Mathematics of the market. Service random flow

© Alexandr Berlin, 2017

ISBN 978-5-4485-9239-3

Created with Ridero smart publishing system

_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‘_receiv s 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 reasons.

For example, the impossibility (or unwillingness) every day to eat burgers, or objective reasons, for example, the capacity of the fuel tank is designed for a week.

WHILE WE EXCLUDE THE FACTOR OF SAVINGS “IN RESERVE” (A DELAY SYSTEM).

The time of consumption is different from the time possession of the goods. For example, a car purchased 3 years ago; the owner can use by but if the owner started looking for another car, the time of consumption is 3 years.

1.3. 3. The dependence of demand in units of relative consumption from the price

We determined the relative consumption as

a=P_real/P_max

We will assume that the unit price the real and maximum consumption is equal; in this case the specific relative consumption does not depend on prices.

If these prices are different, it is possible to introduce coefficients.

Real consumption constitutes the sum of values relative consumption of certain groups of consumers.

Additivity of real consumption will be shown below.

A =Σ ⁿ_ia_i

Where n is the number of groups of consumers.

The increase in demand can have two sources:

1. The increase in the consumption of individual consumers P_real.

2. The increase in consumption due to the increase in the number of consumer groups.

3. Or the increase of both factors

When prices increase, real consumption is reduced, this leads to lower relative consumption. Conservation measures of the market require an increase in numbers buyers – n. If consumption is reducing then this can only be done by increasing the number of buyers, either through advertising or through the expansion of sales territory.

If prices fall, then real consumption is increases, then same number of buyers buys more goods. The limit this growth is P_max.

At the stage when the market is not saturated with goods, the factors of maintaining the input flow are valid.

In case of market saturation (overproduction stage) P_real stops growth (every consumer receives the goods close to the maximum consumption). A numbers of groups of consumption at the saturation point, in most cases, cease to grow. During the production of the goods and preference for certain firms, consumer groups and the influence of other factors, such as the geography of sales is stabilized.

Therefore, manufacturers with capacity of production over a certain size (as we will show below) suffering losses, especially when the prices fall.

In this case, the main exit is the agreement to reduce output. This agreement applies to the type of “Nash equilibrium” [3.4]. This agreement brings to all parties to the losses, but the abandonment leads to even greater losses. Such agreements can be reached with great difficulty. This is true in general for the industry.

However, if we consider individual participants of market, the decline in the prices can increase number of groups of consumers which use product this participants reducing the number of consuming products of the other group manufacturers. Competition in the form of lower prices between manufacturers can limit only prime cost. This contest wins by the party that has the least cost, or he can get outside support (from government, banks, etc.).

Non-price factors that affect real consumption [4.4], and which can be influenced a relative consumption:

• The level of income in the society;

• The size of the market;

• Fashion, seasonality;

• Availability of substitute products (substitutes);

• Inflationary expectations.

Resume

So we have established General properties of the concept “the goods”.

It is despite of the large variety of goods, we reduced quantitative characteristics of the goods up to the two indicators.

1. The maximum possible consumption.

2. The real consumption.

However, if we consider the individual participants of market, the decline the prices individually participants of the market (by manufacturers) can increase number of groups of consumers which use his products by reducing the number consuming of products of the other group manufacturers. Competition in the form of lower prices between manufacturers will is limit only prime cost. This contest wins by the party that has the least cost, or he can get outside support (from government, banks, etc.).

In the future we will divide incoming supply of goods and supplies goods which already were serviced. (by analogy with the concepts of the theory of mass service [2.1] – the offered load and. the carried out load).

Incoming supply of goods in the time interval (t₁, t₂) denoted by A_inc. (t₁, t₂), it is represents the sum of the quantities of goods handled by this consumer group during the period (t_1, t₂).

Supplies of goods which already were serviced. over a period of time (t₁, t₂) is called such the supply of goods if each incoming goods will immediately purchased by the consumer – in moment. (t₁, t₂). Sometimes this supply will be called – a potential supply.

The difference between the incoming supply of goods and supplies goods which already were serviced will be called a” the losses supplies of goods” :

A_lost (t₁, t₂)=A _income (t_1,t₂)– A_served.. (t₁, t₂)

1.4. Numerical characteristics of flows supplies of goods

1.4.1. The intensity of the supplies of goods

By analogy with the concepts of instantaneous and average intensities of random flow may to considered instantaneous and the average intensity of the supply of goods. However, in the theory and practice of calculation of the capacity of the market it is advisable to use the average intensity of the flows offers supply of goods. The intensity the incoming supply of goods means the offer of goods per unit of time. As a unit of measurement of the intensity of the supply of goods adopted the value a=1, i.e. the maximum volume of the supply of goods the (P_real= P_max).

The following theorem facilitates the determination of intensity of the supply goods

It shows that this value has the property of ergodicity, which is that

Average at the time is equal to the average of the ensemble.

In this case, monitoring the arrival of shipments on time can be replaced by monitoring the number of simultaneously incoming groups of consumers.

The theorem on the quantification of the intensity incoming supply of goods:

The intensity of incoming supply of goods, which is expressed in units o relative consumption, quantitatively equal to the average number of simultaneously busy consumer groups serving this load.

