# The Efficiency of Variance Reduction in Manufacturing and Service Systems:  The Comparison of the Control Variates and Stratified Sampling

Authors: Ergün Eraslan; Berna Dengiz
Journal: Mathematical Problems in Engineering
 (2009)

---

## Abstract

There has been a great interest in the use of variance reduction techniques (VRTs) in simulation output analysis for the purpose of improving accuracy when the performance measurements of complex production and service systems are estimated. Therefore, a simulation output analysis to improve the accuracy and reliability of the output is required. The performance measurements are required to have a narrow and strong confidence interval. For a given confidence level, a smaller confidence interval is supposed to be better than the larger one. The wide of confidence interval, determined by the half length, will depend on the variance. Generally, increased replication of the simulation model appears to have been the easiest way to reduce variance but this increases the simulation costs in complex-structured and large-sized manufacturing and service systems. Thus, VRTs are used in experiments to avoid computational cost of decision-making processes for more precise results. In this study, the effect of Control Variates (CVs) and Stratified Sampling (SS) techniques in reducing variance of the performance measurements of M/M/1 and GI/G/1 queue models is investigated considering four probability distributions utilizing randomly generated parameters for arrival and service processes.


---

## Body

## 1. Introduction

Manufacturing systems are processing systems
where raw materials are transformed into finished products through a series of
workstations. It is important to find an alternative design process to obtain
desired performance in a manufacturing system based on management decision. A
service system is also a processing system where one or more service facilities
are provided to customers, patients, and paperworks.The use of simulation for the modeling of
service and manufacturing systems has greatly increased recently in the many
areas of application such as health care systems, restaurants, cafeterias,
banks, and recreation centers (cinemas, theatres), and many manufacturing systems.In systems mentioned above, the most widely
used queue model is M/M/1. The queue model refers to exponential arrivals and
service times with a single server and one line shown in Figure1. M/M/1 is a good approximation for a large number of
queueing systems. There are many systems we encountered in service and
production fields where M/M/1 model can be used for modeling these systems such
as a cashier in a supermarket and a teller in a bank.Figure 1
One server one line queue system (M/M/1).M/M/1
is Kendall's notation of this queuing model. The first M represents the input
process, the second M the service distribution, and 1 the number of server. The
M implies an exponentially distributed interarrival and service time. The M/M/1 queue
system has also unlimited population and First-in First-out (FIFO) queue
discipline. On the other hand, if the distributions of arrival and service
processes are not Markovian, the one server queue system is named GI/G/1.Although there are some analytic solutions for
these systems, the performance measurements of them represent steady-state behavior.
Therefore, in real life applications, the simulation technique is used to
compute system performance measures for any time interval. Simulation is more
relevant and flexible technique to solve the problems of queue systems in manufacturing
and service systems [1].The queues are used for modeling of manufacturing
systems, for example, inventory models, flow line, and JIT production systems. The unbalanced flow line capacity in
the manufacturing systems can constitute a product or semiproduct queue which
can be usually modeled as M/M/1 or GI/G/1. The queues can cause a bottleneck in
front of the machines in the job shop (see Figure2). The delay caused by
bottlenecks in manufacturing systems increases the unit cost, decreases the
productivity, and which in turn affects the competitiveness of the companies in
the market negatively. This is the most common research area of Industrial
Engineering in managerial and operational system analyses.Figure 2
M/M/1 queues of the products/semiproducts in a job shop (□ represents a server◯ represents a manufacturer of a semiproduct).In service systems, the success of a company,
besides using the resources efficiently, depends on winning customers and
keeping them. Any lost time by customers standing in the queues accounts for
loss of profit and usefulness for the service companies.To address such problems, simulation technique
is used as a flexible modeling tool to investigate and solve the queue problems
occuring in manufacturing and service systems. Because random samples from
probability distributions are used to drive a simulation model, outputs of the
simulation model are just particular realization of random variables that may
have large variances. For the reasons mentioned, there has been a rapid growth
