外文翻譯----等待隊列與模式

1、<p><b>　　附件</b></p><p><b>　　外文翻譯</b></p><p><b>　　英文</b></p><p>　　Waiting lines and simulation</p><p>　　The “miss manners” artic

2、le pokes fun at one of life’s realities; having to wait in line.no boubt those waiting in line would all agree that the solution to the problem is obvious;simply add more servers or else do something to speed up service.

3、although both edeas may be potential solutions,there are certain subleties that must be dealt with. For one thing, most service systems have the capacity to process more customers over the long run than they are called o

4、n to process. Hence,the problem of c</p><p>　　The foundation of modern queuing theory is based on studies about automatic dialing equipment made in early part of the twentieth century by Danish telephone eng

5、ineer A.K.Erlang. Prior to World War II,very few attempts were made to apply queuing theory to business problems. Since that time, queuing theory has been applied to a wide range of problems.</p><p>　　The ma

6、thematics of queuing can be complex;for that reason,the emphasis here will not be on the mathematics but the concepts that underlie the use of queuing in analyzing waiting-line problems. We shall rely on the use of formu

7、las and tables for analysis.</p><p>　　Waiting lines are commonly found wherever customers arrive randomly for services. Some examples of waiting lines we encounter in our daily lives include the lines at sup

8、ermarkdt checkouts,fast-food restaurants,aipport ticket counters,theaters, post offices,and toll booths. In many situations, the “customers” are not people but orders waiting to be filled ,trucks waiting to be unloaded,

9、jobs waiting to be processed,or equipment awiting repairs. Still other examples include ships waiting to dock, </p><p>　　One reason that queuing analysis is important is that customers regard waiting as a no

10、n-value-added activity. Customers may tend to associate this with poor service quality，especially if the wait is long. Similarly, in an organizational setting, having work or employees wait is non-value-added—the sort of

11、 waste that workers in JIT systems strive to reduce.</p><p>　　The discussion of queuing begins with an examination of what is perhaps the most fundamental issue in waiting-line theory:why is there waiting? &

12、lt;/p><p>　　Why is there waiting?</p><p>　　Many people are surprised to learn that waiting lines tend to form even though a system is basically underloaded. For example, a fast-food restaurant may

13、have the capacity to handle an average of 200 orders per hour and yet experience waiting lines even though the average number of orders is only 150 per hour. The key word is average. In reality,customers arrives at rando

14、m intervals rather than at evenly spaced intervals,and some orders take longer to fill than others. In other words, both arriv</p><p>　　Managerial Implications of Waiting Lines</p><p>　　Managers

15、 have a number of very good reasons to be concerned with waiting lines. Chief among those reasons are the following:1. The cost to provide waiting space.</p><p>　　2. A possible loss of business should custo

16、mers leave the line before being served or refuse to wait at all </p><p>　　3.A possible loss of goodwill.</p><p>　　4.A possible reduction in customer satisfaction.</p><p>　　5.The re

17、sulting congestion may disrupt other business operations and/or customers.</p><p>　　Goal of Waiting-Line Analysis</p><p>　　The goal of queuing is essentially to minimize total costs. There are t

18、wo basic categories of cost in a queuing situation: those associated with customers waiting for service and those associated with capacity. Capacity costs are the costs of maintaining the ability to provide service. Exam

19、ples include the number of bays at a car wash, the number of chechkouts at a supermarket, the number of repair people to handle equipment breakdowns, and the number of lanes on a highway. When a service facili</p>

20、<p>　　A practical difficulty frequently encountered is pinning down the cost of customer waiting time, especially since major portions of that cost are not a part of accounting data. One approach often used is to t

21、reat waiting times or line lengths as a policy variable: A manager simply specifies an acceptable level of waiting and directs that capacity be established to achieve that level.</p><p>　　The traditional goa

22、l of queuing analysis is to balance the cost of providing a level of service capacity with the cost of customers waiting for service. Figure 1 illustrates this concept. Note that as capacity increases, its cost increases

23、. For simplicity, the increase is shown as a linear relationship. Although a step function is often more appropriate ,use of a straight line does not significantly distort the picture. As capacity increases,the number of

24、 customers waiting and the time they wait</p><p>　　In situations where those waiting in line are external customers(as opposed to employees),the existence of waiting lines can reflect negatively on an organi

