C06 QueuingTheory

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queuing

n
Observed before, Little was first to prove
Time in 3 System 2 1 1 2 3 Packet #
n
Applies to any system in equilibrium, as long as nothing in black box is creating or destroying tasks
J T
J =T
n
Tautology
What is average waiting time before and in the tunnel?
W=
L λ
= (120 33 ) = 3.6min
2
Model Queuing System
Queuing System λ Queue Queuing System λ µ
ρ 1− ρ 0.25 = 1− 0.25 = 0.33
ρ = (λ µ ) = 0.25%
P(n) packets in the gateway = P P = (1 − ρ )( ρ n ) = ( 0.75)(0.25n )
0 n
to limit loss probability to less than 1 in a million:
0
W=
1 µ −λ
Empirical Example
Example
n
Measurement of a network gateway:
n n
mean arrival rate (λ): 125 Packets/s mean response time per packet: 2 ms What is the service rate, µ ? What is the gateway’s utilization? What is the probability of n packets in the gateway? mean number of packets in the gateway? The number of buffers so P(overflow) is <10-6?

lecture 06 consumer theory

• Points A and B should make the consumer equally happy. • Points B and C should make the consumer equally happy. • This implies that A and C would make the consumer equally happy. • But C has more of both goods compared to A.
Figure 2 The Consumer’s Preferences
Quantity of Pepsi
Indifference curve, I1 0 Quantity of Pizza
Four Properties of Indifference Curves • Property 3: Indifference curves do not cross.
Figure 2 The Consumer’s Preferences
Quantity of Pepsi C
B
D I2 A Indifference curve, I1 Quantity of Pizza
0
Representing Preferences with Indifference Curves • The Consumer’s Preferences
250
C Consumer’s budget constraint
A 0 50 100 Quantity of Pizza
THE BUDGET CONSTRAINT: WHAT THE CONSUMER CAN AFFORD
• The slope of the budget constraint line equals the relative price of the two goods, that is, the price of one good compared to the price of the other. • It measures the rate at which the consumer can trade one good for the other.

排队论上课 !

（2）
与时间有关的随机变量的概率，是一个随机过程，即泊松过程。 6
在一定的假设条件下一个泊松过程。
顾客的到达过程就是
若设N(t)表示在时间区间[0,t)内到达的顾客数 (t>0),Pn(t1,t2) 表示在时间区间 [t1,t2)(t2>t1) 内有 n(≥0)个顾客到达的概率。即：
第五章排队论(Queuing Theory)
排队论（queuing),也称随机服务系统理论，是运筹学的一个主要分支。 1909年，丹麦哥本哈根电子公司电话工程师A. K. Erlang的开创性论文“概率论和电话通讯理论” 标志此理论的诞生。排队论的发展最早是与电话，通信中的问题相联系的，并到现在是排队论的传统的应用领域。近年来在计算机通讯网络系统、交通运输、医疗卫生系统、库存管理、作战指挥等各领域中均得到应用。
k0
2
1
1 1 1 0.9197 2
∴ 破碎物品少于3件的概率为91.97 破碎物品多于3件的概率为:
p 3 1 e k!
3 k k0

1 0.98 0.02
3.至少有一件破碎的概率为
P{k1}=1-(1k/k!)e-=1-(10/0!)e-1=0.632
令f(t|n+1)表示wn的概率密度，这是在系统中已有n个顾客时的条件概率，故T的概率密度为:
f (t ) pn f (t | n 1)
n 0

15
因此wn服从爱尔朗分布
f (t | n 1)
(nt ) n e t
n!
n t
所以：
f (t ) (1 )
系统处于空闲状态的概率： P0 1

queues_NG

Assumption 1 The customer inter-arrival times, i.e. the time between arrivals, are independent and identically distributed (usually written as “iid”). Thereby the arrival process is what is called a renewal process. All arriving customers enter the system if there is room to wait. Also all customers wait till their service is completed in order to depart.
9.2. QUEUEING THEORY BASICS
3
and multi-class queues. Most of the notation used in this chapter and the fundamental results are from Kulkarni [7]. If one is interested in a single site with information about various aspects of queues (including Humor!), the place to visit is the page maintained by Hlynka [4]. In fact the page among other things illustrates various books on queueing, course notes, and, a list of software. A few software tools would be pointed out in this chapter, but it would be an excellent idea to visit Hlynka’s site [5] for an up-to-date list of queueing software. In there, software that run on various other applications (such as MATLAB, Mathematica, Excel, etc.) are explained and the most suitable one for the reader can be adopted. This chapter is organized as follows. First some basic results that are used throughout the chapter are explained in Section 9.2 on queueing theory basics. The bulk of this chapter is Section 9.3 which lays the foundation for the other types of systems by initially considering the single-station and single-class queue. Then in Section 9.4, the results for single-station and multiple classes are presented. Following that, the chapter moves from analyzing a single station to a network of queues in Section 9.5 where only one class is considered. This is extended to the most general form (of which all the previous models are special cases) of multi-station and multi-class queue in Section 9.6. Finally some concluding remarks are made in Section 9.7.

运筹学排队论2

现将上式参数引入时间因素 t ,即将
换为 t ,得到
pn
(t)
(t)n
n!
et
,
t
0,
n
0,1,2,.
表示长为t的时间区间内到达n个顾客的概率为 pn (t) ,且服从泊松分布.这称为泊松流或泊松过程或简单流. 设t时间内到达的顾客数为随机变量N(t),则有
E[N(t)] t, D[N(t)] t.
服务台
2.C个服务台,一个公共队伍
服务台1 服务台2 服务台C
3.C个服务台,C个队伍
服务台1 服务台2 服务台C
二.排队系统的三个组成部分
1.输入过程:指顾客按怎样的规律到达. ⑴顾客的总体数或顾客源:指可能到达服务机
构的顾客总数.顾客总体数可以是有限的,也可以是无限的; ⑵顾客到达的类型:顾客是单个到达还是成批到达; ⑶顾客相继到达时间间隔的分布,如按泊松分布,定长分布还是负指数分布.
排队论的创始人是丹麦哥本哈根市电话局的工程师爱尔朗(A.K.Erlang),他早期研究电话理论,特别是电话的占线问题,就是早期排队论的内容.
§2 排队论的基本概念
一.排队现象的共同特征:为了获得某种服务而到达的顾客,如不能立即得到服务而又允许排队等候,则加入等待的队伍,获得服务后离开.我们把包含这些特征的系统称为排队系统. 排队系统的几种情况: 1.单服务台排队系统
例9.1 某仓库全天都可以进行发料业务，假设顾客到达的时间间隔服从均值为1的负指数分布现在有一位顾客正好中午12：00到达领料，试求：
（1）下一个顾客将在下午1：00前到达的概率；（2）在下午1：00与2：00之间到达的概率：（3）在下午2：00以后到达的概率。

