Baxramov Umarkhodja, 1 Kaxarov Bakhodir

 Keywords: Modeling, stochastic, supply, random, autocorrelation, harmonic
 Introduction
The main task of modeling the processes of consumption of the target product in engineering 
networks in this work is to assess the reliability of the mathematical model of stochastic water 
consumption in each water consumption node for a long period of operation of the system proposed 
in [1, 2]. For modeling, a simple scheme for calculating the utility network is used, shown in Figure 
1. It is easy to see that this calculation scheme has a small size, the number of nodes and branches, 
which makes it possible to objectively evaluate the mathematical model for a network of varying 
complexity. With the combined use of the method of modeling the processes of consumption of the 
target product and the hydraulic calculation of the supply and distribution of the target product in the 
networks under various modes of its operation, it is possible to use these models as an expert 
judgment. designed new systems for the supply and distribution of target products at the stages of 
their operation. 
 
 Materials and Methods 
When considering various possible methods for modeling the processes of consumption of the target 
product (heat supply, water supply and gas supply), it was concluded that for the purposes of this 
work, that is, for building models intended for use as part of the general algorithm for simulation 

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Solid State Technology
Volume: 65 Issue: 1 
Publication Year: 2022
64
Archives Available @ www.solidstatetechnology.us 
modeling of stochastic flow distribution of rather large engineering networks including dozens of 
nodes are the most acceptable models that can be represented as a composition of parametrically 
specified functions[1,2]: 
)...]
(
),
(
,...),
,
,
(
,...),
,
,
(
[
)
(
2
1
2
1
1
2
2
1
1
1
t
t
t
x
x
t
F
t
U







(1) 
where
1

and
2

- is a function with random parameters
...;
,
1
i
x

)
(
),
(
2
1
t
t


- random noises with specified properties.
In this paper, a model of the form (1), is concretized by the example of modeling the processes 
of consumption of the target product in water supply systems. 
First of all, we note that in (1) the most acceptable form of the parameter F, which determines 
the method of composition of functions, is its representation as a sum of some harmonic components 
and residual random noise, that is:









i
t
A
F
),
cos(
(2) 
where


,
,
A
- amplitude, frequency and phase shift for the I – th harmony;
t - current time. 
With such a representation of F, which naturally follows from the logical analysis of the 
processes of consumption of the target product in water supply, heat supply and gas supply systems, 
the consumption processes are determined by cyclical fluctuations in the rhythm of the population's 
activity, a simulated random processes of consumption of the target product for any node of the 
system will look like:
,
)
(
)
(
0
Q
t
Q
F
t
U



(3) 
where
Q
- mathematical expectation of processes of consumption of the target product on the 
modeling interval; 
)
(
0
t
Q
- random noise. 
Since the value 
Q
in (3) is determined quite simply from the data and the specific (per person)
consumption of the target product and the number of the served population, the problem of modeling 
the processes of consumption of the target product is reduced to modeling F and 
)
(
0
t
Q
. Analysis of 
real consumption processes shows that any of their daily implementation can be approximated in the 
form (2) with some residual random term, 
)
(
0
t
Q
, having zero mathematical expectation and variance 
2
0
Q

. Moreover, it is known [3,4] , that process 
)
(
0
t
Q
is stationary, that is, its mathematical 
expectation and 
2
0
Q

do not depend on t . Considering a sufficiently large number of realizations of 
processes of consumption of the target product for various real objects, it is possible to determine the 
parameters of the distributions of all random variables in the model(3) - 


,
,
A
, and
0
Q
(for
0
Q
the parameter t may not be specified, since this part of the process
)
( t
U
stationary). 
The above data processing of the processes of consumption of the target product stricts of the 
city of Tashkent were used, where an experimental study of water consumption modes was carried 
out) showed, hat the distribution laws of all random variables required for the model (3),can be fairly 
accurately approximated by the normal probability distribution law.This makes it possible to 
implement the process of modeling the processes of consumption of the target product for all nodes 
of the system using only one pseudo-random number generator, distributed according to the normal 
law [6] therefore, the simulation of random parameters in (2) can be represented in the form:







n
(4) 

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