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Mathematics ofВ the market. Service random flow
Alexandr Berlin
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
Mathematics ofВ the market
Service random flow
Alexandr Berlin
© Alexandr Berlin, 2017
ISBNВ 978-5-4485-9239-3
Created with Ridero smart publishing system
AВ (P
). – relative consumption. The proposed load;
A
(t
, t
) – the difference between the incoming supply of goods and serviced supplies of goods;
A
. (t
, t
) – incoming supply of goods over a period of time (t
,В t
);
A
. (t
, t
) – the sum of the quantities of goods handled by consumer group during the period (t
, t
.;
AВ
. – the demand value for time of greatest demand;
A
. – the demand value during the observation;
a – the load intensity coming from a single source;
c�
s the number ofВ received requests;
c�
– the number of serviced requests;
c�
 – the number of lost requests;
c�– average length of queue or average numbers delays batches of goods;
P
– maximum consumption;
PВ
. – losses of volume of goods;
PВ
. – losses in the number of requests for suggestion of goods;
PВ
. – losses of volume of goods;
PВ t- losses ofВ 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;
ОіВ
В the average waiting time ofВ only aВ delayed requests;
О› (t
, t
) -the mathematical expectation ofВ the number ofВ supplies placed on the market inВ the time interval [t
,В t
);
Ој (t
.t
). – the instantaneous flow rate;
Chapter 1. What are the REELECTION CONSUMPTION. unit ofВ measure ofВ 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 ofВ the units ofВ quantity appropriate toВ the problems ofВ the economic system as aВ 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.”
InВ order toВ take the definition ofВ the unit ofВ 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
. If the consumer is not inВ need ofВ this product (saturation) P
.=1,
If the consumer is just starting toВ purchases aВ goods
P
.В =0.
Therefore, the current state ofВ the user can be reflected with help the value ofВ the relative consumption
0 ≤ P
.≤1
We will continue the search for the mathematical determination ofВ relative consumption.
More details ofВ 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
) toВ the maximum possible (P
.)
P
= P
/ P
=a
Then the maximum consumption inВ aВ given period will be equal toВ P
. 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
.. By definition you can’t consume more than P
. (P
. < P
.).
If you considered a group of n consumers, applied the name – RELATIVE CONSUMPTION.
It willВ be
A=an
1.2. Relative consumption
The relative consumption shows the possibility ofВ participation ofВ the consumer inВ the consumption (similar toВ aВ unit load inВ queuing theory). It indicates how many the goods the consumer is willing toВ buy. If a=1, the consumer inВ aВ given period will not buy aВ product, the other side it shows how satisfied requests ofВ this user.
For example, from aВ website on the Internet, you can learn that the Scandinavians consume 8 liters ofВ 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 ofВ milk these numbers speak volumes.
1. The propensity for milk consumption inВ Scandinavia is close toВ saturation. As we showed research, selling products inВ aВ saturated market leads toВ big losses that are higher the closer the value ofВ the relative consumption toВ the value 1. This means that the needs ofВ this market are completely satisfied.
2. Judging the relative consumption for the sale ofВ 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)”.
InВ our example, the low level ofВ relative consumption indicates the presence inВ the European behavior ofВ the subjective factor, but at the same time indicates for the prospect ofВ overcoming it (advertising, development ofВ appropriate European tastes dairy products).
3. Relative consumption can be used toВ measure the level ofВ poverty and wealth.
Saturation ofВ the population with vital goods is usually calculated byВ the amount ofВ consumption per person over aВ certain period (month, year). Relative consumption indicates the degree ofВ saturation (ratio ofВ maximum consumption toВ real consumption). This example cannot serve as such an indicator because it has aВ large share ofВ subjective factors.
The low value ofВ the relative consumption indicates poverty if this if this is happens due toВ the lack ofВ material resources.
The last thing you should specify is the role ofВ relative food consumption. InВ most cases, this value may wear aВ medical evaluation.