congruence theory

congruence theoryCongruence Theory is a branch of mathematics and sociology which deals with the study of relationships between individuals and groups. It is based on the idea that congruence, or compatibility, is a key factor in the success of relationships. Congruence Theory was first introduced by psychologists Carl Rogers and John G. Renton in the 1950s. It has since been developed and applied in many other fields such as business, education and health.Congruence Theory states that when two individuals or groups are in an interpersonal relationship, there must be a level of congruence between them to ensure it is successful. It states that for a relationship to be effective, the individuals or groups must have a shared understanding of their needs, values, attitudes, and beliefs. It also states that when individuals or groups are congruent, they are better able to communicate with each other, leading to a greater understanding and acceptance of each other.In business, congruence theory is used to assess the compatibility of two or more people in a team. It is used to determine whether a team has the perfect mix of personalities, strengths, and skills in order to successfully achieve the taskat hand. Similarly, in education, congruence theory is used to assess the degree of compatibility between a student and teacher to ensure that a student is learning the correct material in the most effective way. In health, congruence theory is used to assess the compatibility between a patient and a physician, taking into account the patient's individual needs and values.Congruence Theory has been widely applied and studied in many fields. It is a critical concept in the study of relationships, as it provides an evidence-based framework for understanding and improving the dynamics of relationships. It is also useful in problem-solving, as it can be used to identify where problems may arise and how relationships can be improved. As such, Congruence Theory is a key concept in understanding and improving interpersonal relationships.。

QueerTheory,AnIntroduction（AnnamarieJagose）

Queer Theory, An IntroductionAnnamarie JagoseIndeed, as an intellectual model, queer has not been produced solely by lesbian and gay politics and theory, but rather informed by historically specific knowledges which constitute late twentieth-century western thought. Similar shifts can be seen in both feminist and post-colonial theory and practice when, for example, Denise Riley (1988) problematises feminism‟s insistence on “women” as a unified, stable and coherent category, and Henry Louis Gates (1985) d enaturalises “race.” Such conceptual shifts have had great impact within lesbian and gay scholarship and activism and are the historical context for any analysis of queer.Both the lesbian and gay movements were committed fundamentally to the notion of identity politics in assuming identity as the necessary prerequisite for effective political intervention. Queer, on the other hand, exemplifies a more mediated relation to categories of identification. Access to the post-structuralist theorisation of identity as provisional and contingent, coupled with a growing awareness of the limitations of identity categories in terms of political representation, enabled queer to emerge as a new form of personal identification and political organisation. “Identity” is pro bably one of the most naturalised cultural categories each of us inhabits: one always thinks of one‟s self as existing outside all representational frames, and as somehow marking a point of undeniable realness. In the second half of the twentieth century, however, such seemingly self-evident or logical claims to identity have been problematised radically on a number of fronts by such theorists as Louis Althusser, Sigmund Freud, Ferdinand de Saussure, Jacques Lacan and Michel Foucault. Collectively, their work has made possible certain advances in social theory and the human sciences which, in the words of Stuart Hall, have effected “the final de-centring of the Cartesian subject.” Consequently, identity has been reconceptualised as a sustaining and persisten t cultural fantasy or myth. To think of identity as a “mythological” construction is not to say that categories of identity have no material effect. Rather it is to realise--as Roland Barthes does in his Mythologies (1978)--that our understanding of ourselves as coherent, unified, and self-determining subjects is an effect of those representational codes commonly used to describe the self and through which, consequently, identity comes to be understood. Barthes‟s understanding of subjectivity questions that seemingly natural or self-evident “truth” of identity which derives historically from RenèDescartes‟ notion of the self as something that is self-determining, rational and coherent.Reconsidering Karl Marx‟s emphasis on the framework of constraints or hi storical conditions which determine an individual‟s actions, Louis Althusser has argued that we do not pre-exist as free subjects: on the contrary, we are constituted as such by ideology. His central thesis is that individuals are “interpellated” or “called forth” as subjects by ideology, and that interpellation is achieved through a compelling mixture of recognition and identification. This notion is important for any thorough examination of identity politics, because it demonstrates how ideology not only positions individuals in society but also confers on them their sense of identity. In other words, it shows how one‟s identity is already constituted by ideology itself rather than simply by resistance to it.Like the Marxist structuralist approach to subjectivity, psychoanalysis makes culturally available a narrative that complicates the assumption that an identity is the natural property of any individual. Sigmund Freud‟s theorisation of the unconscious further challenges the notion that subjectivity is s table and coherent. In establishing the formative influence of important mental and psychic processes of which an individual is unaware, the theory of the unconscious has radical implications for the common-sense assumption that the subject is both whole and self-knowing. Furthermore, interpretations of Friud‟swork--particularly by the French psychoanalyst, Jacques Lacan-- establish subjectivity as something which must be learned, rather than as something which is always already there. Subjectivity is not an essential property of the self, but something which originates outside it. Identity, then, is an effect of identification with and against others: being ongoing, and always incomplete, it is a process rather than a property.In some influential lectures on structural linguistics which he delivered in 1906-11, Ferdinand de Saussure argues that language does not so much reflect as construct social reality. For Saussure, language is not some second-order system whose function is simply to describe what is already there. Rather, language constitutes and makes significant that which it seems only to describe. Moreover, Saussure defines language as a system of signification that precedes any individual speaker. Language is commonly misunderstood as the medium b y which we express our “authentic” selves, and our private thoughts and emotions. Saussure, however, asks us to consider that our notions of a private, personal and interior self is something constituted through language.The theories of Althusser, Freud, Lacan and Saussure provide the post-structrualist context in which queer emerges. The French historian Michel Foucault has been more explicitly engaged in denaturalising dominant understandings of sexual identity. In emphasising that sexuality is not an essentially personal attribute but an available cultural category--and that it is the effect of power rather than simply its object--Foucault‟s writings have been crucially significant for the development of lesbian and gay and, subsequently, queer activism and scholarship. To say this is not to claim that there is literally a causal connection between Foucault‟s work and queer practice and theory. Yet, as Diana Fuss observes, Foucault‟s work on sexuality resonates with “current disputes amongst gay theorists and activists over the meaning and applicability of such categories as …gay‟, …lesbian‟, and …homosexual‟ in a post-structuralist climate which renders all such assertions of identity problematic”.Foucault‟s argument that sexuality is a discursive produc tion rather than a natural condition is part of his larger contention that modern subjectivity is an effect of networks of power. Not only negative or repressive but also productive and enabling, power is “exercised from innumerable points” to no predetermined effect (Foucault, 1981). Against the popular concept that sex both exists beyond power relations and yet is repressed by them, Foucault (1979) argues that power is not primarily a repressive force:In defining the effects of power by repression, one accepts a purely juridical conception of the power; one identifies power with a law that says no; it has above all the force of an interdict. Now, I believe that this is a wholly negative, narrow and skeletal conception of power which has been curiously shared. If power was never anything but repressive, if it never did anything but say no, do you really believe that we should manage to obey it? What gives power its hold, what makes it accepted, is quite simply the fact that it does not simply weigh like a force which says no, but that it runs through, and it produces things, it induces pleasure, it forms knowledge, it produces discourse; it must be considered as a productive network which runs through the entire social body much more than as a negative instance whose function is repression.In Foucault‟s analysis, marginalised sexual identities are not simply victims of the operations of power. On the contrary, they are produced by those same operations: “For two centuries now, the discourse on sex has been multiplied rather than rarefied; and if it has carried with it taboos and prohibitions, it has also, in a more fundamental way, ensured the solidification and implantation of an entire sexual mosaic” (Foucault, 1981). This emphasis on the productive and enabling aspects of power profoundly alters the models by which traditionally it has been understood. Consequently, Foucault‟s revaluation of power has significantly affected much lesbian and gay analysis.Since he does not think that power is a fundamentally repressive force, Foucault does not endorse such liberationist strategies as breaking prohibitions and speaking out. Indeed, because the idea of modern sexual repression is widely accepted, Foucault speculates that the discursive critique of oppression, far from correctly identifying the mechanisms of power, “is …in fact part of the same historical network as the thing it denounces (and doubtless misrepresents) by calling it …repression.‟” Foucault questions the liberationist confidence that to voice previously denied and silenced lesbian and gay identities and sexualities is to defy power, and hence induce a transformative effect. As Foucault takes a resolutely anti-liberatory position on this matter he is sometimes read--perhaps unsurprisingly given the common currency of what he critiques as “the repressive hypothesis”--as advocating political defeatism.Yet Foucault also argues that “where there is power, there is resistance,” a resistance “coextensive with [power] and absolutely its contemporary” (Fouca ult, 1988). Like power, resistance is multiple and unstable; it coagulates at certain points, is dispersed across others, and circulates in discourse. “Discourse” is the heterogeneous collection of utterances that relate to a particular concept, and thereby constitute and contest its meaning--that “series of discontinuous segments whose tactical function is neither uniform nor stable.” Just as he cautions against thinking that power demarcates only hierarchical relations, so Foucault insists that discourse is not simply for or against anything, but endlessly prolific and multivalent: “we must not imagine a world of discourse divided between accepted discourse and excluded discourse, or between the dominant discourse and the dominated one; but as a multiplicity of discursive elements that can come into play in various strategies.”When describing the relation between discourses and strategies, and demonstrating how a single discourse can be used strategically for oppositional purposes, Foucault specifically instances how the category of homosexuality was formed in relation to structures of power and resistance. The rise of the homosexual as a “species” exemplifies the polyvalent capacities of discourse:There is no question that the appearance in nineteenth-century psychiatry, jurisprudence, and literature of a whole series of discourses on the species and subspecies of homosexuality, inversion, pederasty, and “psychic hermaphrodism” made possible a strong advance of social controls into this area of “perversity;” but it also made possible the formation of a “reverse” discourse: homosexuality began to speak in its own behalf, to demand that its legitimacy or “naturality” be acknowledged, often in the same vocabulary, using the same categories by which it was medically disqualified.Discourse, then, is entirely within (yet not necessarily in the service of) the mechanisms of power. Foucault‟s analysis focuses on discourse as a mode of resistance, not to contest its content but in order to particularise its strategic operations. In so far as homosexuality is one of his key examples, Foucault regards sexual identities as the discursive effects of available cultural categories. Challenging commonly held understandings of power and resistance, his work has obvious appeal for lesbian and gay--and subsequently queer--theory and practice. Although Foucault treats the “author” as a textual effect rather than a real presence, his public identity as a gay man may well have facilitated the gay studies inspired by his work..。

