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Adv Syst Sci Appl 2024; 01; 82-94 
Published online at https://ijassa.ipu.ru. 

Synchronization of Inbound and Outbound Flows 
at Stations in the Model of Freight 

Transportation Organization  

Nerses Khachatryan* 

State Academic University for the Humanities, Moscow, Russia  

Abstract: This article presents a model for organizing railway freight transportation between two node 
stations based on the interaction of neighboring stations depending on their technical capabilities and 
the demand for freight transportation. The main objective of such interaction is to synchronize the 
inbound and outbound flow at the stations, specifically reducing the degree of inconsistency between 
the receipt and dispatch of goods at the stations. This characteristic represents the imbalance between 
the volume of incoming and outgoing goods at the stations per unit of time and is described by a non-
negative function bounded above by unity. Its dynamics are described by a system of differential 
equations containing a set of parameters characterizing the infrastructure of the stations, the mode of 
distribution of goods from the final node station, and the demand for freight transportation. The range 
of parameter variations for which the specified system has a solution has been determined, as well as 
such a set of these parameters for which the task of synchronizing the inbound and outbound flow at the 
stations is best solved. It has been found that regardless of the initial value, starting from a certain point 
in time, the degree of inconsistency between the receipt and dispatch of goods at all stations except the 
initial node station becomes zero. 

Keywords: freight transportation organization, flow synchronization, system of differential equations, 
model parameters, stationary solutions, stability of solutions. 

1. INTRODUCTION 

One of the largest foundational sectors of any country's economy is transportation. It provides 
geographical connectivity across the nation's territories and coordinates the functioning of all 
sectors of the economy. Transportation creates the conditions for the effective operation of a 
country, and its development is a vital component of economic modernization. Additionally, 
transportation contributes to the development of international economic relations, the exploration 
of new economic regions, and ensures the country's defense capabilities. 

Two major groups of mathematical models for transportation systems can be distinguished: 
1. The first group focuses on modeling transportation networks and their utilization. It includes 

models for calculating correspondences, such as the gravity model [63], entropy model [28, 52], 
models within the competing centers framework [21], as well as models for flow distribution 
within the network [57, 47, 3]. 

2. The second group involves modeling the dynamics of transportation flow [56]. It comprises 
fundamental classes of dynamic models: macroscopic (hydrodynamic), kinetic (gas dynamic), and 
microscopic models. Macroscopic models [17, 43, 34, 24] describe averaged characteristics of 
transportation flow and are sometimes referred to as hydrodynamic models because they liken the 
flow itself to the movement of compressible fluid. Macroscopic diagrams, which illustrate the 
relationships between performance parameters such as traffic density, traffic flow, and vehicle 
speed, are used to represent traffic states and system configurations [16, 26, 12]. Microscopic 

 
* Corresponding author: nerses-khachatryan@yandex.ru 

 



             SYNCHRONIZATION OF INBOUND AND OUTBOUND FLOWS AT STATIONS IN THE MODEL…             83 

Copyright ©2024 ASSA                                                                                            Adv. in Systems Science and Appl. (2024) 

models explicitly describe the movement of each individual vehicle. They provide a more detailed 
description of movement on specific segments of the transportation network but require 
significantly greater computational resources for practical implementation. The first microscopic 
models were introduced in the 1950s [51, 56]. Examples of such models include car-following 
models [25, 7], optimal velocity models [2, 59], the Treiber model [60], as well as cellular 
automaton models [15, 14]. Kinetic models occupy an intermediate position between macroscopic 
and microscopic models. In kinetic models, traffic flow is characterized by the density distribution 
of vehicles in phase space, and the dynamics of the phase density are described by kinetic 
equations. These equations result from averaging the effects of interactions among individual 
vehicles [30, 50]. 

It is worth noting that the models mentioned above are most suitable for studying automobile 
traffic. For a vast country like Russia, however, railway transportation plays a pivotal role. It 
ensures reliable and cost-effective delivery of goods, especially when rapid transportation of large 
volumes of cargo is required. Publications related to railway logistics can be categorized into three 
main groups, depending on the types of problems under investigation. 

The first group deals with the design of railway infrastructure [32, 42, 20]. 
The second group focuses on the management of locomotives and wagons. Depending on 

regulatory and market characteristics, different regions may have their own models to account for 
specific factors. For instance, the work of R. Fukasawa et al. [22] presents a model used by one of 
the largest railway operators in Latin America. Another example is the study by A. Chezelli et al. 
[13], which examines multiple optimization models for freight delivery by Swiss Federal 
Railways' Cargo Express Service. Several publications focus on models designed to address the 
specifics of the freight transport market in Italy [49, 11]. Some works present cost-minimization 
models for transporting goods across multiple European countries via railway [1, 33]. There are 
also models created for the Russian railway transport market [54, 45, 6]. 

