摘要：如今，作为消费者越来越不愿意花过多的时间用于排队等待。本文通过收集一手数据、数理分析、数据比较，着重分析了排队理论及相关系统的应用，目的在于帮助管理者解决排队等待时间过长的现象来提升管理质量。此外，本文解释了使用的研究方法、研究过程中的优缺点以及调研的局限性。最后，文章给出了具有建设性的管理意见来有效控制排队现象，达到提高顾客满意度这一目的。
关键词：排队理论 里托定律 M/M/S系统 G/G/S系统
Abstract: Recently, consumers won’t spend much time waiting in the queue. By collecting the primary data, this paper tries to quantify the results and model different queuing systems to solve queue issues. According to the mathematical analysis, the merits and demerits of queuing theory and limitations of the research will be discussed. Meanwhile, some suggestions will be given to enhance queuing management and customers’ satisfaction.
Key words: queue theory Little’s Law M/M/c System G/G/c System
1.Introduction
Today, more and more consumers can’t tolerate waiting in a queue for a long time. In other words, consumers not only demand quality, they also expect speed. The shorter the queue is, the higher satisfaction consumers will have. Thus, it is extremely important to control a queue appropriately. Based on the queuing theory, a survey is conducted to identify and compare the theoretically obtained results with the observed data. By modeling different queuing systems----namely Little’s Law, M/M/c system and G/G/c system, the most effective model will be obtained and meanwhile evaluate its validity. According to the mathematical analysis, this paper will discuss the merits and demerits of queuing theory and limitations of the research, and explain how to improve the objective’s queuing management as well.
2.Methodology
The object of this research is China Construction Bank (CCB), which is located at Yifeng Road at Dongguan in Guangdong province. The bank provides reception service with customers. Observations focus on the two ATMs in the hall of the bank. The whole experient lasts one hour around, starting from 10:45am to 12:00pm on 22, November 2011. Before determination to the bank, both post office and Starbucks coffee were supposed to be studied. However, on one hand, because of 5 servers in the post office there are no enough members to record the data; on the other hand, lack of adequate customers, there is no queue within 30 minutes in Starbucks Coffee, the two objects are not adopted finally. The investigation in bank is run at off-peak period in order to collect the data in a steady state. Furthermore, it is uncertain that whether the initial conditions in a queue affect the results or not, so no inventory is suggested at the beginning of the data collection.
In this paper, assume that customer arrives one at a time and all customers have the same priority. Meanwhile, there is no balking and no reneging in the study. All customers are queuing in a single line and are served by 2 ATMS in term of first-come-first-served. Arrival time, processing time, departure time and inventory in every minute are the objective data in this research. It means that service time equals departure time minus processing time.
There are 5 members in the group to collect raw data, who are four students and I. All the members are allocated into different roles. Student A records the arrival time of each customer, Student B is in charge of inventory in every minute, Student C and D note the process time and departure time at server 1 and server 2 respectively. Last but not least, I concentrate on populating these data into excel forms. According to some references, the survey is driving orderly. Inevitably, there exists limitations when data collection, such as the differences between each stopwatch.
The calculation of all values is obtained by Excel and all data accurate at 4 digits after decimal. The usage of Excel is introduced and referenced by Winston and Albright.
3.Findings and results
According to the collected data, T and I could be stated as the mean flow time and mean inventory respectively. Calculated by Excel, T and I are obtainable from the raw data directly, which is T=1.0182,
I=0.9623. (See figure-1)
3.1 Little’s Law
The Little’s Law can be expressed as
I = RT, R=min (Ri, Rp)
Based on the observed values of I and Ri ( Ri=1/inter-arrival time), so
T= I / Ri=0.9147
Compared the observed value T with theoretical value T (details shown in figure-2), it can be find that the difference is 0.1035.
3.2 M/M/c System
The M/M/c formula is in the exponential distribution. The mean inter-arrival time and mean service time can be obtained directly from the spreadsheet. Referred to Dr. Taka Hosoda, it can be stated that Ri= 1/ mean inter-arrival time, Rp = c/ Tp. According to these values, capacity utilization ρ= Ri/Rp, so inventory in queue Ii can be expressed as
Ii == 0.2402
In term of Little’s Law, Ip = Tp* Rp, Ti = Ii / Ri, so
T = Ti + Tp =1.0662, I = Ii + Ip = 1.1216
All the values have been categorized and can be found in the figure-3.
3.3 G/G/c System
The G/G/c formula is in the general distribution. The Ri, Rp and Tp are the same with those in M/M/c system. According to these values, capacity utilization ρ= Ri / Rp, and Ci and Cp are the coefficients of variation of the inter-arrival and processing time (C= Standard Deviation/ Mean), So inventory in queue Ii can be expressed as
In term of Little’s Law, Ip = Tp* Rp, Ti = Ii / Ri (see figure-4), so
T = Ti + Tp = 1.0050, I = Ii + Ip = 1.0573
All the values have been categorized and can be found in the figure-4.