排队论简述

0.9 1 2 3 0.3(人 / min) 3
八、M/M/S等待制排队模型
• 下表给出了M/M/3/∞和3个M/M/1/∞的比较：
项目空闲的概率顾客必须等待的概率平均队长平均排队长 M/M/3/∞ 0.0748 0.57 3.95 1.70 3个M/M/1/∞ 0.25（每个子系统） 0.75 9（整个系统） 2.25（每个子系统）
二、排队系统模型的基本组成部分
• 排队现象是由两个方面构成，一方要求得到服务，另一方设法给予服务。我们把要求得到服务的人或物（设备）统称为顾客，给予服务的服务人员或服务机构统称为服务员或服务台。顾客与服务台就构成一个排队系统，或称为随机服务系统。显然缺少顾客或服务台任何一方都不会形成排队系统。 • 对于任何一个排队服务系统，每一名顾客通过排队服务系统总要经过如下过程：顾客到达、排队等待、接受服务和离去，其过程如下图所示: 顾客总体输入队伍服务台输出服务系统
五、描述排队系统的主要数量指标

4.根据排队系统对应的理论模型求用以判断系统运行优劣的基本数量指标的概率分布或特征数。平均队长（Ls）:指系统内顾客数（包括正被服务的顾客与排队等待服务的顾客）的数学期望。平均队列长（Lq）:指系统内等待服务的顾客数的数学期望。平均逗留时间（Ws）:顾客在系统内逗留时间（包括排队等待的时间和接受服务的时间）的数学期望平均等待时间（Wq）:指一个顾客在排队系统中排队等待时间的数学期望间忙期（Tb）：指服务机构连续繁忙时间（顾客到达空闲服务机构起，到服务机构再次空闲止的时间）长度的数学期望
Ls Lq ,
Lq

