Computers & Operations Research 35 (2008) 2520 – 2536
www.elsevier.com/locate/cor
Queueing-model based analysis of assembly lines with finite buffers
and general service times
Michael Manitz
Department for Supply Chain Management and Production, University of Cologne, 50923 Köln, Germany
Available online 8 February 2007
Abstract
In this paper, we study the production process on multi-stage assembly lines. These production systems comprise simple processing
as well as assembly stations. At the latter, workpieces from two or more input stations have to be merged to form a new one for
further processing. As the flow of material is asynchronous with stochastic processing times at each station, queueing effects arise
as long as buffers provide waiting room. We consider finite buffer capacities and generally distributed processing times. Processing
is a service operation to customer items in the sense of a queueing system. The arrival stream of customer items is generated by
processing parts at a predecessor station. This paper describes an approximation procedure for determining the throughput of such
an assembly line. Exact solutions are not available in this case. For performance evaluation, a decomposition approach is used. The
two-station subsystems are analyzed by G/G/1/N stopped-arrival queueing models. In this heuristic approach, the virtual arrival
and service rates, and the squared coefficients of variation of these subsystems are determined. A system of decomposition equations
which are solved iteratively is presented. Any solution to this system of equations indicates estimated values for the subsystems’
unknown parameters. The quality of the presented approximation procedure is tested against the results of various simulation
experiments.
Scope and purpose: We consider assembly lines, i.e. flow production systems with a convergent flow of material and synchronization constraints. By considering assembly operations, our paper generalizes the work of Buzacott, Liu and Shanthikumar [Multistage
flow line analysis with the stopped arrival queue model. IIE Transactions 1995;27:444–55], in which flow lines as series production
systems are analyzed. It also extends the work of Helber [Performance analysis of flow lines with non-linear flow of material,
Lecture notes in economics and mathematical systems, vol. 473, Berlin, Heidelberg, New York: Springer; 1999] and Jeong and Kim
[Performance analysis of assembly/disassembly systems with unreliable machines and random processing times. IIE Transactions
1998; 30(1):41–53], who analyze assembly systems, by considering general service times. Station failures can be incorporated with
the completion time concept proposed by Gaver [A waiting line with interrupted service, including priorities. Journal of the Royal
Statistical Society 1962;24(2):73–90. [47]]. The comparison to various simulation results shows that the queueing-model based
approach presented in this paper yields quite good approximations of the throughput.
䉷 2007 Elsevier Ltd. All rights reserved.
Keywords: Flow production; Assembly lines; Assembly stations; Synchronization constraint; Decomposition; Queueing models; G/G/1/N
stopped-arrival queueing system; Throughput
E-mail address: [email protected].
0305-0548/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.cor.2006.12.016
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
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1. Introduction
1.1. Motivation
Flow production systems are installed for products that are produced in high quantities. The layout of these production
systems is determined by the flow of material in accordance with the sequence of the operations to be performed.
Flexibility is needed, if customers can choose variants of a certain product or a product family. From a machine’s
perspective, this results in stochastic processing times. Besides the differences in the variants’ work load, the time
required to complete a job at a production station is stochastic if machine failures occur, manual operations are
performed, or transfer line segments (which can be seen—in an aggregate view—as a single station of a flow production
system) may stop their operations because of disruptions in the flow of material. The stochastics prevent the stations
from being perfectly balanced over time. The flow of material is asynchronous. At a given station of the production
line, available capacity after the completion of a job and its transfer to the succeeding station and the need for capacity
by a job completed at the predecessor station are not perfectly coordinated in time. This is the typical situation, where
queueing effects occur. Using corresponding queueing models, in this paper, we will analyze such flow production
systems with stochastic processing times.
For series flow production systems with exactly one input station (flow lines), procedures for analyzing the number
of items that leave the system per unit of time—and various other performance measures—have been developed. For
systems with generally distributed processing times, latest developments come from Buzacott et al. [1] and Tempelmeier
and Bürger [2].
In manufacturing practice (for instance in car body assembly shops; see [3–5]), there are not only series flow lines
(with stations arranged one behind the other), but some stations at which assembly operations are performed (assembly
lines). The considerable difference from flow lines which can be analyzed by known methods is that a number of
required components are brought together to form a single unit for further processing at the assembly stations. An
assembly operation can begin only if all required parts or subassemblies are available (synchronization constraint).
Methods for analyzing such assembly systems have to consider the synchronization constraint. For systems with exponentially distributed processing times, Helber [6] and Jeong and Kim [7] have presented approaches for performance
evaluation. The restriction to exponentially distributed processing times enables a Markov-chain based analysis.
Performance analysis of flow production systems is generally needed during the planning phase regarding the system
design, when the decision for a concrete configuration of such a system has to be made. The planning problem arises
e.g. with the introduction of a new model or the installation of a new manufacturing plant. Because of the investments
involved, an optimization problem arises, see Spieckermann et al. [5], Tempelmeier [8,9]. The expenditure for new
machines, for buffer or handling equipment, and the holding costs for the expected work-in-process face revenues from
sold products. The performance of a concrete configuration is characterized by the throughput, i.e. the number of items
that are produced per time unit. Other performance measures are the expected work in process or the time an item
spends in the system, respectively.
In this paper, we present an analytical method for the performance evaluation of assembly lines. Simulation is an
obvious alternative evaluation method which is widely used as the only planning tool for configuration. However, a
large amount of time is required to build up a simulation model and to run it until statistically significant results are
achieved. Therefore, considering the huge number of possible configuration alternatives that have to be evaluated during
the optimization process, an industrial system planner is forced to restrict the number of analyzed configurations, which
often results in decisions for suboptimal assembly-line designs. By contrast, analytical methods for evaluating design
alternatives provide their results at the touch of a button. Thus, they can be implemented as an integral part of the
optimization procedure.