The third group addresses railway planning tasks, which traditionally centered on scheduling 
freight train movements [44, 10, 46]. In recent years, publications in this group have been 
complemented by studies that apply macroscopic traffic theory to describe processes in railway 
transportation. N. Weik's work [61], for example, provides a theoretical analysis of traffic flow 
properties on unidirectional railway lines. It constructs macroscopic fundamental diagrams and 
demonstrates how they can be used to determine flow regimes and various phases of train 
movement, which is valuable for system design and operational planning. 

Another area of research that has been actively developing in recent years is related to 
predicting delays in railway systems. Trains in this system follow predefined schedules, which 
allows for the efficient use of routes and tracks. Deviations from such planned operations manifest 
as delays and can reduce the system's efficiency. Minor delays are often absorbed by built-in 
buffers and thus do not impact larger scales [63, 18]. However, logistical disruptions, often caused 
by external factors such as weather, occasionally lead to congestion or even large-scale stoppages 
with negative consequences for society and the economy [48, 9]. Most models that study delays 
are based on railway system schedules and typically use trains as agents that can incur delays [27, 
23, 29]. In contrast, in the work by Dekker et al. [19], delays are treated as variables associated 
not with trains but with nodes (stations) and edges of the railway network, which remain in place. 
The spread of delays between these nodes does not necessarily have to be described in terms of 
discrete trains and events but can be based solely on common (or even systemic) quantities, such 
as network topology and schedules. The authors draw an analogy with hydrodynamics: while 
delays are traditionally considered Lagrangian particles (i.e., following trains like a fluid carrying 
particles), they suggest considering delays from an Eulerian perspective (i.e., defining incoming 
and outgoing delays in a fixed spatial system). They refer to this representation of delays as 
diffusion-like spreading. When examining the microscale, it is expected that this unconventional 
approach to delay handling may be less accurate than more detailed models, but on a large scale, 
the performance of such a model increases. The model contains only basic schedule information 



84                                                              N. KHACHATRYAN 

Copyright ©2024 ASSA                                                                                            Adv. in Systems Science and Appl. (2024) 

(e.g., train frequencies and travel times), and all the model's information is embedded in a single 
matrix, facilitating the analysis of system properties. 

Another crucial issue in railway planning is the study of freight transportation modes and their 
corresponding cargo flows within a dynamic system that describes the transportation process as 
the interaction of key elements of railway infrastructure, primarily stations. This problem has been 
addressed in the works of L.A. Beklaryan and N.K. Khachatryan [4, 5, 35–41]. They present 
dynamic models in which the organization of freight transportation involves forming cargo flows 
based on station interactions. The rules of station interaction depend on the nature of demand for 
freight transportation. In cases of consistently high demand for freight transportation, the focus is 
on utilizing the station's technical capabilities to their fullest. In the absence of consistently high 
demand for freight transportation, the primary goal of station interaction is to synchronize 
incoming and outgoing flows, allowing for more efficient freight transportation, minimizing 
delays, and ensuring a smooth flow of cargo. The work [41] describes a model of organizing freight 
transportation between major node stations when there is no consistently high demand for freight 
transportation. It explores the relationship between the degree of mismatch between cargo arrival 
and departure at stations and the parameters that characterize the demand for freight transportation, 
the technical capacity of stations, and the degree of its utilization. It assumes that the stations are 
identical, meaning that the specified parameters are the same for all stations. This article is 
dedicated to advancing this model and examines the case where stations have different 
characteristics.  

2. PROBLEM STATEMENT 

The movement of cargo on a segment of the railway network between two terminal stations, 
connected by multiple intermediate stations, is being considered. The primary characteristic of 
station 𝑖 at time 𝑡, where 𝑖 ∈ {0, 1, . . , 𝑚, 𝑚 + 1}, is the degree of mismatch between cargo 
reception and dispatch 𝑧 (𝑡), which varies in the range [0, 1]. The precise definition is provided in 
the work [41]. 

The technical capacity of station 𝑖 is determined by the maximum permissible increase in the 
degree of mismatch between cargo reception and dispatch per unit of time and is defined by a non-
negative decreasing function 𝜑 (𝑧) defined on the interval [0, 1], satisfying the condition 𝜑 (1) = 0. 

The initial node station (𝑖 = 0) receives cargo based on the demand for transportation within 
its technical capacity and dispatches it to the next station within its technical capacity. Each of the 
intermediate stations (𝑖 = 1, 2, … 𝑚) receives cargo within its technical capacity and dispatches it 
within the technical capacity of the next station. The final node station (𝑖 = 𝑚 + 1) receives cargo 
within its technical capacity and distributes it according to a specific regime. 

