production systemsis often addressed via simulation. Simulation allows for the construction of very detailedmodels. However, simulation is a very time-consuming and expensive task. Due to the com-putational efficiency of analytical approaches, these methods are well suited to support theearly stages of the planning of production systems. To build a model that fits the real world,
J.A. SchwarzChair of Production Management, University of Mannheim, Schloss, 68131 Mannheim, Germanye-mail: [email protected]
J. Stoll née Matzka (�) · E. ÖzdenInstitute for material handling and logistics, KIT—Karlsruhe Institute of Technology,Gotthard-Franz-Strasse 8, 76128 Karlsruhe, Germanye-mail: [email protected]
we must ensure that stochastic events, such as demand, processing times, machine failures,and scrap, among others, are considered in an appropriate manner. Thus, queueing theoryis a suitable analytical tool for the stochastic modeling of these types of systems (Buzacottand Shanthikumar 1993; Hopp and Spearman 1996). This approach allows key performancemeasures for many different scenarios to be calculated and compared in a short time, partic-ularly during the rough planning phase of production system design.
Using continuous-time queueing models with general distributions, performance mea-sures based on the first two moments can typically be derived. However, these measuresare not sufficient to verify, e.g., whether the requested sojourn times of orders through theplant, a warehouse or a specific process step can be met with an acceptable probability. Theknowledge of the complete probability distribution of performance measures is very use-ful for several production and material flow systems. In the following paragraphs, a list ofillustrative examples is provided.
A warehouse that offers a will-call service must suggest a pick-up date for customers.The warehouse has to ensure that there is a high probability that each customer order can becompleted by its pick-up date. Therefore, the distribution of the throughput time is needed(Schleyer and Gue 2012). Access to the full distribution of throughput times instead ofonly the mean time renders it possible to know, for example, what percentage of orders areprocessed in less than 3 hours or what promised turnaround time will be met 95 % of thetime.
In certain production systems, the time that a part may spend in a buffer between suc-cessive process steps is limited. Parts that are unused for longer than a certain period oftime must be reworked or discarded due to the risk of quality degradation. Therefore,the time that a part must be left waiting for a subsequent operation after a previous op-eration has been completed should remain below a fixed standard to guarantee the qual-ity of the part. This constraint is common and important in several industries, such asthe semiconductor industry (Lee and Park 2005; Kitamura et al. 2006; Shi and Gersh-win 2011) and the food production, chemical production or steel production industries(Yang and Chern 1995). For example, in the semiconductor industry, the time a waferspends in a processing module within a cluster tool should be limited (Kim and Lee 2008;Rostami et al. 2001). If the wafer delay at each process step of low pressure chemical vapordeposition (LPCVD) exceeds 20 seconds, the wafer surface deteriorates because of its ex-cessive exposure to residual gases under high temperature, and the entire wafer is scrapped(Kim et al. 2003). As mentioned by Robinson and Giglio (1999), a baking operation mustalso be started within two hours of a prior cleaning operation. If more than two hours elapsebetween these two operations, the lot must be sent back to be cleaned once again.
Similar problems can be found in health establishments, where reusable medical devicesundergo several sterilization steps after their use (Di Mascolo et al. 2006; Matzka et al.2011). In the pre-disinfection step, for instance, the devices must stay in a disinfection liquidfor approximately 15 minutes to guarantee the achievement of an optimal impact to thematerial. However, the sojourn time in the liquid should not exceed 50 minutes because thedisinfection product attacks the material, causing premature aging of the device.
1.1 The advantages of discrete-time modeling
To design a material flow system that can face requirements similar to the restrictions de-scribed above, it is helpful to have analytical models that calculate the probability distribu-tion of performance measures. Matzka et al. (2011), Schleyer and Gue (2012) and Özdenand Furmans (2011) develop discrete-time queueing network models under general dis-tribution assumptions to calculate the waiting time distribution in queues and to thereby
Ann Oper Res
determine the throughput time distribution of materials/orders in a material flow system.They adapt approaches that were originally developed for the evaluation of asynchronoustransfer mode (ATM) networks (Ackroyd 1980; Tran-Gia 1996; Hübner and Tran-Gia 1995;Hasslinger and Klein 1999) to describe material and information flows in a material han-dling system. In discrete-time models the system state can only change at multiples of tinc ,a fixed time increment. Although the discretization of time serves as an approximation ofa continuous flow of time, discrete-time approaches are known to provide highly accurateresults (Worthington and Wall 1999; Omosigho and Worthington 1988) and high levels ofdetail (Schleyer 2012). The advantage of discrete-time queueing analysis is that in contrastto most continuous-time models, performance measures, such as the waiting time distri-bution of an order/material in a queue, can be calculated efficiently under general distri-bution assumptions (Grassmann and Jain 1989). Furmans (2004), Schleyer (2007, 2012),Schleyer and Furmans (2007), Özden (2011) and Matzka (2011) developed a frameworkfor the stochastic modeling of material and information flows in discrete-time (see Matzka2011 for an overview). Combining these discrete-time methods, several phenomena in mate-rial flow systems can be modeled and analyzed (Matzka et al. 2011; Schleyer and Gue 2012;Özden and Furmans 2011). The main idea of this approach is to divide a complicated ma-terial flow network into single systems that are separately analyzed. The single elementsof a network are linked by the interdeparture time distribution and the departing batch sizedistribution, which represent the arrival stream of the succeeding node of a network. Bysimply combining the obtained system solutions of each node, the behavior of the entirenetwork can be determined. Although this framework has been continuously extended inrecent years, there remains a dearth of appropriate analytical methods in the discrete-timedomain under general distribution assumptions.
1.2 Literature review of batch building models
In the present paper, we will focus on the development of a new discrete-time method for theanalysis of batch building processes in material flow systems. Batch processes are frequentlyencountered in a wide variety of applications. For instance, in manufacturing systems, manyoperations are performed in batches. Batch processes are typically implemented to achievetransportation or processing efficiencies.
Batches are used in transport processes because it is more economical to handle severalmaterial units within one transportation unit (e.g., a pallet or a plastic bin). By reducing setupcosts and maintaining a high overall rate of utilization, batch processing produces reducedunit operating costs. This notion appears to be straightforward to many system operators.However, an in-depth consideration of batch processes reveals an important counter effect;namely, as batch sizes increase, waiting times and, consequently, sojourn times increase. Asmentioned above, keeping the sojourn time low is an extremely crucial consideration formany processing steps in production and service operations. For instance, in semiconductorwafer fabrication, a maximum processing time window exists from the time instance whenthe job exits the upstream processor until the time instance at which it enters the batchprocessor. If a job is not processed before reaching the end of its processing time window,job reworking or validation is required (Tajan et al. 2011). In health establishments, reusablemedical devices are transported from the operation area to a sterilization area in batches andtreated with a disinfection liquid during their transport. If the medical devices must wait toolong for their transport (depending on the transport interval), the disinfection liquid causespremature aging of the material (Di Mascolo et al. 2006). These examples demonstrate whyefficient approaches for the analysis of batch processes under different policies are needed.
Ann Oper Res
In the literature, there are numerous instances in which either batch (bulk) arrivals orbatch services or combinations of both types of occurrences are elaborately studied in eitherthe continuous- or discrete-time domain (Schleyer 2007, 2012; Schleyer and Furmans 2007).There are several models for the analysis of batch building processes in the continuous-time domain. Bitran and Tirupati (1989) use a decomposition approach to investigate theG | G[K,K] | 1-queue with multiple product classes. In this approach, the system is decom-posed into a batch building node, at which batches of fixed size (K) are collected, anda server node. Similarly, Fowler et al. (2002) use the decomposition approach to study abatch building node. In addition, Meng and Heragu (2004) analyze a batch building opera-tion of fixed size; this analysis is also conducted in the continuous-time domain. Schleyerand Furmans (2005) and Schleyer (2007) are the first researchers to analyze batch buildingprocesses in the discrete-time domain. They distinguish between two basic batch buildingmodes of collecting orders until a predetermined number of orders is reached (the capac-ity rule) or collecting orders until a predetermined collecting time interval has elapsed (thetimeout rule). In addition, there are possible modifications of these basic batch buildingmodes. One example is the minimum batch size rule in which the collecting time is at leasttout time units. If the number of customers that have been collected after tout time units haveelapsed is less than a minimum quantity of L, then the batch building process continues un-til the required minimum of L customers is attained. For the aforementioned batch buildingmodes Schleyer (2007) provides mathematical methods for the calculation of the waitingand interdeparture time distribution under general assumptions of the input distributions.Özden (2011) extends this toolbox by contributing an element for a batch building mode inwhich orders are collected until either a predetermined collecting time interval is elapsed ora maximum batch size (e.g., the container capacity) is reached.
In our review of the literature, we find that only restricted models exist for the batchbuilding process in the discrete-time domain. In their queueing network analysis, Matzkaet al. (2011) and Özden and Furmans (2011) identify a limitation of the batch building mod-els under the timeout rule and under the capacity rule; in particular, in the former instance,the existing methods from Schleyer and Furmans (2005) and Schleyer (2007) do not con-sider interarrival times that are greater than the collecting time, whereas in the latter case,these methods do not consider batch sizes of arriving customers that are greater than thecollecting capacity. In addition, the collecting interval in the model for the timeout rule isdeterministic, which is an invalid assumption for most practical applications. Batch buildingis typically the first process in a queueing network of batch processes and therefore occupiesa crucial position. The simplified assumptions of Schleyer’s model can cause deviations inthe results that can build up through the network. This concern motivates the development ofa generalized method for the analysis of batch building processes with relaxed assumptions.Consequently, we have developed a generalized method for analyzing the batch buildingprocess under timeout and capacity rules in the discrete-time domain. Batch collecting pro-cesses that are subject to the timeout rule or the capacity rule finish after a predeterminedtime interval (including stochastic influences) or when a given capacity is fully exploited,respectively. For these batch building rules, we derive the interdeparture time distribution,departing batch size distribution and waiting time distributions.
