Construction of the Mathematical Model of Pricing for Telecommunication Services with Allowance for Congestion in Networks

This paper considers a model of dynamic pricing in the telecommunications market incomplete competition and taking into account overloads in multiservice networks. The model consists in the use of mathematical modeling methods, game theory and queueing theory. It is assumed that telecommunication companies agree on the rules of incoming and outgoing traffic charging in pairs, and this charging is built as a function of the tariffs that companies offer their subscribers for service. Companies are limited the agreement on mutual rules of reciprocal proportional charging for access traffic at first, which subsequently determine the tariffs for the multiservice network users. The reciprocity of the rules means that companies are subject to the same rules for the entire time interval during which the agreement is in force. Taking into account imperfect competition in the telecommunications market and using profit optimization method for each company the equilibrium tariffs and the volume of services are found with subject to congestion in multi-service networks.


Introduction
Methods of mathematical modeling in the economy of telecommunications are being actively developed [1][2][3][4][5][6][7]. Jean Tirole considers the impact of telecommunication technologies on competition in services and goods markets [8][9][10][11][12]. In 2014 he was awarded the Nobel Memorial Prize in Economic Sciences for his analysis of market power and regulation.
In paper [13], Se-Hak Chuna considered optimal access charges for the provision of telecommunication network, mobile commerce, and cloud services. Using theoretical analysis, Se-Hak Chuna investigated, when a regulator can set rational access pricing, considering the characteristics of access demand. Se-Hak Chuna demonstrated that optimal access prices depend on whether the final products or services are independent strategies or substitute strategies. The results have applications for policy makers setting optimal access charges that maximize social welfare.
In this article a mathematical model of pricing for telecommunications services with overloads in networks is built. It generalizes the model that was built earlier [14,15].
It is assumed that telecommunications companies agree in pairs on the rules of charging for access traffic to the network of the other company, and it is considered as a function of the tariffs that companies offer their consumers (subscribers) for services. Thus, these companies have contracts at the first stage by agreements on reciprocal proportional access charge rules (RPACR), which subsequently allow them to determine the subscription rates. The ambiguity of the rules means that companies are subject to the same rules for the entire time interval during which the agreement is valid.
RPACR may be seen as analogous to the regulatory policy of the state of the telecommunications industry. If telecommunication services, provided by different companies, are close substitutes, the use of RPACR by companies leads to competitive prices in industry. However, if it is assumed that competing companies follow the policy of services differentiation, then intervention of the state is required to preclude the use by companies of monopoly power.
It is also assumed that the utility function of subscribers consists of deterministic and stochastic parts. The deterministic part allows to find a linear function of subscribers demand for telecommunications services, which has a constant price elasticity. It allows to avoid unlimited growth of consumption of telecommunication services by subscribers at aspiration the corresponding tariffs to zero and ensures the existence of a saturation point, i.e., for example, there are time limits that the subscriber uses for using telecommunication services. The Weibull distribution is used for the stochastic component of the utility function, which is convenient for further analysis. It is possible to find equilibrium tariffs and equilibrium demand for telecommunication services. This equilibrium is equilibrium in pure strategies and it always exists, and the subscription rates are calculated explicitly. , and the total number of channels is for network . Let be a capacity (bits/sec) of -node ( = 1, ), and a throughput (bits/sec) -link ( = 1, ) of network company in the time period . Two-point connections can be established to transmit information flows between the network nodes of network . Each connection is characterized by a route, i.e. a set of network links , through which connections are established. Let = {1, . . . , } be a set of services that offer companies for potential consumers (subscribers) during the period ∈ {1, 2, . . . , max }. Let ( ∈ (1, 2, . . . , )) be a set of consumers, who want to use the telecommunications services in the market.
Let's assume that the individual consumer demand function for the service = {1, . . . , } has the form: ( ) is a linear function of the price , and > 0 and > 0 is positive coefficients, which are determined from the market research services in the period .
A consumer generates the traffic loading or the load using the service in the period . Let be an individual traffic volume of a consumer , and let =¯ℎ be the average value of , where the parameter¯is the average intensity of the flow of requests and the parameter ℎ is the average duration of service in the period .
We assume that the average load is generated by the consumer when using the service in the period , linearly depends on the corresponding demand function for this service where is the proportionality factor for the service. It links the consumer demand for telecommunication services and the amount of traffic generated by this consumer in the network.
The total network traffic volume that is created by a consumer in the period during using the service , is the sum of consumers network traffic volumes where¯,¯are parameters of the function .
The total consumers demand for the service during the time is the sum of all demand functions for the service of all: where the parameters 0 and 0 are determined from market research of services in the period .
We can get a link between the network traffic volume ( ) and the demand function ( ) of the service during the period : where ( ) is linear price functions and = , = are coefficients.
We can get the network traffic volume that is associated with the consumer ( = 1, ) where¯ 0,¯ 0 are parameters load functions associated with the consumer , and a parameter¯is a tariff for services (service package) during the time period .
A consumer's ( = 1, ) demand for -services in the considered time period has the form: Aggregating the network traffic volume ( ) from (5) for all services = {1, . . . , }, we can get the total network traffic volume ( ) for the period in the form: where¯ 0 and¯ 0 are aggregated parameters of function ( ), and where function of aggregated demand for services (service package) has the form: where the parameters¯ 0 and¯ 0 are aggregated parameters of the demand function ( ).
We can assume that for each company ( = 1, ) there exists a function of consumer demand for services (

