Mathematics of the market. Service random flow

– 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,

ni – the number of sources i —th category.

Then if on the market have k groups of consumers the average number of applications from one group of consumers will be determined from the expression.

c‘= (Σ Ki=1 ci ni) / Σkini

The average duration of consumption (t’).

The duration of one consumption it is to the period of time from the date of acquisition until the moment you start searching the same product.

Time of consumption is always less time of possession of the goods.

For example, the owner of the car bought it 3 years ago. If he started looking for a new car without selling the old, the time of consumption will be 3 years.

The duration of consumption of food products is determined by the physical needs of the person also by the allowable time storage.

The duration of consumption of clothing is determined by its wear resistance, but now mainly accordance to fashion.

The duration of consumption of control automates, computing machines of cars is limited by their reliability and by their main indicators.

Therefore, the duration of consumption, is a random value and its the mean value can be determined only on the basis of observations of existing markets and statistical studies. The impact of advertising and managing consumer behavior in this book is not considered. Therefore, the duration of consumption, is a random value and its the mean value can be determined only on the basis of observations of existing markets and statistical studies.

Sometimes you must take into account the characteristics of the purchase process that affect time of consumption.

It is possible to allocate

1. Immediate purchase as the initial moment of consumption.

2. Purchases when were refunded or replaced

3. Remote buying (online) with delayed time of consumption.

4. The purchase requisition, which is not satisfied at the lack of product.

5. The purchase requisition, which is not satisfied at the lack of goods acceptable to the buyer.

6. The purchase requisition, which is not satisfied by reason of the expiration of the release of goods etc.

In determining the parameters of the market should be taken into account that the duration of consumption of most goods (except commodities first needs) and the number of users significantly to effects on the demand the stage when the market is not saturated and it very much to 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 of quality of service of goods

Quality of service of suggestion on the market of goods is characterized by the number of unsold within a certain time of the goods or length of expectation of sale.

There are two basic approaches, the two disciplines of servicing coming to market goods: apparent losses and conditional losses.

A discipline services with obvious losses is the discipline in which the marketed product needs to 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 a discipline with obvious losses. This discipline services implies that if the manufacturer received the refuse due to lack of demand (all groups of consumers are busy), then abandons further attempts to sell the goods (re-attempts of implementation are considered as a special random flow).

To quantify the quality of service with obvious losses are calculated the following quantities:

loss in the number of requests for suggestion of goods —

R request.;

loss of volume of goods – R goods. ;

loss of implementation time – P t

The loss of the number of requests received for supply of goods in the time interval [t 1, t 2) is the ratio of the number of lost (not serviced) during this period (t 1, t 2) to number received in the same time.

Prequest (t1.t2) =clost (t1.,t2) /c (t1.t2)

Losses in volume of goods in the time interval [ t 1, t 2 ) is the ratio of lost quantity (what not sold) in this period of time of the goods – alost (t 1, t 2 ) to the incoming volume goods during the this time, a (t 1, t 2 ).

Losing time over a period of time [t1, t 2) it is the ratio length of all of time periods to time, when all groups of consumption is available for the group of producers are busy.

If in the expression for the loss of the requests number received for supply of goods, the and in expression, the loss of the amount of goods and loss of time insert the mathematical expectation of the corresponding random variables, and then we can talk about the likelihood of losses these goods.

A discipline of service with the expectation (conditional losses) is a discipline in which incoming goods to the time of the lack of free groups of consumers is not losing and becomes on the storage (discipline of service with waiting).

To quantify the quality of service with the expectation calculated in 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 delay ;

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

– average queue length – C delay.

The main characteristics – p ( γ >0) and p ( γ > t ).

1.5.5. Traffic-carrying capacity market

One of the most important properties of the market is its efficiency.

As indicators of the effectiveness of the market as well as economic (price, cost of goods, etc.) can be used such characteristics as the traffic-carrying capacity market.

Under a traffic-carrying capacity market we will understand the value of the intensity of served supplies of goods (sold goods for a certain period of time) when a certain value of loss (what not sold).

The traffic carrying capacity of market depends on the value of losses that satisfy market participants, the number of consumer groups, distribution of duration of consumption and services discipline ( obvious losses or conditional loss), the type of the input flow.

The quantity of loss which characterizes the condition of the market, different for different categories of goods.

Obviously, the more norm of allowable losses, the more a traffic-carrying capacity market and the worse the quality of producers service, i.e. market can accept from manufacturers a greater of goods and most to lose or will keep on storage more a specified time.

Flow of goods (offer) in mathematical models, often adopted as the simplest flow, flow of Palm or primitive. In these cases it is relatively easy to obtain the solution of the problem using analytical methods.

Real flows of goods have a more complex structure, and the decision of task is usually carried out according results received for the simplest stream in the form of the boundary estimates of indicators (results for simplest stream better or worse).

The most convenient distribution function of service time from the point of view of the analytical description and analysis of the throughput of the market is the exponential distribution, since it has no aftereffect.

Practical widely application founds probability distributions – density distribution, Erlang distribution etc.

Discipline of service has a significant impact on the mathematical model of the market, therefore it is necessary to describe in great detail. For example, in the system with the expectation of 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 of consumer groups may be estimated the ratio of intensity aserv of supplies of goods υ what was serviced.

η =a 0 / υ,

The value η is called the average traffic-carrying capacity market, or average utilization of one group of consumers.

For example, in one people can buy 400 kg of milk per year. By delivery to the dairy market in 1991, was sold to 347 kg of milk per person per year, η = (347/400=0.8675), and in 2014 sold 240 kg of milk per person per year (η =240/400=0.6) [4.5]. Therefore, the capacity of the dairy market fell by 26%/

The capacity market can be analyzed from the point of view of arriving of the supply η =f (y) or in terms of a number of consumer groups η = f ( υ ) ) at fixed values of losses.

