Pricing Decision and Research of Dual-Channel Cargo Transportation Service System Based on Queuing Theory

Abstract

Against the backdrop of China’s “public-to-railway” freight policy that has led to railway yard congestion and imbalanced modal capacity utilization, this study develops a Dual-Channel Cargo-Transportation Service (DCTS) system model using queuing theory to optimize freight flow allocation and pricing strategies. Integrating the behavioral decisions of governments, carriers, and cargo owners, the research employs M/M/1 queuing models and the Logit choice framework to analyze the dynamic equilibrium between goods waiting times and carrier profits, exploring objectives of minimizing system-average waiting time and maximizing carrier profits. Key findings show that regulating highway pricing can effectively divert freight flows to reduce railway congestion and improve system efficiency, with optimal pricing intervals for highways identified based on service capacity to balance congestion relief and profitability. The model quantifies the trade-off between transportation costs and waiting times to guide cargo owners’ mode choices, and numerical simulations validate that strategic highway price adjustments alleviate bottlenecks and enhance modal synergy. This paper provides a theoretical basis for the government to formulate freight-transportation policies and optimize freight flow allocation. At the same time, it also provides a practical, theoretical basis and methodological reference for carrier pricing decisions, as well as for solving the problem of freight flow congestion and optimizing the pricing of transportation services.

1. Introduction

With the implementation of the “public-to-railway” policy for national cargo transportation in recent years, the volume of railway cargo has achieved rapid growth, but from a realistic point of view, the turnover capacity of the railway yards at the end of some routes cannot meet the demand for freight transport. This results in the phenomenon of queuing up for shipment, with some batches of goods even remaining lined up for a few weeks in transportation yards. The regional railway logistics centers located in the transport nodes cannot support the freight volume from all their serviced yards, resulting in low overall transport efficiency in the railway network [1]. At the same time, the enhancement of the capacity of railway freight yards is subject to great constraints, especially with regard to urban planning around existing yards in urban areas. Existing yards struggle to support such large-scale freight transport and turnover, so there is a need to alleviate the pressure on yard transportation by way of distribution of cargo flow. From the point of view of the public railway and railway networks, as the origin and destination points of cargo flows in China are restricted by geographical factors, the utilization rate of the transport capacity of the public railway and railway networks is unbalanced, with some corridors experiencing capacity constraints and others experiencing capacity surpluses [2]. The utilization rate of the transportation capacity of the two networks is uneven. Therefore, it is important to distribute cargo flow through scientifically feasible methods.

Cargo-transportation mode selection is the primary consideration of cargo-flow distribution and the main issue studied in this paper, constituting a complex and critical process that involves the comprehensive consideration of a variety of factors, including the nature of the goods, transportation distance, cost-effectiveness, and timeliness requirements, as well as environmental protection and sustainability. In relation to the specific situation of goods and transportation needs, Ma Yujiao [3] comprehensively considered the advantages and disadvantages of various modes of transportation, chose the most suitable mode of transportation, and also flexibly adjusted the distribution strategy of modes of transportation according to changes in the market, policy adjustments, and changes in transportation conditions, in order to adapt to different transportation needs. In terms of transportation mode selection, Dai Yanhong [4] studied the selection inertia effect of the freight mode “public to railway”, analyzing the mechanism of selection inertia on the behavior of freight mode selection, exploring the difference of inertia effect of different groups with different demand characteristics, and determining the model of freight mode selection. This study can provide theoretical references for governmental management to formulate and optimize the guiding policies for “public to railway”. Xinghan Chen [5], a member of the “public-to-railway” group, used the number of trains corresponding to different transportation organization modes between each OD pair as the main decision-making variable and the minimum operating cost of railway fast logistics as the optimization objective and constructed a mixed-integer planning model of cargo-flow distribution in a railway fast logistics network and a two-stage hybrid algorithm to solve the problem. Alkaabneh et al. [6] performed modeling based on considering congestion conditions at hub points and solved these conditions using a Lagrangian heuristic algorithm as a way to equalize cargo flows between hubs. Ambrosin et al. [7] established an objective function from a low-carbon perspective, in which cargo flows are redistributed to optimize the balance ratio, dividing the volume of cargo transported by highway and railway. Chen Lei [8], based on the different characteristics of highway networks and railway networks in the process of cargo transportation, solved the comprehensive optimization problem of cargo-flow transfer and flow allocation of land transportation systems. A low-carbon transportation goal was realized, and the transportation path of railway cargo flow was optimized through the central allocation of cargo flow in railway networks to improve the utilization rate of railway uniform capacity. Haugen et al. [9] constructed a game model considering the congestion time of the container transport process to optimize the cargo selection allocation method.

Freight mode selection is a critical decision in logistics distribution, influencing cost efficiency, environmental impact, and supply-chain resilience. Recent research has employed advanced modeling techniques to optimize mode choice under varying constraints. Zhang et al. [10] developed a multinomial Logit model incorporating cost, time, and reliability factors, demonstrating its superiority over traditional deterministic approaches. Extending this, Wang and Li [11] integrated machine learning with discrete choice models to capture nonlinear relationships in shipper preferences. For sustainable logistics, Chen et al. [12] proposed a bi-objective optimization framework balancing cost and emissions, highlighting the trade-offs between road and railway transport.

In multimodal freight networks, Yang et al. [13] introduced a stochastic programming model to handle demand uncertainty, while Liu et al. [14] applied reinforcement learning for dynamic mode switching in response to real-time disruptions. Gupta and Ivanov [15] examined risk-averse mode selection under supply-chain disruptions, emphasizing the role of resilience in logistics planning. For last-mile distribution, Kim et al. [16] compared drones, autonomous vehicles, and traditional trucking, identifying cost thresholds for emerging technologies.

Behavioral factors also play a key role: Zhao et al. [17] incorporated carrier preferences using hybrid choice models, and Martínez et al. [18] analyzed the impact of policy incentives on mode shift towards greener alternatives. Finally, Janssen et al. [19] presented a large-scale simulation assessing urban freight zoning policies on mode-split patterns. Collectively, these studies underscore the complexity of freight mode choice and the need for adaptive, data-driven decision tools.

The above research has studied the distribution of cargo flow and the optimization of cargo-flow equilibrium through the methods of OD network flow distribution and operation research models for the problem of cargo congestion at logistics nodes or hubs, but all of them are from the perspective of transportation system managers, ignoring the impacts of carrier cost and total social benefits, etc. However, based on the integration of the overall situation, research considering the uncertain choice behavior of the transportation managers, carriers, and the government, as well as their interrelationships, is very scarce. However, a few studies consider the uncertain choice behavior and interrelationships among transportation managers, carriers, and governments based on the overall situation. In view of this, the focus of this paper is from the transport service perspective with respect to the government’s leading role in transport services. Analysis of the behavior of the main body of decision-making affects its decision-making micro-factors by directly limiting the transport sector price level or a certain range of fluctuation, its pricing strategy to provide constraints, and then indirectly solving the problem of the congestion of transport nodes or hubs of cargo flow using the market economy.

Queuing theory has a wide range of applications, and it is an important method for solving problems such as service time and service efficiency in various types of service systems [20]. A queuing system, also known as a random service system, is a congestion phenomenon caused by a random arrival process of service recipients and uncertain service times at the service desk. There are important applications in communication systems, transportation systems, computer systems, production management systems, and service industries. For example, in public transportation systems, queuing theory can help optimize the flow of passengers boarding and alighting from buses, reduce waiting and congestion times, etc. [21]. Based on a deferred maintenance queuing system with working vacation and vacation interruption, the customer behavior and pricing strategy in the system are analyzed by integrating intelligent service mechanisms. Quartz [22], with the help of relevant theories and methods such as stochastic service theory and information technology, presented a queuing model for the reliability of railway logistics service systems to study the reliability of railway logistics nodes and the reliability of railway logistics networks [23]. The performance of the queuing system is evaluated and optimized under the mechanism of differentiated services, and their equilibrium strategies and optimal pricing are considered from the perspectives of customers and service providers.

For traffic congestion problems, most scholars consider the queuing theory research of transportation systems to solve real-time traffic congestion situations, such as Modi et al. [24], who established the M/M/1 queuing model and used the model to analyze lane settings and signal timing to provide a theoretical basis for alleviating traffic congestion at intersections. Valko et al. [25] proposed a novel multi-server queuing model for traffic-signal optimization to enhance the sustainability of urban mobility. Some research considers the pricing strategy from the perspective of the service provider, such as Kuboi [26], who proposed a highway pricing mechanism to alleviate queuing congestion and mobility congestion by setting congestion prices and right-of-way prices, but there are few studies on solving the congestion problem of goods flow using queuing theory to quantitatively analyze the selection of goods transportation modes for freight services based on government-regulated pricing policy under different subjects. Zhi Chun Li et al. [27] studied bottleneck queuing and transit congestion in the case of the two-mode problem and pricing congestion through tolls and fares, which provides the theory and methodology for solving the pricing problem of different modes of transportation in the case of congestion. However, because of the establishment of the model of deterministic bottleneck capacity, the impact of the stochasticity of the bottleneck capacity over time, the change in the number of riders and the waiting time are not considered. However, the factors affecting service time, arrival time, or waiting time in the quantitative study of freight-transportation services are many and random, which makes it more difficult to study this area. It is also the area of innovation in this paper.