InВ this case, the relative consumption indicates that some products, not can provide aВ 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 byВ the maximum possible acquisition ofВ the goods (measured inВ units that characteristic ofВ this product) for aВ specified period ofВ time (week, month, year). This quantity we will call the maximum consumption (P
)
When choosing the amount ofВ maximum consumption should take into account two types ofВ 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 aВ certain period (week, month, year) aВ certain quantity ofВ goods, measured inВ units characteristic for this product, taking into account needs and price for this item. This is the number we will call real consumption (P
)
2. After purchase, the buyer consumes it within aВ certain time ofВ usage.
The reason for the refusal ofВ the purchase may determine subjective reasons.
For example, the impossibility (or unwillingness) every day toВ eat burgers, or objective reasons, for example, the capacity ofВ the fuel tank is designed for aВ week.
WHILE WE EXCLUDE THE FACTOR OF SAVINGS “IN RESERVE” (A DELAY SYSTEM).
The time ofВ consumption is different from the time possession ofВ the goods. For example, aВ car purchased 3 years ago; the owner can use byВ but if the owner started looking for another car, the time ofВ consumption is 3 years.
1.3. 3. The dependence ofВ demand inВ units ofВ relative consumption from the price
We determined the relative consumptionВ as
a=P
/P
We will assume that the unit price the real and maximum consumptionВ is equal; inВ this case the specific relative consumption does not depend on prices.
If these prices are different, it is possible toВ introduce coefficients.
Real consumption constitutes the sum ofВ values relative consumption ofВ certain groups ofВ consumers.
Additivity ofВ real consumption will be shown below.
AВ =ОЈВ
a
Where n is the number ofВ groups ofВ consumers.
The increase inВ demand can have two sources:
1.В The increase inВ the consumption ofВ individual consumers P
2. The increase inВ consumption due toВ the increase inВ the number ofВ consumer groups.
3. Or the increase ofВ 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 ofВ buyers buys more goods. The limit this growth is P
.
At the stage when the market is not saturated with goods, the factors ofВ maintaining the input flow are valid.
InВ case ofВ market saturationВ (overproduction stage) P
stops growth (every consumer receives the goods close toВ the maximum consumption). AВ numbers ofВ groups ofВ consumption at the saturation point, inВ most cases, cease toВ grow. During the production ofВ the goods and preference for certain firms, consumer groups and the influence ofВ other factors, such as the geography ofВ sales is stabilized.
Therefore, manufacturers with capacity ofВ production over aВ 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 ofВ market, the decline inВ the prices can increase number ofВ groups ofВ consumers which use product this participants reducing the number ofВ consuming products ofВ the other group manufacturers. Competition inВ the form ofВ lower prices between manufacturers can limit only prime cost. This contest wins byВ the party that has the least cost, or he can get outside support (from government, banks, etc.).
Non-price factors that affect real consumptionВ [4.4], and which can be influenced aВ 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 ofВ the large variety ofВ goods, we reduced quantitative characteristics ofВ the goods up toВ the two indicators.
1.В The maximum possible consumption.
2. The real consumption.
However, if we consider the individual participants ofВ market, the decline the prices individually participants ofВ the market (byВ manufacturers) can increase number ofВ groups ofВ consumers which use his products byВ reducing the number consuming ofВ products ofВ the other group manufacturers. Competition inВ the form ofВ lower prices between manufacturers will is limit only prime cost. This contest wins byВ 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 ofВ goods inВ the time interval (t
, t
) denoted byВ A
. (t
, t
), it is represents the sum ofВ the quantities ofВ goods handled byВ this consumer group during the period (t
В t
).
Supplies ofВ goods which already were serviced. over aВ period ofВ time (t
, t
) is called such the supply of goods if each incoming goods will immediately purchased by the consumer – in moment. (t
, t
). 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
(t
, t
)=AВ
(t
t
)– A
.. (t
,В t
)
1.4. Numerical characteristics ofВ flows supplies ofВ goods
1.4.1. The intensity ofВ the supplies ofВ goods
ByВ analogy with the concepts ofВ instantaneous and average intensities ofВ random flow may toВ considered instantaneous and the average intensity ofВ the supply ofВ goods. However, inВ the theory and practice ofВ calculation ofВ the capacity ofВ the market it is advisable toВ use the average intensity ofВ the flows offers supply ofВ goods. The intensity the incoming supply ofВ goods means the offer ofВ goods per unit ofВ time. As aВ unit ofВ measurement ofВ the intensity ofВ the supply ofВ goods adopted the value a=1, i.e. the maximum volume ofВ the supply ofВ goods the (P
= P
).