1.2. Problem description
Let us consider an assembly line with M machines or stations, respectively, like the one in Fig. 1. Between any two
successive stations, there is a buffer with finite capacity. The buffer between station i and station m, where finished
workpieces from station i are stored waiting for service at station m, is denoted by Bi,m . Its capacity is of size Ci,m . At
assembly stations, service is given to groups of workpieces made up of exactly one part from each input buffer. Other
relations can be modeled by identifying batches of arriving workpieces and scaling the size of one buffer place.
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M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
Legend:
station
buffer
assembly stations
Fig. 1. An assembly line with 11 stations, among them two assembly stations.
Because of the synchronization constraint, the assembly operation cannot begin as long as at least one of the required
components is not available. We assume that a part coming from any predecessor station can be loaded onto the server
independently from other components. In each of the queues, the next item to be processed waits at the server of
the assembly station until all required components are available. This mechanism prevents the effective input buffer
capacity from being reduced by 1. The workpieces wait for synchronization at the server. In terms of a queueing
model, this waiting time extends the holding time of an item at the server. This is also the crucial idea to the proposed
approximation procedure in this paper.
When a buffer is full, the station in front of this buffer may be blocked. In production systems, each station usually
continues its service as long as an uncompleted part is available for processing. Upon a service completion, the station
is blocked if it cannot transfer the completed part into the following buffer because it finds the buffer full (transfer
blocking, blocking after service). This blocking mechanism is typical for manufacturing systems. It increases the
effective buffer capacity by 1 including the place at the server of the blocked station.
When all the input buffers of a station are empty, starving occurs if the server is not occupied. In this paper, a station
is defined to be starved if it is empty. The starving time continues until the next part will have arrived. By definition, the
waiting time for synchronization, during which at least one of the required components of an assembly job is available,
does not belong to the starving time.
In order to analyze the performance of an assembly line independently of the effective material supply and of the
finished products’ demand, we assume saturated systems. This means, we start from the assumption that all the input
stations are never starved and that the last station of our assembly system is never blocked by any limited capacity of
an outlet store.
The processing times are stochastic. We do not make an assumption regarding the kind of probability distribution of
the processing times. Utilizing practical data (for the automotive industry see for instance [10]), it can be assumed that
the time TmS needed for processing a part at station m is generally distributed with the mean
E{TmS } =:
1
m
(m = 1, 2, . . . , M),
(1)
and the coefficient of variation
CV{TmS } =: m
(m = 1, 2, . . . , M).
(2)
Usually, the moments can be estimated from given data and, therefore, are assumed to exist. A more detailed analysis,
especially for manual tasks, shows a positive skew nature of the service operation times (see [11–13]). Empirical studies
reveal that the coefficient of variation of processing times is considerably less than 1 (see [8]). It will grow up if short
adjustments, setups, or other service interruptions are considered as a part of the effective processing time; the CVs
may be high (which means beyond 1.33) in the case of extremely long machine failures (see [14]). In this paper, we
describe the probability distribution of service times by only two moments. Powell and Pyke [15] and Lau and Martin
[13] show that for a normal skewness (commonly positive with an unimodal shape of the density function) the influence
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
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of higher moments on the processing time distribution is moderate. In addition, the data collection by the practitioners
usually does not include any information concerning the third or any higher moment.
The production rate X of a manufacturing system is defined as the effective output rate, which gives the mean number
of finished workpieces having left the last station of the system per unit of time (if the stations are numbered from input
stations to the output station, the last station is station M). Because of flow conservation in stable systems, the effective
output rate equals the effective input rate that describes the effective arrival rate at which workpieces enter the system.
For saturated systems, X is the maximum rate at which a system can produce. Thus, X is also called the throughput of
the system.
When a station m is busy, it produces parts at a processing rate m . Still, any station is busy only with a certain
probability. Blocking, starving, waiting for components, or even failures prevent a station from being productive. For
the last station M, it follows:
X = M · P{station M is busy}.
(3)
The remaining question is how to determine the probability that the station M is busy. For the last station, this is the
probability that it is neither under repair nor starved, nor waiting for synchronization. However, the probability of
idleness is a result of the combination of all the stochastic effects within the assembly line, increasing and/or canceling
out each other. Therefore, it is not trivial to compute this probability. Throughout the rest of this paper, however,
we present an algorithm to estimate the blocking and starving probabilities and, with that, the production rate of the
assembly line.
1.3. Literature review
The related work to this paper can be traced back according to four directions.
Markovian systems. First, for very small systems with exponentially distributed processing times, Markov-chain
models may give exact solutions, see Altiok [16]. For a series production system, Hillier and Boling [17] have formulated
a Markov-chain model. Moreover, this approach allows operation-dependent station failures to be modeled explicitly,
if the time to failure and the downtime are exponentially distributed, too. For two-station series systems, Buzacott [18]
and Gershwin and Berman [19] have presented such Markovian models.