Therefore, the main contribution of this paper is the consideration of unlimited interar-rival time distributions with interarrival times that are greater than the collecting interval.In addition, generally distributed stochastic collecting intervals for batch building under thetimeout rule are considered. Moreover, these relaxations require the development of a newset of performance measures to describe the characteristics of the general batch building pro-cess. The proposed approach delivers closed-form solutions. Thus, the discrete-time methodis capable of handling large-scale studies on the influence of different system parameters.
Ann Oper Res
Fig. 1 Batch building under the timeout rule, with a maximum interarrival time amax that is greater than thecollecting time Tout
The remainder of the paper is organized as follows: in Section 2, an analytical modelis proposed for general batch building processes under the timeout rule. In addition, theequivalencies between batch building under the timeout and capacity rules are described.Section 3 first provides a numerical study on the accuracy and runtime performance of theproposed model. This information is then used to analyze two practical cases: a transporta-tion example and a processing example. Neither case can be addressed with the analyticalmethods that are available in the existing literature. Section 4 summarizes the findings of thestudy and provides concluding remarks.
2 Analytical approach to batch building processes
In this section of the paper, we present an analytical model for a general batch building pro-cess. Therefore, the limiting assumptions of existing batch building models in the discrete-time domain are relaxed. In particular, the new model relaxes two significant restrictions: thelimitation of interarrival times to values that are smaller than the collecting interval lengthand the assumption of a deterministic collecting interval length. Before the new model isdeveloped, the process and the characteristic process measures of batch building accordingto the timeout rule are introduced. Then, the concept of residual times as connections be-tween succeeding intervals is presented. Subsequently, we use the residual times distributionto derive the characteristic distributions of the process output: the interdeparture time, thedeparting batch size and the waiting time of an arbitrary customer.
The performance measures are derived for batch building under the timeout rule. How-ever, batch building under the capacity rule behaves similarly. The equivalences betweenthese two batch building modes are presented in Section 2.4.
2.1 Description of the batch building process under the timeout rule
The crucial process parameter for batch building processes under the timeout rule is thecollecting time. All of the customers that arrive during the collecting interval of length Tout
are released together at the end of this collecting interval (see Fig. 1). Please note that werefer to the units in the system in this paper as customers, regardless of the fact that inreality, the units can potentially be orders, products or passengers on a bus. The capacityof the collecting station and therefore the size of the departing batch are assumed to beunlimited.
Ann Oper Res
The arrival stream is described by the interarrival time distribution A and the arrivingbatch size distribution Y . In contrast to existing models, the maximum interarrival time isno longer bounded by the collecting time. The probability of an interarrival time of amax
time units is then given by aamax .
Arrival batch size Y : P (Y = i) = yi ∀i = 1,2, . . . , ymax
Interarrival time A: P (A = i) = ai ∀i = 1,2, . . . , amax
As mentioned above, the batch building process is driven by the collecting time. This paperproposes a new approach in which the interval length is given by the random variable Tout .This approach allows us to explicitly analyze the influence of stochastic effects with respectto the collecting interval length, which is assumed to be independent from A and Y . Ex-isting models neglect stochastic considerations by assuming that the collecting interval isdeterministic. Given that all of the customers must wait until the end of a collecting inter-val, we further identify a waiting process at the collecting station. The waiting time is alsomodeled as a random variable and is denoted as W .
Collecting time Tout : P (Tout = i) = tout,i ∀i = 1,2, . . . , tout,max
Waiting time W : P (W = i) = wi ∀i = 0,1, . . . , tout,max − 1
The output stream is described by the interdeparture time D and the departing batch size Yd .As a direct consequence of relaxing the limiting assumption that amax must be smaller thanthe collecting interval length, it is now possible for batches of size 0 to depart from thecollecting station. Although there is no external limitation on the departing batch sizes, suchas a capacity restriction of the collecting station, there is an implicit limitation that originatesfrom the input stream characteristics. This limit is given by yd,max = lmax · ymax , where lmax
is the maximum number of arrivals in tout,max , the largest collecting interval, and each arrivalis associated with ymax , the maximum number of arriving customers.
Interdeparture time D: P (D = i) = di ∀i = 1,2, . . . , tout,max
The assumption of departures with size 0 might be reasonable for certain systems ofphysical transportation. For instance, consider the situation in which a bus visits a bus stationin accordance with a given schedule. After an interval during which no passengers arrive atthe station, an empty bus may leave the station. However, in-plant transports typically onlyoccur if there is at least one unit to move. Moreover, there are no empty orders to be issuedwith respect to the collecting process of orders before they are issued to a production system.Instead, the time between the issuing of the collected orders increases. Hence, we developthe distributions D∗ and Y ∗
d which are designed for applications in which departures mustinclude a minimum of one customer.
Adjusted departing batch sizes Y ∗d : P
(Y ∗
d = i) = y∗
d,i ∀i = 1,2, . . . , y∗d,max
Adjusted interdeparture time D∗: P(D∗ = i
) = d∗i ∀i = 1,2, . . . , d∗
max
2.2 Derivation of the residual time distribution
The collecting intervals of the batch building process are interrelated. The interarrival timebetween the last arrival in the nth interval and the first arrival in the (n + 1)th interval
Ann Oper Res
Fig. 2 An example of arrivals and resulting residual times
causes a residual time at the beginning of the (n + 1)th collecting interval. This residualtime equals the period of time from the beginning of a collecting interval until the firstcustomer batch arrives. For the purposes of this model, we assume that all of the customersthat arrive at the end of a collecting interval belong to the collecting interval in question (seecustomer arrival at δn+3 in Fig. 2). This assumption sets the lower bound for the residualtimes to 1. The maximum residual time possible equals the maximum interarrival time amax .We consequently define the residual time distribution Ra as follows:
Residual time Ra: P (Ra = s) = ra,s ∀s = 1,2, . . . , amax.
The number of arrivals during a collecting interval depends on the residual time. A longresidual time reduces the number of potential additional arrival instances, given that longerresidual times imply that fewer time units are left for arrivals. Thus, the residual time dis-tribution Ra is a good starting point for the analysis of the distribution of the departingbatch sizes. The residual time itself depends only on the preceding collecting interval andits residual time and therefore follows a Markov process. It is possible to derive a Markovchain for the generalized batch building model and to utilize this Markov chain to derive thesteady state residual time distribution. However, this paper focuses on obtaining a closed-form solution for the residual time distribution. Therefore, the changes that are caused by thegeneralization to the interpretation of the residual times are discussed. We then demonstratethat the residual time distribution equals the distribution of the forward recurrence times ofa renewal process.
We consider an example involving an arrival stream of four consecutive collecting inter-vals (see Fig. 2). The length of the collecting interval varies from two to four time units. Theinterarrival times range from one to four time units. In one case, the collecting interval lengthis smaller than the interarrival time. The interarrival time between the first and the secondcustomer is A = 3 time units, which causes a residual time in collecting interval n + 1 ofRn+1
a = 1 time unit. However, the (n + 2)th collecting interval is more interesting. In thisinterval, we find no arrival at all because the interarrival time between the 3rd and 4th arrivalis A = 4 time units, which exceeds the collecting interval. We find that one interarrival timecan result in more than one residual time. In particular, for the (n + 2)th collecting period,the residual time is Rn+2
a = 3, but for the (n + 3)th collecting interval, the residual time isRn+3
a = 1. We call the residual time from the last departure with a number of customersgreater than 0 direct residual times, and we refer to all of the following residual times untilthe next departure with at least one customer as indirect residual times. Furthermore, wenote that a long interarrival time can cause only one direct residual time but can potentiallygenerate x indirect residuals if the interarrival time exceeds x collecting intervals.
The arrival instances belong to a renewal process, as the interarrival times are positiveand identically independently distributed (i.i.d.). We next need to establish that the residual
Ann Oper Res
Fig. 3 Equal probabilities of arrivals due to the superposition of the sum of k i.i.d. random variables, wherek is large
time distribution equals the forward recurrent time distribution of the renewal process; inother words, we must prove that the timeout instances can be viewed as arbitrary observa-tions between two arrivals. Consequently, we must demonstrate that in the long run, everytimeout incident placed between two consecutive arrivals is equally likely. The long-termbehavior of the arrival process is driven by two properties of k-fold convolutions of randomvariables, where k is a large number: one of these properties refers to the value range, andthe other property relates to the shape of the distribution. Figure 3 illustrates the behaviorfor a continuous rather than a discrete distribution; a continuous distribution is only depictedfor greater ease of drawing and readability.
Starting with the value range, we find that the range for the first interarrival time is limitedto the interval amin to amax . The second arrival is given by the convolution of A with itself,and therefore, the value range for this interval is 2 ·amin to 2 ·amax . Similarly, the third arrivalis given by the 3-fold convolution of A. For a sufficiently large value of k, the value rangeof the (k −1)-fold convolution superimposes the value range of the k-fold convolution. Thissuperposition is shown in Fig. 3. For k = 3, the value range overlaps with the value range ofthe (k − 1) = 2-fold and (k + 1) = 4-fold convolutions.
The insight regarding the shape of the distribution originates from the central limit the-orem (CLT). According to the CLT, the sum of a large number of i.i.d. random variables isnormally distributed if the coefficient of variation is greater than 0. It follows that the fol-lowing findings do not necessarily hold for a single deterministic interarrival time. Notably,if the respective Markov chain for a batch building process with a deterministic interarrivaltime is built, a periodic Markov chain is typically created (Schleyer and Furmans 2005).This periodicity is also associated with issues in calculating the steady state of the residualtime distribution. However, the practical applicability of the developed model is not limited,given that the assumption of a single deterministic interarrival time is relatively unrealistic.
