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Alexandr Berlin

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Alexandr Berlin

Mathematics of the market

Service random flow

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The book offers a new approach to the calculation of economic processes. This approach allows to obtain very interesting data: — to define a generic mathematical description of the goods. In book gives a mathematical market model, it is shown that calculations of the parameters of the market can be carried out according to the formulas of the Queuing theory, in particular according to the formulas of erlang, Engset, etc; to define formulas that reflect the relationship between supply and demand

ISBN 978-5-4485-9239-3

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Mathematics of the market
_Acronyms
Chapter 1. What are the REELECTION CONSUMPTION. unit of measure of supply and demand
1. 1.1. So what is the goods?
2. 1.2. Relative consumption
3. 1.3. Factors affecting relative consumption
4. 1.3. 1.Maximum consumption
5. 1.3. 2.Real consumption (demand)
6. 1.3. 3. The dependence of demand in units of relative consumption from the price
7. 1.4. Numerical characteristics of flows supplies of goods
8. 1.4.1. The intensity of the supplies of goods
9. 1.5. The demand and its fluctuations
10. 1.5.1. The basic definitions. The time of greatest demand
11. 1.5.2. Main parameters and calculation of the intensity of demand
12. 1.5.3. The average number of requests from one consumer per unit of time
13. 1.5.4. Characteristics of quality of service of goods
14. 1.5.5. Traffic-carrying capacity market
15. 1.6. Tests to Chapter 1
Chapter 2. A mathematical model of the market
1. 2. 1.Definition
2. 2.2. Types of SERVICE systems and input flows goods
3. 2.2.1. Types of systems service goods
4. 2.2.2. The types of input flows
5. 2.2.3. Principles of classification of event flows
6. 2.2.4 Basic numerical characteristics of the flow of goods
7. 2.2.5. The simplest flow of goods and its properties
8. 2.2.6. Mathematical expectation and variance of a simplest flow
9. 2.2.7. Distribution law of intervals between applications for the goods of the simplest stream
10. 2.2.8. The duration of consumption. Flow of moments the end of consumption
11. 2.3. Classification non-simplest flows
12. 2.3. 1. flow with simple aftereffect
13. 2.3.2. The flow repetitive supplies of goods
14. 2.3.3. The flow with limited aftereffect
15. 2.4. Tests to Chapter 2
Chapter 3. The market as a system with obvious losses
1. 3.1.Statement of the problem
2. 3.2. Differential equations Erlang
3. 3.3. Stationary mode. Distribution of Erlang
4. 3.4, Losses at service simplest flow with a fully the accessible group
5. 3.5. Average throughput, consumer groups of the market with full accessibility
6. 3.6. Graphical dependence between the parameters of the first formulas of Erlang
7. 3.7. Example. The calculation the market of oil
8. 3.8. Problems
9. 3.9. Tests to Chapter 3
Chapter 4. The market as a system with waiting
1. 4.1. The market model in the system with waiting. The output of the second formula Erlang
2. 4.2. The probability of the system states
3. 4.3. The probability that the time of storage will be more t.
4. 4.4. The average waiting time for batches of goods incoming on the market
5. 4.5. The average waiting realization time for the saved goods
6. 4.6. Probability of queue (probability of presence in the queue at least one batch of the goods)
7. 4.7. The average queue length or the average number of delayed batches of goods
8. 4.8. Example. The calculation of the market of oil with the storage of unsold goods
9. 4.9 Problem
Chapter 5. Serving a fully accessible group of consumers from a group with a limited number of consignments (Engset formula)
1. 5.1. Statement of the problem
2. 5.2. The derivation of formula Engset
3. 5.3. Types of losses in service flow from a limited number of sources
4. 5.4. Problems
5.5. Questions for Tests
1. Glossary
APPENDIX 1 TABLES ENGSET
Reference

_Acronyms

A (P_relative). — relative consumption. The proposed load;

A_loss (t₁, t₂) — the difference between the incoming supply of goods and serviced supplies of goods;

A_inc. (t₁, t₂) — incoming supply of goods over a period of time (t₁, t₂);

A_served. (t₁, t₂) — the sum of the quantities of goods handled by consumer group during the period (t₁, t₂₎.;

A _TGD. — the demand value for time of greatest demand;

A_obs. — the demand value during the observation;

a — the load intensity coming from a single source;

c‘_receivs the number of received requests;

c‘_serv— the number of serviced requests;

c‘_lost. — the number of lost requests;

c‘— average length of queue or average numbers delays batches of goods;

P_max- maximum consumption;

P _goods. – losses of volume of goods;

P _request. — losses in the number of requests for suggestion of goods;

P _goods. — losses of volume of goods;

P t- losses of implementation time;

p (γ> 0) the probability that the waiting time is greater than zero — that is, the probability of queue;

p (γ> t) — the probability of waiting for any incoming product over time t;

p (R> r) — the probability that the queue length exceeds the specified value r);

(γ) — the average waiting time in relation all requests received;

γ _delay the average waiting time of only a delayed requests;

Λ (t₁, t₂) -the mathematical expectation of the number of supplies placed on the market in the time interval [t₁, t₂);

μ (t₁.t₂). – the instantaneous flow rate;

Chapter 1. What are the REELECTION CONSUMPTION. unit of measure of supply and demand

1.1. So what is the goods?

Let’s start with a quote of Keynes [The General Theory of Employment, Interest, and Money by John Maynard Keynes. Book II: Definitions and Ideas]

“The perplexities which most impeded my progress in writing this book, so that I could not express myself conveniently until I had found some solution for them, are:

firstly, the choice of the units of quantity appropriate to the problems of the economic system as a whole;..