The dynamics of the degrees of mismatch between cargo reception and dispatch at the stations 
are described by the following system of differential equations  

�̇� (𝑡) = min 𝑑 , 𝜑 𝑧 (𝑡) − 𝜆 𝜑 𝑧 (𝑡) , 𝑡 ∈ [𝑡 , +∞); (2.1) 

�̇� (𝑡)  = 𝜆 𝜑 𝑧 (𝑡) − 𝜆 𝜑 𝑧 (𝑡) , 𝑖 = 1, 2, … , 𝑚, 𝑡 ∈ [𝑡 , +∞); (2.2) 

�̇� (𝑡) = 𝜆 𝜑 𝑧 (𝑡) − 𝑑  , 𝑡 ∈ [𝑡 , +∞); (2.3) 

0 ≤ 𝑧 (𝑡) ≤ 1 , 𝑖 = 0, 1, … , 𝑚 + 1, 𝑡 ∈ [𝑡 , +∞). (2.4) 
Here 𝑑 > 0, 0 < 𝜆 ≤ 1, 𝑑 > 0 are model parameters: 
𝑑  is a characteristic of the demand for transportation; 
𝜆  is a characteristic of the degree of utilization of the technical potential of station number 𝑖; 
𝑑  is a characteristic of the distribution mode of goods from the final node station. 
Next, consider the function 𝜑 (𝑧), that defines the technical potential of station number 𝑖, of 

the following form 



             SYNCHRONIZATION OF INBOUND AND OUTBOUND FLOWS AT STATIONS IN THE MODEL…             85 

Copyright ©2024 ASSA                                                                                            Adv. in Systems Science and Appl. (2024) 

𝜑 (𝑧) = 𝑎 (1 − 𝑧), 𝑎 > 0. (2.5) 

The parameter 𝑎 > 0, which is involved in defining the function 𝜑 (𝑧), represents the 
capability characteristic of station number 𝑖 in increasing the flow of goods. Parameter 𝑑 , which 
is a characteristic of the demand for transportation and is involved in equation (1), is represented 
as follows: 

𝑑 = 𝜇𝑎 , 0 < 𝜇 ≤ 1. (2.6) 

Let's rewrite the system (2.1)–(2.4), where the function 𝜑 (𝑧) is defined according to (2.5), and 
the parameter 𝑑  is defined according to (2.6). 

�̇� (𝑡) = min 𝜇𝑎 , 𝑎 1 − 𝑧 (𝑡) − 𝜆 𝑎 1 − 𝑧 (𝑡) , 𝑡 ∈ [𝑡 , +∞), (2.7) 

�̇� (𝑡) = 𝜆 𝑎  1 − 𝑧 (𝑡) − 𝜆 𝑎  1 − 𝑧 (𝑡) , 𝑖 = 1, 2, … , 𝑚, 𝑡 ∈ [𝑡 , +∞), (2.8) 

�̇� (𝑡) = 𝜆 𝑎 1 − 𝑧 (𝑡) − 𝑑  , 𝑡 ∈ [𝑡 , +∞), (2.9) 

0 ≤ 𝑧 (𝑡) ≤ 1 , 𝑖 = 0, 1, … , 𝑚 + 1, 𝑡 ∈ [𝑡 , +∞). (2.10) 
Here 𝜇, 𝑎 , 𝜆 , 𝑑  are model parameters: 
𝜇 (0 < 𝜇 ≤ 1) is a characteristic of the demand range for transportation that can be satisfied 

with the existing technical capacity of the stations; 
𝑎  (𝑎 > 0) is a characteristic of the ability of station number 𝑖 to increase freight flow 𝑖; 
𝜆  (0 < 𝜆 ≤ 1) is a characteristic of the degree of utilization of the technical capacity of 

station number 𝑖; 
𝑑  (𝑑 > 0) is a characteristic of the distribution mode of goods from the final node 

station. 
Let's outline the main research tasks: 
- determine the ranges of variation of the parameters 𝜇, 𝑎 , 𝜆 , 𝑑 , for which the cargo 

transportation system can operate smoothly, i.e., the system (2.7)–(2.10) has a solution.  
- for a given value of the demand characteristic for cargo transportation (parameter 𝜇) establish 

the most acceptable achievable levels of inconsistency between receiving and sending goods at all 
stations, by managing the values of the following characteristics: the capabilities of stations to 
increase freight flow (parameter 𝑎 ), the degree of utilization of the technical capacity of stations 
(parameter 𝜆 ) and the mode of distribution of goods from the final node station (parameter 𝑑 ). 

3. INVESTIGATION OF SOLUTIONS OF THE SYSTEM (2.7)–(2.10)  

Studying the solution set of the system (2.7)–(2.10), let's begin by investigating all the solutions 
of the system of differential equations (2.7)–(2.9). 

First and foremost, let's identify the stationary solutions of the system (2.7)–(2.9). By direct 
examination, we can verify the validity of the following statement. 

