Statistical Characteristics of a Flow of Substance in a Channel of Network that Contains Three Arms

Abstract

We study the motion of a substance in a channel that is part of a network. The channel has 3 arms and consists of nodes of the network and edges that connect the nodes and form ways for motion of the substance. Stationary regime of the flow of the substance in the channel is discussed and statistical distributions for the amount of substance in the nodes of the channel are obtained. These distributions for each of the three arms of the channel contain as particular case famous long-tail distributions such as Waring distribution, Yule-Simon distribution and Zipf distribution. The obtained results are discussed from the point of view of technological applications of the model (e.g., the motion of the substance is considered to happen in a complex technological system and the obtained analytical relationships for the distribution of the substance in the nodes of the channel represents the distribution of the substance in the corresponding cells of the technological chains). A possible application of the obtained results for description of human migration in migration channels is discussed too.

Proof that Eq. (10) is a solution of Eq. (7) for main arm of the channel

Let us consider the first equation from Eq. (7) for main arm. In this case $i=0$ and Eq. (10) becomes $x_0^A=x_0^{*A}+b_{00}^A\exp [-(\alpha _0^A+~\gamma _0^A)t]$. The substitution of the last relationship in the first of the Eq. (7) leads to the relationship

$$\begin{aligned} 0=(\sigma _0-\alpha _0^A-\gamma _0^{*A})x_0^* +b_{00}^A\sigma _0\exp [-(\alpha _0^A+\gamma _0^{*A})t]. \end{aligned}$$

(23)

Let us assume $\sigma _0=\alpha _0^A+\gamma _0^{*A}$ and $b_{00}^A=0$. Then Eq. (10) describes the solution of the first of Eq. (7).

Let us now consider Eq. (7) for the main arm of the channel for $i=1,2,\dots$. Let us fix i and substitute the first of Eq. (10) in the corresponding equation from Eq. (7). The result is

$$\begin{aligned} \sum \limits _{j=0}^{i-1} \exp [-(\alpha _j^A+j\beta _j^A+\gamma _j^{*A})t]\big \{-b_{ij}^A(\alpha _j^A+j\beta _j^A+\gamma _j^{*A})-\\\nonumber -b_{i-1,j}^A[\alpha _{i-1}^A+(i-1)\beta _{i-1}^A]+b_{ij}^A(\alpha _i^A+i\beta _i^A+\gamma _i^{*A})\big \}=0 \end{aligned}$$

(24)

As it can be seen from Eqs. (12), (24) is satisfied.