The symmetry of the normal distribution and the superposition of several distributionswith different values of k imply that after a sufficiently long time span, every time instanceis equally likely to become an arrival instance. It is important to note that the long-termbehavior is independent of the generalizations that are set forth in this paper with respect tothe interarrival time distribution A and the distribution of the collecting interval length Tout .
Given that after a sufficient amount of time has passed, every time instance is equallylikely to become an arrival instance, it follows that every time instance between two suc-ceeding arrival instances is equally likely to become a timeout instance, independently ofthe distribution Tout . Therefore, the time from the beginning of a collecting interval to thefirst arrival is equal to the forward recurrence time or the residual lifetime of a renewal pro-cess in which observations are obtained immediately after the occurrence of events. Thisconcept is depicted in Fig. 4.
Ann Oper Res
Fig. 4 An example of a renewal process with residual life times of i and j
According to Tran-Gia (1996), the forward recurrent times of a renewal process indiscrete-time are distributed according to (1):
P (Ra = i) = ra,i = 1
E(A)·(
1 −i−1∑
k=0
ak
)
, i = 1,2, . . . , amax. (1)
The probability for an interarrival time of at least i time units is therefore denoted as follows:
ai = 1 −i−1∑
k=0
ak =amax∑
k=i
ak, i = 1,2, . . . , amax. (2)
From (2), we obtain a simplified expression for the distribution Ra :
ra,i = ai
E(A), i = 1,2, . . . , amax. (3)
The findings from Section 2.2 demonstrate that (3) can be used to calculate the residualtime distribution. In particular, we observed that a large interarrival time can produce multi-ple residual times. Thus, we introduced the concept of direct and indirect residual times. Theperformance measures in the succeeding section are therefore adapted to this new concept.
In the following section, the distributions of the characteristic performance measures ofdeparting batch size, interdeparture time and waiting time are derived from the residual timedistribution. The focus of this section is on demonstrating how the derivation of measuresis affected by the new interpretation of the residual times and stochastic collecting times.Moreover, the derivation of the newly introduced measures D∗ and Y ∗
d is explained.
2.3.1 Departing batch size distribution
In this section, the departing batch size distribution is derived from the residual time distri-bution. First, we determine the conditional probability of a certain number of arrivals duringan arbitrary collecting interval, assuming a given residual time and collecting interval length.
Ann Oper Res
Fig. 5 Correlation of residual times and number of arrivals N during an interval of length t ; the arrivals arerepresented by arrows, and the collecting interval is marked with bold vertical lines
After the dependence of this probability on the residual time and collecting interval lengthhas been resolved, we can derive the distribution of departing batch sizes by combining theprobability of a certain number of arrivals with the distribution of the arrival batch sizes.
The three characteristic cases of N = 0, N = 1 and N = l > 1 arrivals during a collectinginterval of length t are depicted in Fig. 5. First, we consider the probability of l arrivalsduring a collecting interval under the conditions of a given residual time of s time unitsand a collecting interval of length t . As discussed earlier, the residual times that are greaterthan t represent collecting periods without arrivals. Therefore, for all residual times s > t ,the probability for l = 0 arrivals is equal to 1. If there is only one arrival instance duringthe collecting interval, it must be the arrival instance at the end of the residual interarrivaltime. The probability that this arrival remains as the only arrival equals the probability thatthe interarrival time to the next arrival is greater than the remaining collecting interval. Thelikelihood for l > 1 arrivals is given by the sum of the probabilities of all of the combinationswith (l − 1) arrivals and the residual arrival. For the case of N = l arrivals, the sum ofthe interarrival times of the (l − 1) arrivals and the residual time must be smaller than thecollecting interval. Moreover, the collecting interval must be exceeded if the interarrivaltime between the lth and (l + 1)th arrival is added to this sum. Therefore, the conditionalprobability of N = l arrivals is given by the following specification:
P (Na = l | Ra = s, Tout = t) =
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
1, l = 0, s > t,∑amax
j=t−s+1 aj , l = 1, s ≤ t,∑t−s−1
i=0 a(l−1)⊗t−(i+s)
∑amax
j=i+1 aj , l > 1, s < t,
0, otherwise,
(4)
where al⊗i equals the probability that the sum of l i.i.d. random variables A equals i. If the
Bayes theorem and the law of total probability are applied, the distributions Ra and Tout maybe used to resolve the dependence on the residual time and interval length. In particular, wecan derive the following equation:
P (Na = l) =tout,max∑
t=1
amax∑
s=1
P (Na = l | Ra = s, Tout = t) · ra,s · tout,t , l ≥ 0. (5)
We obtain the probability of a departing batch of size i by adding together the probabilitiesthat the sum of the arriving customers of l arrivals will total i; each of these probabili-ties is weighted by the probability of a collecting interval with l arrivals. In this situation,yl⊗
i equals the probability that the sum of l according to Y i.i.d. random variables equals i.Moreover, l is bounded by lmax = � tout,max
amin�, where amin is the smallest interarrival time with
Ann Oper Res
Fig. 6 Generation of an indirect residuum of length s by either a single collecting interval or c collectingintervals
a probability of occurrence greater than 0. Obviously, the probability of a departing batchwith 0 customers equals the probability of a collecting interval without arrivals. This rea-soning results in the following batch size distribution:
P (Yd = i) = yd,i ={
P (Na = 0), i = 0,∑lmax
l=1 P (Na = l) · yl⊗i , i > 0.
(6)
After removing the empty batches from the departing stream, we gain the new departingbatch size distribution Y ∗
d by normalizing the above equation based on the remaining prob-ability mass.
P(Y ∗
d = i) = y∗
d,i = yd,i∑lmax ·amax
i=1 yd,i
, i > 0 (7)
2.3.2 Interdeparture time distribution
The derivation of the interdeparture time distribution is trivial for a departing stream thatallows batches of size 0. In particular, the probability for an interdeparture time of t timeunits is clearly equal to P (Tout = t). Thus, we find:
P (D = i) = P (Tout = i), 1 ≤ i ≤ tout,max. (8)
However, as proposed in Section 2.1, to comply with practical situations, it is necessary toderive an interdeparture distribution D∗ in which collecting intervals without departures arereplaced by longer interdeparture times. A more complex approach is necessary to computethis distribution. According to Section 2.2, there are both direct and indirect residual times.For the following analysis, only direct residual times are taken into account. This approachensures that we consider only one of the several potential residual times that may be gener-ated from the same interarrival time (see Fig. 2 A = 4, Rn+2
a and Rn+3a ). Thus, a corrected
direct residual time distribution R∗a must be developed. The first step in this development
is to subtract the probability of the indirect residual times from the regular residual timedistribution that is given by (3) and to derive Ra .
The following ideas are illustrated in Fig. 6. A residual time of s time units is indirectlyproduced if the direct residual time is exactly s time units longer than the collecting intervallength. In Fig. 2, Rn+2 = 3 and is therefore 1 time unit longer than δn+2
out − δn+1out = 2; thus, the
indirect residual time is Rn+3 = 1. Consequently, the probability of all of the combinationsof residual times and collecting interval lengths, for which the residual time exceeds thecollecting interval length by s units, must be subtracted from the general residual distribu-tion (3). The collecting interval does not need to originate from a single collecting interval,
Ann Oper Res
Fig. 7 An interdeparture time of i time units that is caused by c + 1 consecutive collecting intervals
but can also be based on the combination of c consecutive intervals. Thus, we denote theprobability that the sum of c i.i.d. random variables Tout equals j as t⊗c
out,j . Because an indi-rect residual time can only be caused once by a greater direct residuum, the computation ofra,s begins with the largest value of s. Subsequently, the previously derived values of ra,s areused to compute the probability of smaller values of s. Thus, Ra is given by the followingequation:
P (Ra = s) = ra,s = ra,s −amax−s∑
j=1
ra,j+s
jtout,min
∑
c=1
t⊗cout,j , ∀s = 1,2, . . . , amax. (9)
Normalizing (9) yields the desired direct residual time distribution, R∗a :
P(R∗
a = s) = r∗
a,s = ra,s
∑ra,max
i=1 ra,i
. (10)
Based on (10), the interdeparture time distribution D∗ is then computed. The interdeparturetime is given by the time from the end of the last collecting interval with an arrival to theend of the next collecting interval with at least one arrival. Therefore, we analyze whichcombinations of residual times and collecting interval lengths produce a certain interdepar-ture time i. Assuming that the collecting interval length equals i time units, all residuals R∗
a
that are smaller than or equal to i result in an interdeparture time of i time units. The prob-ability of this occurrence is given by the first summand of (11). Figure 7 demonstrates howan interdeparture time of i time units is created by c + 1 consecutive collecting intervals.The first c collecting intervals have a total length of z time units with a probability of t⊗c
out,z.The probability that the (c + 1)th collecting interval will have a length of i − z is given bytout,i−z. All of the residuals R∗
a that are greater than z but smaller than or equal to i will leadto the desired interdeparture time i. The probability of all of the combinations of z and c isgiven by the second summand of (11).
P(D∗ = i
) = d∗i =
(
tout,i
i∑
s=1
r∗s
)
+(
i−1∑
z=1
tout,i−z · z
tout,min
∑
c=1
t⊗cout,z
)
·i∑
s=z+1
r∗s (11)
2.3.3 Waiting time distribution
In the proposed model, the interval length is no longer deterministic. For the derivationof the waiting time distribution W , we first assume a given collecting interval of length t .
Ann Oper Res
Fig. 8 The derivation of the conditional waiting time t − w for the 1st and lth arrival
This dependency is later resolved by the law of total probability and the Bayes theorem.Collecting intervals without arrivals do not contribute to waiting times, as these intervalsinvolve no customers that can wait. All of the residual times that are greater than t representcollecting intervals without arrivals. Therefore, we simply restrict the equations to residualtimes that are smaller than or equal to t .