of interest in the use of variance reduction techniques (VRTs) for improving
the accuracy of simulation outputs. Thus, the VRTs can be used through the run of
the simulation models to obtain more precise results.Therefore, in this study, we investigated the
effect of two different VRTs on the M/M/1 and GI/G/1 queue models.The common VRTs are Common Random Variables (CRVs),
Antithetic Variables (AV), Control Variates (CV), Stratified Sampling (SS), Importance
Sampling (IS), Indirect Estimation (IE), and Conditional Expectation (CE) [2].The studies on variance reduction (VR) began in
the 1950s. In the years before advance computer technology, AV was used in
Monte Carlo simulation. Kleijnen was interested in CRV and AV [3]. The CV technique
was developed at the end of 1970s and used in a queue simulation by Carson [4],
Lavenberg et al. [5], and Wilson and Pristker [6]. Law’s
formula was the basic of IE and Queue Theory [7]. Cartel and Ignall [8] used
this formula in CRV simulation. In the following years, Nelson [9] tried the
well-known VRTs in dynamic systems. Until the 1990s, there are several authors that
have studied computer simulation of the VRTs, for example, Kleijnen [3], Lavenberg et
al. [5], Carson and Law [10], Carter and Ignall [8], Iglehart and Shedler [11],
and Wilson and Pristker [6].In the last decade, VRTs have been used in
several areas. Statistics and simulation output analyses are the primary fields
and mathematics, chemistry, medicine, biology, quality improvement, portfolio
analysis, pricing, flexible manufacturing systems, scheduling, stochastic
networks, nuclear chemistry, oceanography, and biophysics, and Markov processes
follow them. These studies shortly listed below.Dengiz et al. [12] used the AV in the stochastic
networks, Nava [13] used the VRTs in comparing the simulation models,   and Shih
and Song [14] in regenerative simulation. Crawford and Gallwey [15] took
into account the bias of computerized simulation studies, Dahl [16] in
diffusion with CV technique, Vegas et al. [17] in dichotomous response
variables, Kawrakow and Fippel [18] in calculating photon dose, Plante [19] in
supplier interactions of the companies, Glasserman et al. [20] in estimating of
risk values for the investments, Srikant and Whitt [21] in loss model
simulation, Taylor and Heragu [22] in workshop flows for shortening operations
time in flow shop, Cancela and El Khadiri [23] in increasing the network
reliability, Moreni [24] for establishing the pricing options of the companies,
Jourdain et al. [25] in polymeric fluids in engineering, and Fumera et al. [26] in
bagging optimization. The VRTs have been used in Monte Carlo simulation studies
in the recent years by Skowronski and Turner [27] for statistical tolerance synthesis,
Pacelli and Ravaioli [28] for semiconductor devices in electronics department,
Constantini [29] for reflecting diffusions, Fitzgerald et al. [30] for
determining the dynamic levels of systems, and Baker and Hadjiconstantinou [31]
for Bolztman equation in fluids mechanic.The VR and queuing models being coupled were
examined in only a few studies; Görg and Fuß [32] used ATMs in runtimes
evaluation, Arsham [33] in calculating the score function estimation, Meles [34]
in branching the optimal numbers in mathematical
models, Jocabson [35] in harmonic gradient estimator, and Schmeiser and Taaffe [36]
in queuing network studies. Sabuncuoglu et al. [37] used two input and
two output VRTs to measure the
performance of VRTs under finite simulation run lengths and analyzed their
effects considering three different types of systems: M/M/1, serial production
line, and (S,s) inventory control systems.Previous research in the area mainly focuses on
applications of variance reduction techniques on M/M/1 queues. While the first
objective of this paper deals with comparison and analysis of the performance
of both VRTs (CV, SS) on the simple queue systems such as M/M/1 and GI/G/1, we
also investigate the effects of different distributions used for modeling of
arrival and service processes of these models utilizing experimental design
analysis.In this study, the average waiting time (AWT)
and average number of customers (ANCs) are considered as system performance
measurements. CV and SS techniques are used for the variance reduction of
simulation outputs. The efficiencies of each technique on the simple queue
models are investigated for four different distributions. These distributions
which are exponential (in this case, the queue model is called as M/M/1 using
Kendall’s notation), uniform, triangular, and normal (in these cases the queue
model is named as GI/G/1 using Kendall`s notation) are used in interarrival
times and service times. The randomly selected four parameter sets for four
distributions are stated for experiments. The results of factor analysis are given
in detail.This paper will proceed as follows. The next
section reviews some VRTs, and experimental analyses are described in Section3.
Finally, the research results and conclusion remarks are summarized in Section
4.