“That the units, in terms of which economists commonly work, are unsatisfactory can be illustrated by the concepts of the national dividend, the stock of real capital and the general price-level:”

“The community’s output of goods and services is a non-homogeneous complex which cannot be measured, strictly speaking, except in certain special cases, as for example when all the items of one output are included in the same proportions in another output.”

In order to take the definition of the unit of goods we use the obvious statement.

We define the goods the market, as something that you want to sell or that you want to buy. Let’s try to express this in numbers

Do you love cakes? I will sell them to you. You will buy and will want again. Finally, you can eat 10 cakes a day. And I offer you still. You say, “Enough!” I suggest to you free of charge 12 cakes a day, and you ungrateful say me “No!”. Moreover, buying cakes, you have some time to consume them, and not buy a new one

What is it? It is the overproduction!

I repeat the question: How much can you eat on the day cakes – 10, 12, 15? It doesn’t matter for math, but in principle there is some maximum quantity which you can consume in a certain period (a day, a month, a year).

Let’s try to find the indicator and we will call its – the relative consumption-

P_relative. If the consumer is not in need of this product (saturation) P_relative.=1,

If the consumer is just starting to purchases a goods

P_relative. =0.

Therefore, the current state of the user can be reflected with help the value of the relative consumption

0 ≤ P_relative.≤1

We will continue the search for the mathematical determination of relative consumption.

More details of this figure can be determined from the following reasoning.

Let’s introduce a unit of measure (denoted it a) “specific relative consumption”. It will the ratio of real consumption ( P_relative) to the maximum possible (P_max.)

P_relative = P_real/ P_max=a

Then the maximum consumption in a given period will be equal to P_max. We assume that no material (money, prices) and external factors (season, fashion) don’t limit you.

Next, let’s see how much you are consuming a really – P_real.. By definition you can’t consume more than P_max. (P_real. < P_max.).

If you considered a group of n consumers, applied the name – RELATIVE CONSUMPTION.

It will be

A=an

1.2. Relative consumption

The relative consumption shows the possibility of participation of the consumer in the consumption (similar to a unit load in queuing theory). It indicates how many the goods the consumer is willing to buy. If a=1, the consumer in a given period will not buy a product, the other side it shows how satisfied requests of this user.

For example, from a website on the Internet, you can learn that the Scandinavians consume 8 liters of milk per week, and the Europeans 1 liter per week. One people physically can maximum consume 10 liters per week. Not analyzing for accuracy this information, we take these numbers as truth.

Then the relative consumption is equal for Scandinavians – 8/10, and for Europeans 1/10.

For sale of milk these numbers speak volumes.

1. The propensity for milk consumption in Scandinavia is close to saturation. As we showed research, selling products in a saturated market leads to big losses that are higher the closer the value of the relative consumption to the value 1. This means that the needs of this market are completely satisfied.

2. Judging the relative consumption for the sale of milk are higher Europe prospects because the market is not saturated.

However, it is necessary to make a reservation. Keynes divided the propensity to consume on two categories – objective and subjective factors. He wrote [ THE BOOK THIRD. THE PROPENSITY TO CONSUME. CHAPTER 8 ].

“In order to make our analysis clearer, it is advisable to divide these motives into two big groups: first of them we will include subjective and the second objective factors. Subjective factors, those describe psychological features of human character, and also those public habits and institutes, which though are not invariable, nevertheless hardly whether are subject to significant change over short time periods (excluding any coming out of the ordinary circumstances or revolutionary shocks)”.

In our example, the low level of relative consumption indicates the presence in the European behavior of the subjective factor, but at the same time indicates for the prospect of overcoming it (advertising, development of appropriate European tastes dairy products).

3. Relative consumption can be used to measure the level of poverty and wealth.

Saturation of the population with vital goods is usually calculated by the amount of consumption per person over a certain period (month, year). Relative consumption indicates the degree of saturation (ratio of maximum consumption to real consumption). This example cannot serve as such an indicator because it has a large share of subjective factors.

The low value of the relative consumption indicates poverty if this if this is happens due to the lack of material resources.

The last thing you should specify is the role of relative food consumption. In most cases, this value may wear a medical evaluation.

In this case, the relative consumption indicates that some products, not can provide a normal life, and some goods are consumed excessively,

1.3. Factors affecting relative consumption

1.3. 1.Maximum consumption

Based on statistical data (or on basis common sense) is determined by the maximum possible acquisition of the goods (measured in units that characteristic of this product) for a specified period of time (week, month, year). This quantity we will call the maximum consumption (P_max)

When choosing the amount of maximum consumption should take into account two types of factors.

The OBJECTIVE restriction and FORMAL restriction.

1. Objective restrictions limit the maximum consumption is that it is limited to physiological causes (for example, a person can’t eat more than 10 cakes in a day) or technical reasons (this car consumes 25 miles/gallon. These parameters can be chosen so that the probability of their change for a short period of time will be small.

Certain goods are restricted to natural causes. For example, the maximum of the index of goods – “park the car in a period of 1 hour” is limited to one hour.

2. A formal limit of maximum consumption – consist that its value is arbitrarily is assigned by the researcher or for harmonization (standardization) with other stakeholders (e.g. maximum consumption of cakes one person 100 cakes per day). Further consideration will show that unrealistically high values of maximum consumption does not affect the main results, does not change the resulting estimates, but complicates the calculations by increasing or decreasing the values of the processed variables.

1.3. 2.Real consumption (demand)

The main ideas are the following:

1. The consumer buys for a certain period (week, month, year) a certain quantity of goods, measured in units characteristic for this product, taking into account needs and price for this item. This is the number we will call real consumption (P_real)

2. After purchase, the buyer consumes it within a certain time of usage.

The reason for the refusal of the purchase may determine subjective

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