Decomposition approaches for series systems. The state space of the Markov chains is growing with the number
of buffer places and, even more dramatically, with the number of buffers or stations, respectively. For the reason of
computational tractability, for systems with more than two or three stations, heuristic approximations are used instead
of exact Markovian models. To compute the throughput of a flow line with exponentially distributed processing times,
in addition to their exact approach, Hillier and Boling [17] have developed a decomposition approach. Every station
can be seen as a service facility with a waiting room for arriving items in front of this station. This is equivalent to a
standard queueing system. If the buffers have finite capacity, the station in front of it sometimes is blocked. Then, the
inter-arrival times to the buffers are not exponentially distributed, even if all service times are. Thus, Perros and Altiok
[20] describe the service completion time at the stations with Coxian phase-type distributions. Altiok and Ranjan [21]
have extended this approach to drop the exponentiality assumption for the service times. To get rid of the exponential
distribution, two-moment approximations for general service times can be used. Buzacott et al. [1] and Tempelmeier
and Bürger [2] propose approximation algorithms to compute the throughput of flow lines with generally distributed
processing times. Then, the subsystems are modeled as G/G/1/N queueing systems with stopped arrivals. These two
approaches differ a little in adjusting the virtual service rates of the corresponding queueing models.
All the above mentioned approaches for the analysis of asynchronous flow lines are based on the decomposition
of an M-stage line into M − 1 subsystems. The subsystems consist of one buffer with the succeeding processing
station. They are modeled as queueing systems. Alternatively, if it is appropriate, the subsystems can be modeled as a
Markov chain. In this context, there is substantial literature on the analysis of transfer lines (synchronous systems with
deterministic processing time that is identical for all stations or continuous-material systems with specific machine
speeds; for an overview, see [22,23]). A key early research in this field was done by Zimmern [24] and Sevast’yanov [25],
where continuous material models have been used. They made use of the idea of decomposition, which underlies all
subsequent work both on transfer lines and on asynchronous flow lines. Later, to determine the subsystems’ parameters,
for synchronous transfer lines, the DDX algorithm by Dallery et al. [26] was introduced, see also Papadopoulos et al.
[27]. Based on this algorithm, Burman [28] has developed a faster, more reliable procedure that also allows for different
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M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
machine speeds, the ADDX algorithm. These Markov-chain based procedures are also appropriate to asynchronous
systems in the case of exponentially distributed processing times.
Assembly-like queues. The first pioneering work in analyzing assembly queueing systems with synchronization
constraints was done by Harrison [29]. He studied assembly-like queues with unlimited buffer capacities. Also, as
opposed to common queueing systems, he showed that assembly systems can be unstable even if m < M for all
input stations, m = 1, 2, . . . , M − 1. This is why the time an item has to wait for synchronization may grow without
bound. The limitation of the number of items in the system then works as a control mechanism which ensures stability.
Assembly-like queues with finite buffers are analyzed by Bhat [30], Lipper and Sengupta [31], Hopp and Simon [32],
Altiok [16]. They all assume exponential service times. The analysis of assembly-like queues can be extended onto
fabrication lines as an input for the assembly station. Duenyas and Hopp [33,34] and Rao and Suri [35] consider closed
systems. This analysis is extended onto generally distributed processing times by Duenyas [36] and Rao and Suri [37].
Decomposition approaches for assembly/disassembly networks. As described above, for transfer lines or flow lines
with exponentially distributed processing times, together with a Markov-chain based analysis of the two-station subsystems, the DDX algorithm can be used as an appropriate decomposition approach. Analyzing analogous assembly/disassembly networks can be done with the same methodology, see Gershwin [23,38], Di Mascolo et al. [39],
Gershwin and Burman [40]. In the case of exponentially distributed processing times, such an approach could also be
applied for the flow-line type systems studied in this paper. Latest developments are presented by Jeong and Kim [7]
and Helber [6].
Nevertheless, in our paper, we do not restrict our analysis to the exponential distribution, as the only probability
distribution assumption. We will consider processing time distributions with a coefficient of variation more or less than
that of the exponential distribution (the CV of which is 1), as well. Hence, a more general, queueing-theory based
approach should be used which is done in Section 2.
As the main contribution of this paper, we consider assembly operations together with general service times. Although
omitted in this paper, also disassembly operations could be incorporated into the approach of Section 2. Hence, in terms
of queueing network analysis, the manufacturing systems that are studied in this paper are open folk/join queueing
networks with blocking and general service times. For an overview on queueing networks with blocking see Perros
and Altiok [41], Balsamo et al. [42], including some equivalence theorems for folk/join queueing networks (see also
[43]). A classification of manufacturing systems according to queueing network analysis can be found in Suri et al.
[44], Papadopoulos et al. [27], Altiok [16]. Note, the decomposition approach as described in Section 2 performs
quite well as long as the queueing network remains tree-structured (for a definition see [40]). In this case, the twostation decomposition approach for a network with M nodes leads to M − 1 subsystems. Any loop or cycle in the
network structure would contribute considerable correlations between the subsystems’ buffer levels that would have to
be recognized (see [23]).
2. The decomposition approach
The reminder of this paper is organized as follows: first, we identify subsystems of the assembly line (Section 2.1).
In Section 2.2, we describe how the subsystems are modeled. The parameters of the equivalent queueing models have
to be modified in order to represent the flow of parts throughout the corresponding original subsystem. In Sections
2.3 and 2.4, we describe how the virtual queueing system parameters are adjusted. From standard queueing formulas,
some required performance measures are derived in Section 2.5. The overall procedure is summarized in Section 3. The
numerical accuracy of the algorithm is tested against some simulation experiments, the results of which are documented
in Section 4.
2.1. Two-station subsystems
We virtually decompose an assembly line with M stations (like the one depicted in Fig. 1) into M − 1 buffer-related
subsystems. Each of them consists of two neighboring stations with their intermediate buffer. For a particular subsystem
denoted by (i, m), these are the stations i and m, and the buffer Bi,m . Any such subsystem should reflect the flow of
workpieces throughout the corresponding original buffer. Seen from the buffer, the station in front of it (in upstream
direction) fills the buffer by processing parts, whereas the station in downstream direction causes the buffer to become
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
2525
Detailed view on the original assembly line around station m:
i
m
sm
pmj
Equivalent two-station subsystems (decomposition approach):
Mu(i,m)
Md (i,m)
i
m
Mu( pmj,m)
Md ( pmj,m)
pmj
m
m
sm
Mu(m,sm)
Md (m,sm)
Fig. 2. Decomposition of an assembly line.
empty as well by processing parts. The upstream station of a subsystem (i, m) is denoted by Mu (i, m), the downstream
station by Md (i, m).