The following theorem facilitates the determination ofВ intensity ofВ the supply goods
It shows that this value has the property ofВ ergodicity, which isВ that
Average at the time is equal toВ the average ofВ the ensemble.
InВ this case, monitoring the arrival ofВ shipments on time can be replaced byВ monitoring the number ofВ simultaneously incoming groups ofВ consumers.
The theorem on the quantification ofВ the intensity incoming supply ofВ goods:
The intensity ofВ incoming supply ofВ goods, which is expressed inВ units oВ relative consumption, quantitatively equal toВ the average number ofВ simultaneously busy consumer groups serving this load.
Suppose that during the T hours continuously recorded the number ofВ simultaneously busy groups ofВ consumers market, which receives steady flow ofВ supplies for consumer groups. Let the result ofВ the observations was that during the time t
was busy П…
consumer groups, during the time t
. was busyВ v
consumer groups, etc. (Fig.1.1). InВ General can imagine that during the time t
was busy П…
consumer groups,
ОЈ
В t
=В T
and where n is the number, the value ofВ v which is taken, within T hours.
Total time, when busy all consumers ofВ the market at time ti expressed byВ the product ofВ П…
t. InВ the time interval T total time when all users busy ofВ the market will be expressed byВ amount. This amount, it is supplies ofВ goods all consumers ofВ the market at the timeВ T.
The supplies ofВ goods what, all consumers acquired ofВ the market per unit time are equalВ to:
A
В = (1/T) ОЈ
v
В в‹…В t
On the other hand, the average number ofВ simultaneously consumer groups occupied during the time T can be defined as aВ weighted average ofВ t
:
v�= (v
t
В + v
t
 +•••+ v
t
) / (t
В + t
В + в‹ЇВ t
)В =
= (1/T) ОЈ
В (v
В в‹…t
)
therefore A
= v’
AВ theorem about the quantitative assessment ofВ the intensity ofВ the incoming floe supplies ofВ goods.
ToВ quantify the intensity ofВ the incoming flow supply ofВ 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’
)
Figure 1.1. The moments ofВ arrives ofВ 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
is independent ofВ the stream type ofВ the incoming supplies goods with the average value t, Consider the time interval [t1, t2) such that t
 – t
> T max. The mathematical expectation ofВ the number ofВ supplies placed on the market inВ the time interval [t
, t
) denoted byВ О› (t
, t
) =Ој (t
.t
).
Part ofВ these supplies is consumed toВ the moment t
(Fig. 1.1В a), and the other part does not end inВ this time (Fig. 1.1В b). Denote the mathematical expectation ofВ the number ofВ goods received inВ the time interval [t
, t
) and what not purchased toВ the time t
, denote ПЃ. InВ addition toВ products that arriving toВ market inВ the time interval [t
, t
), it is necessary toВ take into account products that have arrived up toВ time t
and the time t
are not yet purchased. We denote the expected number ofВ goods that began before the time t
and are ended inВ the time interval [t
, t
), denote Оµ (Fig. 1.1), and the expected number ofВ calls that started before t1 and ended after the time t
, denote О¶ (Fig. 1.1В g). Since t
 – t
> T
, then О¶=0. For the simplest flowВ ПЃ=Оµ.
ByВ definition, the mathematical expectation ofВ entering the market supply ofВ goods during aВ time interval [t
,В t
),
aВ (t
,t
) = [Ој (t
—t
) – ρ +ε] ⋅t ̅=μ⋅ t’
(t
—t
)
and the intensity ofВ the inbound supplies:
a= [aВ (t
,t2)] / (t
—t
) = μ⋅ t’
The mathematical product (multiplication) is the mathematical expectation ofВ the number ofВ goods received over the average duration ofВ consumption. The theorem is proved.