25、zation’s quality image. Consequently, some organizations are focusing their attention on providing faster service—speeding up the rate at which service is delivered rather than merely increasing the number of servers. Th

26、e effect of this is to shift the total cost curve downward if the cost of customer waiting decreases by more than the cost o</p><p>　　Figure1: The goal of queuing analysis is to minimize the sum of two costs

27、: customer waiting costs and service capacity cost.</p><p>　　System Characteristics</p><p>　　There are numerous queuing models from which an analyst can choose. Naturally, much of the success of

28、 the analysis will depend on choosing an appropriate model. Model choice is affected by the characteristics of the system under investigation. The main characteristics are:</p><p>　　Population source.</p&

29、gt;<p>　　Number of servers(channels)</p><p>　　Arrival and service patterns.</p><p>　　Queue discipline (order of service).</p><p>　　Figure 2 depicts a simple queuing system.&l

30、t;/p><p>　　Figure 2 A simple queuing system</p><p>　　Population source</p><p>　　The approach to use in analyzing a queuing problem depends on whether the potential number of customers

31、 is limited. There are two possibilities:infinite-source and finitesource populations. In an infinite-source situation,the potential number of customers greatly exceeds system capacity. Infinite-source situations exist w

32、henever service is unrestricted. Examples are supermarkets,drugstores,banks,restaurants,theaters,amusement centers, and toll bridges. Theoretically,large numbers of customers fr</p><p>　　Number of servers (c

33、hannels)</p><p>　　The capacity of queuing system is function of the capacity of each server and the number of servers being used. The terms server and channel are synonymous,and it is generally assumed that

34、each channel can handle one customer at a time. Systems can be either single-or multiple-channel.(A group of servers working together as a team,such as a surgical team,is treated as a single-channel system.) Examples of

35、single-channel systems are small grocery stores with one checkout counter,some theaters,sin</p><p>　　A related distinction is the number of steps or phases in a queuing system. For example,at theme parks, pe

36、ople go from one attraction to another. Each attraction constitutes a separate phase where queues can (and usually do) from.</p><p>　　Figure 3 illustrates some of the most common queuing systems. Because it

37、would not be possible to cover all of these cases in sufficient detail in the limited amount of space available here,our discussion will focus on single-phase systems.</p><p>　　Figure 3 Four common variatio

38、ns of queuing systems</p><p>　　Arrival and service patterns</p><p>　　Waiting lines are a direct result of arrival and service variability. They occur because random, highly variable arrival and

39、service patterns cause systems to be temporarily overloaded. In many instances,the variabilities can be described by theoretical distributions. In fact, the most commonly used models assume that the customer arrival rate

40、 can be described by a Possion distribution and that the service time can be describde by a negative exponential distribution. Figure 4 illustrates these d</p><p>　　The Poisson distribution often provides a

41、reasonably good description of customer arrivals per unit of time(e.g.,per hour). Figure 5A illustrates how poisson-distributed arrivals (e.g., accidents) might occur during a three-day period. In some hours, there are t

42、hree or four arrivals,in other hours one or two arrivals,and in some hours no arrivals.</p><p>　　The negative exponential distribution often provides a reasonably good description of customer service times(e

43、.g.,first aid care for accident victims). Figure 5B illustrates how exponential service times might appear for the customers whose arrivals are illustrated in Figure 5A . Note that most service times are very short—some

44、are close to zero—but a few require a relatively long service time. That is typical of a negative exponential distribution.</p><p>　　Waiting lines are most likely to occur when arrivals are bunched or when s

45、ervice times are particularly lengthy, and they are very likely to occur when both factors are present. For instance, note the long service time of customer 7 on day 1, in Figure 5B. In Figure 5A, the seventh customer ar

46、rived just after10 o’clock and the next two customers arrived shortly after that, making it very likely that a waiting line formed. A similar situation occurred on day 3 with the last three customers: The r</p>&l

47、t;p>　　It is interesting to note that the Poisson and negative exponential distributions are alternate ways of presenting the same basic information. That is ,if service time is exponential ,then the service rate is Po

48、isson..Similarly, if the customer arrival rate is Poisson, then the interarrival time (i.e.,the time between arrivals) is exponential. For example, if a service facility can process 12 customers per hour(rate), average s

49、ervice time is five minutes.and if the arrival rate is 10 per hour, t</p><p>　　The models described here generally require that arrival and service rates lend themselves to description using a Poisson distri