## 2. The Variance Reduction Techniques (VRTs)

The general specifications of CV and SS techniques are reviewed below.


### 2.1. Control Variates (CV) Technique

The basic purpose of CV is to
introduce correlation among observations so as to reduce the variance. Using
“Control Variates”, true estimation statistics based on a secondary estimation
value, and difference between its estimation values are ascertained. With this
technique, instead of direct estimation of the parameter, the possible
relationship between the problem undertaken and the analytic model is considered
(see (2.1)–(2.3) [6, 38, 39].LetXn be a series
of the first 100 customer delays in queue, and let X be an output random variable
representing the average of the first 100 customer delays in queue:
(2.1)X=E[Xn]. The value of X is estimated during
simulation period.Let the secondary random variable,Y,
formed from independent identically distributed random variables (i.e., the
service times of the first 99 customers) and its expectation v=E[Y] be known because service times are generated from some known
input distributions mentioned in Section 1. It is obvious that larger than average service times tend to lead to
longer than average delays and vice versa. Thus Y is
correlated to X, positively [7]. We control the output X using
this relation between X and Y,where Y is the control
variate.The correctedX(Xc)
is obtained from (2.2)Xc=X−a(Y−v), where a>0andY>vifXc<X, a is a constant and takes the same sign with the covariance between X and Y:(2.3)Var(Xc)=Var[X]+a2Var[Y]−2aCov[X,Y].If2aCov(X,Y)>a2Var(Y) inequality is valid, then Xc will have less variability than X. Cov(X,Y) is estimated through the simulation. Xc
is
calculated using coefficient a, then a
confidence interval can be built for Xc (for detailed
information, see Law and Kelton [7]).


### 2.2. Stratified Sampling (SS)

SS technique is such that the heap
is divided into stratums. By converting the heaps to stratums which have smaller
variances, the problems arising from sensitivity due to big variance are
prevented. Here, the determination of the number of stratums is important.
Increasing the stratum number results in smaller variance but decreasing the
number results in loss of the estimating variance because all data cannot be
accounted for in some stratums. Moreover, the more difference between the
averages of heaps and stratums the more benefit is supplied. In literature,
generally, it is expressed that 3–5 stratums are enough.There are four kinds of SS available in
literature. These are Common Random SS which is used in this study, Proportional
SS, Appropriate Sharing Method, and Economical Sharing Method [1].


## 2.1. Control Variates (CV) Technique

The basic purpose of CV is to
introduce correlation among observations so as to reduce the variance. Using
“Control Variates”, true estimation statistics based on a secondary estimation
value, and difference between its estimation values are ascertained. With this
technique, instead of direct estimation of the parameter, the possible
relationship between the problem undertaken and the analytic model is considered
(see (2.1)–(2.3) [6, 38, 39].LetXn be a series
of the first 100 customer delays in queue, and let X be an output random variable
representing the average of the first 100 customer delays in queue:
(2.1)X=E[Xn]. The value of X is estimated during
simulation period.Let the secondary random variable,Y,
formed from independent identically distributed random variables (i.e., the
service times of the first 99 customers) and its expectation v=E[Y] be known because service times are generated from some known
input distributions mentioned in Section 1. It is obvious that larger than average service times tend to lead to
longer than average delays and vice versa. Thus Y is
correlated to X, positively [7]. We control the output X using
this relation between X and Y,where Y is the control
variate.The correctedX(Xc)
is obtained from (2.2)Xc=X−a(Y−v), where a>0andY>vifXc<X, a is a constant and takes the same sign with the covariance between X and Y:(2.3)Var(Xc)=Var[X]+a2Var[Y]−2aCov[X,Y].If2aCov(X,Y)>a2Var(Y) inequality is valid, then Xc will have less variability than X. Cov(X,Y) is estimated through the simulation. Xc
is
calculated using coefficient a, then a
confidence interval can be built for Xc (for detailed
information, see Law and Kelton [7]).