Let us consider in detail the situation around a given station m, see Fig. 2. If this station is not the last station of
the line, station m has a succeeding station denoted by sm . In an assembly line, for each station, there is at most one
successor, but possibly more than one predecessor station. The set Sm of m’s successor stations contains at most one
element, whereas the set Pm of predecessor stations of station m might have a cardinality of more than 1. The jth
predecessor station of an assembly station m is denoted by pmj , j = 1, . . . , |Pm |.
2.2. Virtual queueing systems
The subsystems are modeled as G/G/1/N queueing systems. The arrival process into the buffer of a subsystem
is generated by the preceding processing station. It stops whenever the buffer is full. In the case of blocking after
service, the workpiece that just has been finished has to wait at the server of the upstream station, which then is blocked
and stops further processing. Further workpieces cannot arrive because the arrival process is switched off during that
time. So, the equivalent queueing model of a particular subsystem (i, m) is a stopped-arrival queueing system with a
capacity of Ci,m + 2 waiting items which corresponds to the maximum number of workpieces within the subsystem,
see Fig. 3.
Let us consider in detail a certain subsystem (i, m), m ∈ {1, . . . , M}, i ∈ Pm . An arrival event is generated
by processing a part at the upstream station. Eventually, before such a processing operation can begin, there is an
additional delay due to starving effects or waiting times for synchronization that extends the inter-arrival time of parts
into the buffer. These idle times of the upstream station are a result of stochastic effects that arise from interactions
between the stations within the whole segment of the assembly line in front of the buffer. Thus, in the model of
the subsystem (i, m), the upstream station Mu (i, m) represents the segment of the assembly line (and the overall
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M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
i
m
Bi,m
Fig. 3. The flow of workpieces and the equivalent queueing system with stopped arrivals in the case of blocking after service.
i
m
Fig. 4. The decomposition view to an assembly system, seen from the subsystem (i, m).
effects) upstream of the buffer Bi,m . See Fig. 4 for an illustration of this decomposition view to the assembly line.
The flow into the buffer Bi,m is described by a virtual arrival rate u (i, m) and a coefficient of variation for the time
between two arrivals of parts, u (i, m). The rate u (i, m) will be chosen as to reflect the effective rate at which parts
are moved into the buffer Bi,m , when the arrival process is turned on. Hence, u (i, m) is the virtual arrival rate of
the queueing model of the subsystem (i, m). It differs from i by considering the effects of starving and waiting for
synchronization at the station i which are regarded as random extensions of the inter-arrival times in the equivalent
queueing system.
The virtual service process in the queueing model of subsystem (i, m) is generated by processing parts at the
downstream station. Eventually, before such a processing operation can begin, there is an additional delay due to
the holding time of components from station i that are already loaded onto the server of station m and waiting for
synchronization with other components needed to start an assembly operation. And, upon the processing operation
at station m being completed, blocking can occur. Both the waiting time for synchronization and the blocking time
extend the time needed to finish a service operation at the downstream station in the subsystem (i, m). The blocking
time is a result of stochastic effects that arise from interactions between the stations within the whole segment of the
assembly line that follows the buffer Bi,m . If station m is an assembly station, then it is represented as the downstream
station in several subsystems. The waiting time for synchronization is affected by these parallel subsystems. Thus, the
downstream station Md (i, m) represents the segment of the assembly line downstream of the buffer Bi,m including
parallel subsystems. In Fig. 4, the representation of a particular subsystem (i, m) is depicted. The dependencies between
the parallel subsystems are modeled due to the fact that any assembly station would appear as the downstream station in
more than one subsystem (which is illustrated by dotted lines in Fig. 4). The effective service process in the subsystem
(i, m) is described by a virtual service rate d (i, m) and a coefficient of variation for the time experienced by parts
from station i held at the server of station Md (i, m), d (i, m). The rate d (i, m) will be chosen as to reflect the effective
rate at which completed parts leave the subsystem (i, m), when it is not starved. It differs from m by considering the
effects of blocking and waiting for synchronization at the station m, which are regarded as random extensions of the
service times in the equivalent queueing system.
Note that the virtual parameters of the representing queueing systems generally do not have the same values as the
corresponding original ones. Analyzing the subsystems by a single queueing system neglects the interactions between
the subsystems. The trick of a decomposition approach is to find the subsystem’s right parameter values so that they
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
2527
catch the real dependencies between the subsystems. From the principle of the decomposition approach, actually, the
subsystems are analyzed independently from each other.
2.3. Adjusting the virtual arrival process parameters
Let us consider station m as the upstream machine of the subsystem (m, sm ), which generates the arrival process
into the subsystem. If it is not blocked, a new arrival event occurs by a service completion at station m. Therefore, the
virtual inter-arrival time in the equivalent queueing model should reflect the time until the next service operation is
completed at the server of station m. If the arrival process of workpieces is not interrupted by blocking, the effective
time between two successive service completions at station m is the sum of:
IS : the idle time of station m concerning outstanding parts from the jth preceding processing station p
• Tmj
mj caused
by an empty buffer Bpmj ,m , i.e. the time station m is starved for parts from pmj .