50、bution of ,equivalently, that interarrival and service times lend themselves to description using a negative exponential distribution. In practice, it is necessary to verify that these assumptions are met. Sometimes this

51、 is done by collecting data and plotting them, although the preferred approach is to use a chi-square goodness-of-fit test for that purpose. A discussio</p><p>　　Research has shown that these assumptions are

52、 often appropriate for customer arrivals but less likely to be appropriate for service. In situations where the assumptions are not reasonably satisfied, the alternatives would be to (1) develop a more suitalbe model, (2

53、) search for a better (and usually more complex) existing model, or (3) resort to computer simulation. Each of these alternatives requires more effort or cost than the ones presented here.</p><p>　　Figure 4

54、Poisson and negative exponential distributions</p><p>　　Figure 5 Poisson arrivals and exponential service times</p><p>　　Queue discipline</p><p>　　Queue discipline refers to the ord

55、er in which customers are processed. All but one of the models to be described shortly assume that service is provided on first-come, first-served basis. This is perhaps the most commonly encountered rule. There is frist

56、-come service at banks,store,theaters,restaurants, four-way stop signs, registration lines, and so on. Examples of systems that do not serve on a first-come basis include hospital emergency rooms, rush orders in a factor

57、y, and main frame computer</p><p>　　Measures of system performance</p><p>　　The operations manager typically looks at five measures when evaluating existing or proposed service systems. Those me

58、asures are:</p><p>　　1. The average number of customers waiting, either in line or in the system.</p><p>　　2. The arverage time customers wait, either in line of in the system.</p><p&

59、gt;　　3. System utilization, which refers to the percentage of capacity utiliaed.</p><p>　　4. The implied cost of a given level of capacity and its related waiting line.</p><p>　　5. The probabili

60、ty that an arrival will have to wait for service.</p><p>　　Of these measures, system utilization bears some elaboration. It reflects the extent to which the servers are busy rather than idle. On the surface,

61、 it might seem that the operations manager would want go seek 100 percent utilization. However, as Figure 6 illustrates, increases in system utilization are achieved at the expense of increases in both the length of the

62、waiting line and the average waiting time. In fact, these values become exceedingly large as utilization approaches 100 percent. The</p><p>　　Figure 6 The average number waiting in line and the average time

63、customers wait in line increase ceponentially as the system utilization increases</p><p>　　Queuing models:infinite-source </p><p>　　Many queuing models are available for a manager or analyst to

64、choose from. The discussion here includes four of the most basic and most widely used models. The purpose is to provide an exposure to a range of models rather than an extensive coverage of the field. All assume a Poisso

65、n arrival rate. Moreover, the models pertain to a system operating under steady state conditions;that is, they assume the average arrival and service rates are stable. The four models described are:1. Single channel, &l

66、t;/p><p>　　2. Single channel,constant service time.</p><p>　　3. Multiple channel, exponential service time.</p><p>　　4. Multiple priority service, exponential service time.</p>

67、<p>　　To facilitate your use of queuing models,Table 1 provides a list of the symbols used for the infinite-source models.</p><p>　　Tabel 1 Infinite-source symbols</p><p>　　Basic relationsh

68、ips </p><p>　　There are certain basic relationships that hold for all infinite-source models. Knowledge of these can be very helpful in deriving desired performance measures, given a few key values. Here are

69、 the basic relationships:</p><p>　　System utilization: This reflects the ratio of demand (as measured by the arrival rate) to supply or capacity (as measured by the product of the number of servers, M,and th

70、e service rate, μ)</p><p>　　···········(1)</p><p>　　The average number of customers being served:</p><p>　　···&

71、#183;·······(2)</p><p>　　The average number of customers:</p><p>　　Waiting in line for service: L1(Model dependent. Obtain using a table or formula.)</p>

72、<p>　　In the system (line plus being served): L2=L1+r ···········（3）</p><p>　　The average time customers are:</p><p>　　Waiting i

73、n line: W1= ············（4）</p><p>　　In the system: W2=W1+ ···········（5）</p><

74、;p>　　All infinite-source models require that system utilization be less than 1.0; the models apply only to underloaded systems.</p><p>　　The average number waiting in line ,L1 , is a key value because it

75、is a determinant of some of the other measures of system performance, such as the average number in the system, the average time in line, and the average time in the system. Hence, L1 will usually be one of the first val

76、ues you will want to determine in problem solving.</p><p><b>　　Example</b></p><p>　　Customers arrive at a bakery at an average rate of 18 per hour on weekday mornings. The arrival di

77、stribution can be described by a Poisson distribution with a mean of 18. each clerk can serve a customer in an average of rour minutes; this time can be described by an exponetial distribution with a mean of 4.0 minutes.