## 2.2. Stratified Sampling (SS)

SS technique is such that the heap
is divided into stratums. By converting the heaps to stratums which have smaller
variances, the problems arising from sensitivity due to big variance are
prevented. Here, the determination of the number of stratums is important.
Increasing the stratum number results in smaller variance but decreasing the
number results in loss of the estimating variance because all data cannot be
accounted for in some stratums. Moreover, the more difference between the
averages of heaps and stratums the more benefit is supplied. In literature,
generally, it is expressed that 3–5 stratums are enough.There are four kinds of SS available in
literature. These are Common Random SS which is used in this study, Proportional
SS, Appropriate Sharing Method, and Economical Sharing Method [1].


## 3. Experimental Analysis

We consider a simple queue system which has one
waiting line and one server to perform an experimental analysis. Our aim is to determine
the effectiveness of CV and SS and how VRTs will avoid computational cost of
simulation experiments in obtaining more precise results. The effects of four
different probability distributions having randomly generated parameter(s) for
arrival and service processes on the precision of simulation output are also examined.VRTs have two levels as CV and SS. For the
arrival and service processes, exponential, uniform, triangular, and normal
distributions are selected. Each distribution is assigned to arrival and
service processes with randomly selected parameter values as given in Table3.
Comparison and analysis are carried out using statistical output analysis for a
single system. Two performance measurements, average waiting time (AWT) and average
number of customer (ANC), are considered as system outputs. The reason we use a simple
queue system with one line and one server (called M/M/1 with exponential
distribution for arrival and service process and GI/G/1 with the other general
distributions) is to increase the range of application areas in practice and
also to obtain mathematical models for them [2, 7, 40].


### 3.1. Variance Reduction with CV

To determine the efficiency of CV on M/M/1 and
GI/G/1 models, the simulation code of M/M/1 queue model is used [7]. Some necessary
modifications are performed and subroutines are used for eight design points. The
two levels of VRTs and the four levels of distributions are then tested. The purpose
of CV technique is to combine an appropriate definition of variables, which
depend on the service times. These services are run in a controlled environment.
During this study, the values of Xn (waiting times in queue)
are created and reserved in a hidden file, and then the values are used for the
remaining steps of CV. The same
operations are done for the service times with their mean being v. The constant a is estimated from
sample depending on the calculated covariance between X and Y using (3.1).
Thus, outputs of simulation model, X,
are adjusted using CV [41]:(3.1)a*=Cov(X,Y)Var(Y).


### 3.2. Variance Reduction with SS

The SS technique works under principles of
separation of the heap into stratums and reflects the process of VR of each
stratum itself on the overall variance. The necessary modifications are performed
on the M/M/1 simulation code for stratification. The simulation model is run
ten times for 5 stratums for this study, considering 100 customers for each, to
obtain the sensitivity and small variance. The replications are done via randomly selected 12 seeds shown in
Table1. After the first run of each seed, a repetition is avoided by using the
same initials and a second set of random numbers for the second stratum having 100
customers [1, 4].Table 1
A sample application for 5 stratums of SS.
Random
seedsMinMaxCustomer numbers per
stratum(0–0.5)(0.5–1)(1–1.5)(1.5–2)(2+)Total650000.00188.50932824171120100700000.00088.24432626161220100750000.00036.82702222181122100800000.00116.96702323181121100850000.00525.42302424181119100900000.00126.86002424161420100950000.00188.244024241711201001000000.00456.967025251712191001050000.00095.633023231911191001100000.00188.509023231713181001150000.00128.348523231712211001200000.00128.24432324181120100The five stratums are used in this study as
well as the balance of number of customers. A sample application of SS is shown
in Table1. The first column shows the random seeds, the second and third
columns represent the minimum and maximum values of stratums, and the next five
are the frequency of them for the stratification of 100 customers. Here, one heap
is converted to five stratums which have small variances. Thus,
sensitivity problems based on big variances are prevented. As seen in the table, the application of this
technique is difficult, as it takes a very long run time.