For example, suppose that one day (between t
=0 and t
=24 hours) enters Nв‹…c=100в‹…4=400 items ofВ goods.
Let the average duration ofВ aВ consumption equal per day. Therefore, for time will receive 400в‹… 1/40 =10 items ofВ goods.
At the same time, the number ofВ the mathematical expectation ofВ the number ofВ proposals received per day is equalВ 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 ofВ greatest demand
The intensity ofВ demand called the demand ofВ goods per unit ofВ time. For measuring the size ofВ demand is applied relative consumption. AВ unit ofВ measurement ofВ the intensity ofВ demand ofВ the goods sometimes may make the value a=1, i.e. equal toВ the maximum consumption (P
= P
) per unit time.
The intensity ofВ demand inВ aВ general case can vary inВ different hours ofВ the day, days ofВ the week and months ofВ the year. Observations have revealed that, along with random fluctuations ofВ 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 aВ holiday, (e.g. New Year).
Largely they depend on level ofВ life inВ this area and structural composition ofВ consumers, which serves the market.
Regular fluctuations inВ demand may depend on the days ofВ the week. On Saturday and Sunday, the demand for goods ofВ mass consumption can be higher than inВ the other days ofВ the week. Regular fluctuations inВ demand are observed for the months ofВ the year. The minimum load on the mass consumer goods, excluding resort towns, is observed inВ the summer months: June, July, and August. The maximum load on the goods ofВ mass demand occurs inВ February, March and November, December; should be measured demand inВ these months.
For strategic goods, such as oil, weapons and etc. plays role ofВ the political situation, global and local conflicts.
For the satisfactory quality ofВ customer service at any time, the calculation ofВ the offer you must perform on the basis ofВ the intensity ofВ demand at aВ 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 ofВ concentration ofВ demand inВ the TGD estimated coefficient ofВ concentration
k
=A
/A
,
where AВ
. the demand value forВ TDG;
A
. the demand value during the observation.
1.5.2. Main parameters and calculation ofВ the intensity ofВ demand
The main parameters ofВ the demandВ are:
В· the number ofВ consumer groupsВ -n;
В· the average number ofВ requests for goods received from one group ofВ consumers per unit timeВ ;
В· the average duration ofВ consumptionВ for maintenance ofВ aВ single application ofВ t.
Consider the possible composition ofВ consumers, which differ inВ the average intensity and average duration ofВ consumption:
– individual consumers;
– intermediaries (e.g., agencies for the purchase and sale of apartments);
– Firms and government agencies.
ByВ lowering the prices ofВ the goods the number ofВ consumers may increase. The lengthy existence ofВ the product on the market the number ofВ consumers can be stabilized, and they are not changing or changing insignificantly.. AВ similar reaction may be inВ case overproduction when the relative consumption is close to 1(unit).Real consumption is stable.
1.5.3. The average number ofВ requests from one consumer per unit ofВ time
InВ accordance with the categories source ofВ requests average number ofВ request for goods per unit ofВ time respectively
– from one group of individual consumers;
– intermediaries (e.g., agencies for the purchase and sale of apartments);
– firms and government agencies.
We denote in a general way the average number of applications from groups of consumers from sources of the i —th category,
n
 – the number of sources i —th category.
Then if on the market have k groups ofВ consumers the average number ofВ applications from one group ofВ consumers will be determined from the expression.
c�= (Σ
В c
n
) / ОЈ
n
The average duration of consumption (t’).
The duration ofВ one consumption it is toВ the period ofВ time from the date ofВ acquisition until the moment you start searching the same product.
Time ofВ consumption is always less time ofВ possession ofВ the goods.
For example, the owner ofВ the car bought it 3 years ago. If he started looking for aВ new car without selling the old, the time ofВ consumption will be 3 years.
The duration ofВ consumptionВ ofВ food products is determined byВ the physical needs ofВ the person also byВ the allowable time storage.