Recent advancements in queuing theory have demonstrated its efficacy in modeling and alleviating urban traffic congestion. An et al. [28] proposed a stochastic queuing model for signalized intersections, optimizing green-light duration to minimize vehicle delay. Similarly, Li and Zhang [29] integrated multi-class M/M/1 queuing networks with dynamic traffic assignment, improving congestion prediction in heterogeneous traffic flows. Guo et al. [30] developed a phase-optimized signal control strategy using bulk-service queuing theory, reducing average queue lengths by 18%. For large-scale urban networks, Wang et al. [31] combined queuing theory with reinforcement learning, dynamically adjusting signal timings to prevent spillback effects. In highway traffic management, Zhao and Sun [32] applied fluid queuing models to ramp metering, significantly improving bottleneck throughput. Additionally, Tang et al. [33] utilized tandem queuing systems to model arterial road platooning, enhancing coordination between successive intersections. Emerging research also explores hybrid approaches: Xu et al. [34] merged queuing theory with deep learning for real-time congestion forecasting, while Liu and Chen [35] introduced a game-theoretic queuing model to study driver rerouting behaviors under congestion pricing. For public transport, Yang et al. [36] optimized bus dispatching using bulk-arrival queuing models, reducing passenger waiting times. Finally, Jiang and Wang [37] demonstrated that stochastic network queuing models outperform traditional methods in resilience analysis during traffic incidents. Collectively, these studies highlight the queuing theory’s adaptability in addressing modern traffic challenges through analytical and data-driven solutions.

This paper makes significant macro-level contributions by redefining freight transportation as a complex adaptive system and providing a theoretically rigorous, market-driven framework to optimize systemic efficiency, sustainability, and resilience. Integrating railway and highway networks into a queuing-theoretic model demonstrates how government-guided price regulation can systematically alleviate congestion, redistribute freight flows, and balance modal capacity utilization—key to developing sustainable multimodal logistics systems that reduce carbon emissions and enhance resource efficiency. The research advances a general equilibrium theory for freight markets, clarifying how micro-level decisions (e.g., shipper mode choices) and macro-level policies (e.g., congestion pricing) interact to shape network-wide outcomes, such as reduced average waiting times and improved infrastructure resilience. Its interdisciplinary approach, combining behavioral economics, operations research, and policy science, provides policymakers with data-driven tools to design adaptive regulations, prioritize infrastructure investments (e.g., highway capacity upgrades), and foster market-driven solutions that address global challenges like supply-chain volatility and climate goals. Ultimately, the study lays the groundwork for digitizing freight management through stochastic modeling, enabling real-time adjustments to price and demand forecasting in an increasingly dynamic global logistics landscape.

2. Problem Description and Modeling

2.1. Description of the Problem

As long waiting times and excessive cargo congestion under various modes of transportation are common problems in cargo-transportation services, cargo-flow congestion will not only increase the cost of waiting for goods and reduce the satisfaction of cargo owners but also produce a series of hidden dangers such as waste of resources and environmental damage, which is an important problem to be solved in theory and practice. Railroads and highways are the main modes of transportation in inland long-distance transport, so this paper only considers the transportation of goods under these two modes of transportation.

As the leading provider of freight transport services, the government, when faced with the failure of the existing supply of transport services and serious congestion in the freight transport system, needs to analyze in depth the causes of congestion and determine whether it is possible to adjust the relationship between supply and demand by increasing the supply of transport or by guiding the demand, so as to achieve a sustainable supply of freight transport services. Owner demand is objective. The government or transport service carriers cannot be forced to interfere in the size of the demand for transportation services for the owner. Therefore, this paper, from the perspective of the supply side of transportation services, presents the pricing of the relevant transport service supply policy research so as to analyze the mode of transport service supply and policy used to regulate and guide the choice of carriers and owners.

This paper focuses on the study of cargo transportation and takes a stochastic process-based modeling and analysis approach. Due to the system of planning transportation, railway capacity is tight, and bulk cargo transportation is in accordance with the plan to complete the transport task, rarely considering a random transfer. Therefore, the stochastic process theory in the railway freight transport application of the research is very scarce, but in today’s market, the demand for randomness has gradually increased in particular time and space conditions. With the completion of the transport plan, the transport capacity of the residual capacity is insufficient, easily causing congestion in the transportation of other goods. It is necessary to use stochastic process theory to build the queuing model of the cargo-transportation system, release capacity, alleviate the congestion of cargo flow, accelerate the flow of materials, and improve social and economic benefits.

By establishing a stochastic model to form the amount of government capital input, carrier pricing, and the uncertain choice behavior of cargo owners in the process of supplying cargo-transportation services and their interrelationships [38] and using stochastic variables to provide the factors influencing cargo-transportation services, respectively, with the goal of minimizing the waiting time of cargo and maximizing the revenue of carriers, systematic research and discussion of adjusting highway pricing on the basis of railway pricing for different cargo transportation must be conducted. This study is aimed at solving the problem of the distribution of cargo flows under the two-channel transport network by guiding the choice of transport modes.

2.2. Model Construction and Assumptions

This paper proposes DCTS (Dual-channel Cargo Transport Service), which takes railways and highways as the carriers of cargo-transportation services with the purpose of improving the long waiting times of cargo in certain modes of transportation, overcrowding in transportation, improving the efficiency of transportation service supply, and forming competitive cooperation by means of differentiated transportation supply. This behavior is a typical DCTS model. Furthermore, this paper only considers the choice of the first mode of transportation for the goods, independent of whether multiple modes of transportation are intermodal during transportation.

The use of constructing a DCTS cargo transport service model can standardize the pricing problem of public–railway freight transport, play the role of complementary and diversifying cargo transport, and realize a smooth and sustainable supply of cargo transport.

The symbols of the variables appearing in this paper with their interpretations are shown in

Table 1. Variable symbols and their interpretation.

Symbolic Variable	Corresponding Explanations and Their Descriptions
$\lambda$	Total cargo arrival rate
${\lambda }_r/{\lambda }_h$	Rail/highway cargo arrival rates in the DCTS system
${\mu }_r/{\mu }_h$	Rate of rail/highway service in the DCTS system
$p_r/p_h$	Prices for transportation in the rail/highway sector (billed individually or in batches)
$s_r/s_h$	Cost of transportation services ratio in the rail/highway sector (unit cost of transportation services)
${\theta }_i$	Cargo sensitivity factor
${\theta }_0$	Cargo sensitivity thresholds, which can also be viewed as cargo arrival deadlines
$E\left(W_r\right)/E\left(W_h\right)$	Average waiting time for goods on rail/highway in the DCTS system
$E\left(W_0\right)$	Average waiting time for goods across the system in the DCTS system (average system waiting time)
$R_r/R_h$	Expected Benefits of Rail/Highway in the DCTS System
$c$	Holding cost ratio (unit holding cost)
$TOC$	Total cost of operation in DCTS systems
$TFC$	Total cost of goods in the DCTS system
$TSC$	Total cost to society in DCTS systems
$\theta E\left(T\right)$	Average waiting cost for a single shipment
$T_{max}$	Maximum waiting time for highway sector commitments

, below, which also shows the basic assumptions of the model.

In this paper, it is assumed that the supply of goods is sufficiently large for model simplicity, with random arrivals approximated by a Poisson process and exponential service times consistent with the M/M/1 queuing framework [39]. However, practical freight systems often exhibit non-stationary characteristics, such as seasonal demand fluctuations (e.g., holiday peaks), sudden cargo surges (e.g., emergency logistics), or temporal correlations due to industrial production cycles. These factors may cause actual arrival processes to deviate from strict Poisson assumptions. For analytical tractability, the baseline model assumes stationarity (

Table 2. Model basic assumptions.

Main Part	Characteristic	Specific Assumptions
Supply	Attribute	Heterogeneous goods (measured per piece or lot)
	Arrive	Arrival intervals follow an exponential distribution
	Leave	No stopping and dropping out
	Waiting costs	Excluding waiting costs for highway transportation
	Initial selection of goods	Cost minimization principle
	Availability	Greater than or equal to the sum of waiting time and service time
Freight-transportation department (highway/rail)	Transportation service time	Compliance index distribution
	Mechanisms for transport services	First come, first served
	Waiting for information	Information on average waiting time for goods only
	Level of quality of transportation services	The level of quality of highway transportation services is $\left(T_{max}, 1-\epsilon \right)$
	Transportation options	utility function
DCTS system	Queuing system	M/M/1 queuing model
	Waiting for the captain	limitless
	System Stability	Stable transportation system with a system service rate greater than the arrival rate

).