,
Ws Wq
1

Queueing Theory Basics

Queueing Theory Basics Goal:make an analytical model of customers needing service,and use that model to predict queue lengths and waiting times.May not be accurate for real situations,but we get in-sights needed for studying networks.TerminologyCustomers—independent entities that arrive at random times to a Server and wait for some kind of service,then leave.Server—can only service one customer at a time;length of time to provide service depends on type of service;cus-tomers are served in FIFO order.Time—real,continuous,time.Queue—customers that have arrived at server but are waiting for their service to start are in the queue.Queue Length at time t—number of customers in the queue at time t.Waiting Time—for a given customer,how long that customer has to wait between arriving at the server and when the server actually starts the service(total time is waiting time plus service time).Illustration of customers,queue and server.a9a8a1is being serveda2–a4are in queuea5–a9haven’t yet arrived at server NotationT i is arrival time(at server)of customer a i0T1T2T3∆i is interarrival time deﬁned by∆i T i T i1 S i is service time for customer a iM/M/1QueuesIn order to make analysis possible,some assumptions about in-terarrival and service times.1.The number of arrivals at server in any time interval of lengthτis Poisson distributed with parameterλτ.P A tτA t n eλτλτnµn1λQuestions1.What ifλµ?2.What is expected queue length?Arrival Rate versus Interarrival RateLet A t be the number of arrivals at the server from time zero up to time t.An intuitive deﬁnition of the arrival rateλisλlimt∞A tτ1τ2τkThe average interarrival time(time between arrivals)up to the time when the k-th customer arrives isτ1τ2τk∑k i1τk limk∞1λ1shows that the mean interarrival rate isλ1.If arrival distribution is Poisson,then why is interarrival distri-bution exponential?Actually it’s the other way around:if we start by as-suming that interarrival rates are exponential,then Pois-son distribution of arrival rates can be proved(using re-peated convolution).Mean queue length N is:λNµ1µLittle’s TheoremNλTwhere N is the mean queue length,T is the mean total delay(in-cluding both queuing time and service time),andλis the(Pois-son parameter)arrival rate.IntuitionIf average customer spends time T in system,then aboutλT customers are waiting behind,because arrival rateisλ.QuestionCustomers arrive on average on every two minutes at afast food restaurant.The mean time spent in the restau-rant per customer is20minutes(waiting in line,pay-ing,eating).On average,how many customers are inthe restaurant?Other Types of Queueing ModelsM M m—exponential arrival rate and service times,with m servers(like grocery store with many checkout lanes).M M m m—exponential arrival rate and service times,with m servers,but nobody waits in queue(if all m servers are busy when a customer arrives,that customer gives up and leaves).M M∞—exponential arrival rate and service times,with un-limited number of servers(customers never wait in queue). M D1—service times are deterministic(e.g.a constant,ﬁxed service time regardless of customer).M G1—exponential arrival rate,but service rate has a“gen-eral”(arbitrary)probability distribution,and a single server. M G m—same as above,but with m servers.For each of the above models,the questions are the same:what is mean waiting time,what is mean total time in system,what is mean queue length?How are these factors related?The P-K Formula for M G1The“G”in M G1refers to a general distribution of service times.Let X i be the service time for the i-th arriving customer. Suppose these times X1,X2,...are independent,identitically dis-tributed,and independent of the interarrival times.1E XλE XµThe Pollaczek-Khinchin formula is:λE X2Wµ1ρWhat is P-K formula for M D1?ρWNetworks of Queues We can also model a closed system of cus-tomers that travel from one server to another.A typical example would be a set of“jobs”(the cus-tomers)in a computer system,with a disk drive(oneserver),a CPU(a second server),and a network inter-face(a third server).Using queueing models,one canpredict where bottlenecks occur,if enough is knownabout distributions of service times.11。