### 3.3. Computational Results of VRTs

To ascertain the effects
of the techniques on the considered queuing models,F hypothesis tests are used on the variance data
obtained by applying the CV and SS techniques. The hypotheses are constructed
as follows:HypothesisHo:σ12=σ22,H1:σ12<σ22,Fcal=Y>Ftab=X,
where σ1 and σ2 are the variances of the outputs obtained with and without VRTs, respectively. In the
confidence level of 95%  (α=0.05), the calculated F value (Fcal)
is greater than F table value (Ftab) which means the Ho hypothesis will be
rejected. Thus, the variance of the output of the simulation model (the performance
measurement of the system) obtained with VRT is smaller. The experiments are performed
with a randomly selected parameter set (2 for arrival and 1.5 for service
processes). The exponential distribution is
considered for F tests where m=n=12.As shown in Table2 both of the techniques
reduce variance statistically, and the CV technique is more efficient than SS for
the M/M/1 queue with a randomly selected parameter set. Similar results are also
obtained for GI/G/1 queue system using the uniform, triangular, and normal
distributions.Table 2
The
comparison of the variances of outputs for AWT.
VRT techniquesFcalFtabResultsWithout CV-with CV1392.86Ho: rejectWithout SS-with SS5.642.86Ho: rejectWith SS-with CV24.672.86Ho: rejectTable 3
The
randomly selected parameter sets for four distributions.
Parameter
setsProcessExponential(β)Uniform(a,b)Triangular(a,b,c)Normal(μ,σ2)Set 1Arrival11,21,2,30.5,1Service0.51,1.5Set 2Arrival1.51,32,3,40.5,1.5Service11,2Set 3Arrival22,41,3,41,2Service1.50.5,2Set 4Arrival2.52,31,2,41.5,2Service21,2.5