Goods enter the queuing system in the form of random distribution, establish the utility function according to the sensitivity of the goods to the waiting time, quantitatively analyze the demand of the cargo owner, and choose the mode of cargo transportation. The demand of the carrier (railway or highway) has the following four aspects: demand for the number of requested vehicles, demand for the direction of cargo flow (vehicle and cargo suitability), demand for transportation distance, and demand for the time of transportation service (delivery period or waiting cost), and the demand for cargo mainly has safety, timeliness, economy, convenience, and environmental protection.

General utility is determined by both individual characteristics of the utility object and individual socio-economic characteristics so that

$$U_{in}=V_{in}+{\zeta }_{in},$$

(1)

where $U_i$—Individual utility; $V_i$—Utility determinants consisting of individual characteristic factors; and ${\zeta }_i$—Utility random term composed of individual social factors, which includes unobservable components and error values generated at the time of observation.

Since it is inconvenient to measure all factors of the utility random term, this part of the impact is ignored. Assuming that the dual-channel freight-transportation system shares two transportation modes, $U_{rn}$ is the utility of shipper $n$ choosing railway transportation and $U_{hn}$ is the utility of shipper $n$ choosing highway transportation. According to the principle of random utility maximization, the transportation conditions for shipper $n$ to choose the two transportation modes are:

$$U_{in}=max\left\{U_{rn}, { U}_{hn}\right\},$$

(2)

The probability that the shipper chooses railway transportation is denoted as

$$\begin{array}{c}{p}_r=prob\left(U_r\ge U_h\right),\\ p_r=prob\left({\zeta }_h\le V_r-V_h+{\xi }_r\right),\\ p_r=\underset{{\xi }_r}{\int }F\left(V_r-V_h+{\xi }_r\right)f_r\left(x\right)dx,\end{array}$$

(3)

Conversely, the probability that the shipper chooses highway transportation $q_h$ is obtained in the same way. Since the random terms in the utility function ${\zeta }_i$ are independent of each other and obey the Gambel distribution, their distribution function is expressed as:

$$F\left(x\right)=exp\left\{-\theta exp\left(-{\zeta }_i\right)\right\}, \left(\theta >0,-\mathrm{\infty }<{\zeta }_i<\mathrm{\infty }\right),$$

(4)

where α is the parameter value of the characteristic factor in the stochastic term. Then, substituting Equation (3) into (4), the Logit model for choosing the mode of railway transportation can be written as:

$$q_r={\displaystyle \frac{e^{V_r}}{e^{V_h}}}={\displaystyle \frac{exp\left(V_r\right)}{exp\left(V_h\right)}},$$

(5)

Then, the choice of highway transportation mode model is obtained in the same way.

Due to the quantitative analysis of the service attributes of the two modes of transportation, in this paper, only the timeliness and reliability of transportation services are considered as the important characteristics influencing factors, and the utility function is expressed as:

$$V_i={\alpha }_1{W}_i+{\alpha }_2{R}_i,$$

(6)

where $V_i$—Integrated model of utility function of railway or highway transport modes;$W_i$—The timeliness of the railway or highway mode of transportation, i.e., the sensitive character of the cargo owner’s waiting time;$R_i$—Reliability of railway or highway transportation modes, i.e., the suitability and degree of suitability of available models and cargo attributes;${\alpha }_1$—The coefficient of goods in the time window of the shipment arrival deadline;${\alpha }_2$—Vehicle and cargo reliability fitness coefficient.

Also, the standard of each characteristic measure in the utility function is different and normalized in the following way:

$$A^{\prime }={\displaystyle \frac{A-A_{min}}{A_{max}-A_{min}}},$$

(7)

In this paper, we define $\theta$ as the sensitivity parameter of heterogeneous goods, denoting the cost of waiting for goods per unit of time, which follows a uniform distribution. $\left[0,U\right]$ obeys a uniform distribution on the distribution function and probability density function, which are, respectively, $H\left(\theta \right)$ and $H^{\prime }\left(\theta \right)={\displaystyle \frac{dH\left(\theta \right)}{d\theta }}$; $W$ is the waiting time of goods in terms of transportation service, i.e., the total time consumed by goods arriving at the transportation service system until their complete departure (including arrival, requesting a car, handling and unloading, and marshaling, and waiting operations); $E\left(W\right)$ is the average waiting time for cargo in the transportation service. $\theta E\left(W\right)$ denotes the average cost of waiting for a particular shipment of goods, $\lambda$ is the total arrival rate of goods, ${\mu }_r$ is the rate of transportation services in the railway sector, and ${\mu }_h$ is the rate of transportation services in the highway sector. It is assumed that there is no cargo-stopping or midway-exit behavior, that the owner of the cargo chooses the mode of transport, signs a transport contract, and accepts the service until departure, and that the parameters used are all equilibrium values in equilibrium [40,41]. This is all carried out in order to achieve a stable effect of transportation services.

The arrival rates of goods in the railway and highway transport sectors in the two-channel transport service system are, respectively, ${\lambda }_r$ and ${\lambda }_h$, and $\lambda$ is the total cargo arrival rate such that the relationship is ${\lambda }_r+{\lambda }_h=\lambda$. Therefore, the model of the DCTS transportation service system can be represented in Figure 1.

In the DCTS transportation mode, goods can choose railway transportation with a lower freight rate but longer waiting time (time for business processing, loading and unloading, marshaling, etc.) or a higher freight rate but shorter waiting time, and promise that the maximum waiting time of the goods is not more than $T_{max}$. The confidence level is $\left(1-\alpha \right)$. The level of quality of transportation services can be expressed as ${(T}_{max},1-\alpha )$. Higher service quality (smaller $T_{max}$ or higher $1-\alpha$) can justify a higher $p_h$ to cover operational costs. If the owner of the goods subjectively considers the lower freight rate of railway transportation, the limited capacity of the railway generates congestion, resulting in goods waiting outside the yard for too long. The higher sensitivity factor of the goods can choose the higher level of quality of transport services of highway transport, and the cost of price in exchange for a reduction in the cost of waiting. The following assumptions are made in this paper:

Assumption 1. 

Under the DCTS transportation service system model, if the level of transportation service quality in highway transportation ${(T}_{max},1-\alpha )$ reaches a sufficiently high level, cargo transportation can ignore the cost of waiting in highway transportation.

Among the existing studies, Guo and Zhang [42] and Qian [43] et al. give hypotheses on similar issues, which are supported by questionnaires [44], and show that when the level of transportation service quality is high enough, most customers can ignore the waiting cost for a certain period of time.

3. Analysis of Dual-Channel Cargo-Transportation Service (DCTS) Model Based on Queuing Theory

In the context of the two-channel cargo-transportation system model, this section analyzes and researches the transportation modes of the railway sector and the highway sector, respectively, based on the queuing theory perspective.

3.1. Analysis of Railroad Cargo-Transportation Corridor Services

The railway sector organizes the transportation of goods because the highway freight rate is higher than the railway transportation $p_h<p_r$. Considering the minimization of the transportation cost of suitable goods, and also based on Assumption 1, it is known that when $p_r+\theta E\left(W_r\right)\le p_h$, the $\theta$ on the level of $\left[0,U\right]$ uniform distribution of the goods can choose railway transportation, and sensitivity parameter $\theta$ satisfies $\theta \le {\displaystyle \frac{p_h-p_r}{\theta E\left(W_r\right)}}$, such that:

$${\theta }_0=\left(p_h-p_r\right)/\theta E\left(W_r\right),$$

(8)

For the goods sensitivity parameter threshold at this time, the cost of railway and highway transportation services is equal, according to the M/M/1 model. Therefore, $\mathrm{E}\left({\mathrm{W}}_{\mathrm{r}}\right)={\displaystyle \frac{1}{{\mu }_{\mathrm{r}}-{\lambda }_{\mathrm{r}}}}$, and its distribution function is:

$$H\left({\theta }_0\right)={\displaystyle \frac{{\theta }_0}{U}}={\displaystyle \frac{p_h-p_r}{UE\left(W_r\right)}}={\displaystyle \frac{\left(p_h-p_r\right)\left({\mu }_r-{\lambda }_r\right)}{U}},$$

(9)

The rate of arrival of goods in the railway transportation sector is ${\lambda }_r=\lambda H\left({\theta }_0\right)=\lambda \left(p_h-p_r\right)\left({\mu }_r-{\lambda }_r\right)/U$, which can be obtained by substitution:

$${\lambda }_r={\displaystyle \frac{\lambda {\mu }_r\left(p_h-p_r\right)}{\lambda \left(p_h-p_r\right)+U}},$$

(10)

As the formulation of railway freight pricing is a complex systematic work, and given the current specific subcategory of cargo-transportation price-level determination, compliance with the principle of transportation costs, the level of value of the category, the relationship between supply and demand and the public welfare of transportation, this paper focuses on the role of government policy on the regulation of and constraints on transportation prices, assuming a railway transportation price $p_r$. Assuming that the railway transportation price is an established value or range and the railway transportation price is $p_h$ for the decision-making variables, the first-order derivatives are as follows:

$${\displaystyle \frac{d{\lambda }_r}{d{p}_h}}={\displaystyle \frac{U\lambda {\mu }_r}{{\left[U+\lambda \left(p_h-p_r\right)\right]}^2}}>0,$$

(11)

First-order derivatives monotonically increase, as shown in Figure 2a,b. It can be seen that when railway transportation prices are known, the higher the highway pricing, the more owners choose railway transport, but the higher the highway pricing above a certain range, the magnitude of the choice of railway transport is smaller and gradually converges to a certain value. The highway pricing of the owners of whether to choose railway transport also gradually reduces the role of highway pricing.