The first customers arrive s time units after the beginning of the collecting interval. Con-sequently, their waiting time is t − s for a collecting interval of t units. Thus, the conditionalwaiting time may be expressed by the following equation:
P(W 1 = t − s | Ra = s, Tout = t
) = 1, s ≤ t. (12)
A potential second customer group arrives w time units after the beginning of the collectinginterval of length t , with w > s . The waiting time that is associated with the second groupis t − w. The probability for this second arrival equals the probability of an interarrivaltime between the first and second arrival of w − s, which is given by aw−s . This reasoningproduces the following expression for the conditional waiting time of the second arrival:
P(W 2 = t − w | Ra = s, Tout = t
) = aw−s , w = s + 1, . . . , tout,max. (13)
Generalizing the above equations for the lth arrival, we obtain (14). Figure 8 provides avisualization of this reasonings.
P(Wl = t − w | Ra = s, Tout = t
) = a(l−1)⊗w−s , w = s + 1, . . . , t, l = 2, . . . , lmax (14)
For an arbitrary customer, the waiting time W ∗ = t − w under conditions involving aresidual Ra = s and an interval length of Tout = t time units is then provided by the sum ofthe probabilities of all arrival instance combinations that lead to W ∗ = t − w:
P(W ∗ = t − w | Ra = s, Tout = t
)
∼{
1, w = s,∑lmax
l=2 P (Wl = t − w | Ra = s, Tout = t), w > s.(15)
The unconditioned probability may be obtained by applying Bayes theorem and the law oftotal probability to (15):
P(W ∗ = i
) =tout,max∑
t=1
amax∑
s=1
P(W ∗ = t − w | Ra = s, Tout = t
) · ra,s · tout,t ,
0 ≤ i = t − w < tout,max. (16)
Ann Oper Res
The normalization of the equation above produces the desired waiting time distribution.This normalization is necessary because (16) describes a defective distribution in which w
is bounded by t .
P (W = i) = P (W ∗ = i)∑tout,max−1
i=0 P (W ∗ = i), i = 0,1, . . . , tout,max − 1 (17)
Notably, the waiting time distribution for the generalized model remains independent fromthe batch size distribution Y of the arrival stream. Schleyer’s proof that the waiting timesare uniformly distributed can be adapted for generalized interarrival time distributions and adeterministic interval length td . Consequently, for these cases the following equation musthold:
P (W = i) = wi ={
1td
, i = 1, . . . , td − 1,
0, otherwise.(18)
However, because the proof only remains valid for the special case of deterministic col-lecting interval lengths, we omit this proof from the current paper. For stochastic collectinginterval lengths that are distributed according to the random variable Tout , the waiting timecan be computed by (17).
2.4 The equivalences between batch building under the capacity and timeout rules
The batch building according to the capacity rule behaves similarly to batch building accord-ing to the timeout rule. In the former situation, the input stream is also characterized by A
and Y . However, the end of the collecting process is not determined by a predefined timebut instead dictated by the capacity k of the collecting station. The time it takes to collectk customers, or in other words, the interdeparture time of completed batches, is modeled asa random variable. Whenever the number of customers at the collecting station exceeds k,a batch of size k will depart. In the limited model of Schleyer and Furmans (2005), themaximum size of arrival batches is bounded by ymax ≤ k. This limitation ensures that theminimal interdeparture time between two batches is 1, as it is impossible for more than onebatch to depart at the same time. This restrictive assumption prevents the model from be-ing applied in many practical cases. Therefore, in this paper, the restrictive assumption isrelaxed. We find that multi-batch departures at the same time instant are possible for thegeneralized model. The parameter k, however, is assumed to remain deterministic. Becausek is either a chosen process parameter or a parameter that is implicitly given by a definedprocess characteristic, e.g. the capacity of a collecting container, it is realistic to assume thatk is a fixed value. The generalized process is illustrated in Fig. 9.
For batch building processes under the capacity rule, the collecting intervals are also in-terrelated. However, in contrast to the timeout batch building, residual customers Ry (andnot residual times Ra) are the links between the intervals. For the limited model that is pre-sented by Schleyer and Furmans (2005), the number of residual customers ranges from 0 toymax − 1, with ymax ≤ k. This condition reflects either a last arrival that exactly matches theremaining customers that are necessary to fill the collecting size k, or an arrival of ymax = k
for a situation in which only one customer is needed to complete the batch. Similar to batchbuilding under the timeout rule, the generalization of the model beyond the restricted condi-tions described above leads to an extended range for the residuals; namely, the residuals cannow range from 0 to ymax − 1, with an unrestricted ymax . Because of the analogies betweenbatch building under the timeout and capacity rules, an explicit formulation of the equations
Ann Oper Res
Fig. 9 Batch building under the capacity rule with maximum arrival batch sizes ymax that are greater thanthe collecting size k
Table 1 Equivalences between batch building under the timeout and capacity rules
Timeout rule Capacity rule
Parameters interarrival time ai ⇔ batch size yi
batch size yi ⇔ interarrival time ai
collecting time t ⇔ collecting size k
Results residual time ra ⇔ residual customers ry
waiting time wi ⇔ number of still missing customers
number of still missing customers ⇔ waiting time wi
departing batch size yd,i ⇔ interdeparture time di
interdeparture time di ⇔ departing batch size yd,i
becomes redundant. Instead, a list of the equivalences that is similar to the list presented bySchleyer (2007) can be found in Table 1.
Given the equations for batch building under the timeout rule and the equivalences thatare presented in Table 1, the generalization for batch building under the capacity rule isstraightforward. The only distribution we are not able to generate using the equivalencesfrom Table 1 is the waiting time distribution, as the distribution of missing customers is notdiscussed in Section 2.3. Analyzing the waiting process, we find that the customers mustwait for different amounts of time depending on how many more arrivals are required tocomplete the collecting process. As in Schleyer and Furmans (2005), the residual customersare called “unlucky customers” because they have to wait the longest possible time period.The customers that complete a batch are called “lucky customers” since they do not have towait at all. For the case of ymax > k, the number of lucky customers increases because thedeparture of multiple batches is now possible; thus, there may be complete departing batchesof “lucky customers”. Figure 10 illustrates the occurrence of additional “lucky customers”for arrivals that occur in batches of one to five customers. The capacity is assumed to bek = 4. Only minor changes must be made to adopt the method presented by Schleyer andFurmans (2005) to account for the increased number of “lucky customers”. In essence, theequations must simply be extended to also include the new residual values.
Ann Oper Res
Fig. 10 Extending the maximum arrival batch size to ymax > k generates additional “lucky customers” thathave no waiting time
3 Numerical study
The numerical study is divided into three parts. First, the accuracy and runtime performanceof the developed model are demonstrated by comparing this model to a discrete-event sim-ulation model in continuous-time. To show the relevance of a general batch building model,we also compare the results of the developed model with the results of the limited modelthat was proposed by Schleyer (2007). Subsequently, the general model is applied to twocases: a timeout-based transportation example and a capacity-based processing example.Both problems can be solved using the proposed general batch building model. All of theexperiments were run on a standard desktop computer with a 2.5 GHz CPU and 4 GB mem-ory.
3.1 Accuracy of the approach
The approximation underlying the proposed model is the discretization of time. The follow-ing numerical study shows that for the tested configurations, this approximation leads onlyto minor approximation errors. A discrete-event simulation model in continuous-time servesas a benchmark. To cover a wide range of different configurations, we evaluated 54 differentinput data sets. As input for the interarrival and collecting times, uniform and normal distri-butions were assumed. The characteristic parameters of the inputs are depicted in Table 2,where e.g. n20l corresponds to a normal distribution with expected value of 20 and a lowstandard deviation (m = medium, h = high). For the normal distributions, the probabilitymass that was assigned to negative values was distributed proportionally over the positivevalues. A discretization increment of tinc = 0.5 was selected.
Table 3 presents the results for normally distributed interarrival times, and Table 4 indi-cates the results for uniformly distributed interarrival times. Because the batch size valuesare discrete values, and therefore no approximations are made regarding these distributions,we assumed for simplicity that P (Y = 1) = 1. For the performance measures of interest,
Ann Oper Res
Table 2 Input distributions for the numerical study
exp.value
std.dev.
cv
n20l 20 2.30 0.12
n20m 20 5.77 0.29
n20h 20 10.31 0.52
n15l 15 1.73 0.12
n15m 15 4.33 0.29
n15h 15 7.79 0.52
n10l 10 1.16 0.12
n10m 10 2.89 0.29
n10h 10 5.20 0.52
low. bound;up. bound
exp.value
std.dev.
cv
u20l 16; 24 20 2.31 0.12
u20m 10; 30 20 5.77 0.29
u20h 2; 38 20 10.39 0.52
u15l 12; 18 15 1.73 0.12
u15m 7; 23 15 4.62 0.31
u15h 1; 29 15 8.08 0.54
u10l 8; 12 10 1.15 0.12
u10m 5; 15 10 2.89 0.29
u10h 1;19 10 5.20 0.52
Yd , Y ∗d , D∗ and W , we computed the expected value and the standard deviations. In addi-
tion, the 95 % quantile for the waiting time was analyzed, as maximum waiting times playa crucial role in the context of analyses of manufacturing applications. We depict only theresults for the timeout-based batch building. However, analogous results were obtained forcapacity-based batch building.
To demonstrate the advantages of the generalized batch building model, the tables alsoinclude the results that were obtained from the limited model that was proposed by Schleyer(2007). Please note that our model generates the same results as the limited model for deter-ministic collecting interval length P (Tout = tdetr
out ) = 1 and amax ≤ tdetrout . The analysis there-
fore focuses on cases in which these aforementioned conditions do not hold. In fact, thelimited model is not directly applicable to all of the analyzed configurations because Tout
is a random variable and not a deterministic value; in addition, for several configurations,amax exceeds the collecting interval length. To obtain comparable results, we must modifythe input data for the limited model in accordance with the ideas of Matzka et al. (2011).In particular, the deterministic value for the collecting interval length is set to the expectedvalue of Tout , and the complete probability mass of values of A that are greater than the col-lecting interval length is shifted to the interarrival time that is equal to the collecting intervallength.