queer theory

An Overview/History of Queer Theory:What is Queer Theory?The word 'queer' according to the Merriam-Webster dictionary, is "differing in some odd way from what is usual or normal." The word 'theory' is defined as "the analysis of a set of facts in their relation to one another." Putting the two words together, we can conclude that Queer Theory is the study of the informal norm. Queer theory is not just the study of gays or lesbians, but also the study of transgender, hermaphroditism, and any other sexual orientation that goes against society's formal sexual norms.Queer Theory uprooted from the studies of Feminism. Feminists viewed gender according to the website as a "social construct; something designed and implemented and perpetuated by social organizations and structures, rather than something merely "true," something innate to the ways bodies worked on a biological level." This meaning that gender is portrayed throughout society. Queer Theory took the studies of feminism a little further and began looking at gender in a very broad view.According to the online Science Encyclopedia, the term queer theory came about in 1991 when Teresa de Lauretis edited a feminist studies journal entitled "Queer Theory: Lesbian and Gay Sexualities." She defines queer theory as containing three main points: "a refusal of heterosexuality as the benchmark for all sexual formations; an attentiveness to gender capable of interrogating the frequent assumptions that lesbian and gay studies is a single homogeneous object; and an insistence on the multiple ways in which race crucially shapes sexual subjectives." This quote means that the word "queer" is anything but ordinary. Queer Theory is a study in which traditonal sexual norms are rejected. The studies introduce the fact that there are more than just two types of sexuality.Scholars within the Queer Theory:Michel Foucault is a key scholar that deals with queer theory. His theory is that “sexuality is a discursive production, rather than an essential human attribute”. He believes that sexuality has been repressed since the Western society since the 17th century and also that sexuality is something hard to talk about. Michel sees sexuality in two different ways, one way is called “erotic art” indentified in Japan, China, as well as in India. The other way is called “science of sexuality” identified in the Western society. Foucault identified four major themes that kept occurring within sexuality. The four themes were: 1) Body of women were sexualized from the role of the child bearer. 2) Children should be banned from all dangers about sexuality including masturbation and other sexuality’s. 3) Sexuality is important for the role in reproduction. 4) Adults sexuality can become a danger in forms of perverse actions. Michel thought that it would be best if not to get rid of these themes but to embrace the health and procreation of it./crimtheory/foucault.htmEve Sedgwick wa s one of the founders of the queer theory in the early 90’s. In her most influential book “Epistemology of the Closet” she discussed the social meanings and violent force fields that were created by the hectic crisis of the homosexual and heterosexual definition. She was the leader of a debate that was held on whether sexual identity is inherent or socially constructed in 1990. Another one of Sedgwick’s important essays was “How to Bring Your Kids Up Gay” appearing in 1991 then reprinted by the Duke University Press in 1993 and then again in Routledge shortly after that./article/Eve-Kosofsky-Sedgwick/15045/Judith Butler is yet another theorist who analyzed effects of dominant understandings of sex and gender. Butler argues, “gender, like sexuality is not an essential truth derived from the body’s materiality but rather a regulatory fiction”. Butler looks within the cultural work and looks at the representation of gender and the natural expression of the body. She thinks that gender performativity is a “strategy of resistance” which includes drag and cross-dressing. Butler’s most influential book was “Gender Trouble” where she discussed that women were not just a gro up with common interests. In the book, she also talked about gender relations. Butler is known for her theory that sex should be between a man and a women so it can cause masculine and feminine which would cause desire for the other gender./ctr-butl.htmAnalyzed TextsLady Gaga's song, Pokerface, is about a female initiated sexual game of cat and mouse. Traditionally the male is expected to take the lead when it comes to sexual relations, however; as referenced in the line 'And after he's been hooked I'll play the one that's on his heart,' Gaga implies that she's the one skilled in making the first move.Notable Readings/Links/dept/english/courses/60a/handouts/queer1.html/ctb-quee.htm/pss/3175040/tandf/queer-theory-11-tf/Resources Used/Works CitedKlages, Mary. Queer Theory. University of Colorado at Boulder,1997. Web. 4 March 2011. <>Definition: The Queer Theory/Sexuality Studies is a very inclusive theory that has gained in prominence since the 1980's. Where, at first, the study was originally thought to simply represent homosexual instances, it has been expanded to the point of "queer", where all sorts of odd ball expressions that are outside of society's norm are file. Something can be sexually or socially queer, even in a heterosexual context (Stein, Plummer, 182). The 'Queer Theory' has a very broad scope that refuses to be tied down to the all to simple thought of homosexuality.Example: One might examine Hardy's novel 'Jude the Obscure' with aspects of the Queer Theory. Jude was a rather modest boy who let the women in his life completely dominate his situations. One might purpose that Jude is in fact a homosexual, restrained by the society of his time, playing a submissive, feminine role. Likewise, we could suggest that his dominant lover, Arabella, is playing a masculine role and could also be see through the lens of the Queer Theory."Definition": The queer theory is a theory that was brought about in the early 1970s and is a theory that deals mainly with “queer” readings. At first many thought that it was only about sexual identities and homosexuality. Now it is de fined better as “a set of ideas based around the idea that identities are not fixed and do not determine who we are” (Gauntlett). It is now not only about homosexuality but about different and unusual identities. Many others would say that it is something different all together, this makes the queer theory very versatile."Example": An example of the queer theory is Madonna. She is constantly changing and people often think she is unusual and strange. This just goes to show that her identity is not “fixed” by what she wears or says and does not make her who she is."Definition": The definition of Queer Theory has changed throughout the past decades. In the past, it described the study of homosexual behavior, but today the idea could be extended to unusual actions or behavior in general. Sexual studies, however, are exclusively concerned with the study of sexual behavior and its social connotations."Example": An example of queer theory could be the fascination with the life of Michael Jackson. Michael had a multipolar personality and was often misunderstood. With each passing year, Michael's physical appearance changed, along with his fame and infamy. People watched him just to see what outrageous things he would do next.-The queer theory is more than just homosexual behavior but something that would cover any range of sexual orientations. Theorists evaluate the way the different sexual oriented people would act when they are alone or in contact with one another. They would study the different ways the characteristics affect the personality and actions of a person. It ranges to study any normal or unique behavior determined by sexual behavior. A study could include how a heterosexual girl acts differently around her homosexual male friend compared to the heterosexual male she likes./books?id=U4Qv1j6d03cC&pg=PA20&lpg=PA20&dq=Queer+theory ++clock+without+hands&source=bl&ots=BCT9eKXi2T&sig=slJD3RTVpls02aHs3Q72v_bkEtI& hl=zh-CN&ei=srhQTonVOsWGrAfZ4OCsAg&sa=X&oi=book_result&ct=result&resnum=5&ve d=0CDoQ6AEwBA#v=onepage&q=Queer%20theory%20%20clock%20without%20hands&f=fal se/books?id=U4Qv1j6d03cC&pg=PA20&lpg=PA20&dq=Queer+theory ++clock+without+hands&source=bl&ots=BCT9eKXi2T&sig=slJD3RTVpls02aHs3Q72v_bkEtI& hl=zh-CN&ei=srhQTonVOsWGrAfZ4OCsAg&sa=X&oi=book_result&ct=result&resnum=5&ve d=0CDoQ6AEwBA#v=onepage&q=Queer%20theory%20%20clock%20without%20hands&f=tru e。

第六章排队论

23
例-1 一售货员出售两种商品A和B，每日工作 8 小时。购买每种商品的顾客到达过程为泊松分布，到达率分别为 A=8人/日， B=16人/日，试求：(1) 1小时内来到顾客总数为 3 人的概率；(2) 三个顾客全是购买B类商品的概率。解：(1)总到达率为 A+ B=24人/日，1 小时=1/8 日，故
e
8 1 / 8
例-2
某铁路与公路相交的平面交叉口，当火车通过交叉口时，横木护栏挡住汽车通行。每次火车通过时，平均封锁公路3min，公路上平均每分钟有4辆汽车到达交叉口。求火车通过交叉口时，汽车排队长度超过100m的概率（即排队汽车超过12辆的概率）。
25
Homework
P186
( t t0 )

P h t t 0 P h t 0 P h t 0
t0

1 e
(1 e
t0
)
1e
t
Q.E.D
1 (1 e
)
21
6.3.3 小结
• 如果顾客的到达过程服从最简单流，则顾客单位时间内的到达数服从泊松分布。 • 如果顾客的到达过程服从最简单流，则顾客到达的时间间隔服从负指数分布。 • 从本质上看，泊松分布与负指数分布是同一个过程的不同表现形式。 • 可适用于服务时间分布
– 间隔时间服从爱尔朗分布（Erlang distribution ） – 二项分布（binomial distribution ） – 单位时间 t （或时间区间△t）内到达的顾客数服从泊松分布(法国数学家Poisson, 1836)—最简单流（泊松流）（Poisson Distribution） – 负指数分布（Negative Exponential Distribution）