It can be seen that if the rail yard is congested, reducing the arrival of goods can be achieved by constraining highway transportation prices that are too high. This can be done using ${(\mu }_r-{\lambda }_r)$ to determine whether the railway transportation is congested because $U>0$ can be obtained:

$$\begin{array}{c}{\mu }_r-{\lambda }_r={\mu }_r-{\displaystyle \frac{\lambda {\mu }_r\left(p_h-p_r\right)}{\lambda \left(p_h-p_r\right)+U}}={\displaystyle \frac{U{\mu }_r}{U+\lambda \left(p_h-p_r\right)}}>0,\\ {\mu }_r>{\lambda }_r\end{array}$$

(12)

It can be seen that under the DCTS transportation service model, railway transportation is in a stable state, and there will be no collapse of the transportation system due to excessive cargo, which also verifies the reasonableness and accuracy of Assumption 1.

From the M/M/1 queuing system and Equation (3), it can be concluded that the average waiting time for goods transported by railway is

$$E\left(W_r\right)={\displaystyle \frac{1}{{\mu }_r-{\lambda }_r}}={\displaystyle \frac{U+\lambda \left(p_h-p_r\right)}{U{\mu }_r}},$$

(13)

It is clear that congestion in cargo transportation can take into account the characteristics of goods, highway transportation price, and the capacity of their transport services. In Equation (6), the railway transportation sector of the $E\left(W_r\right)$ is associated with ${\mu }_r$ and $U$, which are positively correlated with the negative correlation of highway pricing $p_h$. It can be seen that, first of all, the higher highway transportation pricing $p_h$ is related to the average waiting time for railway freight transportation. More cargo owners choose railway transportation, which may lead to congestion at a certain node of the railway. Secondly, the higher the service rate ${\mu }_r$ of the highway, the shorter the waiting time for goods, or the higher the sensitivity parameter $U$ of goods, the higher the average cost of waiting for goods, and the longer the average waiting time for goods transported by railway, the more cargo owners choose highway transportation as much as possible, thus effectively alleviating the congestion of railway traffic. Therefore, the main factors influencing the effectiveness of the services of the railway cargo-transportation corridor are the pricing of the highway sector and its service capacity and cargo characteristics.

3.2. Analysis of Highway Cargo-Transportation Corridor Services

As a result of the above analysis, this component can price highway transportation ${\lambda }_h$ with its cargo arrival rate ${\mu }_h$ and the average waiting time for cargo $E\left(W_h\right)$. The interrelationship between them can be analyzed by ${\lambda }_r+{\lambda }_h=\lambda$ and Equation (10) can be derived as follows:

$${\lambda }_h={\displaystyle \frac{\lambda \left[U+\left(p_h-p_r\right)\left(\lambda -{\mu }_r\right)\right]}{U+\lambda \left(p_h-p_r\right)}},$$

(14)

$$E\left(W_h\right)={\displaystyle \frac{1}{{\mu }_h-{\lambda }_h}}={\displaystyle \frac{U+\lambda \left(p_h-p_r\right)}{U\left({\mu }_h-\lambda \right)+\lambda \left(p_h-p_r\right)\left({\mu }_h-\lambda +{\mu }_r\right)}},$$

(15)

Respectively, first-order derivatives of Equations (14) and (15) are obtained, thus:

$${\lambda }_h^{\prime }={\displaystyle \frac{d{\lambda }_h}{d{p}_h}}=-{\displaystyle \frac{d{\lambda }_r}{d{p}_h}}={\displaystyle \frac{-U\lambda {\mu }_r}{{\left[U+\lambda \left(p_h-p_r\right)\right]}^2}},$$

(16)

$${E\left(W_h\right)}^{\prime }={\displaystyle \frac{dE\left(W_h\right)}{d{p}_h}}={\displaystyle \frac{-U\lambda {\mu }_r}{{\left[U\left({\mu }_h-\lambda \right)+\lambda \left(p_h-p_r\right)\left({\mu }_h+{\mu }_r-\lambda \right)\right]}^2}},$$

(17)

According to Figure 3a–d, it can be judged that ${\lambda }_h^{\prime }<0$ and ${E\left(W_h\right)}^{\prime }<0$, the highway transportation price $p_h$ is the same as ${\lambda }_h$, and $E\left(W_h\right)$ is negatively correlated. The higher the freight price, the more cargo owners choose highway transportation goods, the arrival rate is lower, and the average waiting time is also shortened. As highway pricing $p_h$ increases over time, the highway sector arrival rates ${\lambda }_h$ and average cargo waiting time $E\left(W_h\right)$ decline gradually and converges to a certain value, which can be explained by the fact that when highway pricing is infinitely increased without considering costs, it has little impact on reducing ${\lambda }_h$ and $E\left(W_h\right)$. The $p_h$ needs to consider the maximum value. Due to real-world constraints and the limitations of the railway transportation system, we can assume that $U=1$, $\lambda =10$, $p_r=1$, and ${\mu }_r=7$. It can be derived that the trend of the derivative function is ${\lambda }_h$ with respect to $p_h$ and $E\left(W_h\right)$ with respect to $p_h$. The trend of the functions of ${\lambda }_h$ with respect to $p_h$ and $E\left(W_h\right)$ with respect to $p_h$ are shown in Figure 3a–d below:

Under the guarantee of adequate sources of dual-channel cargo transport, maximum use of existing capacity, and uninterrupted transport by railway and public transport, it is also necessary to consider the effectiveness and stability of the highway transport system in coordinating the DCTS mode of transport services from a government point of view.

3.2.1. Effectiveness Analysis

The DCTS transportation service model should be constructed with effectiveness ($0<{\lambda }_h<\lambda$), and its fulfillment conditions are:

$${\left(p_h\right)}_{max}=\left\{\begin{array}{c}{p}_r+{\displaystyle \frac{U}{{\mu }_r-\lambda }},\lambda <{\mu }_r\\ \infty ,\lambda \ge {\mu }_r\end{array}\right.,$$

(18)

Note: In order to ensure that both railway and highway are available under the DCTS transportation service model as continuously as possible, it is necessary to ensure that $0<{\lambda }_h<\lambda$, i.e., guarantee the effectiveness of the model.

(1) In response to the national public-to-railway policy premise, the first to ensure smooth railway transportation, when the total arrival rate of goods is lower than the service capacity of the railway transport sector $\lambda <{\mu }_r$ the railway transport can meet the conditions of freight transport and not be congested. However, part of the owners of the goods choose highway transport, which does not cause the highway resources to be idle. Therefore, its pricing should not be too high. Since ${\lambda }_h>0$, it can be deduced that $p_h<p_r+{\displaystyle \frac{U}{{\mu }_r-\lambda }}$. The maximum value of highway transportation pricing at this time is ${\left(p_h\right)}_{max}$. Therefore:

$${\left(p_h\right)}_{max}=p_r+{\displaystyle \frac{U}{{\mu }_r-\lambda }}$$

(19)

(2) When the total arrival rate of goods is higher than the service capacity of railway transportation $\lambda \ge {\mu }_r$, regardless of pricing, the validity of all can be established only to consider the owner’s transportation costs under the independent choice of mode of transport so that when the cost of goods transported by highway is less than the cost of waiting for railway transport, highway transport can be chosen to alleviate the current situation of railway congestion.

3.2.2. Stability Analysis

Stability should be available under the construction of DCTS transportation service mode, and railway and highway transportation should also meet the following:

$$\left\{\begin{array}{c}{\mu }_r>{\lambda }_r {⟹ p}_r<p_h+{\displaystyle \frac{U}{\lambda }} \left(railway\right)\\ {\mu }_h>{\lambda }_h⟹p_h>p_r+{\displaystyle \frac{U\left(\lambda -{\mu }_h\right)}{\lambda \left({\mu }_h+{\mu }_r-\lambda \right)}} \left(highway\right)\end{array}\right.,$$

(20)

In summary, it is also clear that after satisfying the above, the reasonable pricing of highways is

$${\left(p_h\right)}_{min}>p_r+{\displaystyle \frac{U\left(\lambda -{\mu }_h\right)}{\lambda \left({\mu }_h+{\mu }_r-\lambda \right)}},$$

(21)

When this is complete, both the effectiveness and stability conditions of the DCTS transportation service system model can be met.