Statement 3.1: 

The system (2.7)–(2.9) for any values of the parameters 

0 < 𝜇 ≤ 1, 𝑎 > 0, 0 < 𝜆 ≤ 1, 𝑖 = 0, 1, . . . , 𝑚 + 1, 𝑑 > 0 

such that 𝑑 ≤ 𝜇𝑎  has stationary solutions:  

𝑧 (⋅) ≡ 1 −
𝑑

𝑎
 , 𝑧 (⋅) ≡ 1 −

𝑑

𝜆 𝑎
, 𝑖 = 1, … , 𝑚 + 1, 𝑤ℎ𝑒𝑛 𝑑 < 𝜇𝑎  (3.1) 

𝑧 (⋅) ≤ 1 − 𝜇, 𝑧 (⋅) ≡ 1 −
𝑑

𝜆 𝑎
, 𝑖 = 1, … , 𝑚 + 1, 𝑤ℎ𝑒𝑛 𝑑 = 𝜇𝑎  (3.2) 



86                                                              N. KHACHATRYAN 

Copyright ©2024 ASSA                                                                                            Adv. in Systems Science and Appl. (2024) 

For 𝑑𝑚+1 > 𝜇𝑎0 the system (2.7)–(2.9) has no stationary solutions. 

Let's proceed to investigate the remaining solutions of the system (2.7)–(2.9). 
Theorem 3.1: 

Any solution of the system (2.7)–(2.9) with 𝑑 < 𝜇𝑎  eventually converges to a stationary 
trajectory (3.1), and when 𝑑 = 𝜇𝑎  it converges to one of the stationary trajectories (3.2). For 
𝑑 > 𝜇𝑎  the coordinates 𝑧 (⋅), 𝑖 = 1, . . . , 𝑚 + 1 of the solution of the system (2.7)–(2.9) 
eventually converge to the stationary mode described in (3.1), while the function 𝑧 (⋅) decreases 
linearly. 
Proof.  

Let's find the general solution of the system (2.7)–(2.9). We will start with the last equation, 
rewritten as follows  

�̇� (𝑡) + 𝜆 𝑎 𝑧 (𝑡) = 𝜆 𝑎 − 𝑑 . (3.3) 

The linear equation (3.3) has the following general solution 

𝑧 (𝑡) = 1 −
𝑑

𝜆 𝑎
+ 𝑐 𝑒 . (3.4) 

Using this, let's find the solution to the penultimate equation of the system (2.7)–(2.9). It is 
easy to verify that if 𝜆 𝑎 ≠ 𝜆 𝑎 , then it has the following form 

𝑧 (𝑡) = 1 −
𝑑

𝜆 𝑎
+

𝜆 𝑎 𝑐

𝜆 𝑎 − 𝜆 𝑎
𝑒 + 𝑐 𝑒 . (3.5) 

Otherwise (𝜆 𝑎 = 𝜆 𝑎 ) the solution will be as follows  

𝑧 (𝑡) = 1 −
𝑑

𝜆 𝑎
+ 𝜆 𝑎 𝑐 𝑡𝑒 + 𝑐 𝑒 . (3.6) 

Similarly, we can find solutions to all the other equations of the system (2.7)–(2.9), except for 
the initial one. If all 𝜆 𝑎 , 𝑖 = 1, 2, … , 𝑚 + 1 are pairwise distinct, then we obtain the following 
solutions:  

𝑧 (𝑡) = 1 −
𝑑

𝜆 𝑎
+

𝜆 𝑎 𝜆 𝑎 𝑐

(𝜆 𝑎 − 𝜆 𝑎 )(𝜆 𝑎 − 𝜆 𝑎 )
𝑒 + 

𝜆 𝑎 𝑐

𝜆 𝑎 − 𝜆 𝑎
𝑒 + 𝑐 𝑒 ; 

… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 



             SYNCHRONIZATION OF INBOUND AND OUTBOUND FLOWS AT STATIONS IN THE MODEL…             87 

Copyright ©2024 ASSA                                                                                            Adv. in Systems Science and Appl. (2024) 

𝑧 (𝑡) = 1 −
𝑑

𝜆 𝑎
+

𝜆 𝑎 𝜆 𝑎 … 𝜆 𝑎 𝑐 𝑒

(𝜆 𝑎 − 𝜆 𝑎 )(𝜆 𝑎 − 𝜆 𝑎 ) … (𝜆 𝑎 − 𝜆 𝑎 )
+

𝜆 𝑎 𝜆 𝑎 … 𝜆 𝑎 𝑐 𝑒

(𝜆 𝑎 − 𝜆 𝑎 )(𝜆 𝑎 − 𝜆 𝑎 ) … (𝜆 𝑎 − 𝜆 𝑎 )
 +

𝜆 𝑎 𝜆 𝑎 … 𝜆 𝑎 𝑐 𝑒

(𝜆 𝑎 − 𝜆 𝑎 )(𝜆 𝑎 − 𝜆 𝑎 ) … (𝜆 𝑎 − 𝜆 𝑎 )
 +

… +
𝜆 𝑎 𝑐 𝑒

(𝜆 𝑎 − 𝜆 𝑎 )
+ 𝑐 𝑒 ;