78、</p><p>　　What are the arrival and service rates?</p><p>　　Compute the average number of customers being sered at any time.</p><p>　　Suppose it has been determined that the average

79、number of customers waiting in line is 3.6. Compute the average number of customers in the system (i.e.,waiting in line or being served),the average time customers wait in line, and the average time in the s

80、ystem.</p><p>　　Determine the system utilization for M=2,3 and 4 servers.</p><p><b>　　Solution</b></p><p>　　a. The arrival rate is given in the problem: customers per

81、hour. Change the service time to a comparable hourly rate by first restating the time in hours and then taking its reciprocal. Thus, (4 minutes pre customer)/(60 minutes per hour)=1/15=1/μ . Its reciprocal is μ=15 custom

82、ers per hour.</p><p>　　b. = = 1.2 customers.</p><p>　　c. Given: L1=3.6 customers.</p><p>　　L2=L1+r=3.6+1.2=4.8 customers</p><p>　　hours per customer,

83、 or 0.2 hours *60 minutes/hour =12 minutes</p><p>　　W2=waiting in line plus service</p><p>　　=W1+=0.2+ =0.267 hour, or approximately 16 minutes</p><p>　　System utilization is <

84、;/p><p>　　For M=2, =0.6</p><p>　　For M=3, =0.4</p><p>　　For M=4, =0.3</p><p>　　Hence, as the system capacity as measured as measured by Mμ in

85、creases, the system utilization for a given arrival rate decreases.</p><p>　　Reference literature:</p><p>　　Operations Management William J. Stevenson Seventh Edition</p><p>&l

86、t;b>　　中文</b></p><p><b>　　等待隊列與模式</b></p><p>　　“Miss Manners”文章嘲笑一種生活現(xiàn)實：必須在隊里等待。毫無疑問，在隊里等待的人都同意得到問題的解答：增加更多服務(wù)器或者提高服務(wù)速度。盡管這兩種方法都能解決問題，但一些敏銳的問題必然得以解決。首先，多數(shù)系統(tǒng)從長遠的角度來看，它能提供比實際要處理

87、的服務(wù)還要多，因此顧客等待問題是一種短期現(xiàn)象。另外，有些機器處于等待顧客的服務(wù)，因此要是采取通過增加系統(tǒng)的服務(wù)能力并不能解決這類問題，這樣只會增加服務(wù)系統(tǒng)的閑置時間。因此，在設(shè)計系統(tǒng)時，設(shè)計者要衡量成本與提供服務(wù)容量水平，要考慮潛在客戶的需求的同時減少多余服務(wù)的浪費。對服務(wù)能力分析與計劃是基于數(shù)學方法中的對等候行列分析的排隊理論。</p><p>　　現(xiàn)代排隊理論是在20世紀早期丹麥電話工程師A.K.Erlang

88、研究自動撥號設(shè)備的基礎(chǔ)上發(fā)展而成的。第二次世界大戰(zhàn)之前，排隊理論好少應(yīng)用于處理商業(yè)業(yè)務(wù)所碰到的問題，二次大戰(zhàn)后，排隊理論被廣泛應(yīng)用于各種各樣的問題。</p><p>　　數(shù)學排隊問題可以是很復(fù)雜，因此，這里重點不在于數(shù)學，而是強調(diào)使用排隊理論在分析等待線問題的概念。我們將依靠使用數(shù)學公式和表格圖來分析。</p><p>　　排隊等候經(jīng)常見到，盡管顧客所需的服務(wù)是隨機的。如在我們?nèi)粘Ｉ钏?/p>