From Tables 3 and 4, it can be observed that the type of distribution (normal or uniform)does not influence the approximation quality. Both distribution types demonstrate deviationsfrom the simulation results that range from 1 % to 3 %, with few exceptions. Approxima-tion errors greater than 3 % were found for input configurations with low coefficients ofvariation; this result reflects the fact that in these configurations, the probability mass is dis-tributed among a small number of values. The approximation quality obviously depends onthe length of the discretization increment tinc . Smaller intervals will produce higher qualityapproximations. Figure 11 shows that for Tout = u20l and A = u15l, a rescaling generatesbetter results. The computation time remains very low compared with the simulation time,whereas the computed values for the standard deviation of D∗ and the expected waiting timeE[W ] tend to match the simulation value.
The contribution of the proposed model becomes apparent if the results of the generalmodel are compared with the results of the limited model. The limited model will generateapproximations of lower quality in situations involving high variability of Tout and largeamax values relative to the mean collecting times. By contrast, the proposed general modeldelivers highly accurate results for all of the analyzed configurations. The limited model
Ann Oper ResTa
ble
3A
ccur
acy
ofth
elim
ited
mod
el(L
.M.)
and
the
prop
osed
gene
raliz
edm
odel
(G.M
.)fo
rno
rmal
lydi
stri
bute
din
puts
Tout
AYd
Y∗ d
D∗
W
exp.
valu
est
d.de
v.ex
p.va
lue
std.
dev.
exp.
valu
est
d.de
v.ex
p.va
lue
std.
dev.
95%
quan
tile
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
n20
ln
10l
2.00
−2.4
5−1
.23
0.47
−12.
405.
102.
00−2
.45
−1.2
320
.01
−12.
405.
0920
.01
−0.0
41.
212.
300.
7310
.14
−3.8
9−1
.36
6.00
−3.8
21.
1119
.84
−6.7
50.
81
n20
mn
10l
2.00
−2.5
8−1
.33
0.72
−42.
540.
522.
02−3
.56
−1.2
320
.23
−40.
450.
3320
.23
−1.1
51.
245.
650.
3510
.84
−10.
02−1
.27
7.02
−17.
780.
8823
.35
−20.
770.
64
n20
hn
10l
2.07
−5.6
3−1
.97
1.08
−61.
40−3
.75
2.20
−11.
24−2
.08
21.9
9−5
7.25
−4.0
621
.99
−9.0
30.
369.
13−2
.05
12.6
1−2
2.69
−2.3
79.
00−3
5.89
−1.5
729
.56
−37.
42−1
.89
n20
ln
10m
2.00
−2.4
1−1
.19
0.62
−9.0
4−1
.66
2.00
−2.4
4−1
.19
20.0
1−8
.92
−1.6
820
.01
−0.0
41.
242.
330.
0910
.12
−3.6
7−1
.13
6.00
−3.8
51.
0719
.84
−6.7
50.
81
n20
mn
10m
2.00
−2.3
7−1
.12
0.82
−30.
53−1
.94
2.03
−3.9
8−1
.16
20.3
4−2
7.35
−1.9
020
.34
−1.6
71.
225.
700.
0910
.83
−10.
01−1
.25
7.02
−17.
830.
8223
.34
−20.
740.
69
n20
hn
10m
2.07
−5.5
4−1
.87
1.13
−49.
84−3
.60
2.21
−11.
87−2
.04
22.1
5−4
4.44
−3.8
422
.15
−9.7
20.
359.
13−1
.83
12.6
0−2
2.61
−2.2
89.
00−3
5.85
−1.5
029
.57
−37.
44−1
.93
n20
ln
10h
1.94
−1.7
7−1
.12
0.81
−7.8
7−2
.31
1.95
−2.6
5−1
.16
20.1
8−5
.83
−2.2
820
.18
−0.9
01.
222.
840.
1210
.12
−3.6
6−1
.12
6.00
−3.7
81.
1419
.82
−6.6
60.
91
n20
mn
10h
1.93
−1.7
5−1
.07
0.96
−21.
89−2
.17
2.01
−5.2
0−1
.10
20.7
3−1
6.86
−2.3
120
.73
−3.5
21.
255.
950.
2310
.84
−10.
07−1
.32
7.02
−17.
730.
9523
.35
−20.
770.
64
n20
hn
10h
2.00
−4.8
9−1
.78
1.22
−38.
84−3
.37
2.20
−13.
40−1
.98
22.6
9−3
2.07
−3.6
322
.69
−11.
840.
369.
25−1
.83
12.5
8−2
2.50
−2.1
38.
98−3
5.75
−1.3
529
.47
−37.
22−1
.59
n20
ln
15l
1.33
−1.6
4−0
.42
0.48
−3.2
4−0
.48
1.34
−1.9
7−0
.43
20.0
6−2
.12
−0.4
420
.06
−0.3
21.
252.
460.
1810
.12
−3.7
0−1
.16
6.00
−3.7
31.
2019
.82
−6.6
60.
91
n20
mn
15l
1.33
−1.6
7−0
.43
0.58
−19.
83−0
.90
1.40
−6.1
8−0
.53
20.9
7−9
.21
−0.7
220
.97
−4.6
21.
135.
850.
4610
.83
−10.
01−1
.25
7.02
−17.
790.
8723
.34
−20.
740.
69
n20
hn
15l
1.38
−4.7
0−1
.01
0.78
−40.
32−2
.41
1.55
−15.
52−1
.20
23.2
9−2
7.27
−2.8
923
.29
−14.
110.
479.
07−1
.85
12.5
9−2
2.56
−2.2
18.
98−3
5.72
−1.3
129
.50
−37.
29−1
.69
n20
ln
15m
1.33
0.41
−0.2
90.
55−1
0.75
−0.9
21.
37−2
.25
−0.3
820
.54
−3.8
2−0
.74
20.5
4−2
.65
1.16
3.72
−0.2
510
.14
−3.8
6−1
.33
5.99
−3.6
91.
2419
.82
−6.6
60.
91
n20
mn
15m
1.33
0.24
−0.4
30.
65−2
5.35
−1.1
11.
43−6
.68
−0.5
621
.49
−13.
30−1
.07
21.4
9−6
.95
1.11
6.35
0.44
10.8
4−1
0.08
−1.3
47.
03−1
7.85
0.80
23.3
8−2
0.87
0.51
n20
hn
15m
1.38
−2.7
9−0
.97
0.84
−41.
88−2
.58
1.58
−15.
57−1
.22
23.7
7−2
9.65
−3.1
623
.77
−15.
860.
389.
33−1
.90
12.6
1−2
2.68
−2.3
79.
01−3
5.95
−1.6
629
.60
−37.
50−2
.03
n20
ln
15h
1.29
8.80
3.29
0.70
−16.
69−2
.74
1.42
−0.8
30.
8021
.94
−2.6
30.
2821
.94
−8.8
5−1
.19
6.07
−11.
6210
.14
−3.8
9−1
.36
6.00
−3.7
71.
1619
.84
−6.7
50.
81
n20
mn
15h
1.29
8.94
3.45
0.78
−25.
55−1
.79
1.47
−4.6
40.
9522
.86
−10.
900.
2022
.86
−12.
50−1
.20
7.63
−6.4
310
.82
−9.8
9−1
.13
7.02
−17.
820.
8423
.37
−20.
840.
56
n20
hn
15h
1.33
5.37
2.63
0.93
−37.
64−1
.54
1.61
−12.
940.
5025
.01
−24.
89−0
.98
25.0
1−2
0.03
−1.5
410
.03
−4.7
512
.62
−22.
72−2
.42
9.01
−35.
91−1
.60
29.5
4−3
7.37
−1.8
3
n20
ln
20l
1.00
4.17
0.01
0.36
−44.
64−0
.61
1.07
−2.6
1−0
.08
21.3
9−2
1.49
−0.5
821
.39
−6.5
11.
165.
090.
5810
.14
−3.8
2−1
.28
6.00
−3.7
81.
1419
.84
−6.7
50.
81
n20
mn
20l
1.00
4.09
−0.0
40.
50−5
9.90
−0.6
51.
14−8
.78
−0.2
022
.83
−42.
80−0
.69
22.8
3−1
2.40
1.08
7.00
0.77
10.8
3−9
.97
−1.2
17.
02−1
7.78
0.88
23.3
5−2
0.77
0.64
n20
hn
20l
1.03
0.77
−0.7
50.
65−6
9.10
−2.0
51.
27−1
7.90
−0.9
725
.37
−56.
89−3
.13
25.3
7−2
1.18
0.27
9.49
−1.7
712
.63
−22.
80−2
.52
9.02
−36.
00−1
.73
29.6
1−3
7.52
−2.0
6
n20
ln
20m
1.00
12.2
2−0
.02
0.50
−34.
14−0
.74
1.14
−1.7
1−0
.20
22.8
3−6
.52
−0.8
222
.83
−12.
391.
096.
980.
5810
.13
−3.7
9−1
.25
5.99
−3.7
21.
2119
.83
−6.7
10.
86
n20
mn
20m
1.00
12.1
9−0
.01
0.58
−42.
85−0
.59
1.19
−6.0
2−0
.22
23.8
9−1
9.05
−0.6
123
.89
−16.
300.
998.
060.
6010
.82
−9.8
7−1
.11
7.02
−17.
730.
9423
.34
−20.
740.
69
n20
hn
20m
1.03
8.69
−0.6
40.
70−5
3.01
−1.8
51.