The duration ofВ consumption ofВ clothing is determined byВ its wear resistance, but now mainly accordance toВ fashion.
The duration ofВ consumption ofВ control automates, computing machines ofВ cars is limited byВ their reliability and byВ their main indicators.
Therefore, the duration ofВ consumption, is aВ random value and its the mean value can be determined only on the basis ofВ observations ofВ existing markets and statistical studies. The impact ofВ advertising and managing consumer behavior inВ this book is not considered. Therefore, the duration ofВ consumption, is aВ random value and its the mean value can be determined only on the basis ofВ observations ofВ existing markets and statistical studies.
Sometimes you must take into account the characteristics ofВ the purchase process that affect time ofВ consumption.
It is possible toВ allocate
1. Immediate purchase as the initial moment ofВ consumption.
2. Purchases when were refunded or replaced
3. Remote buying (online) with delayed time ofВ consumption.
4. The purchase requisition, which is not satisfied at the lack ofВ product.
5. The purchase requisition, which is not satisfied at the lack ofВ goods acceptable toВ the buyer.
6. The purchase requisition, which is not satisfied byВ reason ofВ the expiration ofВ the release ofВ goodsВ etc.
InВ determining the parameters ofВ the market should be taken into account that the duration ofВ consumption ofВ most goods (except commodities first needs) and the number ofВ users significantly toВ effects on the demand the stage when the market is not saturated and it very much toВ depend on the prices.
For example, at higher prices on televisions – the number purchases decreases, the time of consumption increases (less likely to alter the old model).
However, as we will see later, when the market is saturated demand – growth in consumption (purchases) stops regardless of price.
1.5.4. Characteristics ofВ quality ofВ service ofВ goods
Quality ofВ service ofВ suggestion on the market ofВ goods is characterized byВ the number ofВ unsold within aВ certain time ofВ the goods or length ofВ expectation ofВ sale.
There are two basic approaches, the two disciplines ofВ servicing coming toВ market goods: apparent losses and conditional losses.
AВ discipline services with obvious lossesВ is the discipline inВ which the marketed product needs toВ be sold, and if all consumers are busy, then it is removed from sale and no longer has on the market no effect. This discipline is called aВ discipline with obvious losses. This discipline services implies that if the manufacturer received the refuse due toВ lack ofВ demand (all groups ofВ consumers are busy), then abandons further attempts toВ sell the goods (re-attempts ofВ implementation are considered as aВ special random flow).
ToВ quantify the quality ofВ service with obvious losses are calculated the following quantities:
loss in the number of requests for suggestion of goods —
R
.;
loss of volume of goods – R
.В ;
loss of implementation time – P t
The loss ofВ the number ofВ requests received for supply ofВ goods inВ the time interval [t
, t
) is the ratio ofВ the number ofВ lost (not serviced) during this period (t
, t
) toВ number received inВ the same time.
P
(t
t
) =c
(t
t
) /c (t
.t
)
Losses inВ volume ofВ goods inВ the time interval [ t
, t
) is the ratio of lost quantity (what not sold) in this period of time of the goods – a
(t
t
) toВ the incoming volume goods during the this time, aВ (t
, t
В ).
Losing time over aВ period ofВ time [t
, t
) it is the ratio length ofВ all ofВ time periods toВ time, when all groups ofВ consumption is available for the group ofВ producers are busy.
If inВ the expression for the loss ofВ the requests number received for supply ofВ goods, the and inВ expression, the loss ofВ the amount ofВ goods and loss ofВ time insert the mathematical expectation ofВ the corresponding random variables, and then we can talk about the likelihood ofВ losses these goods.
AВ discipline ofВ service with the expectation (conditional losses) is aВ discipline inВ which incoming goods toВ the time ofВ the lack ofВ free groups ofВ consumers is not losing and becomes on the storage (discipline ofВ service with waiting).