89、現(xiàn)的排隊現(xiàn)象：離開超市時的結(jié)算、快餐店、飛機售票臺、電影院、郵局和所有收費通行的服務(wù)等。在很多情況下，等待的“顧客”并不是人員而是等待其他。如卡車等待卸載，工作等待處理、或者設(shè)備等待維修。還有其他例子，如船等待停泊、飛機等待登陸、醫(yī)院的病人等待護士，汽車等待通行證等等。</p><p>　　排隊問題分析的一個重要原因是顧客認為等待不能創(chuàng)造價值。等待會影響服務(wù)質(zhì)量，尤其是過久的等待會使顧客產(chǎn)生服務(wù)質(zhì)量差的印象。類似

90、的，在組織設(shè)置時，要工作或雇員等待都是非增值活動――JIT系統(tǒng)所努力減少的浪費。</p><p>　　關(guān)于排隊理論的討論源于一個問題或者說是等候線的基本問題：那兒為什么要等候？</p><p><b>　　為什么要等候？</b></p><p>　　很多市民很驚訝去知道等候線的形成原因，即使系統(tǒng)未負荷。例如，快餐館有能力每小時處理200份菜單，

91、但等候現(xiàn)象還是會產(chǎn)生，即使每小時只有150份菜單。關(guān)鍵問題是“平均”，顧客的到來是隨機的，而不是平均間隔到來；同時菜單所花費的時間不一致，有些會比其他的要長。換句話說，到來的隨機性以及服務(wù)時間是不確定的。因此系統(tǒng)隨時面臨超載，增加了隊伍的長度。而其他沒有顧客時間，系統(tǒng)無所事事。因此，盡管系統(tǒng)在宏觀立場上看是未超載，但由于顧客到來的隨機性以及服務(wù)時間的不一致，在微觀立場上來，系統(tǒng)出現(xiàn)超載現(xiàn)象。因此，可以說，在變化性很少的系統(tǒng)（到來的時間可

92、以預(yù)測和服務(wù)時間是恒定）一般不存在排隊等候的現(xiàn)象。</p><p>　　管理人員需要掌握充分的有關(guān)排隊等候的資料，主要包括以下幾點：</p><p>　　提供等候排隊的空間所花費的成本。</p><p>　　顧客離開而造成的可能存在的商業(yè)損失。</p><p><b>　　信譽可能損失。</b></p>&

93、lt;p>　　顧客滿意度可能降低。</p><p>　　發(fā)生的阻塞可能會打亂商業(yè)活動運作和（或）顧客。</p><p><b>　　等候線分析的目標</b></p><p>　　排隊理論的本質(zhì)是使總成本達到最小。在排隊過程存在著兩種基本費用：與顧客等待有關(guān)以及與系統(tǒng)容量有關(guān)。系統(tǒng)容量成本是維護所提供的服務(wù)、穩(wěn)定服務(wù)水平所支出的成本。如汽車

94、服務(wù)中的汽車道的多少、超級市場付賬服務(wù)臺數(shù)量、維修機器的人員數(shù)量、在高速公路上車道的數(shù)量等。由于這些容量是不能儲存的，故當設(shè)施、設(shè)備等處于空閑的狀態(tài)時，容量就會消失。有關(guān)顧客等待的成本包括支付給等待服務(wù)的雇員的薪金（如等待維修用的工具或設(shè)備、等待卡車卸載的司機）和等待所占用空間所花費的成本（候診室的大小、洗車位的多少、在飛機著落前所要消耗原料所占的空間）以及由于顧客拒絕等候或由于等候過長而將來失去顧客所造成的商業(yè)損失。</p>

95、;<p>　　常常遇到一些實實在在的困難，不得不采取花費顧客等待時間，尤其是大部分費用不是可數(shù)的數(shù)據(jù)。應(yīng)對這類問題，常用的方法就是把等待的時間和隊列的長度看作是可變的，管理人員確定顧客可接受的等候時間以及隊列長度，并建立一個服務(wù)水平達到這一要求的系統(tǒng)。</p><p>　　排隊理論的傳統(tǒng)分析目標在于平衡系統(tǒng)提供一定服務(wù)水平所花的費用與顧客排隊所支付的成本。圖1充分說明了這一原理。注意到，當容量增加會

96、伴隨著費用的增加，明顯它們間的關(guān)系是線性關(guān)系。在實際情況下，階梯函數(shù)比直線更適當應(yīng)用。 當容量增量，傾向于減少等待的顧客的數(shù)量和時間，從而減少等待成本。 在典型的交易關(guān)系中，總成本可以用Ｕ形曲線代表。 分析的目標是在獲得一定服務(wù)能力水平的情況下如何使總成本最小。(它不同于在存貨EOQ模型的情況，在總成本曲線的極小的點通常不是兩條成本線相交的地方。)</p><p>　　在隊伍內(nèi)等候的外部顧客（與雇員即內(nèi)部顧客相對