 (3.7) 

… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 

𝑧1(𝑡) = 1 −
𝑑𝑚+1

𝜆1𝑎1
+

𝜆2𝑎2𝜆3𝑎3 … 𝜆𝑚+1𝑎𝑚+1𝑐𝑚+1𝑒−𝜆𝑚+1𝑎𝑚+1𝑡

(𝜆𝑚𝑎𝑚 − 𝜆𝑚+1𝑎𝑚+1)(𝜆𝑚−1𝑎𝑚−1 − 𝜆𝑚+1𝑎𝑚+1) … (𝜆1𝑎1 − 𝜆𝑚+1𝑎𝑚+1)
+

𝜆2𝑎2𝜆3𝑎3 … 𝜆𝑚𝑎𝑚𝑐𝑚𝑒−𝜆𝑚𝑎𝑚𝑡

(𝜆𝑚−1𝑎𝑚−1 − 𝜆𝑚𝑎𝑚)(𝜆𝑚−2𝑎𝑚−2 − 𝜆𝑚𝑎𝑚) … (𝜆1𝑎1 − 𝜆𝑚𝑎𝑚)
+

𝜆2𝑎2𝜆3𝑎3 … 𝜆𝑚−1𝑎𝑚−1𝑐𝑚−1𝑒−𝜆𝑚−1𝑎𝑚−1𝑡

(𝜆𝑚−2𝑎𝑚−2 − 𝜆𝑚−1𝑎𝑚−1)(𝜆𝑚−3𝑎𝑚−3 − 𝜆𝑚−1𝑎𝑚−1) … (𝜆1𝑎1 − 𝜆𝑚−1𝑎𝑚−1)
+

… +
𝜆2𝑎2𝑐2𝑒−𝜆2𝑎2𝑡

(𝜆1𝑎1 − 𝜆2𝑎2)
+ 𝑐1𝑒−𝜆1𝑎1𝑡.

 

If 𝜆 𝑎 = 𝜆 𝑎 = ⋯ 𝜆 𝑎 , then the solutions take the following form: 

𝑧 (𝑡) = 1 −
𝑑

𝜆 𝑎
+ 𝑒

𝜆 𝑎 𝑐

2
𝑡 + 𝑐 𝑡 + 𝑐 ,

… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

𝑧 (𝑡) = 1 −
𝑑

𝜆 𝑎
+ 𝑒

𝜆 𝑎 𝑐

(𝑘 + 1)!
𝑡 + 𝑐 𝑡 + ⋯ + 𝑐 𝑡 + 𝑐 ,

… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …

𝑧 (𝑡) = 1 −
𝑑

𝜆 𝑎
+ 𝑒

𝜆 𝑎 𝑐

𝑚!
𝑡 + 𝑐 𝑡 + ⋯ + 𝑐 𝑡 + 𝑐 .

 (3.8) 

In all other cases, the solutions have the following form 

𝑧 (𝑡) = 1 −
𝑑

𝜆 𝑎
+ 𝐺 (𝑡), (3.9) 

where 𝐺 (𝑡) is the sum of functions of the form 

𝑏 𝑒  and 𝑐 𝑒 𝑡𝑘, 𝑝 = 𝑖, … , 𝑚 + 1; 𝑗 = 𝑖, … , 𝑚; 𝑘 = 1, … , 𝑚 − 𝑖 + 1 (3.10) 

From (3.4)–(3.10), it follows that  

lim
𝑡→+∞

𝑧𝑖(𝑡) = 1 −
𝑑𝑚+1

𝜆𝑖𝑎𝑖
, 𝑖 = 1, 2, … , 𝑚. 

We now need to solve the first equation of the system (2.7)–(2.9). To do this, let's rewrite it in 
the following form: 



88                                                              N. KHACHATRYAN 

Copyright ©2024 ASSA                                                                                            Adv. in Systems Science and Appl. (2024) 

�̇� (𝑡) =
𝜇𝑎 − 𝜆 𝑎 1 − 𝑧 (𝑡) , if 𝑧 (𝑡) < 1 − 𝜇, 𝑡 ∈ [𝑡 , +∞),

𝑎 1 − 𝑧 (𝑡) − 𝜆 𝑎 1 − 𝑧 (𝑡) , if 𝑧 (𝑡) ≥ 1 − 𝜇 , 𝑡 ∈ [𝑡 , + ∞).
 (3.11) 

Let's consider the following two equations 
�̇� (𝑡) = 𝜇𝑎 − 𝜆 𝑎 1 − 𝑧 (𝑡) , 𝑡 ∈  𝑡, +∞ , (3.12) 

�̇�0(𝑡) = 𝑎0 1 − 𝑧0(𝑡) − 𝜆1𝑎1 1 − 𝑧1(𝑡) , 𝑡 ∈  𝑡, +∞ , (3.13) 

where 𝑡 ≥ 𝑡0. 