IW : the waiting time for synchronization, i.e. the time a workpiece from station p
• Tmj
mj has to wait at the server of
station m, until all required components are available for an assembly operation.
• TmS : the processing time, i.e. the time required to carry out a service operation at station m.
Then, having in mind the equivalence between the service completion time at station m and the virtual inter-arrival
time in the corresponding queueing system, the expected value of the virtual inter-arrival time in the queueing system
(m, sm ) and its squared coefficient of variation can be described as follows:
1
IS
IW
= E{Tmj
+ Tmj
+ TmS }
u (m, sm )
(m ∈ {{1, . . . , M}|Sm = ∅}; j = 1, . . . , |Pm |),
IS
IW
+ Tmj
+ TmS }
2u (m, sm ) = 2u (m, sm ) · Var{Tmj
(m ∈ {{1, . . . , M}|Sm = ∅}; j = 1, . . . , |Pm |).
(4)
(5)
At some stations, not all the three phases mentioned above are possible to pass through. Any input station of the
assembly line is assumed to be never starved. And, at a simple processing station (a station without any or with only
one predecessor station), there are no synchronization constraints.
The phase lengths of the service completion time are assumed to be i.i.d. for all workpieces. And, for a certain job,
they are stochastically independent of each other. Then, in order to determine the expressions in Eqs. (4) and (5), one
can sum up the expected values and variances of the phase lengths. For the processing time, it can be calculated from
given data:
E{TmS } =
1
m
Var{TmS } =
2m
2m
(m ∈ {1, . . . , M}),
(m ∈ {1, . . . , M}).
(6)
(7)
IS and T IW , we have to find an appropriate representation in the queueing models. If the station m is idle because
For Tmj
mj
of an empty buffer Bpmj ,m , the corresponding queueing system (pmj , m) is seen to be empty, too. The station m is
starved for parts from pmj until the next service completion will succeed at station pmj . Again, using the same modeling
correspondance as above, this remaining service completion time is represented by the remaining inter-arrival time in
the subsystem (pmj , m). Assuming that the probability distribution of any residual inter-arrival time is the same as the
full virtual inter-arrival time (which is exact only for the exponential distribution), it follows for the starving time at
station m for parts from the jth predecessor station:
IS
E{Tmj
} = pS∗ (pmj , m) ·
1
(m ∈ {{1, . . . , M}|Pm = ∅}; j = 1, . . . , |Pm |),
u (pmj , m)
2u (pmj , m) + 1
1
IS
∗
∗
Var{Tmj } = pS (pmj , m) ·
− pS (pmj , m) · 2
2u (pmj , m)
u (pmj , m)
(m ∈ {{1, . . . , M}|Pm = ∅}; j = 1, . . . , |Pm |),
(8)
(9)
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M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
pS∗ (pmj , m) is the probability that at the instant of a service completion at station m this station is starved for
parts from pmj . This probability is described by the time-instant probability that the queueing system (pmj , m)
IS is positive only with probability p ∗ (p , m), 0 otherwise. Eqs. (8) and (9) folis empty. The starving time Tmj
S mj
low from the fact that, for any random variable Y which takes positive values only with probability p ∗ according to a corresponding random variable X, the moments are: E{Y r } = p ∗ · E{Xr }. Furthermore, for a common
random variable X, it is Var{X} = E{X2 } − E{X}2 , and E{X 2 } = (2 + 1)/2 by defining E{X} = 1/
and CV{X} = .
At an assembly station m, an item that has already been delivered from station pmj has to wait at the service facility
of station m until the very last one of all required components of an assembly job will have arrived. For such an item
from pmj , a positive synchronization time arises if at least one of the subsystems (i, m) is empty, i ∈ Pm \{pmj }. Thus,
the synchronization time corresponds to the longest remaining starving time among the parallel subsystems. Again,
the remaining starving time in any queueing system is equal to the time until the next arrival takes place. Then, the
longest remaining starving time can be described by the maximum of residual inter-arrival times. In addition to the
memorylessness assumption that we already used for (8) and (9), we assume that the moment of a maximum of some
random variables is approximated by the maximum of the moments of these random variables. Actually, the maximum
of moments is only a lower bound to the (true) moment of a maximum of random variables. With these two assumptions,
it follows for the waiting time for synchronization:
⎛
IW
E{Tmj
} = P∗ ⎝
⎞
{(i, m)empty}⎠ · max
i=pmj
i∈Pm , i=pmj
1
u (i, m)
(m ∈ {{1, . . . , M}||Pm |2}; j = 1, . . . , |Pm |),
⎛
(10)
⎞ 2
(i,
m)
+
1
u
IW
∗⎝
Var{Tmj } = P
{(i, m)empty}⎠ · max
i=pmj
2u (i, m)
i∈Pm , i=pmj
⎛
⎞
⎞
2
1
⎠
−P ∗ ⎝
{(i, m)empty}⎠ · max
i=pmj u (i, m)
i∈Pm , i=pmj
(m ∈ {{1, . . . , M}||Pm |2}; j = 1, . . . , |Pm |)
(11)
with P ∗ (·) as the probability that at the instant of loading a workpiece from station pmj onto the server at station m
at least one parallel subsystem is empty. With the well-known formula for the probability of the union of some events
(exclusion-inclusion principle), it follows:
⎛
P∗ ⎝
i∈Pm , i=pmj
⎞
{(i, m)empty}⎠ =
i∈Pm
i=pmj
+
pS∗ (i, m) −
pS∗ (i, m) · pS∗ (k, m)
i,k∈Pm
i,k=pmj
i<k
pS∗ (i, m) · pS∗ (k, m) · pS∗ (, m)
i,k,∈Pm
i,k,=pmj
i<k<
..