97、）通常會對組織的質(zhì)量存在著消極的影響。結(jié)果很多組織把注意力放在如何提供更快的服務(wù)速度——快速服務(wù)并不是僅僅增加服務(wù)器的數(shù)量。減少顧客等待比提供更快的服務(wù)速度能總成本曲線向下滑移更多。</p><p>　　圖1 排列理論分析的目標在于尋找最低成本時的最佳容量</p><p><b>　　系統(tǒng)特征</b></p><p>　　有許多排隊理論模型可供

98、選擇分析。顯然，能否分析成功很大程序取決于是否選擇了合適的模型。模型的選擇受系統(tǒng)特征的影響，主要特征如下：</p><p><b>　　人口來源</b></p><p>　　服務(wù)器（渠道）的數(shù)量</p><p><b>　　到來和服務(wù)的方式</b></p><p>　　隊列的制度（服務(wù)命令）<

99、/p><p>　　如圖2描述了一個簡單的排隊系統(tǒng)。</p><p>　　圖2 一個簡單的隊列系統(tǒng)</p><p><b>　　人口來源</b></p><p>　　在這方面分析排隊問題在于潛在顧客來源是否有限制。有兩種可能情況：無限制的顧客來源與有限制的顧客來源。在無限制來源情況下，顧客的潛在的數(shù)量很大地超出系統(tǒng)容量。無限制

100、顧客來源存在于服務(wù)是無限制的情況下，如超級市場、藥房、銀行、餐館、劇院、娛樂中心和收費橋。理論上，在“號召顧客群”中的大量顧客的服務(wù)請求能得以實現(xiàn)。某一時候可能出現(xiàn)需要修理的機器超過預(yù)計修理的數(shù)量而得不到修理，或者超出于應(yīng)分配給每個維修工的數(shù)量。同樣，一個操作工可能同時要負責四臺機器的操作，護士同時接收病人入院和退房的要求，秘書要負責記錄三個董事長的工作指令，一個公司的銷售店可能要求需要20輛卡車。</p><p&g

101、t;　　服務(wù)器（渠道）的數(shù)量</p><p>　　排隊系統(tǒng)容量是指每臺服務(wù)器的功能與能提供服務(wù)需求的數(shù)量。服務(wù)器和渠道是同義的。通常假設(shè)，每一種渠道在每一次幾乎只處理一名顧客。系統(tǒng)可以是單一或多渠道的（一組服務(wù)人員組成服務(wù)團體，如一個外科醫(yī)療小組共同處理一個單一系統(tǒng)）。單一系統(tǒng)的例子還有只有一個結(jié)算臺的小雜貨店，某些劇院，只有一條汽車道的服務(wù)和只有一位出納的銀行。多渠道系統(tǒng)（擁有多臺服務(wù)器）常見于銀行系統(tǒng)、售飛機

102、票系統(tǒng)、自動服務(wù)系統(tǒng)、加油站等。</p><p>　　它們的分別聯(lián)系在于服務(wù)層次的數(shù)量或一個排隊系統(tǒng)的所處的階段。例如，人們往往從一種吸引力轉(zhuǎn)移到另一種吸引力，每個階段解釋了隊伍的形成的一般理由。</p><p>　　圖3說明一些最常見的的排隊系統(tǒng)。由于在有限的空間，不可能包括所有的細節(jié)，在這里也不能一一述說，因此我們集中討論單階系統(tǒng)。</p><p>　　圖3 基

103、于四種形式變化的隊列系統(tǒng)</p><p><b>　　到來和服務(wù)的形式</b></p><p>　　等候隊列是顧客到來的隨機性和服務(wù)方式多樣化的直接結(jié)果。它們的發(fā)生是隨機的，顧客到來的高度不確定性以及服務(wù)要求的多樣化導(dǎo)致系統(tǒng)臨時超載。很多情況下，可變性可由用分布理論描述。實際上，常用假設(shè)模型來分析，顧客的到達率可以用泊松分布描述。服務(wù)時間可以用消極指數(shù)來描述。圖4說明