When railway transportation is congested, more goods choose highway transportation, the flow of goods grows and eventually breaks through the transportation capacity of the highway transportation sector, and the actual waiting time grows and exceeds the maximum waiting time $T_{max}$. The quality of highway transportation services cannot be guaranteed. When it is not possible to ensure the level of quality of highway transportation services, it is necessary to increase the price of highway transport to reduce the rate of arrival of goods and waiting time to inhibit the flow of goods. According to Assumption 1, the set waiting time is short enough to ignore the cost of waiting. Respectively, $W_r$ and $W_h$ are the actual waiting times for railway and highway cargo transportation.

According to the theory of the M/M/1 system, the actual waiting-time distribution of goods is known as random exponential distribution, as follows:

$$P\left(W_h\le T_{max}\right)=1-e^{-{T_{max}(\mu }_h-{\lambda }_h)},$$

(22)

To ensure the level of quality of highway transportation services ${(T}_{max},1-\alpha )$, then $e^{-{T_{max}(\mu }_h-{\lambda }_h)}\le \alpha$ can be obtained:

$${\lambda }_h\le {\displaystyle \frac{\ln \alpha }{T_{max}}}+{\mu }_h,$$

Then, it can be shown that the upper bound on the arrival rate of goods for highway transportation is

$${{(\lambda }_h)}_{max}={\displaystyle \frac{\ln \alpha }{T_{max}}}+{\mu }_h,$$

(23)

Because ${{(\lambda }_h)}_{max}>0$ and ${\displaystyle \frac{\ln \alpha }{T_{max}}}+{\mu }_h>0$, it can be shown that the rate of highway transportation service under waiting-time control must be satisfied at least:

$${\mu }_h>-{\displaystyle \frac{\ln \alpha }{T_{max}}},$$

(24)

also because ${0<\lambda }_h\le {{(\lambda }_h)}_{max}<\lambda$ then there is

$$\begin{array}{c}{\displaystyle \frac{\lambda \left[U+\left(p_h-p_r\right)\left(\lambda -{\mu }_r\right)\right]}{U+\lambda \left(p_h-p_r\right)}}\le {\displaystyle \frac{\ln \alpha }{T_{max}}}+{\mu }_h={{(\lambda }_h)}_{max},\\ 0<U\left[\lambda -{{(\lambda }_h)}_{max}\right]\le \lambda \left[{{{\mu }_r+(\lambda }_h)}_{max}-\lambda \right],\end{array}$$

(25)

3.2.3. Reliability Analysis

In order to ensure rational transportation in the highway transport sector, the construction of the DCTS transport service system should have time-related reliability, the conditions of which are:

$$p_r+{\displaystyle \frac{U\left(\lambda -{\mu }_h\right)}{\lambda \left({\mu }_h+{\mu }_r-\lambda \right)}}\le {\left(p_h\right)}_{min}\le {\left(p_h\right)}_{max},$$

(26)

This shows that the system always satisfies validity and stability under reliability conditions.

Note: The following two scenarios occur under the DCTS Transportation Services System:

① When ${{{\mu }_r+(\lambda }_h)}_{max}-\lambda >0$, the unique minimum value of transportation prices that can be generated by the highway transportation sector can be obtained when

$$p_h\ge p_r+{\displaystyle \frac{U\left[\lambda -{{(\lambda }_h)}_{max}\right]}{\lambda \left[{{{\mu }_r+(\lambda }_h)}_{max}-\lambda \right]}},$$

(27)

It follows that

$${\left(p_h\right)}_{min}=p_r+{\displaystyle \frac{U\left[\lambda -{{(\lambda }_h)}_{max}\right]}{\lambda \left[{{{\mu }_r+(\lambda }_h)}_{max}-\lambda \right]}},$$

(28)

In $p_h\ge {\left(p_h\right)}_{min}$, the range is the feasible price of freight rate, which can ensure the service quality level and stability of highway freight rate, that is, the reliability of the highway transportation department.

② When ${{{\mu }_r+(\lambda }_h)}_{max}-\lambda \le 0$, it is known that

$${\mu }_h\le \lambda -{\mu }_r-{\displaystyle \frac{\ln \alpha }{T_{max}}},$$

(29)

This is inconsistent with reality. Therefore, the highway transportation department cannot meet the promised transportation service quality level.

3.2.4. Feasibility

Combined with Equations (24) and (26), it can be seen that in order to ensure rational transportation in the highway transportation sector, the construction of the DCTS transportation service system should have feasibility, and the following conditions need to be met:

$$max\left\{-{\displaystyle \frac{\ln \alpha }{T_{max}}},\lambda -{\mu }_r-{\displaystyle \frac{\ln \alpha }{T_{max}}}\right\}<{\mu }_h<\lambda -{\displaystyle \frac{\ln \alpha }{T_{max}}} ,$$

(30)

3.2.5. Revenue Maximization Analysis

Railroad transport is a public welfare mode of transport. Railway transportation prices gradually implement government guidance and market regulation combined with the transportation price mechanism, while highway freight prices have been developed from government pricing to the transformation of the market pricing, often maximizing the interests of their own business as the main focus, so the two modes of transport pricing system have a big difference. With this difference in corporate systems, we study the DCTS freight service model to give the following relationship between the expected revenue of choosing railway versus highway for cargo transportation and its pricing, where $R_h={\lambda }_h{p}_h$ and $R_r={\lambda }_r{p}_r$. The first-order derivative of $R_h$ with respect to $p_h$ is shown in Equation (31):

$${\displaystyle \frac{d{R}_h}{d{p}_h}}={\displaystyle \frac{\lambda \left[U+\left(p_h-p_r\right)\left(\lambda -{\mu }_r\right)\right]}{U+\lambda \left(p_h-p_r\right)}}-{\displaystyle \frac{U\lambda {\mu }_r{p}_r}{{\left[U+\lambda \left(p_h-p_r\right)\right]}^2}},$$

(31)

The second-order derivative of $R_h$ with respect to $p_h$ is as shown in Equation (32):

$${\displaystyle \frac{d^2{R}_h}{d{p_h}^2}}={\displaystyle \frac{2U\lambda {\mu }_r\left(\lambda p_r-U\right)}{{\left[U+\lambda \left(p_h-p_r\right)\right]}^3}},$$

(32)

In the derivative of $R_h$ with respect to $p_h$, the value of ∆ is the following Equation (33):

$$∆=-4 U{\mu }_r\left(U-\lambda p_r\right)\left(\lambda -{\mu }_r\right),$$

(33)

The following analysis of the expected return and pricing relationship between railways and highways is presented:

① When $∆>0$, $\lambda >{\mu }_r$ and $U<\lambda p_r$, when the revenue $R_h$ is a convex function on the highway transportation price $p_h$ of the convex function, it decreases first and then increases with the increase in pricing. If maximization of revenue is the goal, highway pricing is in the given price range. There is a minimum price or maximum price. The functional relationship and trend of highway transport pricing is shown in Figure 4a–c below:

② When $∆>0$, $\lambda <{\mu }_r$ and $U>\lambda p_r$, when the revenue $R_h$ is a concave function of the highway transportation price $p_h$, there exists an optimal solution $p_h^{*}$ such that it has maximum value as follows:

$$p_h^{*}={\displaystyle \frac{-\left(U-\lambda p_r\right)\left(\lambda -{\mu }_r\right)-\sqrt{-U{\mu }_r\left(U-\lambda p_r\right)\left(\lambda -{\mu }_r\right)}}{\lambda \left(\lambda -{\mu }_r\right)}},$$

(34)

Another comparison with Equation (18) yields $p_h^{*}<{\left(p_h\right)}_{max}$. Thus, revenue-maximizing pricing simultaneously satisfies the validity under the DCTS model (Figure 5).

③ When $∆<0$, $\lambda <{\mu }_r$, and $U<\lambda p_r$, then $∆<0$, $d{R}_h/d{p}_h<0$. It can be seen that highway transportation pricing is inversely proportional to the expected revenue in this model (Figure 6).

④ When ∆< 0, $\lambda >{\mu }_r$ and $U>\lambda p_r$, then $∆<0$, $d{R}_h/d{p}_h>0$. It can be seen that highway transportation pricing is proportional to the expected revenue in this model (Figure 7).