.
pS∗ (i, m)
+ (−1)|Pm |−1
i∈Pm
i=pmj
(m ∈ {{1, . . . , M}||Pm | 2}; j = 1, . . . , |Pm |).
(12)
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
2529
The expected values in (8), (10), and (6) can be added to get Eq. (4). Then, the expected value of the virtual inter-arrival
time of the subsystem (m, sm ) is
1
1
1
=
+ pS∗ (pmj , m) ·
u (m, sm ) m
u (pmj , m)
⎛
⎞
+ P∗ ⎝
{(i, m)empty}⎠ · max
i=pmj
i∈Pm , i=pmj
1
u (i, m)
(m ∈ {{1, . . . , M}||Pm |2, Sm = ∅}; j = 1, . . . , |Pm |).
(13)
In order to get a second-moment expression according to (5), we sum up the phase lengths’ variances that are described
in (9), (11), and (7). For technical reasons of derivation we make use of the fact that, for a common random variable X, the
relation E{X 2 }=Var{X}+E{X}2 applies. Furthermore, with E{X}=1/ and CV{X}=, it follows: E{X 2 }=(2 +1)/2 .
Using these expressions, squaring out, placing some probabilities outside the brackets, an approximation formula for
the squared coefficient of variation of the virtual inter-arrival time can be derived:
2u (m, sm ) = 2u (m, sm )
⎛
2u (pmj , m) + 1
2m + 1
2
∗
⎝
·
+
+ pS (pmj , m) ·
2u (pmj , m)
m · u (pmj , m)
2m
⎛
⎞
+ P∗ ⎝
{(i, m)empty}⎠
i∈Pm , i=pmj
2u (i, m) + 1
2
· max
+ max
i=pmj
i=pmj m · u (i, m)
2u (i, m)
⎛
⎞
+ 2 · pS∗ (pmj , m) · P ∗ ⎝
{(i, m)empty}⎠ ·
i∈Pm , i=pmj
1
u (pmj , m)
· max
i=pmj
⎞
1
⎠−1
u (i, m)
(m ∈ {{1, . . . , M}||Pm |2, Sm = ∅}; j = 1, . . . , |Pm |).
(14)
2.4. Adjusting the virtual service parameters
Now, let us consider station m as the downstream machine of the subsystem (pmj , m), whose processing operations
cause the virtual service process in the representing queueing model.
The service time in any queueing system is the time during which a customer item is held at the service facility.
Therefore, the virtual service time in the queueing model of the subsystem (pmj , m) should reflect the holding time at
IW for
the server of station m experienced by a workpiece delivered from station pmj . In addition to the waiting time Tmj
synchronization and the original processing time TmS as described in Section 2.3, the effective holding time contains the
blocking time TmB , i.e. the time experienced by a finished job at the server of station m while it waits for queue space
in the buffer in front of station sm . With all three phases, the expected value of the virtual service time in the queueing
model of subsystem (pmj , m) and its squared coefficient of variation can be described as follows:
1
d (pmj , m)
IW
= E{Tmj
+ TmS + TmB }
(m ∈ {{1, . . . , M}||Pm | 2, Sm = ∅}; j = 1, . . . , |Pm |),
IW
2d (pmj , m) = 2d (pmj , m) · Var{Tmj
+ TmS + TmB }
(m ∈ {{1, . . . , M}||Pm |2, Sm = ∅}; j = 1, . . . , |Pm |).
(15)
(16)
IW and, of course, for T S the above developed descriptions can be used. Now, let us find a queueing-model based
For Tmj
m
expression for the blocking time.
If station m is blocked, a completed item is held at station m, as long as the buffer between the station m and sm is full.
The blocking lasts until the next completed part at station sm can be transferred out of this station. Thus, the blocking
2530
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
time at station m is equal to the remaining holding time of an item at station sm , which, in turn, can be described by the
remaining virtual service time in the corresponding queueing system. Hence, assuming that the probability distribution
of the remaining virtual service time is the same as the full virtual service time, it follows for the blocking time at
station m:
1
(m ∈ {{1, . . . , M}|Sm = ∅}),
d (m, sm )
2d (m, sm ) + 1
1
B
∗
∗
Var{Tm } = pB (m, sm ) ·
− pB (m, sm ) · 2
2d (m, sm )
d (m, sm )
(m ∈ {{1, . . . , M}|Sm = ∅}).
E{TmB } = pB∗ (m, sm ) ·
(17)
(18)
pB∗ (m, sm ) is the probability that at the instant of a service completion at station m this station is blocked. This probability
is described by the time-instant probability that the queueing system (m, sm ) is completely full. The blocking time TmB
is positive only with probability pB∗ (m, sm ), 0 otherwise. The derivation of Eqs. (17) and (18) then works with similar
arguments than that for Eqs. (8) and (9).
Using descriptions (10), (6), and (17), summing up, it follows from (15):
⎛
⎞
1
1
1
1
+ pB∗ (m, sm ) ·
{(i, m)empty}⎠ · max
=
+ P∗ ⎝
i=pmj u (i, m)
d (pmj , m) m
d (m, sm )
i∈Pm , i=pmj
(m ∈ {{1, . . . , M}||Pm |2, Sm = ∅}; j = 1, . . . , |Pm |).
