Alexandr Berlin - Mathematics of the market. Service random flow

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 t 1was busy υ 1consumer groups, during the time t 2. was busy v 2consumer groups, etc. (Fig.1.1). In General can imagine that during the time t iwas busy υ iconsumer groups,

Σ n 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) Σ n 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‘= (v 1t 1 + v 2t 2 +•••+ v nt n) / (t 1 + t 2 + ⋯ 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 maxis independent of the stream type of the incoming supplies goods with the average value t, Consider the time interval [t1, t2) such that t 2 – t 1> T max. The mathematical expectation of the number of supplies placed on the market in the time interval [t 1, t 2) denoted by Λ (t 1, t 2) =μ (t 1.t 2).

Part of these supplies is consumed to the moment t 2(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 [t 1, t 2) and what not purchased to the time t 2, denote ρ. In addition to products that arriving to market in the time interval [t 1, t 2), it is necessary to take into account products that have arrived up to time t 1and the time t 2are not yet purchased. We denote the expected number of goods that began before the time t 1and are ended in the time interval [t 1, t 2), denote ε (Fig. 1.1), and the expected number of calls that started before t1 and ended after the time t 2, denote ζ (Fig. 1.1 g). Since t 2 – t 1> 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 [t 1, t 2),

a (t 1,t 2) = [μ (t 2—t 1) – ρ +ε] ⋅t ̅=μ⋅ t’ consum(t 2—t 1)

and the intensity of the inbound supplies:

a= [a (t 1,t2)] / (t 2—t 1) = 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 t 1=0 and t 2=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

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.

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);