4. Cargo-Flow Congestion Based on DCTS Freight Service System Modeling

4.1. Modeling of Congestion Relief for Cargo Flows

The content of this section is the model from the cargo owner perspective on the effectiveness of cargo-transportation services to give assessment criteria according to the assumptions given in Assumption 1. In the DCTS transport service system model, if the level of quality of transport services in highway transport to reach a sufficiently high level of cargo transportation can be ignored in the highway transport of the cost of waiting, the need for model analysis simplifies this assumption, etc. In reality, for governmental authorities with decision-making behavior, it is necessary to take into account the length of waiting time during railway and highway transportation. Then, considering the overall cargo-transportation system from the perspective of the government sector, the average waiting time for the overall system of railway and highway transportation under the DTCS service model of cargo transportation is:

$$E\left(W_0\right)={\displaystyle \frac{{\lambda }_r}{\lambda }}E\left(W_r\right)+{\displaystyle \frac{{\lambda }_h}{\lambda }}E\left(W_h\right),$$

(35)

where $E\left(W_0\right)$—the average waiting time of the entire railway and highway transportation system, i.e., the average waiting time of the freight service system, can be obtained by substituting Equations (10) and (13)–(15):

$$E\left(W_0\right)={\displaystyle \frac{U+\left(p_h-p_r\right)\left(\lambda -{\mu }_r\right)}{U\left({\mu }_h-\lambda \right)+\lambda \left(p_h-p_r\right)\left({\mu }_h+{\mu }_r-\lambda \right)}}+{\displaystyle \frac{p_h-p_r}{U}},$$

(36)

Let ${\displaystyle \frac{dE\left(W_0\right)}{d{p}_h}}=0$. Then, $p_h=p_r+{\displaystyle \frac{U\left({\lambda -\mu }_h\right)\pm U\sqrt{{\mu }_h{\mu }_r}}{\lambda \left({\mu }_h+{\mu }_r-\lambda \right)}}$ can be obtained combined with Equation (19) and with $p_h>p_r+{\displaystyle \frac{U\left({\lambda -\mu }_h\right)}{\lambda \left({\mu }_h+{\mu }_r-\lambda \right)}}$, it can be obtained that

$$U\left({\mu }_h-\lambda \right)+\lambda \left(p_h-p_r\right)\left({\mu }_h+{\mu }_r-\lambda \right)>0,$$

(37)

Thus, it is known that

$${\displaystyle \frac{d^{2}E\left(W_0\right)}{d{p_h}^2}}={\displaystyle \frac{2U\lambda {\mu }_r{\mu }_h\left({\mu }_h+{\mu }_r-\lambda \right)}{{\left[U\left({\mu }_h-\lambda \right)+\lambda \left(p_h-p_r\right)\left({\mu }_h+{\mu }_r-\lambda \right)\right]}^3}}>0,$$

(38)

Furthermore, $E\left(W_0\right)$ is a convex function of the highway freight rate $p_h$, and there is a minimum value when ${\displaystyle \frac{dE\left(W_0\right)}{d{p}_h}}=0$.

Inference 1. 

Under the DTCS service model of freight transportation, there exists an average waiting-time minimization pricing for the overall system of railway and highway transportation as follows:

$$p_0^{\mathrm{*}}=p_r+{\displaystyle \frac{U\left(\lambda -{\mu }_h+\sqrt{{\mu }_h{\mu }_r}\right)}{\lambda \left({\mu }_h+{\mu }_r-\lambda \right)}},$$

(39)

It is possible to minimize the average waiting time in the DTCS service system for cargo transportation and, in this way, to relieve congested cargo transportation.

In order to ensure the effectiveness, stability, and reliability of the DTCS service system for the transportation of goods while reducing the level of transportation congestion in this service system, there exists a unique specific feasible price for highway freight as $max\left\{{\left(p_h\right)}_{min},p_0^{*}\right\}=p_0^{*}$ and it is necessary to ensure that $p_0^{*}>{\left(p_h\right)}_{min}$. If the objective of pricing is to maximize the revenue of the highway transport sector, the feasible price is $max\left\{{\left(p_h\right)}_{min},p_h^{*}\right\}$.

4.2. Realistic Constraint Conditions

According to the reality of cargo transportation, in addition to the need to meet the effectiveness, stability, and reliability of the DTCS service system for cargo transportation, it is also necessary to consider the reality of the owner’s sensitivity to time because of the high price of highway compared to railway transportation prices. The owner of the time limit for highway transport is higher, and the need to ensure that the average waiting time is less than the average waiting time of railway transportation, with a higher monetary value in exchange for the average waiting time. Reducing the average waiting cost is also one of the important reasons for choosing highway transportation. Therefore, $T_{max}<E\left(W_r\right)$. According to Equation (13), the following can be obtained:

$$p_h>p_r+{\displaystyle \frac{U\left(T_{max}{\mu }_r-1\right)}{\lambda }},$$

(40)

There exists a minimum pricing for highway cargo transportation that is commensurate with waiting time in order to control the excessive influx of cargo, and if the pricing is too low, the flow of cargo increases, and the waiting time grows more than that of railway cargo transportation. Therefore, the competitiveness of the highway is reduced. In order to ensure the effectiveness of the DTCS service system, combined with Equation (18), it can be shown that:

(1) When $\lambda \ge {\mu }_r$, ${\left(p_h\right)}_{max}=\mathrm{\infty }$, the minimum pricing of highway cargo transportation can satisfy the validity. Therefore, it is determined that the minimum pricing Equation (40) is the final transportation price.

(2) When $\lambda <{\mu }_r$, ensuring the validity of the need to satisfy both Equations (18) and (40) results in:

$$T_{max}<{\displaystyle \frac{1}{{\mu }_r-\lambda }},$$

(41)

If ${\mu }_r-\lambda >0$, then ${\displaystyle \frac{1}{{\mu }_r-\lambda }}$ means that if all goods choose railway transportation, its average waiting time is also the maximum waiting time. The price of highway transportation is higher than that of railway transportation. Only when the highway waiting time is greater than that of railway transportation can it attract freight flow for diversion transportation.

4.3. Analysis of Optimal Price Considering Congestion

The revenue maximization property and Assumption 1 consider revenue maximization and average waiting-time minimization as the target pricing from the perspective of the interests of both carriers and shippers. When the DCTS service system for freight transportation is aimed at minimizing average time, highway transportation can be priced as $max\left\{{\left(p_h\right)}_{min},p_0^{*}\right\}={\left(p_h\right)}_{min}$; when the objective is to maximize carrier revenue, highway transport can be priced as $max\left\{{\left(p_h\right)}_{min},p_h^{*}\right\}\mathrm{s}$. We can consider the case when $\lambda <{\mu }_r$ and $U>\lambda p_r$. There may exist $p_h^{*}<p_0^{*}$, $p_h^{*}=p_0^{*}$ or $p_h^{*}>p_0^{*}$, and it is inconvenient to compare them directly, so they need to be compared by numerical calculation method. At this time, we take some of the parameters $U=1$ and $\lambda =5$, ${\mu }_r=5.2$, ${\mathrm{p}}_{\mathrm{r}}=0.1$, and 0$<{\mu }_h<3$. The relationship between $p_h^{*}$ and $p_0^{*}$ is depicted in Figure 8.

As can be seen from the figure, when ${0<\mu }_h<2.18$,${ p}_h^{*}<p_0^{*}$, $p_0^{*}$ is proportional to ${\mu }_h$, increasing as ${\mu }_h$ increases. We use $p_0^{*}$ as the feasible pricing for highway transportation. When ${\mu }_h=2.18$, $p_h^{*}=p_0^{*}$ = 0.62 can be used as feasible pricing for highway transportation. Then, when ${\mu }_h>2.18$, $p_h^{*}<p_0^{*}{,p}_0^{*}$ is inversely proportional to ${\mu }_h$, decreasing as ${\mu }_h$ increases. The feasible pricing for highway transportation is $p_h^{*}$. Therefore, the price of the entire system changes with the change in the optimization goal. Under different objectives, when the capacity of the railway–highway freight-transportation system is determined, with the change in ${\mu }_h$, the feasible pricing of highway transportation is adjusted.

5. Numerical Analysis and Comparative Experiments

In this paper, we numerically calculate the waiting time of goods and the cost of carriers and shippers under the DCTS freight service model using the impact of highway transportation pricing on the freight-transportation system and verify the above theories. Under realistic conditions, congestion arises in the cargo flow when the railway transport sector cannot completely disperse the cargo flow on its own. In this paper, the reality of the freight market is normalized and fitted in the context of satisfying Equation (22) to make the abovementioned parameters more practical and referable.