(19)
An approximation for the coefficient of variation of the virtual service time can be given by
⎛
2
2
+
1
(m,
s
)
+
1
2
m
2d (pmj , m) = 2d (pmj , m) · ⎝ m 2 + pB∗ (m, sm ) · d 2
+
m
m · d (m, sm )
d (m, sm )
⎛
⎞ 2
2
(i,
m)
+
1
u
∗⎝
+P
+ max
{(i, m)empty}⎠ · max
i=pmj
i=pmj m · u (i, m)
2u (i, m)
i∈Pm , i=pmj
⎛
⎞
⎞
1
1
⎠−1
+ 2 · pB∗ (m, sm ) · P ∗ ⎝
· max
{(i, m) empty}⎠ ·
d (m, sm ) i=pmj u (i, m)
i∈Pm , i=pmj
(m ∈ {{1, . . . , M}||Pm |2, Sm = ∅}; j = 1, . . . , |Pm |).
(20)
2.5. Performance measures
With updated parameter values of the virtual queueing systems, some performance measures can be calculated. First
of all, we are interested in the production rate of the assembly line, representing the effective output rate at the last
station. From the equivalent queueing model, we know that the last subsystem of the line produces with a certain virtual
service rate if it is not empty. The last subsystem consists of the station M, and (chosen for the general case of one or
more predecessor stations) the station pM1 . Hence, with the description of the virtual arrival and service process of the
equivalent queueing system by u (pM1 , M), d (pM1 , M), u (pM1 , M), and d (pM1 , M), the production rate X can be
calculated as follows:
X = d (pM1 , M) · (1 − pS (pM1 , M))
(21)
pS (pM1 , M) = P0 (u (pM1 , M), u (pM1 , M), d (pM1 , M), d (pM1 , M), CpM1 ,M + 2)
(22)
with
as the probability that the subsystem (pM1 , M) is empty. From queueing theory, P0 (, cA , , cS , N ) is the time-average
probability that an appropriate queueing system with an arrival rate , a service rate , a coefficient of variation of the
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
2531
inter-arrival time, cA , and of the service time, cS , and a capacity of N customers is empty. In general, the probability
Pn (, cA , , cS , N ) that exactly n jobs are in a G/G/1/N queueing system can be approximated as proposed in Buzacott
and Shanthikumar [4].
Likewise for any other subsystem (i, m), the effective output rate can be calculated by
Xd (i, m) = d (i, m) · (1 − pS (i, m)) (m ∈ {1, . . . , M}; i ∈ Pm )
(23)
pS (i, m) = P0 (u (i, m), u (i, m), d (i, m), d (i, m), Ci,m + 2)
(24)
with
as the time-average probability that the equivalent queueing system to the subsystem (i, m) is empty. The reciprocal
value of the rate Xd (i, m) gives the mean time between two successive departures of workpieces from the subsystem
(i, m). The difference between the inter-departure time and the virtual service time in the equivalent queueing system
is the starving time. Then, from (8), it follows for the expected values:
1
1
1
−
= pS∗ (i, m) ·
Xd (i, m) d (i, m)
u (i, m)
(m ∈ {1, . . . , M}; i ∈ Pm ).
(25)
Eq. (25) gives an expression for the time-instant probability that the queueing system (i, m) is empty. Together with
(23), we get
1
1
(i, m)
pS (i, m)
∗
pS (i, m) = u (i, m) ·
−
= u
·
(m ∈ {1, . . . , M}; i ∈ Pm ). (26)
Xd (i, m) d (i, m)
d (i, m) 1 − pS (i, m)
Similar ideas can be tried on the blocking probabilities. In accordance to the equivalent queueing model for a particular
subsystem (i, m),
pB (i, m) = PCi,m +2 (u (i, m), u (i, m), d (i, m), d (i, m), Ci,m + 2)
(27)
is the time-average probability that the subsystem (i, m) is filled up with Ci,m + 2 items so that the upstream station is
blocked. The arrival process into the subsystem is active with an arrival rate u (i, m) unless blocking occurs. Thus, the
effective arrival rate is
Xu (i, m) = u (i, m) · (1 − pB (i, m)) (m ∈ {1, . . . , M}; i ∈ Pm ).
(28)
Then, the reciprocal value describes the mean effective time between two successive arrivals of parts into the subsystem
(i, m). It differs from the virtual arrival time in the equivalent queueing system by the blocking time. Hence, it is
according to (17):
1
1
1
−
= pB∗ (i, m) ·
Xu (i, m) u (i, m)
d (i, m)
(m ∈ {1, . . . , M}; i ∈ Pm ).
(29)
From expression (28), it follows for the time-instant probability that a subsystem is blocked:
1
1
(i, m)
pB (i, m)
pB∗ (i, m) = d (i, m) ·
−
= d
·
(m ∈ {1, . . . , M}; i ∈ Pm ). (30)
Xu (i, m) u (i, m)
u (i, m) 1 − pB (i, m)
3. The algorithm
Eqs. (13), (14), (19), and (20) give an expression for the virtual parameters of the corresponding queueing models for
the subsystems. They form a system of 4·(M −1) decomposition equations, relating successive and parallel subsystems.
A solution to this system is found by an iterative procedure. After initializing the virtual parameters with the values of
the original ones, the upstream parameters are updated in a forward pass from station 1 to M (if numbered in topological
order). In this way, the required upstream parameter values of preceding subsystems just have been actualized during
the same iteration. Using these modified upstream parameters, the downstream parameters are updated in a backward
pass. Again, the direction of the calculations ensures that all the required downstream parameter values of succeeding
2532
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
Fig. 5. The basic procedure ALRMGEN.
subsystems have just been updated in the calculation step before. In the next iteration, the subsystem parameters are
recalculated in the same way using the most current values from the preceding calculations. The procedure terminates
if the change in the estimated production rate between two successive iterations is small enough to be within a given
tolerance.
