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Mathematics of the market. Service random flow. Alexandr Berlin
Читать онлайн.Название Mathematics of the market. Service random flow
Год выпуска 0
isbn 9785448592393
Автор произведения Alexandr Berlin
Жанр Математика
Издательство Издательские решения
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.