### 3.4. The Effects of VRTs and Distributions on the Output Variance

As stated in the previous sections, two factors
are considered, and the effects of these factors are investigated on the system
performance measurements. Factor settings are as follows.(1)
VRTs: this factor is tested in
experimental design in two levels being CV and SS.(2)
The distribution type of arrival and
service processes: this factor is tested in four levels: exponential (for this case queue system is called M/M/1), uniform,
triangular, and normal (for these three distributions, queue systems are
called GI/G/1).The
considered system performance measurements are AWT and ANC.Since this study contains two
factors with two and four levels, respectively,2×4=8 design points are
required in case of full factorial design. Four replications are made for each
design point, so 32 experiments are performed. The results of the
experimentation are analyzed by ANOVA. The validation of ANOVA results depends
on normality and independence for the error components. This is performed by
MINITAB by observing the standardized residual plot graphs. The assumptions are
obtained using relevant transformations to the variance data.These operations are performed
for each considered performance measurements of the queue model. The four variance
values obtained from different parameter sets for AWT and ANC are stated for
four distributions used with CV and SS in Tables4 and 5, respectively.Table 4
The variances of AWT.
Levels1.Exponential2.Uniform3.Triangular4.Normal1.Control
variates0.0000100.07150000.00003980.00001550.0007800.00638400.00104000.00010100.0394000.00001880.01248000.00002541.5030000.08248000.00071700.00009392.Stratified
sampling0.1050.3950.4321.5411.0120.4920.3861.2240.9721.4150.6641.2382.4300.1620.5311.127Table 5
The variances
of ANC.
Levels1.Exponential2.Uniform3.Triangular4.Normal1.Control
variates0.00000010.00034590.00000010.00000020.00002230.00006630.00000070.00000320.01295000.00000000.00001910.00000030.08282000.00019320.00000230.00000102.Stratified
sampling0.5360.1670.1135.8691.5920.5370.0452.6640.6690.1670.0882.1770.2230.0260.1071.924The ANOVA results given in Table6 for AWT and
Table 7 for ANC provide the followings.Table 6
The ANOVA output for AWT.
General linear model: Var versus Tech; DistFactorType levelsValuesTechFixed2 cv ssDistFixed4 ex un tr noAnalysis
of variance for Var, using adjusted SS for testsSourceDFSeq SSAdj SSAdj MSFPTech14.293304.293304.2933048.73.000Dist30.420130.420130.140041.59.218Tech*Dist30.421650.421650.140551.60.217Error242.114322.114320.08810Total317.24940Table 7
The ANOVA output for ANC.
General linear model: Var versus Tech; DistFactorType levelsValuesTechFixed2 cv ssDistFixed4 ex un tr noAnalysis
of variance for Var, using adjusted SS for testsSourceDFSeq SSAdj SSAdj MSFPTech14.98234.98234.982397.65.000Dist32.51632.51630.838816.44.000Tech*Dist32.54592.54590.848616.63.000Error241.22451.22450.0510Total3111.2689(i)
The main factor
VRTs are statistically significant for AWT, others are not.(ii)
The main factor
VRTs, the type of distributions, and their interactions are statistically
significant for ANC.(iii)
Conversely, the
effect of CV technique on the performance measurements is stronger than
the effect of SS technique. The CV technique results in smaller variance
for both considered performance measurements; the difference in VR can be
easily seen (P<.05).(iv)
The
investigation of interaction between distribution types and the VRTs show
that interaction is efficient in VR technique, only for the ANC. The
smallest mean belongs to the first level of the first factor, that is, CV, and the
third level of the second factor, that is, triangular distribution, shown in
Tables4 and 5.


## 3.1. Variance Reduction with CV

To determine the efficiency of CV on M/M/1 and
GI/G/1 models, the simulation code of M/M/1 queue model is used [7]. Some necessary
modifications are performed and subroutines are used for eight design points. The
two levels of VRTs and the four levels of distributions are then tested. The purpose
of CV technique is to combine an appropriate definition of variables, which
depend on the service times. These services are run in a controlled environment.
During this study, the values of Xn (waiting times in queue)
are created and reserved in a hidden file, and then the values are used for the
remaining steps of CV. The same
operations are done for the service times with their mean being v. The constant a is estimated from
sample depending on the calculated covariance between X and Y using (3.1).
Thus, outputs of simulation model, X,
are adjusted using CV [41]:(3.1)a*=Cov(X,Y)Var(Y).


## 3.2. Variance Reduction with SS

The SS technique works under principles of
separation of the heap into stratums and reflects the process of VR of each
stratum itself on the overall variance. The necessary modifications are performed
on the M/M/1 simulation code for stratification. The simulation model is run
ten times for 5 stratums for this study, considering 100 customers for each, to
obtain the sensitivity and small variance. The replications are done via randomly selected 12 seeds shown in
Table1. After the first run of each seed, a repetition is avoided by using the
same initials and a second set of random numbers for the second stratum having 100
customers [1, 4].Table 1
A sample application for 5 stratums of SS.
Random
seedsMinMaxCustomer numbers per
stratum(0–0.5)(0.5–1)(1–1.5)(1.5–2)(2+)Total650000.00188.50932824171120100700000.00088.24432626161220100750000.00036.82702222181122100800000.00116.96702323181121100850000.00525.42302424181119100900000.00126.86002424161420100950000.00188.244024241711201001000000.00456.967025251712191001050000.00095.633023231911191001100000.00188.509023231713181001150000.00128.348523231712211001200000.00128.24432324181120100The five stratums are used in this study as
well as the balance of number of customers. A sample application of SS is shown
in Table1. The first column shows the random seeds, the second and third
columns represent the minimum and maximum values of stratums, and the next five
are the frequency of them for the stratification of 100 customers. Here, one heap
is converted to five stratums which have small variances. Thus,
sensitivity problems based on big variances are prevented. As seen in the table, the application of this
technique is difficult, as it takes a very long run time.