The value of η by increasing the intensity of the incoming of supply asymptotically approaches unity. This is due to the decrease in the influence of temporary fluctuations of demand. This property shows the usefulness of combining groups of consumers (larger market), which reduces these oscillations.

1.6. Tests to Chapter 1

1. The intensity of the supply of goods is it_________

a. the number of calls per unit of time

b. the total duration of calls per unit of time

c. the demand of products per unit time

d. amount of goods that passed system

2. Serviced market offer for time period (t1, t2) is it_________

a. the number of products sold over time (t1, t2)

b. the difference between the received and a serviced supply of goods for

the considered period of time.

c. the sum of the quantities of goods handled by this group

consumers for the period (t1, t2)

d. the sum of the requirements for a certain number of goods, served during (t1, t2)

3. The unit of measure of the intensity of demand of the goods is accepted _________

a. the number of purchases per hour

b. the relative value of consumption at the maximum consumption (Preal= Pmax) per unit time.

c. the amount of goods delivered

d. the amount of applications received for the goods

4. The time of greatest demand – THD _________

a. time of the season with the highest demand

b. time from sampling successive periods of time with maximum load

c. time of maximum demand for each month

d. continuous given period of time, with a maximum value of the intensity of demand

5.The main parameters of the demand are: __________

a. the number of consumer groups -n; the average number of bids on the goods unit of time; the average duration of consumption of service the one application t.+

b. the number of consumer groups -n; the average number of bids on the goods, price per time unit; price per unit;

c. the quantity of the goods; the average number of bids on the goods, price per unit of time; the average duration of consumption of service the one application t.

d. the number of consumers is n; the number of served requests for goods; the average duration of consumption of service the one application t.

6. When overproduction of goods and lowering the prices of the goods the number of consumers ______

a. increases

b. reduced

c.do stabilize

7. The average number of requests from one consumer per unit time from the k groups of consumers is determined by ______,

a.. according the group with a large number of requests

b. according the weighted average number of

c. according the most frequently used number

d. randomly

8. The time consumption of the goods can be _________ time of possession

a. equally

b. less

c. more

d. dependent on the storage conditions

9. Under a traffic-carrying capacity market we will understand the value. ________

a. intensity serviced supplies for a certain period of time when a certain amount of losses

b. intensity serviced supplies for a certain period of time

c. the amount of goods sold over a certain period of time

d. the number of applications for a certain product for a certain period of time

10. The more allowable are losses, then_________

a. less a traffic-carrying capacity market and the worse the quality services for producers

b. more a traffic-carrying capacity market and the worse the quality of service manufacturers

c. better quality of service manufacturers

d. stable quality maintenance of producers

11. A traffic-carrying capacity market is estimated —

.a number of consumers

b. value of losses

c. relation the intensity of services supply to the number of groups of consumers

d. number of suppliers

Chapter 2. A mathematical model of the market

2. 1.Definition

For the mathematical estimate of processes in the economy it is necessary to create a model that would reflect them in mathematical form. In conjunction with the definition of units of measure of supply, demand and also definition of relative consumption, this model allows to obtain and to provide a quantitative calculation, analysis and give assessment of possible market behavior. The proposed model is shown in Fig.2.1.

Figure 2.1.A mathematical model of the market

The market contains a 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 of 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 of 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 “not full accessibility system”.

If the order is accepted, the distribution system seeks deliver its in the system consumption.

Consumption system serves the flow of goods. This is reflected in the fact that individual consumer or group of consumers can buy the offered goods or to refuse from the purchase.

If all consumer groups are busy, then the system of consumption is refuses from the purchase of goods, i.e. occur losses.

And not purchased goods more is not offered for example can be removed or in distribution system can be supplied to the queue for waiting the implementation (the accumulation and storage of goods). In this case it is important to know the waiting time t

2.2. Types of SERVICE systems and input flows goods

Flow of events (in our case products) is a sequence of goods coming through any interval or any moment of 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 of events can be determined by one or more of the three equivalent ways:

1. A sequence of moments of occurrence of the event t1, t2,…,tn,,

For example, the probability of admission of a particular product, and at the time t1.

This may be a possibility that mobile phones will be shipped to the store every Friday.

In the lottery is, for example, the probability of occurrence in the draw of lottery with numbers 5, 7, 15 will be the number 6. These events i can be described as moments of time.

2 The sequence of intervals between the events – the moments of z1, z2,…,zn

Example, the probability that mobile phones will be shipped to the store every 7 days.

In the lottery is, for example, the probability of occurrence number 6 every 5 lottery draws.

3. A sequence of numbers k1, k2,…,kn, determining the quantity of events arriving within the time intervals [t0, t1), [t0, t2),…, [t0, tn).

Example, the probability that during 5 months (counted from the initial moment t0 – t15) fresh fish will goes on sale 5,6, … 10 times.

2.2.1. Types of systems service goods

Below show a classify service goods, it is widely used in queuing theory [2.1], but in the economy has not yet is applied.

Types of systems service the goods is shown on Fig.2.2

Figure 2. 2. Types of systems service the goods

We will explain the figure

System with obvious losses

System, in which if the goods is not sold, then it is removed from the market, is called system with obvious losses

System of service with the expectation

System, in which if the goods is not sold, then it will stand is queued for implementation is called system of service with the expectasion

2.2.2. The types of input flows

The flow of goods is a sequence of goods incoming through any interval or at through any moments of 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 of events differs from the deterministic, those and only, that moments of supply of goods and intervals between deliveries are random variables, but not strictly fixed values. Deterministic flow is a special case of random flows and is rare in practice. Strictly speaking, even in such streams often occurs a random event.

The flow of events can be determined by one or more of the three equivalent ways: