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

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Introduction Methods of mathematical modeling in the economy of telecommunications are being actively developed [1-7]. Jean Tirole considers the impact of telecommunication technologies on competition in services and goods markets [8-12]. 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. 1. The Model of the Telecommunications Industry in the Case of Multiservice Network Let’s consider a network NW (NW = в‹ѓ i=1nNWi) consisting of n equivalent multiservice network (numbered in a certain order multiservice network SR = в‹ѓ s=1mSRs ) belonging to different telecommunication companies Ti (i = 1,nВЇ), and it is assumed that in between all the networking companies there are switching nodes. Let t О 1,2,†,Tmax be time intervals (for example, the time period equals a week, a month or a year) equal to the length of time periods during which companies Ti independently decide on pricing for their services, and tmax is the maximum planning horizon. Let’s assume that the network NW consists of a set of nodes Jt = в‹ѓ i=1sjJit and a set of channels Lt = в‹ѓ i=1slL it, and NW = Jt И lt. In the time period t each network NWi of the company Ti (i = 1,nВЇ) is represented by the set of nodes Jijt (j = 1,†,siJ) and channel set Lijt (j = 1,†,siL), numbered in a certain way, where Jit = в‹ѓ j=1siJ Jijt, Lit = в‹ѓ k=1siL Likt and NWi = Jit И Lit, and the total number of nodes is SNWJ(t) = е i=1nsiJ, and the total number of channels is SNWL(t) = е i=1nsiL for network NW. Let Hijt be a capacity (bits/sec) of j-node (j = 1,JsiJВЇ), and Sikt a throughput (bits/sec) k-link (k = 1,LsiLВЇ) Ti of network NWi company Ti in the time period t. Two-point connections can be established to transmit information flows between the network nodes of network NW. Each connection is characterized by a route, i.e. a set of network links NW, through which connections are established. Let s = {1,†,m} be a set of services that offer companies for potential consumers (subscribers) during the period t О 1,2,†,Tmax. Let b (b О 1,2,†,Bt) 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 s = {1,†,m} has the form: Dbst(p st) = rbst - p st 2sbst = abst - b bstp st,a bst = rbst 2sbst,bbst = 1 2sbst, (1) Dbst(pst) is a linear function of the price pst, and rbst > 0 and sbst > 0 is positive coefficients, which are determined from the market research services SR in the period t. A consumer b generates the traffic loading or the load using the service s in the period t. Let Y bst be an individual traffic volume of a consumer b, and let Y bst = lМ„bsthbst be the average value of Y bst, where the parameter lМ„bst is the average intensity of the flow of requests and the parameter hbst is the average duration of service in the period t. We assume that the average load is generated by the consumer b when using the service s in the period t, linearly depends on the corresponding demand function for this service s Y bst = lМ„ bsth bst = θ sDbst(p st) = θ s abst - b bstp st , (2) where θs is the proportionality factor for the s 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 t during using the service s, is the sum of consumers network traffic volumes Y st = е b=1Bt Y bst = е b=1Bt θs abst - b bstp st = ДЂ st -BМ„ stpМ„ st, ДЂst = е b=1Bt θsabst,BМ„ st = е b=1Bt θsbbst, (3) where ДЃst, BМ„st are parameters of the function Y st. The total consumers demand for the service s during the time t is the sum of all demand functions for the service s of all: Dbst(p st) = е b=1Bt Dbst(p st) = е b=1Bt abst - b bstp st , Dbst(p st) = a st - b stp st ,a st = е b=1Bt abst,b st = е b=1Btbbst, (4) where the parameters ast і 0 and bst і 0 are determined from market research of services in the period t. We can get a link between the network traffic volume Y st(pst) and the demand function Dbst(pst) of the service s