## 3.3. Computational Results of VRTs

To ascertain the effects
of the techniques on the considered queuing models,F hypothesis tests are used on the variance data
obtained by applying the CV and SS techniques. The hypotheses are constructed
as follows:HypothesisHo:σ12=σ22,H1:σ12<σ22,Fcal=Y>Ftab=X,
where σ1 and σ2 are the variances of the outputs obtained with and without VRTs, respectively. In the
confidence level of 95%  (α=0.05), the calculated F value (Fcal)
is greater than F table value (Ftab) which means the Ho hypothesis will be
rejected. Thus, the variance of the output of the simulation model (the performance
measurement of the system) obtained with VRT is smaller. The experiments are performed
with a randomly selected parameter set (2 for arrival and 1.5 for service
processes). The exponential distribution is
considered for F tests where m=n=12.As shown in Table2 both of the techniques
reduce variance statistically, and the CV technique is more efficient than SS for
the M/M/1 queue with a randomly selected parameter set. Similar results are also
obtained for GI/G/1 queue system using the uniform, triangular, and normal
distributions.Table 2
The
comparison of the variances of outputs for AWT.
VRT techniquesFcalFtabResultsWithout CV-with CV1392.86Ho: rejectWithout SS-with SS5.642.86Ho: rejectWith SS-with CV24.672.86Ho: rejectTable 3
The
randomly selected parameter sets for four distributions.
Parameter
setsProcessExponential(β)Uniform(a,b)Triangular(a,b,c)Normal(μ,σ2)Set 1Arrival11,21,2,30.5,1Service0.51,1.5Set 2Arrival1.51,32,3,40.5,1.5Service11,2Set 3Arrival22,41,3,41,2Service1.50.5,2Set 4Arrival2.52,31,2,41.5,2Service21,2.5


## 3.4. The Effects of VRTs and Distributions on the Output Variance

As stated in the previous sections, two factors
are considered, and the effects of these factors are investigated on the system
performance measurements. Factor settings are as follows.(1)
VRTs: this factor is tested in
experimental design in two levels being CV and SS.(2)
The distribution type of arrival and
service processes: this factor is tested in four levels: exponential (for this case queue system is called M/M/1), uniform,
triangular, and normal (for these three distributions, queue systems are
called GI/G/1).The
considered system performance measurements are AWT and ANC.Since this study contains two
factors with two and four levels, respectively,2×4=8 design points are
required in case of full factorial design. Four replications are made for each
design point, so 32 experiments are performed. The results of the
experimentation are analyzed by ANOVA. The validation of ANOVA results depends
on normality and independence for the error components. This is performed by
MINITAB by observing the standardized residual plot graphs. The assumptions are
obtained using relevant transformations to the variance data.These operations are performed
for each considered performance measurements of the queue model. The four variance
values obtained from different parameter sets for AWT and ANC are stated for
four distributions used with CV and SS in Tables4 and 5, respectively.Table 4
The variances of AWT.
Levels1.Exponential2.Uniform3.Triangular4.Normal1.Control
variates0.0000100.07150000.00003980.00001550.0007800.00638400.00104000.00010100.0394000.00001880.01248000.00002541.5030000.08248000.00071700.00009392.Stratified
sampling0.1050.3950.4321.5411.0120.4920.3861.2240.9721.4150.6641.2382.4300.1620.5311.127Table 5
The variances
of ANC.
Levels1.Exponential2.Uniform3.Triangular4.Normal1.Control
variates0.00000010.00034590.00000010.00000020.00002230.00006630.00000070.00000320.01295000.00000000.00001910.00000030.08282000.00019320.00000230.00000102.Stratified
sampling0.5360.1670.1135.8691.5920.5370.0452.6640.6690.1670.0882.1770.2230.0260.1071.924The ANOVA results given in Table6 for AWT and
Table 7 for ANC provide the followings.Table 6
The ANOVA output for AWT.
General linear model: Var versus Tech; DistFactorType levelsValuesTechFixed2 cv ssDistFixed4 ex un tr noAnalysis
of variance for Var, using adjusted SS for testsSourceDFSeq SSAdj SSAdj MSFPTech14.293304.293304.2933048.73.000Dist30.420130.420130.140041.59.218Tech*Dist30.421650.421650.140551.60.217Error242.114322.114320.08810Total317.24940Table 7
The ANOVA output for ANC.
General linear model: Var versus Tech; DistFactorType levelsValuesTechFixed2 cv ssDistFixed4 ex un tr noAnalysis
of variance for Var, using adjusted SS for testsSourceDFSeq SSAdj SSAdj MSFPTech14.98234.98234.982397.65.000Dist32.51632.51630.838816.44.000Tech*Dist32.54592.54590.848616.63.000Error241.22451.22450.0510Total3111.2689(i)
The main factor
VRTs are statistically significant for AWT, others are not.(ii)
The main factor
VRTs, the type of distributions, and their interactions are statistically
significant for ANC.(iii)
Conversely, the
effect of CV technique on the performance measurements is stronger than
the effect of SS technique. The CV technique results in smaller variance
for both considered performance measurements; the difference in VR can be
easily seen (P<.05).(iv)
The
investigation of interaction between distribution types and the VRTs show
that interaction is efficient in VR technique, only for the ANC. The
smallest mean belongs to the first level of the first factor, that is, CV, and the
third level of the second factor, that is, triangular distribution, shown in
Tables4 and 5.