Multiservice Demand Function
Suppose that each consumer can use telecommunication multiservice network of companies ( ∈ {1, . . . , }) at any time period . Let's assume that each consumer has individual tastes and preferences in relation to these services . We assume that the consumer ( ∈ {1, . . . , }), which is ready to choose one service from the set ∈ {1, . . . , } of the company ( ∈ {1, . . . , }), has the following utility function: where the random parameter characterizes individual tastes and preferences of the consumer. Let's consider that has a Weibull distribution. The value of gives the characteristic measures of the dispersion of tastes and preferences of the consumers, that is, allows us to estimate the substitutability telecommunication services ∈ {1, . . . , } Since the values are independent and have a Weibull distribution we have that where = 2/ . Similarly for the company we have the same Thus, each consumer chooses one service in the company with probability and in the company with probability . We can generalize this approach for the case when the consumer chooses one company from the set of companies { 1 , . . . , } to obtain the service , and we can get the probability in case the consumer gives preference to the company : The probability that the consumer chooses one company from a set of companies { 1 , . . . , } to receive service package has the form: The expected value of consumers ( ) who chooses a company is determined by the probability , which can be considered as the market share of a company , and has the form The demand of consumers for services ∈ {1, . . . , } of the company ( ∈ {1, . . . , }) has the form: Demand function of the consumers who have plan to use the service of a company , which may be implemented within network , and demand function of the consumer who has plan to use the service implemented with resources of the networks and , have the form: where the aggregated -service demand has the form: and the total network traffic volume demand for company has the form: , and the total network traffic volume for a company has the form: where is an "average" linking parameter for function and . Revenue function of companies ( ∈ {1, . . . , }) at the period ( = 1, 2, . . . , max ) has the form: where ∈ [0, 1] is a parameter to be defined during negotiations between companies and . We assume that the cost of an access service to the competitor's network is a value proportional to the cost of servicing by this company of its consumers. Profit function Π of companies ( ∈ {1, . . . , }) at the period ( = 1, 2, . . . , max ) has the form: where is a total costs function and is a fix cost.

Profit Company Control Problem and Overloads in Networks
We can formulate an optimization problem for each company ( ∈ {1, . . . , }) at any time ∈ {1, 2, . . . , max }: The following theorem holds true. Proof. Let's write out the profit function of company in the form of: We can calculate the derivatives of¯and equal them to zero, thus we obtain a system of algebraic equations of the form: and the equilibrium value of the tariff has the form: We can obtain for 2 Π /¯2, The theorem is proved.
After substituting the corresponding equilibrium tariffs¯* in the profit function, we obtain the following equation and differentiating by and equaling to zero, we have a system of algebraic equations, by solving which, we obtain an equilibrium value of * = 0.5.
The equilibrium tariff¯* for the services of company , taking into account the optimal value * = 0.5 during the period , has the form: The equilibrium demand function for the company ( ∈ {1, . . . , }) services at any can be represented as follows: and the total network traffic volume for a company with the equilibrium tariff has the form: The total equilibrium market demand function * and the total equilibrium traffic volume * for services at any has the form: and we can show that with a uniform distribution of customers between all companies ( ∈ {1, . . . , }) the total equilibrium traffic volume for services reaches maximum. If the network bandwidth of companies is less than the traffic volume that subscribers generate, then companies can manage the overload by creating such tariffs that reduce the overload on the network.

Conclusions
In this paper a mathematical model of the telecommunications market is constructed taking into account overloads in networks. The analysis of equilibrium tariffs for telecommunications services for this type of market is carried out.
The most important result of this paper is the following: when the companies follow the reciprocal proportional access charge rules (PACR) then there always exist equilibrium tariffs for services. The applied value of the model is that the use of PACR telecommunication companies does not require detailed information market telecommunications, as the number of parameters of the model is minimized. This model proved to be effective in analysing the dynamics of the telecommunications market, as it allows companies to respond flexibly to external changes, which allows to change the strategy at every moment of time. The proposed model can serve as a tool for analyzing the existence of collusion between companies in the telecommunications industry market.