掌握电信交通模型：队列理论与随机过程简介

需积分: 15 109 浏览量更新于2024-08-01 收藏 739KB PDF 举报

《排队论与随机电信交通模型入门》是一本旨在向电信工程领域学生介绍基本随机模型的教材，涵盖了交通建模、资源分配和流量管理等关键研究主题，统称为电信交通。作者Moshe Zukerman强调，读者应具备一定的编程语言基础、数学知识，以及概率论和随机过程的基础，这些通常在电气工程课程中教授。本书首先会回顾那些对不具备足够强概率和随机过程背景的学生来说至关重要的概念，确保他们对后续章节的理解。作者试图通过直观和物理性的解释，来增强理论概念的学习体验。正如著名数学家Pierre-Simon Laplace所言，“概率是常识的计算化”，这表明这本书将理论与实际相结合，注重实用性和理解力的培养。在内容上，排队论作为核心部分，探讨了服务系统中的等待、服务和容量限制，如何通过随机性来理解和预测系统性能。学生将学习到诸如马尔科夫过程、泊松过程、排队模型（如M/M/1、M/M/c等）和排队网络等概念，这些都是电信网络设计和优化的重要工具。此外，书中还将涉及电信网络的负载均衡、流量控制策略，以及如何用随机过程理论分析电话交换机、互联网数据包传输等问题。通信系统中的随机特性，比如信道容量、信号衰落、干扰等，也是本书关注的重点。通过这些模型，学生们能了解到如何预测并应对通信系统的不确定性和波动性，这对于优化网络资源利用率，提高服务质量至关重要。《排队论与随机电信交通模型入门》不仅提供理论知识，还鼓励读者运用概率思维去解决实际问题，培养他们将理论知识转化为实际操作的能力。对于电信工程专业的学生而言，这是一本不可或缺的参考书，帮助他们在电信交通领域的研究和实践中取得成功。

Queueing Theory and Stochastic Teletraﬃc Models

° Moshe Zukerman 16

successes among the ﬁrst i −1 is equal to the probability of having a binomial random variable

with parameters p and i − 1 equal to k −1, for i ≥ k ≥ 1, which is equal to

i − 1

k − 1

k−1

(1 − p)

i−k

k = 1, 2, . . . , i

and since the two random variables here, namely, the Bernoulli and the Binomial are indepen-

dent (because the underlying Bernoulli trials are independent), we can multiply their probabil-

ities to obtain

P (X = 1) =

i − 1

k − 1

(1 − p)

i−k

i = k, k + 1, k + 2, . . . . (27)

1.9 Continuous Random Variables and their Probability Functions

Continuous random variable are related to cases whereby the set of possible outcomes is un-

countable. A continuous random variable X is a function that assigns a real number to outcome

of an experiment, and is characterized by the existence of a function f(·) deﬁned for all x ∈ R,

which has the property that for any set A ⊂ R,

P (X ∈ A) =

f(x)dx. (28)

Such function is the probability density function (or simply the density) of X. Since the con-

tinuous random variable X must take a value in R with probability 1, f must satisfy,

+∞

−∞

f(x)dx = 1. (29)

If we consider Eq. (28), letting A = [a, b], we obtain,

P (a ≤ x ≤ b) =

f(x)dx. (30)

An interesting point to notice is that the probability of a continuous random variable taking a

particular value is equal to zero. If we set a = b in Eq. (30), we obtain

P (x = a) =

f(x)dx = 0. (31)

As a result, the distribution function F (x) is equal to both P (X ≤ x) and to P (X < x).

Similarly, the complementary distribution function is equal to both P (X ≥ x) and to P (X > x).

By Eq. (30), we obtain

F (x) = P (X ≤ x) =

−∞

f(s)ds. (32)

Hence, the probability density function is the derivative of the distribution function.

An important concept which gives rise to a continuous version of the law of total probability

is the continuous equivalence of Eq. (12), namely, the joint distribution of continuous random

Queueing Theory and Stochastic Teletraﬃc Models

° Moshe Zukerman 18

The latter is the convolution of the densities f

(u) and f

(v).

As in the discrete case the convolution f

(y), of k densities f

), i = 1, 2, 3, . . . , k, of

random variables X

, i = 1, 2, 3, . . . , k, respectively, is given by

(y) =

, x

, ..., x

: x

, x

, ..., x

≤y

(y − Σ

i=2

)

i=2

)

. (42)

And again, in the sp ecial case where all the random variable X

, i = 1, 2, 3, . . . , k, are IID,

the density f

is the k-fold convolution of f

1.10 Selected Continuous Random Variables

We will now discuss several continuous random variables and their corresponding probability

distributions: uniform, exponential, hyper-exponential, Erlang, hypo-exponential Gaussian,

multivariate Gaussian and Pareto. These are selected because of their applicability in teletraﬃc

and related queueing models and consequently their relevance to the material in this book.