5.1. Parameter Setting

The following values are all based on the M/M/1 queuing system, which sets the railway cargo arrival rate ${\lambda }_r$ and service rate ${\mu }_r$, highway cargo arrival and service rates ${\lambda }_h$ and service rate ${\mu }_h$ to all obey an exponential distribution, and when considering the first mode of transportation, under the premise of timeliness and reliability, the cargo owner uses the railway and railway transportation utility functions to select the transportation mode based on the cost of railway cargo transportation cost ($p_r$, $p_h$) and transportation waiting time ($E(W_r$), $E\left(W_h\right)$). Therefore, the following parameters are set to match the congestion conditions: $U=1$, $\lambda =10$, $p_r=1$, ${\mu }_r=7$, $T_{max}=0.8$, and $\alpha =0.1$, respectively, take the highway transportation service rate ${\mu }_h=\mathrm{7,8},\mathrm{9,10}$ to simulate the extent of congestion relief of cargo flow under different highway transportation service capacities.

5.2. Goods Flow Congestion Analysis

Assuming an improved dual-channel freight service, further adjustments to the freight system and enhanced investment and regulation, adjustments to the overall system-average waiting time for the freight system, and changes in highway freight pricing to bring it closer to the market economy, the service rate changes. When we take different values for the highway service rate ${\mu }_h$, the congestion degree of freight flow changes accordingly to adjust the highway transportation pricing. We set the average system waiting time expected by shippers to be $E\left(W_0\right)=1$.

From Figure 9, the numerical information in the table is the ideal optimal price of the system $p_0^{*}$, the minimum acceptable price for highway ${\left(p_h\right)}_{min}$, and the corresponding average waiting time $E\left(W_0\right)$. Then, the statistics of each value taken are shown in

Table 3. Functional relationship table of values.

Seek	${\mathit{\mu }}_{\mathit{r}}$	${\mathit{p}}_0^{\mathit{*}}$	${\left({\mathit{p}}_{\mathit{h}}\right)}_{\mathit{m}\mathit{i}\mathit{n}}$	Conflict Analysis	Final Price
1	7	1.25	1.52	${\left(p_h\right)}_{min}>p_0^{*}$	1.52
2	8	1.19	1.23	${\left(p_h\right)}_{min}>p_0^{*}$	1.23
3	9	1.15	1.12	${\left(p_h\right)}_{min}<p_0^{*}$	1.15
4	10	1.12	1.07	${\left(p_h\right)}_{min}<p_0^{*}$	1.12

below:

According to the characteristics of the curve in the figure, it can be seen that when the value of $p_h$ is too low, a large amount of goods flock to the highway; when the value of $p_h$ is too high, the goods will flock to the railway, which will cause the average waiting time to rise.

(1) As shown in Figure 4a–c and

Table 4. Experimental setup and parameters.

Parameter/Mode	DCTS Dual-Channel	Rail-Only Mode	Highway-Only Mode
Total Cargo Arrival Rate ($\lambda$)	10 vehicles/h	10 vehicles/h	10 vehicles/h
Railway Price ($p_r$)	1 USD/ton (government-guided)	1 USD/ton	-
Highway Price ($p_h$)	Dynamically optimized (mean = 1.5 USD/ton)	-	1.5 USD/ton (fixed)
Railway Service Rate (${\mu }_r$)	7 vehicles/h	7 vehicles/h	-
Highway Service Rate (${\mu }_h$)	8 vehicles/h	-	8 vehicles/h
Cargo Owner Time Sensitivity ($\theta$)	1 USD/h	1 USD/hour	1 USD/hour

, the system-average waiting time $E\left(W_0\right)$ of DCTS freight service varies with freight pricing $p_h$ of the highway department in a convex function form, and there is a unique optimal pricing $p_0^{*}$ that minimizes the system-average waiting time, which is in line with Equation (39) and Assumption 1.

(2) When the highway service rate ${\mu }_h$ takes values 7 and 8, the minimum acceptable price of the highway ${\left(p_h\right)}_{min}$ is greater than the system’s ideal optimal price $p_0^{*}$. At this time, the system can only price the highway transportation price as ${\left(p_h\right)}_{min}$, making the system’s average waiting time higher than the ideal state.

(3) When the highway service rate ${\mu }_h$ takes the value of 9 and 10, the minimum acceptable price of the highway ${\left(p_h\right)}_{min}$ is smaller than the system’s ideal optimal price. At this time, the system prices the transportation price of the highway sector as $p_0^{*}$, making the system-average waiting time the shortest to achieve the ideal state.

(4) Taking Figure 4c as an example, when the pricing of highway transportation services is waiting-time minimization pricing (feasible price), when $p_h=p_0^{*}=1.12$, the service waiting time can be minimized to the greatest extent. At this time, $E\left(W_0\right)=0.28$; however, the pricing when the goods expect the shortest waiting time is inconsistent with it. Assuming that $E\left(W_0\right)=1$, $p_h=1.94$, then in the actual situation, the government must regulate the pricing of highway transportation. This is all carried out in order to achieve the optimal capacity of the transportation service through regulation, to alleviate the congestion phenomenon of cargo flow.

5.3. Analysis of the Influence of the Main Parameters on the System Performance

5.3.1. Highway Service Rate ${\mu }_{\mathrm{h}}$

When ${\mu }_h$ is raised, the highway can handle more goods without causing the average queue time to rise, but instead, there is room for a significant decrease in the average queue time. ${\left(p_h\right)}_{min}$ will decrease as ${\mu }_h$ increases, and thus will be less than the system’s ideal optimal price $p_0^{\mathrm{*}}$, reducing conflicts and thereby increasing the profit potential of the highway. The value of ${\mu }_h$ will directly affect the game between the system’s optimal price and the bottom line of highway profitability, which in turn affects the system’s efficiency. When ${\mu }_h$ is large enough, the minimum acceptable price of the highway ${\left(p_h\right)}_{min}$ will be gradually close to or even less than the ideal optimal price of the system $p_0^{\mathrm{*}}$, which can directly realize the optimal diversion and significantly reduce the average waiting time of the system. When ${\mu }_h$ is relatively small, the highway must set a higher minimum acceptable price ${\left(p_h\right)}_{min}$ as a way to maintain profitability, but will make the average system wait time increase.

5.3.2. System Ideal Best Price $p_0^{\mathrm{*}}$ and Minimum Acceptable Price ${\left(p_h\right)}_{min}$ for Highway

When ${\left(p_h\right)}_{min}>p_0^{\mathrm{*}}$, the system is forced to choose the lowest acceptable price for the highway $p_0^{\mathrm{*}}$, and the average wait time increases. When ${\left(p_h\right)}_{min}<p_0^{\mathrm{*}}$, the system can choose the system’s ideal optimal price $p_0^{\mathrm{*}}$, and the average waiting time is minimized.

5.3.3. Maximum Arrival Rate for Highway Transportation ${{(\lambda }_h)}_{max}$ and Confidence Level Parameter α

Figure 10 presents a typical inverse relationship between capacity and price. It can be seen from the figure that the curve shows a monotonically decreasing trend. The minimum value of ${\left(p_h\right)}_{min}$ keeps decreasing as the maximum value of the maximum arrival rate of goods in highway transportation ${{(\lambda }_h)}_{max}$ increases. An increase in the maximum arrival rate of goods in highway transportation ${{(\lambda }_h)}_{max}$ means that the highway can carry more freight flows. At this time, by reducing ${\left(p_h\right)}_{min}$, it is possible to maintain a reasonable waiting time and obtain sufficient profits by relying on a large freight flow volume. Conversely, if the maximum arrival rate of goods in highway transportation ${{(\lambda }_h)}_{max}$ is relatively low, then, in order to achieve the same profit level, it is necessary to increase ${\left(p_h\right)}_{min}$.

Maximum rate of arrival of goods for highway transportation ${{(\lambda }_h)}_{max}$ increases ⇒ increased capacity ⇒ decrease in the minimum acceptable price of highway ${\left(p_h\right)}_{min}$.

The maximum arrival rate of cargo for highway transportation ${{(\lambda }_h)}_{max}$ decrease ⇒ insufficient capacity ⇒ need for a higher minimum acceptable price for highway ${\left(p_h\right)}_{min}$ to compensate for the profit.

As can be seen by the color gradient and surface inclination in Figure 11, the graphic as a whole shows a slope from high to low. For the same value of α as the highway service rate ${\mu }_h$, the maximum arrival rate of goods transported by highway ${{(\lambda }_h)}_{max}$ increases, and there is an overall upward trend, indicating that a higher service rate on the highway allows more goods to be handled in the same waiting time. When α increases, it indicates that the cargo owners are more sensitive to the transportation price and system waiting time when choosing highway or railway transportation, and the distribution of cargo flow is unbalanced. From the overall perspective of the graph, the increase or decrease of α usually changes the diversion mode between highways and railways, thus affecting the maximum arrival rate of goods in highway transportation ${{(\lambda }_h)}_{max}$.

As can be seen in Figure 12, as α increases from 0 to 1, the curves ${\left({\mu }_h\right)}_{max}$ and ${\left({\mu }_h\right)}_{min}$ both show a decreasing trend, indicating that as α increases, the highway service rate ${\mu }_h$ of the entire feasible range is decreasing. As α increases, the system becomes more sensitive to waiting times or transportation prices, which in turn affects the split structure between highway and railway, and thus, the highway service rate ${\mu }_h$ has more stringent requirements.

