2009 International Conference on Electronic Commerce and Business Intelligence
Simulation of Queuing Systems with Different Queuing Disciplines Based on Anylogic Ni Zhiwei, Lu Xiaochun, Liu Dongyuan School of Economics and Management Beijing Jiaotong University Beijing 100044,China e-mail:
[email protected] Monte Carlo (MC) simulation. The result showed that the method was better than the MC. Jeong, Hae-Duck J., Lee, Jong-Suk R., McNickle, Don, Pawlikowski, Krzysztof [2] studied the simulation of M/M/1/ infinity queuing systems in the method based on the batch means, conducted for estimating steady-state mean waiting times. Rossetti, Manuel D., Delaney, Patrick J. [3] investigated the use of analytical queuing approximations to assist in mitigating the effects of the initial transient period in steady state GI/G/m queuing simulations. And they investigated using queuing approximations to stochastically set the initial conditions of the simulation and developed a new set of truncation heuristics based on GI/G/m queuing approximations. Wiendahl, Hans-Peter, Lorenz, Wolfram [4] introduced two new queuing models for the calculation of the average inventory and of average processing time specifically for workshop manufacturing. A comparison with measured operational data and deterministic simulation results showed that the model is suitable for practical conditions. Bahr, H.A., DeMara, R.F. [5] developed and evaluated low runtime overhead, self-adapting storage policies for priority queues, called smart priority queue (SPQ) techniques. Results indicated that optimizing storage to the spatial distribution of queue access could decrease a lost of operation cost. At home, Xue Yongji, Li Xuhong, Cheng Risheng [6] established a model, and achieved the optimal dock number with queuing theory and introduced the simulation based on Arena. Zhang Bo, Wang Yachao [7] applied Witness 2004 to simulate and optimize a money-collecting system in a supermarket.Based on queuing theory, Cao Yuhua, Qiao Liang, Huang Xuefei [8] transferred the mathematical model into computer simulation language to realize simulation by Delphi language. Their method could provide reference for the decision-making of logistics system planning. Zhang Dianzhong, Tan Xiaohong [9] introduced a model that the weigh factor was used to calculate the queuing length for various types of vehicles. And they wrote a program in Matlab software to simulate the complicated vehicles’ queuing system. Moreover, They found the rule that the vehicles’ capacity of tollbooths changed with the number of tollbooth and average queuing length. Wang Jun, Yu Qiaoying [10] introduced the basic model of queuing theory in operation research and the main features of programming with Flash. And they analyzed the kernel of developing simulation with Flash for machine repair problem in queuing
Abstract—In this paper, based on Anylogic6.0 simulation software, some simulation models of queuing systems with different queuing disciplines are established, including First Come First Service, Last Come First Service and Random Service. Compared with the theoretical values, the accuracy of the experiment data is verified. Finally, with Comparative Analysis of experiment data, we show that under a special condition, the difference of the performance of the queuing systems with different queuing disciplines is limited. Keywords-Anylogic; simulation; queuing theory; queuing discipline;
I.
INTRODUCTION
Queuing is the most common phenomenon in people’s daily work and life. In the case of waiting, queuing disciplines have many forms, including First Come First Service, Last Come First Service and Random Service and so on. First Come First Service is the most common form such as the queue waiting for buses and for payment in supermarkets. Previous studies on queuing theory are mostly based on this queuing discipline. Last Come First Service and Random Service are also common disciplines. Information processing of intelligence system and weather forecast are two kinds of Last Come First Service. Connection of telephone by operator is a case of Random Service. However, scholars have had fewer studies on these two disciplines. Computer simulation is a crossed and marginal research field, relating to Computer Science, Computing Science, Probability Statistics and Operations Research and so on. Through establishing mathematical models and simulating by computer, the original issue can be solved by using simulation result as approximate solution. It has become a powerful tool for system analysis and design. It can be tested and quantified economically and quickly. Due to these, it is more persuasive and intuitive than the other analysis tools. Queuing theory was once studied by mathematical methods. However, with the development of computer simulation technology, more and more scholars study on queuing theory by simulation. In overseas, Nakagawa, Kenji [1] investigated an importance sampling (IS) simulation of MMPP/D/1 queuing to obtain an estimate for the survivor function P(Q>q) of the queue length Q in the steady state. They compared the simulation results of the IS simulation with the ordinary
978-0-7695-3661-3/09 $25.00 © 2009 IEEE DOI 10.1109/ECBI.2009.52
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theory. Moreover, they proposed the structure and methods of key technique(such as the time simulation model etc). Zhang Jian [11] carried out a simulation for a hospital clinic queue system with a program written in Siman simulation language and Fortran language in a microcomputer. Through the experiments, optimal queue structures and disciplines demonstrated their effectiveness to modify service quality and clinic efficiency even though there was no increase of manpower and equipment. Wu Qingbiao [12] studied the simulation for the queuing system based on LIFO with a limited Capacity. Zhang Ying, Guo Jingtian [13] established some simulation models with different queuing disciplines with Extend simulation software package. Based on these works the inherent rules of the queuing system with different queuing disciplines were discussed Anylogic is an innovative modeling tool. It’s based on the latest progress of simulation science and information technology developed in the past decade. With the characteristics of interactivity, visibility and capacity of excellent analysis and optimization, researcher could focus on the process of modeling. However, the studies in which domestic and foreign scholars used Anylogic are less. Domestic scholars are unfamiliar with it. And there is few influential study. Therefore, in order to study on operation efficiency of queuing systems with different queuing disciplines, this paper use computer simulation technology to establish some simulation models based on Anylogic6.0 simulation software, which is very feasible and meaningful. II.