## 4. Discussion and Conclusions

Queue systems are widely used in various fields
in manufacturing and the service industry. The system analysis of the queues
for both industries is one of the most highly research problems in Industrial
Engineering. These analyses are mainly performed by simulation technique.
Simulation output analysis is used to improve the accuracy and the reliability
of the performance measures of systems. For a given confidence level, a smaller
confidence interval is supposed to be better than the larger one. The wide of
the confidence interval will depend on variance. Generally, increased
replication of the simulation model seems to be the easiest way to reduce
variance but this increases the simulation costs. Therefore, VRTs are used in
experiments to avoid computational cost.In this study, the effects of CV and SS
techniques were investigated for queues (with one waiting line and one service)
occurring in manufacturing and service. The effects of the two factors are
investigated using the experimental design analysis in reducing variance. The
first factor, VRT, with two levels (CV and SS) and the second factor,
distributions, with four levels (exponential, uniform, triangular, and normal) are considered with
ANOVA.The ANOVA results show that the main factor
VRTs, the type of distributions, and their interactions are statistically
significant for ANC. Conversely, VRTs are statistically significant for AWT; the
other factors are not.The effect of the CV technique on the
performance measurements is stronger than the effect of the SS technique. The
CV technique results in smaller variance for both considered performance
measurements; the difference in VR can be easily seen (P<.05). It can
be concluded that distribution types and VRTs jointly affect variance of the ANC
measure but do not affect AWT.The smallest mean belongs to the first level of
the first factor (i.e., CV) and the third level to the second factor (i.e.,
triangular distribution), a combination resulting in higher efficiency.The results underline that both CV and SS VRTs
reduce variance quite efficiently in the 95% confidence level. 80% of the overall
variance reduction is obtained using CV technique and 43% of using SS
technique.The further results based on the design of
experiment demonstrate that if the considered system is M/M/1, CV technique is efficient.
If the considered model of a system is GI/G/1 and its source of randomness
(arrival and service distributions) is fitted using triangular distribution,
then the CV technique is preferable to obtain more beneficial results with smaller
variance. The results are only valid under the current experiments for the
selected two VRTs and four distributions.It is supposed that it is more useful to extend
this research considering other VRTs to investigate and solve the problems of
queuing systems in the manufacturing and service systems area as future
research.



---

*Source: 791750-2009-05-10.xml*