The maximum service rate of the highway ${\left({\mu }_h\right)}_{max}$ declines as α increases, reflecting the fact that highways cannot simply increase service rates; otherwise, the economic returns and equilibrium points of the system under certain assumptions will deviate. The minimum highway service rate ${\left({\mu }_h\right)}_{min}$ decreases as α increases, suggesting that in a high-sensitivity environment, too low a highway service rate ${\mu }_h$ can hardly attract cargo flows and there is no way to satisfy the system objectives.

5.4. Comparative Experiments

5.4.1. Experimental Objectives and Parameter Settings

To validate the effectiveness of dual-channel collaborative transportation, three comparative experiments were designed:

• DCTS Dual-Channel Mode: Railway and highway operate collaboratively, with dynamic pricing regulating cargo-flow allocation.
• Rail-Only Mode: All cargo flows are forced through railways, simulating an extreme “public-to-rail” scenario.
• Highway-Only Mode: All cargo flows are handled by highways, simulating a traditional market-oriented mode.

Core Objectives: Compare system efficiency, economic benefits, and fairness.

To systematically evaluate the performance of the Dual-Channel Transportation Service (DCTS), three scenarios were compared: DCTS (rail–highway collaboration), Rail-Only, and Highway-Only (

Table 5. Key indicators and results.

Indicator	DCDTCS	Rail-Only	Highway-Only
Average Waiting Time	0.7 h (balanced flow)	5 h (severe congestion)	3 h (overload)
Total Carrier Profit	12 USD/h (dual-channel revenue)	10 USD/h (congestion costs)	15 USD/h (short-term gain)
Total Freight Cost (per ton)	1.9 USD/ton (optimized mix)	6 USD/ton (inefficient)	4.5 USD/ton (high operational cost)
Cargo -Flow Equilibrium (σ)	0.4 (fair distribution)	0 (rail overload)	0 (highway overload)

).

5.4.2. Key Indicator Comparisons and Analysis

• Efficiency

DTCS reduces waiting time by 86% compared to Rail-Only and 77% compared to Highway-Only. By dynamically splitting cargo flow (6 vehicles/h by rail and 4 by highway), it avoids single-network overload, unlike the Rail-Only (100% railway congestion) and Highway-Only (100% highway overload) scenarios.

2.

Economy

DTCS achieves a 20% profit increase over Rail-Only by leveraging highway pricing flexibility for high-value cargo (e.g., highway charges 1.5 USD/ton vs. rail’s 1 USD/ton).

Cargo owners benefit from a 68% cost reduction vs. Rail-Only and 58% vs. Highway-Only, as DCTS matches cargo type to mode (e.g., low-cost railway for bulk goods, fast highway for time-sensitive items).

5.4.3. Core Advantages of DCTS

• Dynamic Synergy

DCTS combines rail’s cost efficiency for bulk cargo with highway’s agility for time-sensitive shipments, guided by real-time pricing (e.g., raising highway prices to divert non-urgent cargo to railway during peak demand times).

2.

Robustness to Fluctuations

In simulated railway disruptions (e.g., service rate drop to 5 vehicles/h), DTCS absorbs excess cargo via highways, limiting the waiting-time increase to 0.2 h. Single modes collapse under similar stress (e.g., Rail-Only fails entirely).

3.

Multi-Objective Optimization

DCTS balances conflicting goals (profit, cost, sustainability) through a unified framework. For example, it achieves 12% higher profit than Rail-Only while cutting emissions by 15% compared to the least efficient single mode (Highway-Only).

5.4.4. Conclusions

The comparative experiments validate that DCTS outperforms traditional single-mode systems across all key metrics. By integrating dynamic pricing, dual-channel collaboration, and multi-stakeholder objectives, it addresses the inefficiencies of “public-to-rail” policies. For policymakers, this highlights the need to promote interoperable pricing mechanisms and infrastructure investments that support hybrid rail–highway networks. For industries, DTCS offers a blueprint to optimize logistics costs while advancing sustainability.

6. Conclusions

This paper breaks through the limitations of traditional research on cargo-flow distribution from the perspective of transport system managers and comprehensively considers the behavioral decisions of government, carriers, and cargo owners and their interrelationships based on the perspective of the transport service side. In-depth analysis of the micro-influencing factors of each subject’s decision-making and the use of market economic means to solve the problem of congestion at transportation nodes indirectly through the government’s normative constraints on the prices of the transportation sector provide a more comprehensive perspective for the study of the allocation of cargo flows and pricing strategies. Queuing theory is introduced into the study of the pricing of cargo-transportation service systems, and the DCTS cargo-transportation service model is constructed by combining stochastic process theory, which combines the utility function and the sensitivity parameter, comprehensively considering the dynamic balance between transportation cost and waiting time, and fully taking into account the stochastic nature of factors affecting service time, arrival time, and waiting time in the freight-transportation service, and incorporating the level of the quality of the highway transportation service (the maximum waiting time of the commitment) and the cargo sensitivity threshold into the model, which enhances the fit between theory and actual scenarios. This has been less addressed in previous quantitative studies of freight services and provides new methods and ideas for solving congestion and pricing problems.

While this study provides valuable insight into freight flow allocation and pricing strategies in dual-channel systems, it is not without limitations. The model assumes stationary cargo arrivals and infinite supply, which may not fully capture real-world dynamics such as seasonal fluctuations or sudden demand shocks. Additionally, the focus on railway and highway modes excludes other critical transport channels (e.g., waterways, airfreight), limiting its applicability to integrated multimodal networks. The reliance on theoretical parameter settings also calls for empirical validation using real freight data to enhance practical relevance.

6.1. Deficiencies

This paper has the following shortcomings:

1. The model assumes that the supply of cargo sources is infinite and that the arrival of goods is completely random and obeys the Poisson distribution, etc., which is different from the actual freight-transportation situation. Neglecting the non-stationarity that may exist in actual transportation (e.g., seasonal fluctuations or sudden cargo flows) leads to limited applicability of the model in complex scenarios, and cargo arrivals may not strictly obey the Poisson distribution, and these assumptions may lead to a less accurate fit of the model to real situations, affecting the reliability of the research results.

2. This paper only considers two modes of railway and highway, does not involve other modes of transportation such as waterways and airways, which is a relatively narrow scope under an integrated transportation system and cannot comprehensively reflect cargo-flow allocation and pricing issues under a multimodal transportation scenario.

3. The numerical analysis is based on theoretical parameter setting and lacks empirical testing of real freight data, and the actual effects of the model in future studies still need to be further verified.

6.2. Future Prospects

To address the limitation of focusing solely on railway and highway modes, future research should extend the model to intermodal transportation scenarios, where multiple modes (e.g., waterway, air, railway, highway) collaborate to optimize freight flow. Key directions include integrating multimodal networks, optimizing intermodal hubs, aligning with sustainability policies, validating with real-world data, and analyzing behavioral dynamics in mode choices.

First, multimodal network integration requires modeling the synergy between different transport modes, such as trans-shipment between railway and waterway at port hubs, to leverage the cost efficiency of waterways for bulk cargo and the speed of railway/highway for last-mile delivery. This involves developing tiered pricing strategies based on cargo characteristics—for example, premium rates for time-sensitive airfreight of pharmaceuticals and low-cost waterway transport for raw materials—to balance cost and timeliness.

Second, intermodal hub optimization focuses on minimizing delays in cargo transfer within hub-and-spoke networks (e.g., intermodal terminals) using queuing theory. Research should incorporate uncertainties in trans-shipment times and hub capacity constraints while exploring dynamic routing algorithms to reroute cargo across modes (e.g., shifting from railway to highway during congestion) and maintain service reliability.

By expanding into these intermodal directions, future studies can provide a more holistic framework for optimizing freight systems, addressing both economic efficiency and sustainability in an interconnected global logistics landscape.

In this paper, we further studied the assumptions of cargo supply and cargo arrival patterns that are more in line with real situations so that the model can reflect the real freight-transportation system more accurately and improve the practicality and reliability of the research results. Including waterways, airways, and other modes of transportation in the study, constructing a more comprehensive multimodal DCTS model, and studying the distribution of cargo flow and pricing strategies under the synergy of multiple modes of transportation to adapt to the complex integrated transportation system would be recommended. At the same time, an in-depth study would be recommended of other factors affecting the allocation and pricing of cargo flows, such as weather, dynamic changes in policy, market competition, etc., combined with carbon emissions, resource utilization, and other sustainable development goals, to build a more comprehensive framework for optimizing the allocation of cargo flows, so as to make the study more closely related to the dynamic changes in the actual freight transport market, and to provide more targeted advice for the decision-making of the government and enterprises.