Using the expressions for 𝑧1(𝑡), we obtain the solutions to equations (3.12) and (3.13). They can 

be represented as follows 

𝑧 (𝑡) = (𝜇𝑎 − 𝑑 )𝑡 + 𝐹 (𝑡) + 𝑐  where 𝐹 (𝑡) ∈ С  𝑡, +∞ , lim
→

𝐹 (𝑡) = 0, (3.14) 

𝑧 (𝑡) = 1 −
𝑑

𝑎
+ 𝐹 (𝑡) where 𝐹 (𝑡) ∈ С  𝑡, +∞ , lim

→
𝐹 (𝑡) = 0. (3.15) 

Using these solutions, let's investigate the asymptotic behavior of the solution to equation 
(3.11). When 𝑑 < 𝜇𝑎  the asymptotics of the solution to equation (3.11) is determined by 

relation (3.15), i.e., lim
→

𝑧 (𝑡) = 1 − . When 𝑑 > 𝜇𝑎  the asymptotics of the solution to 

equation (3.11) is determined by relation (3.14), i.e., 𝑧 (⋅) decreases linearly with lim
→

𝑧 (𝑡) = − ∞. 

However, if 𝑑 = 𝜇𝑎  then the asymptotics of the solution to equation (3.11) may be 
determined by either relation (3.14) or relation (3.15), depending on the initial conditions. In other 

words, either lim
→

𝑧 (𝑡) = с , where с < 1 − 𝜇 or lim
→

𝑧 (𝑡) = 1 − . ∎ 

4. INVESTIGATION OF SOLUTIONS OF THE SYSTEM (2.7)–(2.10) 

Let's proceed to study the solutions of the system (2.7)–(2.9) that satisfy the constraints (2.10).  
Lemma 4.1:  

For all parameter values 

𝑎 > 0, 0 < 𝜆 ≤ 1, 𝑖 = 1, 2, … , 𝑚 + 1, 𝑑 > 0 

satisfying the condition 

𝜆 𝑎 ≥ 𝜆 𝑎 ≥ ⋯ 𝜆 𝑎 ≥ 𝜆 𝑎 ≥ 𝑑  

the components 𝑧 (⋅), 𝑧 (⋅), … 𝑧 (⋅) of any solution of the system (2.7)–(2.10) that satisfy 
the constraints (2.10) at the initial moment in time will continue to satisfy them at subsequent 
moments in time. 

Proof. 
Let's begin by examining the last component of the solution to the system (2.7)–(2.9), namely 

𝑧 (⋅). It takes the form (3.4), where с  is determined by the condition 

1 −
𝑑

𝜆 𝑎
+ 𝑐 𝑒 = 𝑧 , where 0 ≤ 𝑧 ≤ 1, 

i.e. 

𝑐 =
𝑑

𝜆 𝑎
− 1 + 𝑧 𝑒 , where 0 ≤ 𝑧 ≤ 1 (4.1) 



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Copyright ©2024 ASSA                                                                                            Adv. in Systems Science and Appl. (2024) 

From (4.1), it follows that 
𝑑

𝜆 𝑎
− 1 𝑒 ≤ 𝑐 ≤

𝑑

𝜆 𝑎
𝑒 . 

Using this estimate for с  and the expression (3.4), we obtain an estimate for 𝑧 (⋅). 
It will take the following form  

1 −
𝑑

𝜆 𝑎
− 1 −

𝑑

𝜆 𝑎
𝑒 ( ) ≤ 𝑧 (𝑡) ≤

1 −
𝑑

𝜆 𝑎
1 − 𝑒 ( )

 (4.2) 

From (4.2), it follows that when the condition 𝑑 ≤ 𝜆 𝑎  is met, the inequality holds 

0 ≤ 𝑧 (𝑡) ≤ 1, 𝑡 ∈ [𝑡 , +∞). (4.3) 

Let's demonstrate that similar inequalities to (4.3) hold for the other components of the solution 
to the system (2.7)–(2.9). We'll start with the component 𝑧 (⋅). To do this, consider the equation 
(2.8) for 𝑖 = 𝑚: 

�̇� (𝑡) = 𝜆𝑚𝑎𝑚 1 − 𝑧𝑚(𝑡) − 𝜆𝑚+1𝑎𝑚+1 1 − 𝑧𝑚+1(𝑡) , 𝑡 ∈ [𝑡0, +∞). 