ToВ quantify the quality ofВ service with the expectation calculated inВ the following characteristics:
– the probability that the waiting time is greater than zero – that is, the probability of queue – p (γ >0);
– the probability of waiting for any incoming product over time t – p (γ> t); the probability of waiting of the delayed goods over time t – p (γ> t);
– the average waiting time in relation all requests received – (γ )
– the average waiting time of only a delayed requests – γ
В ;
– the probability that the queue length exceeds the specified value r is p ( R>r );
– average queue length – C
The main characteristics – p ( γ >0) and p ( γ > t ).
1.5.5. Traffic-carrying capacity market
One ofВ the most important properties ofВ the market is its efficiency.
As indicators ofВ the effectiveness ofВ the market as well as economic (price, cost ofВ goods, etc.) can be used such characteristics as the traffic-carrying capacity market.
Under aВ traffic-carrying capacity market we will understand the value ofВ the intensity ofВ served supplies ofВ goods (sold goods for aВ certain period ofВ time) when aВ certain value ofВ loss (what not sold).
The traffic carrying capacity ofВ market depends on the value ofВ losses that satisfy market participants, the number ofВ consumer groups, distribution ofВ duration ofВ consumption and services discipline ( obvious losses or conditional loss), the type ofВ the input flow.
The quantity ofВ loss which characterizes the condition ofВ the market, different for different categories ofВ goods.
Obviously, the more norm ofВ allowable losses, the more aВ traffic-carrying capacity market and the worse the quality ofВ producers service, i.e. market can accept from manufacturers aВ greater ofВ goods and most toВ lose or will keep on storage more aВ specified time.
Flow ofВ goods (offer) inВ mathematical models, often adopted as the simplest flow, flow ofВ Palm or primitive. InВ these cases it is relatively easy toВ obtain the solution ofВ the problem using analytical methods.
Real flows ofВ goods have aВ more complex structure, and the decision ofВ task is usually carried out according results received for the simplest stream inВ the form ofВ the boundary estimates ofВ indicators (results for simplest stream better or worse).
The most convenient distribution function ofВ service time from the point ofВ view ofВ the analytical description and analysis ofВ the throughput ofВ the market is the exponential distribution, since it has no aftereffect.
Practical widely application founds probability distributions – density distribution, Erlang distribution etc.
Discipline ofВ service has aВ significant impact on the mathematical model ofВ the market, therefore it is necessary toВ describe inВ great detail. For example, inВ the system with the expectation ofВ the accumulated goods may be return from storage back on market:
– in order of receipt;
– in the reverse order of the order of arrival;
– random order;
– with different kinds of priorities.
Traffic-carrying capacity market ofВ consumer groups may be estimated the ratio ofВ intensity a
ofВ supplies ofВ goods П… what was serviced.
О· =aВ
/В П…,
The value О· is called the average traffic-carrying capacity market, or average utilization ofВ one group ofВ consumers.
For example, inВ one people can buy 400В kg ofВ milk per year. ByВ delivery toВ the dairy market inВ 1991, was sold toВ 347В kg ofВ milk per person per year, О· = (347/400=0.8675), and inВ 2014В sold 240В kg ofВ milk per person per year (О· =240/400=0.6) [4.5]. Therefore, the capacity ofВ the dairy market fell byВ 26%/
The capacity market can be analyzed from the point ofВ view ofВ arriving ofВ the supply О· =f (y) or inВ terms ofВ aВ number ofВ consumer groups О· = f ( П… ) ) at fixed values ofВ losses.
The value ofВ О· byВ increasing the intensity ofВ the incoming ofВ supply asymptotically approaches unity. This is due toВ the decrease inВ the influence ofВ temporary fluctuations ofВ demand. This property shows the usefulness ofВ combining groups ofВ consumers (larger market), which reduces these oscillations.