1.10.1 Uniform

The probability density function of the uniform random variable takes nonnegative values over

the interval (a, b) and is given by

f(x) =

(

b−a

if a < x < b

0 otherwise.

(43)

Of particular interest is the special case - the uniform (0,1) random variable. Its probability

density function is given by

f(x) =

(

1 if 0 < x < 1

0 otherwise.

(44)

The uniform (0,1) random variable is very important in simulations. Almost all computers

programs have a function which generates uniform (0,1) random deviates. By a simple trans-

formation such uniform (0,1) random deviates can be translated to sequence of random deviates

of any distribution as follows. Let U

(0, 1) be the ﬁrst uniform (0,1) random deviate, and let

F (x) be a distribution function of an arbitrary random variable. Set,

(0, 1) = F (x

) (45)

so x

= F

−1

(0, 1)) is the ﬁrst F (·) random deviate. Then generating the second uniform

(0,1) random deviate, the second F (·) random number is obtained in the same way, etc.

To see why this metho d works, let U be a uniform (0,1) random variable. Let F (x) be an

arbitrary cumulative distribution function. Let the random variable Y be deﬁned by: Y =

−1

(U). That is, U = F (Y ). We will now show that the distribution of Y , namely P (Y ≤ x),

is equal to F (x). Notice that P (Y ≤ x) = P [F

−1

(U) ≤ x] = P [U ≤ F (x)]. Because U is a

uniform (0,1) random variable, then P [U ≤ F (x)] = F (x). Thus, P (Y ≤ x) = F (x). 2

Queueing Theory and Stochastic Teletraﬃc Models

° Moshe Zukerman 20

Homework 1.4

Write a computer program that generates a sequence of 100 random deviates from an exponen-

tial distribution with µ = 1. 2

Let X

and X

be exponentially distributed random variables with parameters λ

and λ

. We

are interested to know the distribution of X = min[X

, X

]. In other words, we are interested

in the distribution of the time that passes until the ﬁrst one of the two random variables X

and X

occurs. This is as if we have a competition between the two and we are interested in

the time of the winner whichever it is. Then

P (X > t) = P (min[X

, X

] > t) = P (X

> t, X

> t) = e

−λ

= e

−(λ

+λ

. (51)

Thus, the distribution of X is exponential with parameter λ

+ λ

Another interesting question related to the competition between two exponential random vari-

ables is what is the probability that one of them, say X

, wins. That is, we are interested in

the probability of X

< X

. This is obtained using the continuous version of the law of total

probability (41) as follows:

P (X

< X

) =

∞

(1 − e

−λ

)λ

−λ

dt =

+ λ

. (52)

In the following table we explain how to obtain the latter from the continuous version of the

law of total probability (41) by pointing out the equivalence between the corresponding terms

in the two equations.

term in (41) equivalent term in (52)

event A event {X

< X

}

random variable Y random variable X

event {Y = y} event {X

= t}

event {A | Y = y} event {X

< t}

P (A | Y = y) P (X

< t) = 1 − e

−λ

density f

(y) density f

(t) = λ

−λ

In a similar way,

P (X

> X

) =

+ λ

. (53)

As expected, P (X

< X

) + P (X

> X

) = 1. Notice that as X

and X

are continuous-time

random variables, the probability that they are equal to each other is equal to zero.

1.10.3 Hyper-Exponential

Let X

for i = 1, 2, 3, . . . , k be k independent exponential random variables with parameters

, i = 1, 2, 3, . . . , k, respectively. Let p

for i = 1, 2, 3, . . . , k be k nonnegative real numbers

such that

i=1

= 1. A random variable X that is equal to X

with probability p

is called

Hyper-exponential. By the Law of total probability, its density is

(x) =

i=1

(x). (54)

剩余192页未读，继续阅读

damho

粉丝: 0
资源: 1

掌握电信交通模型：队列理论与随机过程简介

Introduction to Queueing Theory

Introduction to Queueing Theory Cooper 2nd

排队论（Queueing_Theory）概述

basic queueing theory

排队论 matlab

排队系统matlab

mmc排队论matlab

操作系统电梯调度算法

排队论模型matlab

v4l2 android hal

最新资源