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.

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 Figure

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 [

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 Figure

M/M/1 queues of the products/semiproducts in a job shop (

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) [

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 [

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. [

The VR and queuing models being coupled were
examined in only a few studies; Görg and Fuß [

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 Section

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

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 (

Let

Let the secondary random variable,

The corrected

If

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 [

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 Table

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 [

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
Table

A sample application for 5 stratums of SS.

Random seeds | Min | Max | Customer numbers per stratum | |||||
---|---|---|---|---|---|---|---|---|

( | ( | ( | ( | ( | Total | |||

65000 | 0.0018 | 8.5093 | 28 | 24 | 17 | 11 | 20 | 100 |

70000 | 0.0008 | 8.2443 | 26 | 26 | 16 | 12 | 20 | 100 |

75000 | 0.0003 | 6.8270 | 22 | 22 | 18 | 11 | 22 | 100 |

80000 | 0.0011 | 6.9670 | 23 | 23 | 18 | 11 | 21 | 100 |

85000 | 0.0052 | 5.4230 | 24 | 24 | 18 | 11 | 19 | 100 |

90000 | 0.0012 | 6.8600 | 24 | 24 | 16 | 14 | 20 | 100 |

95000 | 0.0018 | 8.2440 | 24 | 24 | 17 | 11 | 20 | 100 |

100000 | 0.0045 | 6.9670 | 25 | 25 | 17 | 12 | 19 | 100 |

105000 | 0.0009 | 5.6330 | 23 | 23 | 19 | 11 | 19 | 100 |

110000 | 0.0018 | 8.5090 | 23 | 23 | 17 | 13 | 18 | 100 |

115000 | 0.0012 | 8.3485 | 23 | 23 | 17 | 12 | 21 | 100 |

120000 | 0.0012 | 8.2443 | 23 | 24 | 18 | 11 | 20 | 100 |

The five stratums are used in this study as
well as the balance of number of customers. A sample application of SS is shown
in Table

To ascertain the effects
of the techniques on the considered queuing models,

Hypothesis

As shown in Table

The comparison of the variances of outputs for AWT.

VRT techniques | Results | ||
---|---|---|---|

Without CV-with CV | 139 | 2.86 | |

Without SS-with SS | 5.64 | 2.86 | |

With SS-with CV | 24.67 | 2.86 |

The randomly selected parameter sets for four distributions.

Parameter sets | Process | Exponential | Uniform | Triangular | Normal |
---|---|---|---|---|---|

Set 1 | Arrival | 1 | 1,2 | 1,2,3 | 0.5,1 |

Service | 0.5 | 1,1.5 | |||

Set 2 | Arrival | 1.5 | 1,3 | 2,3,4 | 0.5,1.5 |

Service | 1 | 1,2 | |||

Set 3 | Arrival | 2 | 2,4 | 1,3,4 | 1,2 |

Service | 1.5 | 0.5,2 | |||

Set 4 | Arrival | 2.5 | 2,3 | 1,2,4 | 1.5,2 |

Service | 2 | 1,2.5 |

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.

VRTs: this factor is tested in experimental design in two levels being CV and SS.

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,

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 Tables

The variances of AWT.

Levels | 1.Exponential | 2.Uniform | 3.Triangular | 4.Normal |
---|---|---|---|---|

1.Control variates | 0.000010 | 0.0715000 | 0.0000398 | 0.0000155 |

0.000780 | 0.0063840 | 0.0010400 | 0.0001010 | |

0.039400 | 0.0000188 | 0.0124800 | 0.0000254 | |

1.503000 | 0.0824800 | 0.0007170 | 0.0000939 | |

2.Stratified sampling | 0.105 | 0.395 | 0.432 | 1.541 |

1.012 | 0.492 | 0.386 | 1.224 | |

0.972 | 1.415 | 0.664 | 1.238 | |

2.430 | 0.162 | 0.531 | 1.127 |

The variances of ANC.

Levels | 1.Exponential | 2.Uniform | 3.Triangular | 4.Normal |
---|---|---|---|---|

1.Control variates | 0.0000001 | 0.0003459 | 0.0000001 | 0.0000002 |

0.0000223 | 0.0000663 | 0.0000007 | 0.0000032 | |

0.0129500 | 0.0000000 | 0.0000191 | 0.0000003 | |

0.0828200 | 0.0001932 | 0.0000023 | 0.0000010 | |

2.Stratified sampling | 0.536 | 0.167 | 0.113 | 5.869 |

1.592 | 0.537 | 0.045 | 2.664 | |

0.669 | 0.167 | 0.088 | 2.177 | |

0.223 | 0.026 | 0.107 | 1.924 |

The ANOVA results given in Table

The ANOVA output for AWT.

General linear model: Var versus Tech; Dist | ||||||
---|---|---|---|---|---|---|

Factor | Type levels | Values | ||||

Tech | Fixed | 2 cv ss | ||||

Dist | Fixed | 4 ex un tr no | ||||

Analysis of variance for Var, using adjusted SS for tests | ||||||

Source | DF | Seq SS | Adj SS | Adj MS | ||

Tech | 1 | 4.29330 | 4.29330 | 4.29330 | 48.73 | .000 |

Dist | 3 | 0.42013 | 0.42013 | 0.14004 | 1.59 | .218 |

Tech*Dist | 3 | 0.42165 | 0.42165 | 0.14055 | 1.60 | .217 |

Error | 24 | 2.11432 | 2.11432 | 0.08810 | ||

Total | 31 | 7.24940 |

The ANOVA output for ANC.

General linear model: Var versus Tech; Dist | ||||||
---|---|---|---|---|---|---|

Factor | Type levels | Values | ||||

Tech | Fixed | 2 cv ss | ||||

Dist | Fixed | 4 ex un tr no | ||||

Analysis of variance for Var, using adjusted SS for tests | ||||||

Source | DF | Seq SS | Adj SS | Adj MS | ||

Tech | 1 | 4.9823 | 4.9823 | 4.9823 | 97.65 | .000 |

Dist | 3 | 2.5163 | 2.5163 | 0.8388 | 16.44 | .000 |

Tech*Dist | 3 | 2.5459 | 2.5459 | 0.8486 | 16.63 | .000 |

Error | 24 | 1.2245 | 1.2245 | 0.0510 | ||

Total | 31 | 11.2689 |

The main factor VRTs are statistically significant for AWT, others are not.

The main factor VRTs, the type of distributions, and their interactions are statistically significant for ANC.

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 (

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
Tables

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 (

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.