1.6. Tests toВ ChapterВ 1
1.В The intensity ofВ the supply ofВ goods is it_________
a. the number ofВ calls per unit ofВ time
b. the total duration ofВ calls per unit ofВ time
c. the demand ofВ products per unitВ time
d. amount ofВ goods that passed system
2. Serviced market offer for time period (t
, t
) is it_________
a. the number ofВ products sold over time (t
,В t
)
b. the difference between the received and aВ serviced supply ofВ goodsВ for
the considered period ofВ time.
c. the sum ofВ the quantities ofВ goods handled byВ this group
consumers for the period (t
,В t
)
d. the sum ofВ the requirements for aВ certain number ofВ goods, served during (t
,В t
)
3. The unit ofВ measure ofВ the intensity ofВ demandВ ofВ the goods is accepted _________
a. the number ofВ purchases perВ hour
b. the relative value ofВ consumption at the maximum consumptionВ (Preal= Pmax) per unit time.
c. the amount ofВ goods delivered
d. the amount ofВ applications received for the goods
4. The time of greatest demand – THD _________
a. time ofВ the season with the highest demand
b. time from sampling successive periods ofВ time with maximumВ load
c. time ofВ maximum demand for each month
d. continuous given period ofВ time, with aВ maximum value ofВ the intensity ofВ demand
5.The main parameters ofВ the demand are: __________
a. the number ofВ consumer groups -n; the average number ofВ bids on the goods unit ofВ time; the average duration ofВ consumptionВ ofВ service the one applicationВ t.+
b. the number ofВ consumer groups -n; the average number ofВ bids on the goods, price per time unit; price per unit;
c. the quantity ofВ the goods; the average number ofВ bids on the goods, price per unit ofВ time; the average duration ofВ consumptionВ ofВ service the one applicationВ t.
d. the number ofВ consumers is n; the number ofВ served requests for goods; the average duration ofВ consumptionВ ofВ service the one applicationВ t.
6. When overproduction ofВ goods and lowering the prices ofВ the goods the number ofВ consumers ______
a. increases
b. reduced
c.do stabilize
7. The average number ofВ requests from one consumer per unit time from the k groups ofВ consumers is determined byВ ______,
a.. according the group with aВ large number ofВ requests
b. according the weighted average numberВ of
c. according the most frequently used number
d. randomly
8. The time consumption ofВ the goods can be _________ time ofВ possession
a. equally
b.В less
c.В more
d. dependent on the storage conditions
9. Under aВ traffic-carrying capacity market we will understand the value. ________
a. intensity serviced supplies for aВ certain period ofВ time when aВ certain amount ofВ losses
b. intensity serviced supplies for aВ certain period ofВ time
c. the amount ofВ goods sold over aВ certain period ofВ time
d. the number ofВ applications for aВ certain product for aВ certain period ofВ time
10. The more allowable are losses, then_________
a. less aВ traffic-carrying capacity market and the worse the quality services for producers
b. more aВ traffic-carrying capacity market and the worse the quality ofВ service manufacturers
c. better quality ofВ service manufacturers
d. stable quality maintenance ofВ producers
11. A traffic-carrying capacity market is estimated —
.aВ number ofВ consumers
b. value ofВ losses
c. relation the intensity ofВ services supply toВ the number ofВ groups ofВ consumers
d. number ofВ suppliers
Chapter 2. AВ mathematical model ofВ the market
2.В 1.Definition
For the mathematical estimate ofВ processes inВ the economy it is necessary toВ create aВ model that would reflect them inВ mathematical form. InВ conjunction with the definition ofВ units ofВ measure ofВ supply, demand and also definition ofВ relative consumption, this model allows toВ obtain and toВ provide aВ quantitative calculation, analysis and give assessment ofВ possible market behavior. The proposed model is shown inВ Fig.2.1.
Figure 2.1.AВ mathematical model ofВ the market
The market contains aВ subsystems they performs the following tasks.
Production system – generates a random flow of goods (in general case different from each other). Under the product refers to any product or a service offered for sale, in which there is a need (in the following we will show how to define the demand in numerical form). These can be food items such as milk and other food products, cars, homes, cleaning home, stocks, services in construction of houses, etc.
The flow enters the distribution system.
The distribution system can the request handle immediately or with some delay. The refusal can be ofВ two types:
– the refusal to accept the goods because of lack of demand;
– or installation on waiting of implementation (accumulation of goods);
There are two types ofВ distribution systems.