Suppose that during the T hours continuously recorded the number of simultaneously busy groups of consumers market, which receives steady flow of supplies for consumer groups. Let the result of the observations was that during the time t₁ was busy υ₁ consumer groups, during the time t₂. was busy v₂ consumer groups, etc. (Fig.1.1). In General can imagine that during the time t_i was busy υ_i consumer groups,

Σⁿ_i=1 t_i= T

and where n is the number, the value of v which is taken, within T hours.

Total time, when busy all consumers of the market at time ti expressed by the product of υ_it. In the time interval T total time when all users busy of the market will be expressed by amount. This amount, it is supplies of goods all consumers of the market at the time T.

The supplies of goods what, all consumers acquired of the market per unit time are equal to:

A_serv = (1/T) Σⁿ_i=1v_i ⋅ t_i

On the other hand, the average number of simultaneously consumer groups occupied during the time T can be defined as a weighted average of t_i:

v‘= (v₁t₁ + v₂t₂ +•••+ v_nt_n) / (t₁ + t₂ + ⋯ t_n) =

= (1/T) Σ ⁿ_i=1 (v_i ⋅t_i)

therefore A_serv= v’

A theorem about the quantitative assessment of the intensity of the incoming floe supplies of goods.

To quantify the intensity of the incoming flow supply of goods you can use the following theorem:

The intensity of the incoming flow supplies of goods, which expressed in terms of relative consumption, creates a simplest flow of goods, which quantitatively equal to the mathematical expectation of the number of goods (c’), received for a time equal to the average duration of one consumption of one batch products (t’_cons)

Figure 1.1. The moments of arrives of goods

Let the inputs of the market comes a simple flow of goods with the intensity μ. We assume that the duration of consumption – T is a finite random variable

0≤T≤ T_max is independent of the stream type of the incoming supplies goods with the average value t, Consider the time interval [t1, t2) such that t₂ – t₁> T max. The mathematical expectation of the number of supplies placed on the market in the time interval [t₁, t₂) denoted by Λ (t₁, t₂) =μ (t₁.t₂).

Part of these supplies is consumed to the moment t₂ (Fig. 1.1 a), and the other part does not end in this time (Fig. 1.1 b). Denote the mathematical expectation of the number of goods received in the time interval [t₁, t₂) and what not purchased to the time t₂, denote ρ. In addition to products that arriving to market in the time interval [t₁, t₂), it is necessary to take into account products that have arrived up to time t₁ and the time t₂ are not yet purchased. We denote the expected number of goods that began before the time t₁ and are ended in the time interval [t₁, t₂), denote ε (Fig. 1.1), and the expected number of calls that started before t1 and ended after the time t₂, denote ζ (Fig. 1.1 g). Since t₂ – t₁> T_max, then ζ=0. For the simplest flow ρ=ε.

By definition, the mathematical expectation of entering the market supply of goods during a time interval [t₁, t₂),

a (t₁,t₂) = [μ (t₂—t₁) – ρ +ε] ⋅t ̅=μ⋅ t’_consum (t₂—t₁)

and the intensity of the inbound supplies:

a= [a (t₁,t2)] / (t₂—t₁) = μ⋅ t’_cosum

The mathematical product (multiplication) is the mathematical expectation of the number of goods received over the average duration of consumption. The theorem is proved.

For example, suppose that one day (between t₁=0 and t₂=24 hours) enters N⋅c=100⋅4=400 items of goods.

Let the average duration of a consumption equal per day. Therefore, for time will receive 400⋅ 1/40 =10 items of goods.

At the same time, the number of the mathematical expectation of the number of proposals received per day is equal to:

A= N⋅ c’(T) /T=400⋅ (1/40) =10 items of goods per day

1.5. The demand and its fluctuations

1.5.1. The basic definitions. The time of greatest demand

The intensity of demand called the demand of goods per unit of time. For measuring the size of demand is applied relative consumption. A unit of measurement of the intensity of demand of the goods sometimes may make the value a=1, i.e. equal to the maximum consumption (P_real= P_max) per unit time.

The intensity of demand in a general case can vary in different hours of the day, days of the week and months of the year. Observations have revealed that, along with random fluctuations of demand, there are exist, relatively regular fluctuations that must be considered when predicting the quantity demanded.

The most significant fluctuations with the seasons.

For some goods the greatest demand falls on a holiday, (e.g. New Year).

Largely they depend on level of life in this area and structural composition of consumers, which serves the market.

Regular fluctuations in demand may depend on the days of the week. On Saturday and Sunday, the demand for goods of mass consumption can be higher than in the other days of the week. Regular fluctuations in demand are observed for the months of the year. The minimum load on the mass consumer goods, excluding resort towns, is observed in the summer months: June, July, and August. The maximum load on the goods of mass demand occurs in February, March and November, December; should be measured demand in these months.

For strategic goods, such as oil, weapons and etc. plays role of the political situation, global and local conflicts.

For the satisfactory quality of customer service at any time, the calculation of the offer you must perform on the basis of the intensity of demand at a time when he is the greatest.

This time will be called the time of greatest demand – TGD (similar to the Busy-Hour of Greatest Traffic – HGT in theory queueing).

The time of greatest demand – TGD it is a continuous time interval during which the average intensity of the demand is the greatest.

The degree of concentration of demand in the TGD estimated coefficient of concentration

k_TGD=A_TGD/A_obs,

where A _TGD. the demand value for TDG;

A_obs. the demand value during the observation.