31−1
4.23
−0.9
126
.18
−35.
73−2
.75
26.1
8−2
3.60
0.29
10.1
3−1
.53
12.6
2−2
2.72
−2.4
19.
01−3
5.92
−1.6
229
.57
−37.
44−1
.93
n20
ln
20h
0.97
25.2
10.
750.
65−3
1.44
−0.9
81.
23−1
.25
−0.1
725
.36
−3.7
3−0
.73
25.3
6−2
1.14
0.32
9.23
−1.7
110
.13
−3.7
5−1
.21
6.00
−3.7
81.
1419
.83
−6.7
10.
86
n20
mn
20h
0.97
25.1
30.
720.
69−3
6.06
−0.7
41.
27−4
.31
−0.1
626
.16
−11.
61−0
.54
26.1
6−2
3.55
0.36
9.98
−1.2
010
.83
−10.
01−1
.26
7.02
−17.
790.
8723
.36
−20.
800.
60
n20
hn
20h
1.00
21.0
6−0
.06
0.79
−43.
99−1
.80
1.36
−11.
21−0
.86
28.1
7−2
5.91
−2.5
228
.17
−29.
01−0
.26
11.5
9−2
.47
12.6
2−2
2.71
−2.4
19.
01−3
5.96
−1.6
829
.58
−37.
46−1
.96
Ann Oper ResTa
ble
4A
ccur
acy
ofth
elim
ited
mod
el(L
.M.)
and
the
prop
osed
gene
raliz
edm
odel
(G.M
.)fo
run
ifor
mly
dist
ribu
ted
inpu
ts
Tout
AYd
Y∗ d
D∗
W
exp.
valu
est
d.de
v.ex
p.va
lue
std.
dev.
exp.
valu
est
d.de
v.ex
p.va
lue
std.
dev.
95%
quan
tile
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
Sim
abs.
L.M
.de
v.%
G.M
.de
v.%
u20
lu
10l
2.00
−0.0
1−0
.01
0.48
−20.
034.
092.
00−0
.01
−0.0
120
.00
−20.
034.
0920
.00
0.02
0.02
2.31
6.17
10.1
4−3
.82
−2.3
45.
99−3
.71
0.50
19.9
7−7
.36
0.15
u20
mu
10l
2.00
0.10
0.10
0.73
−47.
051.
862.
00−0
.06
0.21
20.0
2−4
6.76
1.54
20.0
2−0
.08
0.19
5.79
2.38
10.8
2−9
.92
−1.8
37.
02−1
7.80
0.75
23.6
8−2
1.88
−0.7
6
u20
hu
10l
2.00
−0.1
7−0
.17
1.13
−65.
921.
532.
20−9
.10
0.49
22.0
0−6
0.87
1.00
22.0
0−9
.11
0.48
9.72
0.93
12.7
1−2
3.31
−1.4
88.
94−3
5.47
0.69
29.5
1−3
7.31
−0.0
3
u20
lu
10m
2.00
−0.0
8−0
.08
0.63
−5.0
52.
702.
00−0
.08
−0.0
820
.01
−5.0
52.
7020
.01
−0.0
5−0
.05
2.31
6.12
10.1
4−3
.84
−2.3
66.
00−3
.82
0.38
20.0
1−7
.55
−0.0
5
u20
mu
10m
2.00
0.12
0.12
0.82
−26.
862.
552.
02−0
.90
0.38
20.1
8−2
4.89
2.01
20.1
8−0
.91
0.37
5.85
2.56
10.8
3−9
.95
−1.8
77.
02−1
7.78
0.77
23.6
8−2
1.88
−0.7
6
u20
hu
10m
2.00
−0.1
2−0
.12
1.19
−49.
652.
052.
22−9
.99
0.64
22.2
4−4
2.37
1.68
22.2
4−1
0.07
0.55
9.73
1.02
12.7
0−2
3.25
−1.4
08.
95−3
5.48
0.68
29.5
4−3
7.37
−0.1
4
u20
lu
10h
2.00
−0.0
8−0
.08
0.89
−1.3
22.
442.
01−0
.38
0.06
20.0
6−0
.70
2.19
20.0
6−0
.30
0.15
2.49
7.82
10.1
3−3
.79
−2.3
16.
00−3
.83
0.37
20.0
0−7
.50
0.00
u20
mu
10h
2.00
0.06
0.06
1.04
−15.
382.
482.
07−3
.29
0.48
20.6
9−1
0.87
2.14
20.6
9−3
.34
0.43
6.09
2.60
10.8
3−9
.95
−1.8
77.
02−1
7.76
0.81
23.6
3−2
1.71
−0.5
5
u20
hu
10h
2.00
0.02
0.02
1.35
−35.
101.
652.
29−1
2.79
0.50
22.9
2−2
6.43
1.78
22.9
2−1
2.76
0.53
9.76
1.15
12.6
9−2
3.17
−1.3
08.
95−3
5.48
0.67
29.4
9−3
7.27
0.03
u20
lu
15l
1.33
−0.0
2−0
.02
0.48
−0.8
70.
471.
34−0
.21
0.09
20.0
4−0
.21
0.09
20.0
4−0
.19
0.11
2.42
7.78
10.1
3−3
.80
−2.3
26.
00−3
.79
0.42
19.9
9−7
.45
0.05
u20
mu
15l
1.33
0.00
0.00
0.57
−17.
261.
761.
40−4
.60
0.46
20.9
6−5
.79
0.85
20.9
6−4
.60
0.47
6.15
2.47
10.8
3−9
.93
−1.8
57.
01−1
7.69
0.88
23.6
3−2
1.71
−0.5
5
u20
hu
15l
1.33
−0.0
6−0
.06
0.82
−42.
471.
091.
59−1
6.01
0.40
23.8
0−2
5.15
1.05
23.8
0−1
5.98
0.43
9.50
1.05
12.7
1−2
3.30
−1.4
78.
94−3
5.47
0.69
29.5
0−3
7.29
0.00
u20
lu
15m
1.33
2.13
−0.0
40.
56−9
.42
1.81
1.37
−0.9
10.
3320
.62
−1.6
81.
1020
.62
−3.0
00.
343.
835.
1210
.12
−3.6
6−2
.18
6.00
−3.7
70.
4319
.98
−7.4
10.
10
u20
mu
15m
1.33
2.13
−0.0
30.
67−2
4.54
1.92
1.44
−5.6
00.
5121
.64
−11.
561.
4421
.64
−7.5
80.
546.
642.
1310
.84
−10.
03−1
.96
7.02
−17.
740.
8323
.65
−21.
78−0
.63
u20
hu
15m
1.33
2.18
0.01
0.88
−43.
031.
231.
62−1
5.89
0.52
24.3
0−2
7.76
1.30
24.3
0−1
7.71
0.47
9.75
0.91
12.7
1−2
3.30
−1.4
78.
96−3
5.55
0.57
29.5
6−3
7.42
−0.2
0
u20
lu
15h
1.33
11.1
40.
020.
79−1
3.35
1.62
1.49
−0.3
50.
4322
.30
−0.0
31.
6522
.30
−10.
310.
436.
451.
7310
.14
−3.8
4−2
.36
6.00
−3.8
50.
3520
.00
−7.5
00.
00
u20
mu
15h
1.33
11.0
9−0
.02
0.87
−21.
141.
241.
55−4
.15
0.42
23.1
6−7
.48
1.28
23.1
6−1
3.66
0.51
7.74
1.32
10.8
3−9
.99
−1.9
17.
02−1
7.83
0.72
23.6
5−2
1.78
−0.6
3
u20
hu
15h
1.33
11.1
80.
061.
04−3
3.79
1.29
1.70
−12.
990.
6625
.56
−20.
381.
4725
.56
−21.
750.
5910
.23
1.05
12.7
0−2
3.22
−1.3
78.
95−3
5.51
0.64
29.5
1−3
7.31
−0.0
3
u20
lu
20l
1.00
5.63
0.04
0.37
−37.
222.
681.
07−1
.46
0.41
21.4
4−1
0.87
2.78
21.4
4−6
.72
0.37
5.17
2.43
10.0
9−3
.38
−1.8
96.
01−3
.99
0.21
20.0
2−7
.59
−0.1
0
u20
mu
20l
1.00
5.57
−0.0
20.
51−5
5.29
1.27
1.15
−8.3
50.
3823
.04
−36.
041.
1823
.04
−13.
190.
407.
181.
3010
.83
−10.
00−1
.93
7.02
−17.
830.
7223
.67
−21.
84−0
.72
u20
hu
20l
1.00
5.52
−0.0
70.
68−6
6.15
0.77
1.30
−18.
620.
3825
.95
−50.
110.
7525
.95
−22.
920.
389.
640.
9312
.71
−23.
28−1
.45
8.95
−35.
490.
6629
.51
−37.
31−0
.03
u20
lu
20m
1.00
14.6
8−0
.01
0.51
−31.
051.
461.
15−0
.41
0.43
23.0
3−1
.45
1.43
23.0
3−1
3.16
0.44
7.17
1.33
10.1
3−3
.79
−2.3
16.
00−3
.78
0.42
19.9
7−7
.36
0.15
u20
mu
20m
1.00
14.6
7−0
.01
0.58
−39.
171.
371.
20−4
.44
0.47
24.0
1−1
3.16
1.52
24.0
1−1
6.69
0.46
8.09
1.42
10.8
4−1
0.06
−1.9
97.
02−1
7.83
0.72
23.6
8−2
1.88
−0.7
6
u20
hu
20m
1.00
14.9
30.
210.
72−5
1.17
1.17
1.33
−13.
650.
6626
.59
−30.
661.
5026
.59
−24.
790.
5610
.18
0.95
12.7
0−2
3.22
−1.3
78.
94−3
5.45
0.72
29.4
9−3
7.27
0.03
u20
lu
20h
1.00
29.4
6−0
.06
0.72
−25.