3.4 Eye balling
In this case, the histogram of inter-arrival times appears to be consistent with the exponential density in figure-5(left side). The highest bar is at the left, and then fall off gradually from left to right. However, it can be seen visually that the histogram of the service times in figure-6 (right side) is not figured like exponential distribution. To summarize, the exponential assumption for inter-arrival times is reasonable, but it is questionable for service.
Figure-5 Histogram of inter-arrival time for ATM in CCM bank (left side)
Figure-6 Histogram of service time for ATM in CCM bank (right side)
3.5 Assessment of the validity
It can be seen from the underlying figure-7 that the theoretical values of Flow time and Inventory obtained from Little’s formula, M/M/c and G/G/c are basically approximate with the raw data. The survey is studied in a steady state, so Little’s Law follows this satiation. According to the eye balling analysis, the exponential assumption for inter-arrival time is appropriate, so the values from M/Mc formula are valid. Moreover, the G/G/c model allows any inter-arrival time and service time distribution, thus, the validity of values is available. However, by comparison, differences between the values from G/G/c model and raw data are least, which can be expressed as following
ΔT= │T0- Tg│=│1.0182-1.0050│=0.0132,
ΔI = │I0 - Ig │=│0.9623-1.0573│ =0.0950
Figure-7 comparison with values obtained by three different models
4.Conclusion and recommendation
According to the Figure-7, there is a gap between the observed data and the theoretically obtained results. For instance, the differences about Flow Time (T) from Little’s law, M/M/c model and G/G/c are 0.1035, 0.0480 and 0.0132 respectively. There are some reasons for data gap existence. Firstly, only the inter-arrival time follows the exponential distribution neither the service time in the M/M/c system, so mean inter-arrival time and mean service time may give the misleading results when the actual service time distributions are not exponential (Winston). Secondly, in the G/G/c system, although this models allow any distribution, the SCV= (standard deviation/mean)2 will affect the variations whether it follows exponential distribution or not (+bid). However, all the theoretical values are approximation to the raw data which are assessed to be validity.
Therefore, it can be found that the queuing theory appears advantage and disadvantage. The merit of the theory is that it is helpful for employees to control the queue. From the theory, the queue manager could know the expected time in a queue and the expected customers in a queue as well as in the system. However, to some extent, it is difficult to the reality. In the theory, assume that the queue is in a steady state without short-run behavior and overcome the initial conditions. In addition, there is no balking and reneging. But, in the real world, when customers find that the queue is too long, they may not enter the queue or leave the queue before starting service. Or, if at the peak lunchtime period in a takeaway restaurant, many customers come in the same time, the queue may become quite long immediately, it is hard for the researchers to identify whether the long initial conditions effect the results or not. So the queue theory seems to be idealization. Additionally, the queuing models typically require that all customs wait in a single line and are served in FCFS principle. In the daily life, there are many queues and several servers provide service together for example in supermarket. Customer may switch queues when they find the queue next to them goes faster than theirs. Under this situation, the queue theory is not appropriate. Therefore, the queue theory has limitations. Also, it is the limited point about this report. This research undertaken is based on the typical requirement of the queue theory. The data recorded when there is no inventory, and ignore the balking and reneging. When doing further research next time, simulation models can be recommended to analyze the queue problem which seems to be more integrated.
It can be figured out from this survey that the inventory for ATMs in this bank is near one person in every minute, which is supposed to be acceptable. However, when facing long queues, queue manager can adopt some measures to improve the customer’s experience. Katz (1991) et al stated that managers should pay more attention about some important issues like fairness, interest level, customer attitudes, environment and value of service to manage customer perceptions of waiting. For instance, is there any interesting things happening to appeal the customers? Is waiting comfortable? Does the customer have to freeze in the cold or bake in the sun? Good management could improve customer’s satisfaction. The manager could establish an electronic blackboard to broadcast some business news or weather report to distract customer’s attention in order to prevent from customers’ irritable mood.
In conclusion, Queuing is everywhere. The efficiency can be improved when the queue is controlled well.
参考文献:
[1]Albright, S.C. & Winston W.L. (2005) Spreadsheet modeling and applications: essentials of practical management science. Thomson south-western
[2]Katz, K., Larson B. & Larson, R. (1991) Prescription for the waiting-in-line blues: entertain, enlighten and engage. Sloan management review. Vol:32, no:2, pp:44-53.
作者简介：
朱佳俐,女，1985-12-25，江苏省宜兴市，东莞职业技术学院，邮编：523000，广东省，研究方向：物流与供应链管理。