Figure 1. Simulation model of queuing system based on Anylogic 6.0
This model contains four objects. They are Source Object which generates entities according to the need ， Queue Object which simulates customers queue ， Delay Object which simulates service desk，and Sink Object which mean that the end of the service and the customer leave off. Entities are transferred in this model. And the transfer process is customer arrival – queue– service –customer departure. Entities’ behavior can be defined by Java Class. In this paper, a Java Class named Order is established in the model. Its parameters contain time of customer arrival (arrivetime), time of customer departure (leavetime), customer’s level (vip). Figure 2 lists the creation code of Order Class.
SIMULATION MODEL
A. Simulation model of queuing system based on Anylogic6.0 In this paper, the queuing systems with different queuing disciplines will be studied, including First Come First Service, Last Come First Service and Random Service. The input process and the service organization refer to the following two kinds of situations: System 1: Under the assumption that the input process is Possion flow and the service time is an negative exponential distribution, with one service counter in the system whose capacity is infinite(This is a M/M/1/∞ queuing system). System 2: Under the assumption that the input process is Possion flow and the service time is independent, which subjects to the general probability distribution G(t), with the ∞ 1 0 < = ∫ tdG (t ) 0 μ , and there average service time denoted by is one service counter in the system whose capacity is infinite(This is a M/G/1/∞ queuing system). The simulation model of queuing system based on Anylogic6.0 is established. It is shown in Figure 1 below:
Figure 2. The creation code of Order Class
B. The function and operation of the Simulation model (1)The definition of simulation experiment. Before running a model, length of experiment time and time units should be defined. This paper defines that the time units are hours and the length of experiment time is 1000 hours. It should be paid attention that the length of experiment time should be long enough long in order to simulate the system operating under a state of balance. (2)The definition of queuing system. In order to study on operation efficiency of queuing systems with different queuing disciplines, kinds of queuing systems should be considered. We define the parameters of the abovementioned queuing systems as follows show: 165
System 1: Under the assumption that the input process
λ = 3 and the service time μ =4. is a negative exponential distribution with parameter
average sojourn time
is Possion flow with parameter
1
μ
+
λ E[ X 2 ] 2(1 − ρ )
(3)
(3) Nq =
System 2: Under the assumption that the input process
λ 2 E[ X 2 ] 2(1 − ρ )
average queue length (4) Through the above conclusions, we get the theoretical values of the two queuing systems. They are shown as follows: System1: W =1.00 hour, Nq =2.25 persons.
is Possion flow with parameter λ = 3 and the service time is a triangular distribution—triangular(0,0.25,0.5), which
μ=4
means parameter . (3)The definition of queuing disciplines in the model. Queuing disciplines in the model are set through priority which is a parameter in Queue Object. The queuing discipline is First Come First Service when Priority is set as 0. Last Come First Service and Random Service can be set up through setting as entity.vip in Priority. Thereby, queuing disciplines can be set up by vip of Order Class. The queuing discipline is Last Come First Service when vip shows a gradually increasing tendency. The queuing discipline is Random Service when vip is a random number. (4)The definition and collection of quantitative indexes of queuing system. There are many quantitative indexes to describe queuing system. This paper chose average sojourn time ( W ) and average queue length ( N q ) to analyze the performance of queuing systems with different queuing disciplines. In order to observe the changes of average sojourn time and average queuing length in the model, we should collect data. This paper uses Data Set of Anylogic6.0 to collect the two quantitative indexes through establishing avtimeDS and avlengthDS. And it use Time Plot which is an analysis tool in Anylogic6.0 to load the two Data Set. Then while the model is running, data curves which are reflecting the trends of the two quantitative indexes will be shown in Time Plot. III.
W=
N
System2: W =0.69 hour, q =1.31 persons. After running the simulation model, the data curves which are reflecting the trends of the two quantitative indexes are shown in Time Plot. Figure 3 and Figure 4 show the data curves of the quantitative indexes of the two systems.
Figure 3. The data curves of the quantitative indexes of System 1
THE ANALYSIS OF THE SIMULATION RESULTS
The number of simulation experiments on per queuing system with each queuing discipline is 20. A. Data verification At present, the application of mathematical methods to study queuing theory is relatively mature. And the theoretical values of quantitative indexes of queuing system under a certain condition has been got [14]. With First Come First Service, Last Come First Service and Random Service, ρ=
Figure 4. The data curves of the quantitative indexes of System 2
Though with different queuing disciplines, through many simulation experiments, the data curves of the two quantitative indexes of the two queuing systems reflect a common phenomenon that the two quantitative indexes are fluctuating at the beginning of the experiment, and they become stable when it is at a certain time. It shows that when the running time of these two queuing systems are enough long, the average sojourn time and the average queue length are individully close to a fixed value. It agrees well with the theory. Through data comparative, we can further verify the validity of the experiment data. The experiment data and the theoretical values are shown in Table 1 below:
λ