14排队论

32
例（p310 (321)）某储蓄所只有一个服务窗口。根据统计分析，顾客的到达过程服从泊松分布，平均每小时到达顾客 36人；储蓄所的服务时间服从负指数分布，平均每小时能处理 48 位顾客的业务。试求这个排队系统的数量指标。解：已知平均到达率 = 36/60 = 0.6，平均服务率 = 48/60 = 0.8。
时间控制在一定的限度内，在服务质量的提高和成本的降低之间取得平衡，找到最适当的解。排队论就是解决这类问题的一门科学，它被广泛地应用于解决诸如电话局的占线问题，车站、码头、
机场等交通枢纽的堵塞与疏导，故障机器的停机待修，
水库的存贮调节等有形无形的排队现象的问题。
5
排队论模型是由一些数学公式和它们相应之间的关系所组成，这些数学公式使我们可以
13
顾客到达过程：顾客的到达时间是随机的。
本教材主要考虑顾客的到达服从泊松分布的排队问题。例这个储蓄所根据统计分析得知顾客的到达过程服从泊松分布，并且平均每小时到达顾客 36人，即平均每分钟到达的顾客人数为 36/ 60 = 0.6。若把时间单
位定为分钟，则平均到达率 = 0.6 ，每分钟有 x 个
第十四章排队论(Queuing Theory)
1. 排队过程的组成部分 2. 单服务台泊松到达、负指数服务时间的排队模型 3. 多服务台泊松到达、负指数服务时间的排队模型 4. 排队系统的经济分析
5. 单服务台泊松到达、任意服务时间的排队模型
6. 单服务台泊松到达、定长服务时间的排队模型
1
7. 多服务台泊松到达、任意的服务时间、损失制排队模型 8. 顾客来源有限制排队模型 9. 单服务台泊松到达、负指数服务时间、系统容量有限制的排队模型 10.多服务台泊松到达、负指数服务时间、系统容量有限制的排队模型 11. 生灭过程及生灭过程排队系统

