Week 2.13: Elements of the queueing theory. M/G/k systems-1

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来自 National Research University Higher School of Economics 的课程

Stochastic processes

28 个评分

National Research University Higher School of Economics

Stochastic processes

28 个评分

The purpose of this course is to equip students with theoretical knowledge and practical skills, which are necessary for the analysis of stochastic dynamical systems in economics, engineering and other fields. More precisely, the objectives are 1. study of the basic concepts of the theory of stochastic processes; 2. introduction of the most important types of stochastic processes; 3. study of various properties and characteristics of processes; 4. study of the methods for describing and analyzing complex stochastic models. Practical skills, acquired during the study process: 1. understanding the most important types of stochastic processes (Poisson, Markov, Gaussian, Wiener processes and others) and ability of finding the most appropriate process for modelling in particular situations arising in economics, engineering and other fields; 2. understanding the notions of ergodicity, stationarity, stochastic integration; application of these terms in context of financial mathematics; It is assumed that the students are familiar with the basics of probability theory. Knowledge of the basics of mathematical statistics is not required, but it simplifies the understanding of this course. The course provides a necessary theoretical basis for studying other courses in stochastics, such as financial mathematics, quantitative finance, stochastic modeling and the theory of jump - type processes.

从本节课中

Week 2: Poisson Processes

Upon completing this week, the learner will be able to understand the definitions and main properties of Poisson processes of different types and apply these processes to various real-life tasks, for instance, to model customer activity in marketing and to model aggregated claim sizes in insurance; understand a relation of this kind of models to Queueing Theory

Week 2.8: Non-homogeneous Poisson processes-14:24

Week 2.9: Non-homogeneous Poisson processes-24:23

Week 2.10: Relation between renewal theory and non-homogeneous Poisson processes-14:11

Week 2.11: Relation between renewal theory and non-homogeneous Poisson processes-27:10

Week 2.12: Relation between renewal theory and non-homogeneous Poisson processes-34:29

Week 2.13: Elements of the queueing theory. M/G/k systems-19:47

Week 2.14: Elements of the queueing theory. M/G/k systems-25:14

Week 2.15: Compound Poisson processes-16:05

Week 2.16: Compound Poisson processes-26:52

Week 2.17: Compound Poisson processes-33:18

与讲师见面

Vladimir Panov
Assistant Professor
Faculty of economic sciences, HSE

Next I would like to provide some basic elements of the so-called Queuing theory.

As I already mentioned for a homogeneous Poisson process,

the following three equalities are fulfilled.

The equalities tell us that the probability that only one,

more than one, or zero events occur during

some infinitely small interval from T to T plus hash H are equal to lambda H,

zero, and one minus lambda H plus some factors

which tend to zero once that we divide them by H. Actually,

the same formulas hold also for non-homogeneous Poisson process.

In fact, in this case,

this lambda should be changed to its intensity function lambda of T. Actually,

all other elements of this formula remain the same.

Moreover, we can define and

a non-homogeneous Poisson process as a process which starts from zero,

has independent increments and these properties are fulfilled.

Homogeneous and non-homogeneous Poisson processes are

widely used for describing the queuing systems.

The most popular annotation for different types of queue is the following.

So this notation is based on three characters which are separated by slashes.

The first character means the arrival process.

It can be of the following types.

First of all, it can be memoryless and in this case,

we we'll write a capital letter

M. This basically means that the arrival process is Poisson.

Secondly, it can be deterministic, and in this case,

we'll write a capital letter D. And thirdly,

it can be from any distribution and in this case,

we'll write letter G,

which stands for general.

We will assume that interarrival times are independent, identically distributed.

Generally, this process can be only of these three types.

The second character is used for the service times.

And it can be also as M or D or G. And as for the third character,

this is a number of servers.

It can be either a positive integer value or infinitive.

Let me consider more precisely the system M, G, and Infinitive.

According to our definition,

this is a system where arrival processes are modeled by Poisson process.

Also, we know that service times are modeled by some distribution which I

also denote by G. And the infinitive means that the amount of zeros is infinite.

It is any arrival,

it's starting to be soft immediately after arrival.

So no elements are ever queued.

You may think that you have a call center and when you get new

your calls that one operator is starting to work with this call.

Of course, this system is a bit unrealistic but it is very

useful for showing the most important issues of the queuing theory.

You might have the following picture.

So the calls are arriving.

According to some Poisson process N of T,

and then let me fix some time moment torque,

and consider two processes.

The first process N1 of T. This process indicates the amount of arrivals who

are still being served at time moment torque.

And secondly, N2 of T is an amount of arrivals who

are already completed before by torque.

Let me consider more precisely this process N1

of T. If we consider the increment of this process,

it is N1 T plus Delta minus N1 T. This increment

in fact indicates how many new arrivals come into the system between T and T plus

torque and how many arrivals from this amount are still being served.

Therefore, the probability that N1 T plus

delta minus N1 T is equal to some number,

let me write here one,

is in fact equal to the product of two probabilities.

The first probability is that one customer arrives between T and T plus Delta,

so this is N T plus delta minus N T is equal to one,

multiplied by the probability that this customer is still being served at time torque.

It is a probability that a certain variable Y which

indicates the service time for this customer

is larger than torque minus T. It

can be some other terms which are the effect of other small O of torque.

The first term is equal to delta lambda plus some terms which have

smaller rate of convergence to zero and the second item is equal to

one minus G and the argument is

torque minus T plus maybe some terms which have smaller order of convergence to zero.

To sum up, we get that this probability is equal to

delta lambda one minus G torque minus T plus small O of delta.

These formulas are important and if you

consider the probabilities of these different cycles to zero or larger than two,

you will get very similar expressions.

Finally, you may think about this in the following way, In fact,

we have a homogeneous Poisson process which is splitted into two subprocesses.

The first subprocess N1 is in fact non-homogeneous Poisson,

because for this process,

all formulas of this period are fulfilled.

I mean all formulas was here, one,

zero, and larger equals than two.

Moreover, the process N1 is zero at

zero and it possesses a property of having independent increments.

Therefore, N1 is non-homogeneous process whose intensity function equals to

lambda multiplied by one minus G torque minus

T. And similar arguments work well also with the second process.

And I guess this process is also non-homogeneous Poisson and the intensity function is

lambda multiplied by G torque minus T. This is a very interesting observation,

but what is more important and what is a little bit on logic is

that these N1 and N2 are in fact independent.

This seems to be very surprising because if we consider these two processes,

condition on the process N T as in one completed as legitimizes second.

But, nevertheless this is true and I would like

to show this now.

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