For every subsystem under consideration, the blocking and starving probabilities are (re-)calculated, using (27) and
(24) for the time-average state probabilities, and (30) and (26) for the time-instant probabilities. The production rate
of the assembly line is given by the effective output rate of one of those subsystems where the last station M is the
downstream station. For the first predecessor station (may be the only one), it can be calculated by (21), updated
throughout the approximation procedure with every iteration. The entire algorithm to determine the production rate is
summarized by the schematic depiction in Fig. 5.
The convergence of the proposed algorithm cannot be proven in general. This would be possible only for special
cases, for instance, if the coefficients of variation are not affected by modifications, see Buzacott et al. [1] and Dallery
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
2533
and Frein [45]. Although it cannot be guaranteed, in all tests, the algorithm converged to a solution. The speed of
convergence was usually very fast.
4. Numerical tests
The approximations have been tested on a number of assembly line configurations. For a system with 13 stations
(among them are four assembly stations) as depicted in Fig. 6, the agreements with simulation results are almost always
within 4% and often much better, see Table 1. For the configurations recorded in Table 1, the system is unbalanced
in the sense that the station’s characteristics in the middle of the system (containing stations 5–7, and 10) differ from
those at the edge of the station network. The processing rates , the coefficients of variations (), and the buffer sizes
C vary between two values of each.
The approximations perform worse if the coefficients of variation increase beyond a level of 2; see also Buzacott et al.
[1] for their approximation of the queue length probabilities. Nevertheless, in manufacturing systems it is quite unusual
to have variances of that size. The approximations become better if the buffer levels increase. The average deviation
from simulation results is below 1% for five or more buffers, and still below 6% even in the case of small buffers;
9
B9.10
8
B8.10 10
7
B7.10
11
B11.12 12
B1.3
1
3
B3.5
B10.13
B2.3
2
5
4
B5.6
6
B6.7
13
B4.5
B12.13
Fig. 6. Example: A 13-station assembly line (see [4]).
Table 1
Approximation results in comparison to simulation
C
Edge
Middle
Edge
Middle
Edge
Middle
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.25
1.25
1.25
1.25
1.25
1.25
1.25
1.25
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.5
0.5
1.5
1.5
0.5
0.5
1.5
1.5
0.5
0.5
1.5
1.5
0.5
0.5
1.5
1.5
1.5
1.5
0.5
0.5
1.5
1.5
0.5
0.5
1.5
1.5
0.5
0.5
1.5
1.5
0.5
0.5
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
5
10
10
5
10
5
5
10
5
10
10
5
10
5
XSim
X
X − XSim
(%)
XSim
0.8195
0.9175
0.6798
0.7785
0.8673
0.8531
0.6871
0.7750
0.5535
0.6285
0.6443
0.7311
0.6230
0.5552
0.6541
0.7157
0.7899
0.9038
0.6985
0.7994
0.8460
0.8295
0.6989
0.7988
0.5412
0.6226
0.6590
0.7445
0.6132
0.5446
0.6647
0.7291
−3.61
−1.49
2.75
2.68
−2.46
−2.76
1.71
3.08
−2.23
−0.94
2.28
1.83
−1.56
−1.91
1.63
1.86
2534
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
Deviation from Simulation Results
2%
0%
-2%
-4%
-6%
CV{T(S)} = 0.5
CV{T(S)} = 1.0
CV{T(S)} = 1.5
CV{T(S)} = 2.0
-8%
1
2
4
Buffer capacity
7
10
Fig. 7. Average deviations from simulation results for balanced configurations.
Deviation from Simulation Results
2%
1%
0%
-1%
-2%
-3%
-4%
-5%
-6%
-7%
-8%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Number of tested configuration
Fig. 8. Deviations from simulation for 50 particular configurations with non-identical, unreliable machines.
see Fig. 7 for a balanced system (as shown in Fig. 6) with varying buffer sizes and various coefficients of variation (CV). As one can see, the deviations are significantly higher if the coefficients of variation are equal to 2
(or larger).
Practical data about assembly lines reveal that the CV’s of the processing times usually are below 1. Higher CV’s
may be the result of the presence of station failures. But, even in this case, the algorithm gives quite good results for
moderate coefficients of variation. Fig. 8 shows the results for an assembly system (structured like the one depicted
in Fig. 6) with unbalanced configurations, and unreliable machines. For test reasons, the configuration parameters are
randomly varied within the interval [0.85, 1.15] for the processing rates and a range of [0.3, 1] for the CV’s of the
processing times. Due to the random generation of analyzed configurations, the buffers have been chosen sufficiently
large, C lies between 5 and 15. The mean time to failure lies between 20 and 50, the mean time to repair between 0.5
and 1.5. The coefficients of variation of the downtime of a machine is assumed to be between 0.5 and 1.5. In all the
cases, the deviation from simulation is below 2% and usually much better (below 1%). The simulation experiments
run over 101 000 time units with a warm-up phase of 1000 time units. The statistics are based on 20 replications which
guarantees half-widths of confidence intervals for XSim of less than 0.001 (if the processing rates are normalized to 1).
For the simulation models, the probability distributions are generated assuming Gamma distributions according to the
stations’ data. For more detailed information on the experiments’ settings and the results, see Manitz [46].
5. Conclusions
In this paper, we have proposed a decomposition approach for performance evaluation of assembly lines both
with either simple processing stations and assembly stations. The generally distributed service times are described by a
M. Manitz / Computers & Operations Research 35 (2008) 2520 – 2536
2535
two-moment approximation. As simulation experiments reveal, this method seems to be accurate enough to be included
in an optimization procedure. Further research should be devoted to optimization, see Tempelmeier [9], and to extensions
to manufacturing systems with disassembly operations, and queueing networks with loops.
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