Let's demonstrate that the function 𝑧 (⋅) cannot take a value greater than 1. Indeed, otherwise, 
due to the continuity of the function 𝑧 (⋅) there must exist a point  𝑡∗ > 𝑡 , such that 𝑧 ( 𝑡∗) = 1. 
In such a case 

�̇�𝑚( 𝑡∗) = −𝜆𝑚+1𝑎𝑚+1 1 − 𝑧𝑚+1( 𝑡∗)  

and due to inequality (4.3), it follows that �̇� ( 𝑡∗) ≤ 0, i.e. 𝑧 (⋅) is upper-bounded by one. 
Let's now show that the function 𝑧 (⋅) cannot take a value less than 0. Indeed, otherwise, due to 
the continuity of the function 𝑧 (⋅) there must exist a poin  𝑡∗∗ > 𝑡 , such that 𝑧 ( 𝑡∗∗) = 0. Then  

�̇�𝑚( 𝑡∗∗) = 𝜆𝑚𝑎𝑚 − 𝜆𝑚+1𝑎𝑚+1 1 − 𝑧𝑚+1( 𝑡∗∗)  

and when the condition 𝑑 ≤ 𝜆 𝑎 ≤ 𝜆 𝑎  is satisfied, the inequality �̇� ( 𝑡∗∗) ≥ 0, 
holds, i.e., 𝑧 (⋅) is lower-bounded by zero. 

Similarly, the validity of all the other inequalities (2.10) can be proven, except for the first one 
(related to the function 𝑧 (⋅)).∎ 

Let's formulate a similar lemma for the zeroth component of the solution to the system (2.7)–
(2.9). 

Lemma 4.2:  

For all parameter values 0 < 𝜇 ≤ 1, 𝑎 > 0, 𝑖 = 0, 1, … 𝑚 + 1, satisfying the condition 𝑎 ≤

𝑎 , there exists 𝜆 (𝜇, 𝑎 , 𝑎 , 𝑧 (𝑡 ), 𝑧 (𝑡 ), . . . , 𝑧 (𝑡 )), 𝜇 ≤ 𝜆 ≤ 1 such that for all 𝜆  in the 

interval 𝜇, 𝜆 , 0 < 𝜆 ≤ 1, 𝑖 = 2, 3, … , 𝑚 + 1, 𝑑 > 0, satisfying the condition  

𝜆 𝑎 ≥ 𝜆 𝑎 ≥ ⋯ 𝜆 𝑎 ≥ 𝜆 𝑎 ≥ 𝑑  

the zeroth component 𝑧 (⋅) of any solution to the system (2.7)–(2.9) that satisfies the constraint 
(2.10) at the initial moment in time will continue to satisfy it at subsequent moments in time.  

Proof. 
Similarly to the other components, let's show that the function 𝑧 (⋅) cannot take a value greater 

than 1. Indeed, otherwise, due to the continuity of the function 𝑧 (⋅) there must exist a point �̃� >
𝑡 , such that 𝑧 (�̃�) = 1. Then, from (2.7), it follows that 

�̇�0(𝑡) =  𝜆1𝑎1(𝑧1(𝑡) − 1), 



90                                                              N. KHACHATRYAN 

Copyright ©2024 ASSA                                                                                            Adv. in Systems Science and Appl. (2024) 

i.e., according to Lemma 1, �̇� (�̃�) ≤ 0. This means that the function 𝑧 (⋅) is upper-bounded by 
one. 

Now, let's estimate the function 𝑧 (⋅) from below. To do this, we'll investigate the behavior of 
its derivative as 𝑧 (⋅) → 0 +. According to (2.7), it is described by the equation 

�̇�0(𝑡) = 𝜇𝑎0 − 𝜆1𝑎1 1 − 𝑧1(𝑡) . 

Let's analyze the inequality  

𝜇𝑎 − 𝜆 𝑎 (1 − 𝑧 (𝑡)) ≥ 0. 

We will show that for any 0 < 𝜇 ≤ 1, 𝑡 ∈ [𝑡 , +∞) there exists a range of 𝜆  values on the 
half-open interval (0, 1], for which this inequality holds. To do this, rewrite it as 

𝑧1(𝑡) ≥ 1 −
𝜇𝑎0

𝜆1𝑎1
. (4.4) 

According to Lemma 4.1, 

0 ≤ 𝑧 (𝑡) ≤ 1, 𝑡 ∈ [𝑡 , +∞). (4.5) 

From (4.5), it follows that for any 𝜇, satisfying 0 < 𝜇 ≤ 1, and for any 𝑎 , 𝑎 , satisfying 𝑎 ≤

𝑎  there exists 𝜆 , 𝜇 ≤ 𝜆 ≤ 1 such that for any value of 𝜆  from the interval 𝜇, 𝜆  the 

inequality (4.4) will hold for all 𝑡 ∈ [𝑡 , +∞), i.e. �̇� (𝑡) ≥ 0 as 𝑧 (𝑡) → 0 +. This demonstrates 
the lower bound of the function 𝑧 (⋅) as 0. It is evident that 𝜆  depends on the parameters 𝜇, 𝑎 , 𝑎 , 
as well as the initial conditions, so we denote it as 𝜆 (𝜇, 𝑎 , 𝑎 , 𝑧 (𝑡 ), 𝑧 (𝑡 ), . . . , 𝑧 (𝑡 )). ∎  