621.
111.
30−0
.08
0.46
25.9
1−0
.13
1.43
25.9
1−2
2.82
0.52
9.19
0.83
10.1
4−3
.82
−2.3
46.
00−3
.83
0.38
20.0
0−7
.50
0.00
u20
mu
20h
1.00
29.6
90.
110.
76−2
9.53
1.21
1.33
−2.5
40.
5926
.62
−6.6
01.
6526
.62
−24.
870.
459.
990.
9010
.85
−10.
13−2
.07
7.03
−17.
870.
6623
.70
−21.
94−0
.84
u20
hu
20h
1.00
29.6
80.
100.
86−3
7.67
1.12
1.43
−9.2
60.
6628
.54
−19.
451.
5428
.54
−29.
910.
7211
.48
0.82
12.6
8−2
3.12
−1.2
38.
94−3
5.43
0.75
29.4
8−3
7.25
0.07
Ann Oper Res
Fig. 11 Impact of tinc , the discretization increment length, on the accuracy of the proposed model
always delivers a standard deviation of 0 for D∗, which leads to 100 % relative deviation,and is therefore not included in Table 3 and 4. Furthermore, the average computation time forthe 54 input configurations is 0.07 s for the limited model, 2.19 s for the general model and114.56 s for the simulation model. Although the generalized model involves slightly greatercomputation times than the limited model, the proposed model remains over 52 times fasterthan the simulation model.
3.2 A transportation example for the general batch building model under the timeout rule
As mentioned in the introduction, discrete-time calculation methods are suitable for the anal-ysis of many production or service operations. In the following section, we will demonstratethe applicability of the developed method to the example of sterilization processes in healthestablishments.
The sterilization of reusable medical devices occurs through the following procedures(Di Mascolo et al. 2006; Matzka et al. 2011). Directly after their use in the operation room,the medical devices (MDs) are placed in a disinfectant liquid to decrease the populationof micro-organisms that are present on the soiled equipment, protecting the medical staffduring their manipulations of these MDs and facilitating the subsequent washing of theMDs. The used MDs from several operations are collected before they are transported tothe sterilization area by an elevator. During the collection and transport of these MDs, theyremain immersed in the disinfectant liquid. At the sterilization area, the MDs are rinsed.The MDs are washed then in machines to eliminate stains and to obtain clean MDs. Afterthis washing step, the MDs are assessed to ensure that no deterioration may affect theirsecurity and functionality. Subsequently, the MDs are packed to create a barrier againstmicro-organisms. In a sterilization step, the MDs are placed in an autoclave, where they aretreated with saturated steam. Finally, the sterile MDs are transferred from the sterilizationservice back to a storage area that is near to the operating rooms.
We will now focus on the analysis of the first step, the pre-disinfection step that occursdirectly after the use of the MDs. In the pre-disinfection step, the devices must stay in thedisinfectant liquid for approximately 15 minutes to guarantee an optimal impact of the disin-fectant on the material. However, the sojourn time in the liquid should not exceed 50 minutesbecause the disinfectant product attacks the material and causes premature aging. The so-journ time of the MDs in the disinfectant liquid is the sum of the waiting time during thecollecting process, the time that is required for the transport from the operation area to thesterilization area and the waiting time of the MDs in front of the rinsing station.
Ann Oper Res
Fig. 12 A queueing networkmodel of the pre-disinfectionprocess
As illustrated in Fig. 12, we can model the pre-disinfection process using discrete-timequeueing models (this model may be compared to Matzka et al. 2011). At the input of themodel, the entities arriving at the pre-disinfection step are called “operations” and repre-sent a batch of containers and bags of medical devices that were used for a single surgicaloperation.
We assume that the MDs of several operations are collected within a certain time inter-val tout before they are transferred to the sterilization area. Thus, the pre-disinfection andtransfer step can be modeled by a combination of the general batch building model underthe timeout rule that is presented in this paper and a subsequent G|G|1-queue (Grassmannand Jain 1989). At the rinsing step, a batch of operations arrives, and single operations areserved. We can model this step with a Gx |G|1-queue with batch arrivals and single service(Schleyer and Furmans 2007).
In the present system, the transfer of MDs to the sterilization area should be performed atregular intervals of tout = 30 minutes. The personnel are advised to transfer batches of MDsat the top and bottom of every hour. As this process is triggered by consulting a watch, theintervals between two transfers (tout ) are not deterministic but stochastic with the distributionshown in Table 5 with E(Tout ) = 30 minutes and a standard deviation of 4 minutes and17 seconds. The interarrival time of MDs, the transport time and the rinsing time are alsoprovided in terms of discrete probability functions (see Table 5).
The unsteadiness of the collecting interval tout can cause durations in pre-disinfectionthat lead to undesirable effects on the material. Using the queueing network model, we canderive the sojourn time distribution of MDs in the disinfectant liquid. As observed in Table 6,for the original scenario, the duration of the pre-disinfection process exceeds 50 minutes in18 % of the cases.
To improve the performance of this system, different scenarios for the transfer of MDsto the sterilization area can be analyzed. We will now present two different strategies to im-prove the system performance. First, we will reduce the expected collecting interval E(Tout )
from 30 minutes to 20 minutes to reduce the sojourn time of MDs in the disinfectant liquid.The standard deviation of Tout remains the same, as the transport is still triggered by consult-ing a clock. A second strategy is to implement an acoustic signal that triggers the transportevery 30 minutes, thus, leading to a reduction in the variability of the collecting time. Thedistributions of Tout for both strategies can be found in Table 5.
In Table 6, we observe that the percentage of orders that remain in the liquid for morethan 50 minutes decreases enormously for both strategies. Planners in health establishmentswould intuitively decide to increase the transport frequency if they sought to reduce the so-journ time of MDs in the disinfectant liquid. As we determined in our analysis, reducing the
Ann Oper Res
Table 5 Input parameters for the pre-disinfection process
t Original scenario Strategy 1 Strategy 2
P(A = t) P (Tout = t) P (transport = t) P (rinsing = t) P (Tout = t) P (Tout = t)
0 0 0 0 0 0 0
1 0.022 0 0 0.018 0 0
2 0.025 0 0 0.009 0 0
3 0.042 0 0 0.027 0 0
4 0.022 0 0.2 0.027 0 0
5 0.172 0 0.6 0.100 0 0
6 0.009 0 0.2 0.018 0 0
7 0.04 0 0.072 0 0
8 0.023 0 0.148 0 0
9 0.012 0 0.200 0.003 0
10 0.112 0 0.083 0.008 0
11 0.002 0 0.063 0.012 0
12 0 0 0.050 0.015 0
13 0.011 0 0.045 0.026 0
14 0.012 0 0.090 0.035 0
15 0.062 0 0.050 0.045 0
16 0.01 0 0.055 0
17 0.013 0 0.072 0
18 0.015 0 0.084 0
19 0 0.003 0.095 0
20 0.042 0.008 0.100 0
21 0 0.012 0.095 0
22 0.002 0.015 0.084 0
23 0 0.026 0.072 0
24 0 0.035 0.055 0
25 0.021 0.045 0.045 0.005
26 0 0.055 0.035 0.007
27 0.032 0.072 0.026 0.008
28 0 0.084 0.015 0.055
29 0.112 0.095 0.012 0.150
30 0.027 0.100 0.008 0.550
31 0 0.095 0.003 0.150
32 0.017 0.084 0.055
33 0 0.072 0.008
34 0.024 0.055 0.007
35 0.02 0.045 0.005
36 0 0.035
37 0.03 0.026
38 0 0.015
39 0 0.012
40 0 0.008
41 0 0.003
Ann Oper Res
Table 5 (Continued)
t Original scenario Strategy 1 Strategy 2
P(A = t) P (Tout = t) P (transport = t) P (rinsing = t) P (Tout = t) P (Tout = t)
42 0
43 0.002
44 0
45 0.02
46 0.02
47 0.002
48 0
49 0.01
50 0.002
51 0
52 0.002
· · · 0
55 0.002
56 0
57 0.001
· · · 0
73 0.001
74 0.001
75 0
76 0.001
77 0
78 0.001
· · · 0
82 0.001
· · · 0
86 0.001
· · · 0
90 0.002
exp. value 17.82 30.00 5.00 9.04 20.00 30.00
std. dev. 14.29 4.17 0.63 3.32 4.17 1.17
Table 6 Percentage of MDs thatstay in the disinfectant liquid formore than 50 minutes
P(sojourn time > 50 minutes)
Original scenario 18.6 %
Strategy 1: higher frequency 1.6 %
Strategy 2: lower variability 4.9 %
process variability by simply implementing an acoustic signal, can also reduce the sojourntime to acceptable levels. In strategy 2, labor costs do not increase. However, the percentageof material that can be negatively affected is greater than it is if strategy 1 is utilized. The
Ann Oper Res
Fig. 13 Deviations of the interdeparture and waiting time distributions between limited and general model
queueing network model can be used to find a cost-optimal operating point that accounts forboth labor and material costs.
3.3 A processing example for the general batch building model under the capacity rule
Batch building under the capacity rule is considered through the following example, whichexamines the collecting process that occurs prior to the use of batch processes in manufac-turing systems. The input must be collected until the capacity of the processor is reached.The output of the collecting process then serves as the input for the batch processor. Ex-amples from the semiconductor industry are mentioned in the introductory section of thispaper.
We assume that the size of the arriving batches is distributed as a discretized normaldistribution with mean 15 and cv = 0.25. We further choose ymax = 30, given that the prob-ability of greater batches is negligible. The common practices for manufacturing systemsinvolve processor capacities that are designed to match the average arriving batch size: thus,we assume that k = 15. The interarrival times are assumed to be uniformly distributed acrossthe intervals from 5 to 20 minutes.