MicroeconomicTheoryII

Econ 752Microeconomic Theory IIProfessor: Douglas NelsonOffice: Tilton 108 (Murphy Institute), Phone: 865-5317Office Hours: Tuesday and Thursday, 3:30-5:30Phone: 865-5317email: ******************Webpage: /~dnelson/This course provides an overview of equilibrium analysis for competitive markets. The course is organized in four sections. An introductory section illustrates the main themes of the course in simple partial and general equilibrium environments. The second part of the course develops the main positive results from abstract general equilibrium theory. The third and fourth part of the course introduces students to the analysis of general equilibrium systems. Specifically, part III introduces positive analysis in terms of comparative statics, while part IV introduces students to welfare economics.Evaluation: Your performance in this course will be evaluated on the basis of two examinations (worth 100 points each). All students are expected to do all the expected reading and actively participate in all classes.Readings and exercises for the course will be drawn from the following core texts: Andreu Mas-Colell, Michael Whinston, and Jerry Green (1995). Microeconomic Theory.New York: Oxford University Press. [MWG]Hal Varian (1992). Microeconomic Analysis. New York: Norton. [Varian]Eugene Silberberg and Wing Suen (2001). The Structure of Economics: A Mathematical Analysis. Boston: Irwin/McGraw Hill. [Silberberg and Suen]Alan Woodland (1982). International Trade and Resource Allocation. Amsterdam: North Holland.Gareth Myles (1995). Public Economics. Cambridge: Cambridge University Press.In addition, there will be a large number of articles available electronically.The main substantive material of this course has been covered in a number of excellent texts. On pure general equilibrium theory, at a relatively elementary level the following are excellent: Peter Newman (1965). The Theory of Exchange. Englewood Cliffs: Prentice-Hall.James Quirk and Rubin Saposnik (1968). Introduction to General Equilibrium Theoryand Welfare Economics. New York: McGraw Hill.Werner Hildenbrand and Alan Kirman (1988). Equilibrium Analysis: Variations onThemes by Edgeworth and Walras. Amsterdam: North-Holland.Ross Starr (1997). General Equilibrium Theory: An Introduction. Cambridge: CUP.Bryan Ellickson (1993). Competitive Equilibrium: Theory and Applications. Cambridge: CUP.Alan Kirman, ed. (1998). Elements of General Equilibrium Analysis. Oxford: Blackwell. At a more advanced level, the following are excellent:Kenneth Arrow and Frank Hahn (1971). General Competitive Analysis. Amsterdam:North-Holland.Lionel McKenzie (2002). Classical General Equilibrium Theory. Cambridge: MIT Press.Andreu Mas-Colell (1985). The Theory of General Equilibrium: A DifferentiableApproach. Cambridge: CUP/Econometric Society.Yves Balasko (1988). Foundations of the Theory of General Equilibrium. San Diego:Academic Press.C. Aliprantis,D. Brown, and O. Burkinshaw (1990). Existence and Optimality ofCompetitive Equilibrium. Berlin: Springer-Verlag.On the application to public economics, texts emphasizing modern general equilibrium methods include:David Starrett (1988). Foundations of Public Economics. Cambridge: CambridgeUniversity Press.Jean-Jacques Laffont (1988). Fundamentals of Public Economics. Cambridge: MIT Press.Roger Guesnerie (1995). A Contribution to the Pure Theory of Taxation. Cambridge:Cambridge University Press.On the application to trade:Avinash Dixit and Victor Norman (1980). Theory of International Trade. Cambridge:Cambridge University Press.Kar-yiu Wong (1995). International Trade in Goods and Factor Mobility. Cambridge:MIT Press.Those interested in computational methods of general equilibrium analysis may want to consult: John Shoven and John Whalley (1992). Applying General Equilibrium. Cambridge:Cambridge University Press.Victor Ginsburgh and Michiel Keyzer (1997). The Structure of Applied GeneralEquilibrium Models. Cambridge: MIT Press.Joseph Francois and Kenneth Reinert, eds. (1997). Applied Methods for Trade PolicyAnalysis: A Handbook. Cambridge: Cambridge University Press.Finally, for those with an interest in the historical and philosophical background to general equilibrium theory, the place to start is a series of excellent books by E. Roy Weintraub:E.R. Weintraub (1979). Microfoundations. Cambridge: CUP.E.R. Weintraub (1986). General Equilibrium Analysis: Essays in Appraisal. Cambridge:CUP.E.R. Weintraub (1991). Stabilizing Dynamics: Constructing Economic Knowledge.Cambridge: CUP.E.R. Weintraub (2002). How Economics Became a Mathematical Science. Durham: DukeUniversity Press.Examination format. Both exams will be made up of problems drawn from material covered in the lectures and reading. These problems will generally be in the nature of extensions of that material, not simply replication of the relevant content. Exams must be written in blue books, which you must supply.Policy on examinations. The midterm exam will be given on tba. Unless you have a standard university accepted excuse for missing the exam (e.g. health with standard university form), you must take the exams at their scheduled time. The final examination will only be given on the scheduled date: tba (there will be no exceptions so do not make travel plans that conflict with this).Policy on examinations. The midterm exam will be given on 9 March. Unless you have a standard university accepted excuse for missing the exam (e.g. health with standard university form), you must take the exams at their scheduled time. The final examination will only be given on the scheduled date: 10 May, 8:00-12:00 (there will be no exceptions so do not make travel plans that conflict with this).Econ 752SYLLABUS Fall 2006 Topic I. Introduction! Partial Equilibrium Analysis of Competitive Equilibrium# Varian, Chapter 13.# MWG, Chapter 10 a-d and f.! General Equilibrium: Applying Microeconomic Tools toMacroeconomic Questions, Pure Exchange# MWG, Chapter 15, sections a-b# Varian, Chapter 17 and section 21.1.# Shapley and Shubik (1977). “An Example of a Trading Economywith Three Competitive Equilibria”. Journal of Political Economy;V.85-#4, pp. 873-875.# Debreu and Scarf (1963). “A Limit Theorem on the Core of anEconomy”. International Economic Review; V.4-#3, pp. 235-246.# Aumann (1964). “Markets with a Continuum of Traders”.Econometrica; V.32-#1/2, pp. 39-50.# Shubik (1984). “Two-Sided Markets: The Edgeworth Game”.Chapter 10 of A Game Theoretic Approach to Political Economy.Cambridge: MIT Press, pp. 252-285. [optional]# Hildenbrand and Kirman (1988). “Introduction”. In EquilibriumAnalysis. Amsterdam: North-Holland, pp. 1-49. [optional] ! General Equilibrium: Applying Microeconomic Tools toMacroeconomic Questions, Simple Economies with Production# MWG, Chapter 15, section c.# Varian, Chapter 18.# Koopmans (1957). “Allocation of Resources and the Price System”.Essay 1 of Three Essays on the State of Economic Science. NewYork: McGraw Hill, pp. 3-126. [especially pp. 1-66.]Topic II. Pure General Equilibrium Theory! Characterizing Equilibrium and Proving Existence# MWG, Chapter 17, sections a-d and f, appendix B.# Geanakoplos (2003). “Nash and Walras Equilibrium via Brouwer”.Economic Theory; V.21-#2/3, pp. 585-603.# Debreu (1998). “Existence”. Chapter 2 in A.P. Kirman, ed.,Elements of General Equilibrium Analysis. Oxford: Basil Blackwell,pp. 10-37. [optional]! Problems/Extensions: Nonconvexities# MWG, Chapter 17, section I# Chipman (1970). “External Economies of Scale and CompetitiveEquilibrium”. Quarterly Journal of Economics; V.84-#3, pp. 347-363.# Mayer (1974). “Homothetic Production Functions and the Shape of the Production Possibility Locus”. Journal of Economic Theory; V.8-#2, pp. 101-110. [ERes]# Starrett (1971). “Fundamental Non-Convexities in the Theory ofExternalities”. Journal of Economic Theory; V.4-#2, pp. 180-199.[ERes]# Cornes (1980). “External Effects: An Alternative Formulation”.European Economic Review; V.14-#3, pp. 307-321. [optional]! Problems/Extensions: Uncertainty# Varian, Chapter 20.# MWG, Chapter 19# Hens (1998). “Incomplete Markets”. Chapter 5 in A.P. Kirman, ed., Elements of General Equilibrium Analysis. Oxford: Basil Blackwell,pp. 139-210. [optional]! General Equilibrium Comparative Statics?: The Sonnenschein-Debreu-Mantel Result# MWG, Chapter 17, section d-f.# Saari (1995). “The Mathematical Complexity of SimpleEconomies”. Notices of the American Mathematical Society; V.42-#2, pp. 222-230.# Kirman (1989). “The Intrinsic Limits of Modern Economic Theory: The Emperor Has No Clothes”. Economic Journal; V.99-#395, pp.126-139.# Sonnenschein (1973). “Do Walras' Indentity and ContinuityCharacterize the Class of Community Excess Demand Functions?”.Journal of Economic Theory; V.6-#4, pp. 345-354.# Debreu (1974). “Excess Demand Functions”. Journal ofMathematical Economics; V.1-#1, pp. 15-23. [optional]# Mantel (1979). “Homothetic Preferences and Community ExcessDemand Functions”. Journal of Economic Theory; V.12-#2, pp.197-201. [optional]# Mas-Colell (1977). “On the Equilibrium Price Set of an ExchangeEconomy”. Journal of Mathematical Economics; V.4-#2, pp.117-126. [optional]# Kemp and Shimomura (2002). “The Sonnenschein-Debreu-MantelProposition and the Theory of International Trade”. Review ofInternational Economics; V.10-#4, pp. 671-679. [optional]# Brown and Matzkin (1996). “Testable Restrictions on theEquilibrium Manifold”. Econometrica; V.64-#?, pp. 1249-1262.[optional]# Nachbar (2002). “General Equilibrium Comparative Statics”.Econometrica; V.70-#5, pp. 2065-2974. [optional]Midterm: Tuesday, 9 March.No Class: Thursday, 11 MarchTopic III. Applied General Equilibrium Theory: Positive Analysis ! Introduction to Comparative Statics for Applied GE# Silberberg and Suen, Chapter 18. [ERes]# Woodland (1982). “The Production Sector”. Chapter 3 ofInternational Trade and Resource Allocation. Amsterdam: North-Holland, pp. 39-65. [ERes]# MWG, Chapter 15, section d# Jones (1965). “The Structure of Simple General EquilibriumModels”. Journal of Political Economy; V.73-#6, pp. 557-572.[optional]# Mussa (1979). “The Two Sector Model in Terms of Its Dual: AGeometric Exposition”. Journal of International Economics; V.9-#4,pp. 513-526. [optional]# Hale, Lady, Maybee, and Quirk (1999). “The CompetitiveEquilibrium: Comparative Statics”. Chapter 7 in NonparametricComparative Statics and Stability. Princeton: Princeton UniversityPress, pp. 170-205. [optional]! Maximum Value Functions and Comparative Statics for GeneralEquilibrium Analysis# MWG, Chapter 17, section g# Woodland (1982). “Comparative Statics of the Production Sector”.Chapter 4 of International Trade and Resource Allocation.Amsterdam: North-Holland, pp. 67-103. [ERes]# Woodland (1982). “Intermediate Inputs and Joint Outputs”.Chapter 5 of International Trade and Resource Allocation.Amsterdam: North-Holland, pp. 105-146. [ERes]! Applied General Equilibrium Theory: The Stolper-SamuelsonTheorem, from 2 × 2 to m × n.# Chipman (1969). “Factor Price Equalization and the Stolper-Samuelson Theorem”. International Economic Review; V.10-#3, pp.399-406.# Jones and Scheinkman (1977). “The Relevance of the Two-SectorProduction Model in Trade Theory”. Journal of Political Economy;V.85-#5, pp. 909-935.# Ethier (1982). “The General Role of Factor Intensity in theTheorems of International Trade”. Economics Letters; V.10-#3/4, pp.337-342. [ERes]# Ethier (1984). “Higher Dimensional Issues in Trade Theory”. in R.Jones and P. Kenen, eds. Handbook of International Economics--Vol.1. Amsterdam: North-Holland, 131-184. [optional]Topic IV. Welfare Economics: Pure and Applied! Fundamental Theorems of Welfare Economics# Silberberg and Suen, Chapter 19, sections 1-3. [ERes]# MWG, Chapter 16# Hammond (1998). “The Efficiency Theorems and Market Failure”.Chapter 6 in A.P. Kirman, ed., Elements of General EquilibriumAnalysis. Oxford: Basil Blackwell, pp. 211-240. [ERes]! Applied Welfare Economics,1: Introduction# MWG, Chapter 10 e.# Silberberg and Suen, Chapter 19, section 7. [ERes]# Blackorby and Donaldson (1985). “Consumers’ Surpluses andConsistent Cost-Benefit Tests”. Social Choice and Welfare; V.1-#4,pp. 251-262. [optional]# Blackorby and Donaldson (1990). “The Case Against the Use ofthe Sum of Compensating Variations in Cost-Benefit Analysis”.Canadian Journal of Economics; V.23-#3, pp. 471-494.# Blackorby and Donaldson (1999). “Market Demand Curves andDupuit-Marshall Consumers’ Surpluses: A General EquilibriumAnalysis”. Mathematical Social Sciences; V37-#2, pp. 139-163.# Ahlheim (1998). “Measures of Economic Welfare”. In Barberà,Hammond, and Seidl, eds. Handbook of Utility Theory. Dordrecht:Kluwer, pp. 483-568. [Optional: covers one person theory]! Applied Welfare Economics, 2: Commodity Taxation# Myles, Chapter 4. [ERes]# Diamond and Mirrlees (1971). “Optimal Taxation and PublicProduction, I: Production Efficiency”. American Economic Review;V.61-#1, pp. 8-27. [optional]# Diamond and Mirrlees (1971). “Optimal Taxation and PublicProduction, II: Tax Rules”. American Economic Review; V.61-#3, pp.261-278. [optional]# Diamond and McFadden (1974). “Some Uses of the ExpenditureFunction in Public Finance”. Journal of Public Economics; V.3-#1,pp. 3-21.# Greenberg and Denzau (1988). “Profit and Expenditure Functionsin Basic Public Finance: An Expository Note”. Economic Inquiry;V.26-#1, pp. 145-158.# Deaton (1981). “Optimal Taxes and the Structure of Preferences”.Econometrica; V.49-#5, pp. 1245-1260.# Stern (1986). “A Note on Commodity Taxation: The Choice ofVariable and the Slutsky, Hessian and Antonelli Matrices (SHAM)”.Review of Economic Studies; V.53-#2, pp. 293-299.! Applied Welfare Economics, 3: Distortions, Second-best, and Policy# Silberberg and Suen, Chapter 19, sections 5 and 6. [ERes]# MWG, Chapter 22, sections a-d# Myles, Chapter 10. [ERes]# Hammond (1998). “The Efficiency Theorems and Market Failure”.Chapter 6 in A.P. Kirman, ed., Elements of General EquilibriumAnalysis. Oxford: Basil Blackwell, pp. 240-260. [ERes]# Hurwicz (1999). “Revisiting Externalities”. Journal of PublicEconomics Theory; V.1-#2, pp. 225-245.! Social Choice Theory: A (Very) Brief Introduction# MWG, Chapter 21# Fleurbaey and Mongin (2005). “The News of the Death of WelfareEconomics is Greatly Exaggerated”. Social Choice & Welfare;V.25-#2/3, pp. 381-418.# Mongin and d’Aspermont (1998). “Utility Theory and Ethics”. InBarberà, Hammond, and Seidl, eds. Handbook of Utility Theory.Dordrecht: Kluwer, pp. 371-481. [optional]Final Examination: 10 May, 8:00-12:00.。