Let's state the main result of this study.  
Theorem 4.1: 

For any initial values 0 ≤ 𝑧 (𝑡 ) ≤ 1, parameters 0 < 𝜇 ≤ 1, 𝑎 > 0, 𝑖 = 0, 1, . . . , 𝑚 + 1, 
satisfying the condition 𝑎 ≤ 𝑎 , there exists 𝜆 (𝜇, 𝑎 , 𝑎 , 𝑧 (𝑡 ), 𝑧 (𝑡 ), . . . , 𝑧 (𝑡 )), 𝜇 ≤

𝜆 ≤ 1 such that for all 𝜆  in the interval 𝜇, 𝜆 , 0 < 𝜆 ≤ 1, 𝑖 = 2, 3, … , 𝑚 + 1, 𝑑 > 0, 

satisfying the condition  

𝜆 𝑎 ≥ 𝜆 𝑎 ≥ ⋯ 𝜆 𝑎 ≥ 𝜆 𝑎 ≥ 𝑑  

the solution to the system (2.7)–(2.10) exists and converges to either the stationary solution 
(3.1) (when 𝑑 < 𝜇𝑎 ) or to one of the stationary solutions (3.2), which is the same for all 𝑑  
and 𝑎  (when 𝑑 = 𝜇𝑎 ).  

Proof. 
The proof directly follows from Theorem 3.1, Lemma 4.1, and Lemma 4.2. ∎ 
Corollary 1: 

The system of differential equations (2.7)–(2.10) has a globally stable stationary solution (3.1) 
and a family of stable solutions of the form (3.2). 

Proof.  
The proof directly follows from Theorem 4.1. ∎  
Corollary 2:  

For any initial values 0 ≤ 𝑧 (𝑡 ) ≤ 1, parameters 𝑎 > 0 , 𝑖 = 0, 1, . . . , 𝑚 + 1, satisfying the 
condition 𝑎 ≥ 𝑎 , 0 < 𝜇 ≤ 1, �̅� = 𝜇, 𝑖 = 1, . . . , 𝑚 + 1 and 𝑑 = 𝜇𝑎  the solution to the 

system (2.7)–(2.10) exists and converges to one of the stationary solutions of the form 

𝑧 (⋅) ≤ 1 − 𝜇, 𝑧 (⋅) ≡ 0, 𝑖 = 1, . . . , 𝑚 + 1. 

Proof.  



             SYNCHRONIZATION OF INBOUND AND OUTBOUND FLOWS AT STATIONS IN THE MODEL…             91 

Copyright ©2024 ASSA                                                                                            Adv. in Systems Science and Appl. (2024) 

The proof directly follows from Theorems 3.1 and 4.1. ∎ 
Let's move on to the interpretation of the main results obtained in this study. Before doing that, 

let's remind ourselves that the research on this model, described by the system (2.7)–(2.10), boils 
down to solving two fundamental tasks. 

The first task is to determine the ranges of model parameter variations within which the 
mentioned system has a solution (ensuring uninterrupted freight flow). Theorem 4.1 defines these 
specified ranges. 

The second task is to determine the most acceptable achievable levels of inconsistency between 
cargo reception and dispatch on all stations for a given value of the freight demand characteristic 
using control parameters. The solution to this task is presented in Corollary 2. According to it, for 
any values of inconsistency levels between cargo reception and dispatch at the initial time, if the 
capabilities of all stations to increase freight flow, starting from the first one, are not less than the 
capabilities of the zero station (initial node station), it is always possible to activate station 
potentials in such a way (by selecting values for the parameter 𝜆 ) and align the cargo distribution 
mode with the final node station's demand for freight transport (by selecting the value for the 
parameter 𝑑 ), such that the degree of inconsistency at all stations except the initial node station 
will gradually become zero over time. The corresponding characteristic at the initial node station 
will depend on both the freight demand characteristic and the initial conditions 

5. CONCLUSION 

A dynamic model for organizing cargo transportation has been investigated on a segment of the 
railway network, which represents a railway line between two node stations. This model is 
represented by a system of differential equations that describe the dynamics of the discrepancy 
between cargo reception and dispatch at the stations. It includes a set of parameters that define the 
characteristics of cargo transportation demand, the technical capabilities of the stations, the extent 
of their utilization, as well as the mode of cargo distribution from the node station. Ranges of 
parameter variations have been determined within which the cargo transportation system can 
operate smoothly without interruptions. For each station, the most suitable level of utilizing its 
technical potential has been identified, allowing for a smooth and efficient flow of cargo and the 
ability to respond to changes in cargo transportation demand. 

ACKNOWLEDGEMENTS 

The article was prepared at the State Academic University for the Humanities as part of the 
state assignment of the Ministry of Science and Higher Education of the Russian Federation (topic 
No. FZNF-2023-0004 "Digitalization and formation of modern information society: cognitive, 
economic, political and legal aspects"). 

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