The limited model by Schleyer and Furmans (2005) is not applicable to this basic prob-lem, as 30 = ymax > k = 15. We adjust the input in accordance with the ideas presented inSection 3.1 to demonstrate how misleading the application of the limited model is for thisfundamental practical problem. The complete probability mass of arrivals greater than 15 isshifted to arrivals of size 15, which corresponds to the largest value that the limited modelcan handle. Figure 13 indicates the superiority of the proposed general batch building modelcompared with the approximative approach of the limited model.
The results that were computed with the two models demonstrated significant deviationswith respect to the interdeparture and waiting time distributions. The limited model over-estimates the expected interdeparture time by 12.4 %. Deviations of the higher momentsare clearly observable directly from Fig. 13. The limited model with ymax < k allows forno simultaneous departures of multiple batches. The general model correctly considers theprobability of P (D = 0), whereas the limited model sets the probability of P (D = 0), to 0.This simplification results in the aforementioned overestimation of the expected interdepar-ture time.
In addition, the waiting time distributions differ significantly. The limited model drasti-cally overestimates the chance of no waiting time at all. This effect is due to the shiftingof the probability mass to the collecting capacity k. If batches of a size that is equal to the
Ann Oper Res
capacity arrive, no collecting process is necessary. However, the unmodified input containsa number of arrivals that is greater than the capacity. For these arrivals, only a part of thearriving units requires no collecting process, whereas units exceeding the collecting capacitymust wait for the following arrivals to become a part of complete batch. The underestimatingof waiting times becomes particularly visible if the 95 %-quantiles of the distributions arecompared. The limited model underestimates this quantile by 42.1 %. Given the exampleof the semiconductor industry in which long waiting times lead to the reworking or scrap-ping of the batch components, the relevance of the developed general batch building modelbecomes obvious.
4 Conclusion
In this paper, general models for the batch building process under the timeout and capacityrules are proposed. These models relax the limiting assumptions of the existing batch build-ing models in the discrete-time domain. In particular, the new model for the batch buildingprocess under the timeout rule relaxes two significant restrictions: the limitation of interar-rival times to values that are smaller than the collecting interval length and the assumptionof a deterministic collecting interval length. As a result, the investigated batch building pro-cess may yield collecting periods in which no arrival is observed. Moreover, the relaxationsrequire a new set of performance measures, which are developed to describe all of the char-acteristics of the general batch building process. The suggested closed-form solutions ofthe model outperform simulation models with respect to computation times. Thus, the newdiscrete-time method is capable of handling large-scale studies on the influence of differ-ent system parameters. In addition, the accuracy of the discrete-time approach is shown bycomparing its results to the results of a discrete-event simulation model in continuous-time.
Because of the relaxed assumptions, our analytical method is suitable for modeling realcases. For example, the case of the sterilization process in health establishments is consid-ered. We demonstrate how a reduction in the variability of the collecting times can lead toa reduction in waiting times that is similar to the reduction that is produced by a highercollection frequency. In addition, the batch building process that occurs prior to the batchprocesses in manufacturing systems is analyzed. The numerical study indicates that the ex-isting method significantly underestimates the waiting times. The proposed model thereforeadds value to the analysis of manufacturing contexts in which waiting times are crucial, suchas the semiconductor industry.
Acknowledgements The authors would like to thank the reviewers for their valuable comments and sug-gestions to improve the quality of the paper. This research is supported by the research project “QuantitativeAnalyse stochastischer Einflüsse auf die Leistungsfähigkeit von Produktionssystemen mittels analytischerund simulativer Modellierung”, which is funded by the Deutsche Forschungsgemeinschaft (DFG) (referencenumber FU-273/8-1)
References
Ackroyd, H. M. (1980). Computing the waiting time distribution for the G/G/1 queue by signal processingmethods. IEEE Transactions on Communications, 38(1), 52–58.
Bitran, G. R., & Tirupati, D. (1989). Approximation for product departures from a single-server station withbatch processing in multi-product queues. Management Science, 35(7), 851–878.
Buzacott, J. A., & Shanthikumar, G. (1993). Stochastic models of manufacturing systems. Englewood Cliffs:Prentice Hall.
Ann Oper Res
Di Mascolo, M., Gouin, A., & Ngo Cong, K. (2006). Organization of the production of sterile medical de-vices. In Proceedings of the 12th IFAC symposium on information control problems in manufacturing—INCOM 2006: Saint Etienne, France (pp. 35–40).
Fowler, J., Phojanamongkolkij, N., Cochran, J., & Montgomery, D. (2002). Optimal batching in a waferfabrication facility using a multiproduct G/G/c model with batch processing. International Journal ofProduction Research, 40(2), 275–292.
Furmans, K. (2004). A framework of stochastic finite elements for models of material handling systems,progress. In Material handling research: 2004 (Vol. 8). International Material Handling Research Col-loquium, Graz.
Grassmann, W. K., & Jain, J. L. (1989). Numerical solutions of the waiting time distribution and idle timedistribution of the arithmetic GI/G/1 queue. Operations Research, 37(1), 141–150.
Hasslinger, G., & Klein, T. (1999). Breitband-ISDN und ATM-Netze. Leipzig: Teubner.Hopp, W. L., & Spearman, M. L. (1996). Factory physics: foundation of manufacturing management. New
York: McGraw-Hill.Hübner, F., & Tran-Gia, P. (1995). Discrete-time analysis of cell spacing in ATM systems. Telecommunica-
tions Systems, 3, 379–395.Kim, J.-H., Lee, T.-E., Lee, H.-Y., & Park, D.-B. (2003). Scheduling analysis of time-constrained dual armed
cluster tools. IEEE Transactions on Semiconductor Manufacturing, 16(3), 521–534.Kim, J.-H., & Lee, T.-E. (2008). Schedulability analysis of time-constraint cluster tools with bounded time
variation by an extended petri net. IEEE Transactions on Automation Science and Engineering, 5(3),490–503.
Kitamura, S., Mori, K., & Ono, A. (2006). Capacity planning method for semiconductor fab with time con-straints between operations. In Proceedings of the 2006 SICE-ICASE international joint conference,Busan, Korea (pp. 1100–1103).
Lee, T.-E., & Park, S.-H. (2005). An extended event graph with negative places and tokens for time windowconstraints. IEEE Transactions on Automation Science and Engineering, 2(4), 319–332.
Matzka, J., Di Mascolo, M., & Furmans, K. (2011). Queueing analysis of the production of sterile medicaldevices by means of a hybrid model. In 8th conference on Stochastic Models of Manufacturing andService Operations (SMMSO 2011), Kusadasi, Turkey.
Matzka, J. (2011). Discrete time analysis of Multi-Server Queueing Systems in Material Handling and Ser-vice. Ph.D. thesis, Karlsruhe Institute of Technology, Institut für Fördertechnik und Logistiksysteme.
Meng, G., & Heragu, S. (2004). Batch size modeling in a multi-item, discrete manufacturing system via anopen queueing network. IIE Transactions, 36(8), 743–753.
Omosigho, S., & Worthington, D. (1988). An approximation of known accuracy for single server queues withinhomogeneous arrival rate and continuous service time distribution. European Journal of OperationalResearch, 33(3), 304–313.
Özden, E., & Furmans, K. (2011). Discrete time analysis of takted milk-run systems. In 8th conference onStochastic Models of Manufacturing and Service Operations (SMMSO 2011), Kusadasi, Turkey.
Özden, E. (2011). Discrete time analysis of consolidated transport processes. Ph.D. thesis, Karlsruhe Instituteof Technology, Institut für Fördertechnik und Logistiksysteme.
Robinson, J. K., & Giglio, R. (1999). Capacity planning for semiconductor wafer fabrication with time con-straints between operations. In Proceedings of the 1999 winter simulation conference, Phoenix, Arizona,USA (pp. 880–887).
Rostami, S., Hamidzadeh, D., & Camporese, D. (2001). An optimal periodic scheduler for dual-arm robotsin cluster tools with residency constraints. IEEE Transactions on Robotics and Automation, 17(5), 609–618.
Schleyer, M. (2007). Discrete time analysis of batch processes in material flow systems. Ph.D. thesis, Univer-sität Karlsruhe, Institut für Fördertechnik und Logistiksysteme.
Schleyer, M. (2012). An analytical method for the calculation of the number of units at the arrival instant ina discrete time G/G/1-queueing system with batch arrivals. OR Spektrum, 34(1), 293–310.
Schleyer, M., & Furmans, K. (2005). Analysis of batch building processes. In Proceedings of the 20th IARannual meeting, Mulhouse, France.
Schleyer, M., & Furmans, K. (2007). An analytical method for the calculation of the waiting time distributionof a discrete time G/G/1-queueing system with batch arrivals. OR Spektrum, 29(4), 745–763.
Schleyer, M., & Gue, K. (2012). Throughput time distribution analysis for a one-block warehouse. Trans-portation Research. Part E, Logistics and Transportation Review, 48(3), 652–666.
Shi, C., & Gershwin, S. B. (2011). Part waiting time distribution in a two-machine line. In 8th conference onStochastic Models of Manufacturing and Service Operations (SMMSO 2011), Kusadasi, Turkey.
Ann Oper Res
Tajan, J., Sivakumar, A., & Gershwin, S. (2011) Controlling job arrivals with processing time windows intobatch processor buffer. Annals of Operations Research, 191(1), 193–218.
Tran-Gia, P. (1996). Analytische Leistungsbewertung verteilter Systeme. Berlin: Springer.Worthington, D., & Wall, A. (1999). Using the discrete time modelling approach to evaluate the time-
dependent behaviour of queueing systems. Journal of the Operational Research Society, 50(8), 777–788.
Yang, D.-L., & Chern, M.-S. (1995). A two-machine flowshop sequencing problem with limited waiting timeconstraints. Computers & Industrial Engineering, 28(1), 63–70.