Measure-Theory

测度论测度论是研究一般集合上的测度和积分的理论。

它是勒贝格测度和勒贝格积分理论的进一步抽象和发展，又称为抽象测度论或抽象积分论，是现代分析数学中重要工具之一。

测度理论是实变函数论的基础。

定义测度理论是实变函数论的基础。

测度论所谓测度，通俗的讲就是测量几何区域的尺度。

我们知道直线上的闭区间的测度就是通常的线段长度；平面上一个闭圆盘的测度就是它的面积。

定理形成纵观勒贝格积分和勒贝格－斯蒂尔杰斯积分理论，不难发现它们都有三个基本要素。

第一，一个基本空间（即n维欧几里得空间Rη）以及这个空间的某些子集构成的集类即L（勒贝格）可测集或某L－S（勒贝格-斯蒂尔杰斯）可测集全体，这个集类对集的代数运算和极限运算封闭。

第二,一个与这个集类有关的函数类（即L可测函数或某L-S可测函数全体）。

第三,一个与上述集类有关的测度（即L测度或某L-S 测度）。

在三个要素的基础上，它们都是运用完全类似的定义和推理过程获得完全类似的一整套测度、可测函数、积分的定理（见勒贝格积分、贝尔函数）。

测度论正是基于这些基本共同点所形成一般理论一般定义对于更一般的集合，我们能不能定义测度呢？比如直线上所有有理数构成的集合，它的测度怎么衡量呢？一个简单的办法，就是先在每个有理点上找一个开区间覆盖它，就好比给它带个“帽子”。

因为有理数集是可列集（就是可以排像自然一样排好队，一个个数出来，也叫可数集，见集合论），所以我们可以让第n个有理数上盖的开区间长度是第一个有理数（比方是1）上盖的开区间长度的2^n分之一。

这样所有那些开区间的长度之和是个有限值（就是1上的开区间长度的2倍）。

现在我们让1上的开区间逐渐缩小趋向于一个点，那么所有区间的总长度也相应缩小，趋向于长度0。

这样我们就说有理数集的测度是0。

用上面这种方法定义的测度也叫外测度。

一个几何区域有了测度，我们就可以定义上面的函数的积分，这是推广的黎曼积分。

比如实数上的狄利克雷函数D(x)=1(如果x是有理数)，0（如果x是无理数）。