Mathematics of the market. Service random flow. Alexandr Berlin
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Название: Mathematics of the market. Service random flow
Автор: Alexandr Berlin
Издательство: Издательские решения
Жанр: Математика
isbn: 9785448592393
isbn:
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≤ Tmax 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> Tmax, 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 (Preal= Pmax) 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
kTGD=ATGD/Aobs,
where A TGD. the demand value for TDG;
Aobs. the demand value during the observation.
1.5.2. Main parameters and calculation of the intensity of demand
The main parameters of the demand are:
· the number of consumer groups -n;
· the average number of requests for goods received from one group of consumers per unit time ;
· the average duration of consumption for maintenance of a single application of t.
Consider the possible composition of consumers, which differ in the average intensity and average duration of consumption:
– individual consumers;
– intermediaries (e.g., agencies for the purchase and sale of apartments);
– Firms and government agencies.
By lowering the prices of the goods the number of consumers may increase. The lengthy existence of the product on the market the number of consumers can be stabilized, and they are not changing or changing insignificantly.. A similar reaction may be in case overproduction when the relative consumption is close to 1(unit).Real consumption is stable.
1.5.3. The average number of requests from one consumer per unit of time
In accordance with the categories source of requests average number of request for goods per unit of 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,
ni – the number of sources i —th category.
Then СКАЧАТЬ