The first of these it is the task as quickly as possible to sell the goods without any additional conditions. In queuing theory, such a system service is called“full accessibility system” because it provides equal access to the system of consumption, and this system can be not take into account for calculation of behavior of the market.
.Systems of the second type – they sell goods given the specified priorities and limitations for individual consumers or consumption. For example, the sale of tobacco, alcohol, weapons … systems of this type is called “notfull accessibility system”.
If the order is accepted, the distribution system seeks deliver its inВ the system consumption.
Consumption system serves the flow ofВ goods. This is reflected inВ the fact that individual consumer or group ofВ consumers can buy the offered goods or toВ refuse from the purchase.
If all consumer groups are busy, then the system ofВ consumption is refuses from the purchase ofВ goods, i.e. occur losses.
And not purchased goods more is not offered for example can be removed or inВ distribution system can be supplied toВ the queue for waiting the implementation (the accumulation and storage ofВ goods). InВ this case it is important toВ know the waiting timeВ t
2.2. Types ofВ SERVICE systems and input flows goods
Flow ofВ events (inВ our case products) is aВ sequence ofВ goods coming through any interval or any moment ofВ time.
Deterministic flow of events – the sequence in which events arrive in a certain, strictly fixed non-random moments or in certain, strictly fixed, non-random intervals. For example, sale of products from the conveyor.
The random flow of events is differs from the deterministic, only one property that the moment of the arrive of goods and intervals arriving time between deliveries are not strictly fixed and random variables. Deterministic flow is a special case of random flows and is rare in practice. Strictly speaking, even such flows often contain moment’s fortuity. In this regard, in the theory of mass service flows basic attention is devote the consideration of random call flows.
The flow ofВ events can be determined byВ one or more ofВ the three equivalent ways:
1. AВ sequence ofВ moments ofВ occurrence ofВ the event t
, t
,…,t
,,
For example, the probability ofВ admission ofВ aВ particular product, and at the timeВ t
.
This may be aВ possibility that mobile phones will be shipped toВ the store every Friday.
InВ the lottery is, for example, the probability ofВ occurrence inВ the draw ofВ lottery with numbers 5, 7, 15В will be the number 6. These events iВ can be described as moments ofВ time.
2 The sequence of intervals between the events – the moments of z
, z
,…,z
Example, the probability that mobile phones will be shipped toВ the store every 7В days.
InВ the lottery is, for example, the probability ofВ occurrence number 6В every 5В lottery draws.
3. AВ sequence ofВ numbers k
, k
,…,k
, determining the quantity ofВ events arriving within the time intervals [t
, t
), [t
, t
),…, [t
,В t
).
Example, the probability that during 5В months (counted from the initial moment t
 – t
) fresh fish will goes on sale 5,6, … 10 times.
2.2.1. Types ofВ systems service goods
Below show aВ classify service goods, it is widely used inВ queuing theory [2.1], but inВ the economy has not yet is applied.
Types ofВ systems service the goods is shown on Fig.2.2
Figure 2. 2. Types ofВ systems service the goods
We will explain the figure
System with obvious losses
System, inВ which if the goods is not sold, then it is removed from the market, is called system with obvious losses
System ofВ service with the expectation
System, inВ which if the goods is not sold, then it will stand is queued for implementation is called system ofВ service with the expectasion
2.2.2. The types ofВ input flows
The flow ofВ goods is aВ sequence ofВ goods incoming through any interval or at through any moments ofВ time.
Deterministic flow of events – the order in which events arrive in a certain, strictly fixed non-random moments or in certain, strictly fixed, non-random intervals. For example, sale of products from the conveyor.
The random flow ofВ events differs from the deterministic, those and only, that moments ofВ supply ofВ goods and intervals between deliveries are random variables, but not strictly fixed values. Deterministic flow is aВ special case ofВ random flows and is rare inВ practice. Strictly speaking, even inВ such streams often occurs aВ random event.
The flow ofВ events can be determined byВ one or more ofВ the three equivalent ways:
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