Mathematics / 5. Mathematical design

The candidate of the physicist - mathematical sciences Kalzhanov M. U.

Kostanay State University name A.Baitursynov

The definition of simulation experiments

The main advantage of the simulation model from the analytical is due to its details can be made close to the object being modeled. Although the approach is associated with the increasing complexity and more time developing a simulation model. As a result, there may be situations where the simulation model development time is comparable with the development of the system itself and, as a consequence, reduces the relevance derived from the simulation results. At the development of simulation models is necessary and important a creation of basic software modules which describe typical situations arising in simulation modeling. The process of drafting aggregation models on the basis of the available modules accelerates the process of simulation modeling of the investigated system.

The volume of modeling is determined by the nature of the investigated variables (random events, values or processes), the required accuracy, and the specific conditions of the simulation.

We define the required number of experiments in assessing the likelihood of an event, which is the estimate of the frequency of P * = L / N, where L - the number of successful experiments in the simulation, N - total number of tests. The frequency of P * is a random variable, as it will take on different values in repeating series of experiments, N. According to the limit theorem of probability theory, random variable P * is distributed approximately normal distribution.

We define a discrete random variable Z with the law of the distribution P (Z = 1) = P *, P (Z = 0) = 1 - P *.

The expectation and variance of Z are

M = 1 • P * + (1 - P *) • D = P *,

D = P * (1 - P *) 2 + (1 - P *) • (0 - P *) 2 - P * (1 - P *).

Then, the expectation and variance of

, (1)

(2)

Let's find number of tests, at which the meaning Х differs from probability P^* less, than on the given size with the given reliability.

For the normalized size Х

(3)

In view of the normal law of distribution of size the expression (3) will look like

(4)

where β= (1-α) /2 — significance value;

U_β — quintile, corresponding to the value

Meanings β and U_β are tabulated. For example, at β = 0,61, U_β = 1,28 , β = 0,025 , U_β = 1,96 etc.

From (4) we shall receive (5)

At modeling the size P^* is usually unknown. Therefore modeling of volume N = 50 – 100 samples in the beginning is spent, on which is defined P^*, and then from (5) finally is N.

In tab. 1 the necessary number of realization for reception of an estimation L/ N with accuracy is resulted ε and reliability a = 0,95 for various meanings P^*.

The table 1

Р*		ε
		0,05	0,02	0,01
0.2	0.8	250	1500	6200
0,3	0,7	330	2100	8400
0,4	0.6	380	2300	9400
0,5		390	2400	9800

The meanings, given in tab. 1, the values of the required volume of simulation experiments themselves even simple estimates indicate the need for a large number of implementation on a computer.

For specific cases, modeling is necessary to make such an assessment and determine the accuracy of calculations, depending on the traded volume modeling.

References:

1. Buslenko N.P simulation of complex systems. - Moscow: Nauka, 1978.

2. Lukyanov V.S Meeting the challenges in engineering simulation methods: Studies. benefit VolgPI. - Volgograd, 1989.