during the period t: Y st(p st) = Q bst(p st)θ sDbst(p st) = θ s ast - b stp st = A st - B stp st, (5) where Y st(pst) is linear price functions and Ast = θsast, Bst = θsbst are coefficients. We can get the network traffic volume that is associated with the consumer b (b = 1,BtВЇ) Y bt = е s=1mY bst = е s=1mθ s abst - b bstp st Ј ДЂ bt - BМ„ btpМ„t, ДЂbt = е s=1mθ sabst,BМ„ bt = е s=1mb bst,pМ„t = е s=1mp st,BМ„ btpМ„t Је s=1mθ sbbstp st. (6) where ДЂbt і 0, BМ„bt і 0 are parameters load functions Y bt associated with the consumer b, and a parameter pМ„t is a tariff for services SR (service package) during the time period t. A consumer’s b (b = 1,BtВЇ) demand for SR-services in the considered time period t has the form: Qbt(p bt) = е s=1mD bst(p st) = е s=1m a bst - b bstp st Ј a bt - b btpМ„t , abt = е s=1ma bst,b bt = е s=1mb bst,b btpМ„t Ј е s=1mb bstp s. (7) Aggregating the network traffic volume Y st(pst) from (??) for all services s = {1,†,m}, we can get the total network traffic volume Y (t) for the period t in the form: Y (t) = е s=1mY st(p st) = е s=1m a st - b stp st = е s=1mθ s as - bstp st = ДЂt -BМ„tpМ„t, ДЂt = е s=1mθ sast,BМ„tpМ„t і е s=1mθ sbstp st,BМ„t = е s=1mθ sbst, (8) where ДЂt і 0 and BМ„t і 0 are aggregated parameters of function Y (t), and where function of aggregated demand for services SR (service package) has the form: D(t) = е s=1m a st - b stp st = ДЃt - bМ„tpМ„t, ДЃt = е s=1ma st,bМ„tpМ„t іе s=1mb stpst,bМ„t = е s=1mb st, (9) where the parameters ДЃt і 0 and bМ„t і 0 are aggregated parameters of the demand function D(t). We can assume that for each company Ti (i = 1,nВЇ) there exists a function of consumer demand for services SR (service package) during the time period t. Let Dsii (i О {1,†,n}) be a demand function of services SR = в‹ѓ s=1mSRs provided by the company Ti using its NWi network resource only, and let Dsijt (i,j О {1,†,n},i№j) be a demand function of services provided together with a network NWi of a company Ti and a network NWj of a company Tj (i,j О {1,†,n},i№j). Thus, there is a question of access of one company to resources of a network of the other company. We assume that the companies Ti and Tj (i,j О {1,†,n},i№j) agree on the charges Гўijt and Гўjit, where Гўijt is a charge, which company Ti pays the company Tj (i,j О {1,†,n},i№j) for the use of its network resources in connection with the service of s О {1,†,m} (traffic from the network NWi to the network NWj or outgoing traffic for the company Ti and incoming traffic for the company Tj), and Гўjit is a corresponding charge at which the company Tj pays the company Ti (i,j О {1,†,n},i№j) for the use of network resources in connection with the provision of a similar service s О {1,†,m} (traffic from the network NWj to the network NWi or outgoing traffic for the company Tj and incoming traffic for the company Ti) during the time period t. Suppose that any two companies Ti and Tj (i,j О {1,†,n},i№j) charges Гўijt and Гўjit depend on tariffs pМ„it and pМ„jt, and Гўijt = ait(pМ„it,pМ„jt) for any (i,j О {1,†,n},i№j) and s О {1,†,m} at any time t О 1,2,†,Tmax. We assume that there is the proportional dependence between Гўijt and pМ„it, then Гўijt = aitpМ„it, where the proportionality factor is 0 Ј ait Ј 1 for i О {1,†,n} and s О {1,†,m}. 2. Multiservice Demand Function Suppose that each consumer can use telecommunication multiservice network of companies Ti (i О {1,†,n}) at any time period t . Let’s assume that each consumer has individual tastes and preferences in relation to these services SR. We assume that the consumer b (b О {1,†,Bt}), which is ready to choose one service from the set s О {1,†,m} of the company Ti (i О {1,†,n}), has the following utility function: uibst = U ibstehs ϵibst = Ubst(Q bst(p ist),p is)ehs ϵibst , (10) Uibst = r bst - s bstQ bst(p st)Q bst(p st) - p stQ bst(p st), where the random parameter ϵibst characterizes individual tastes and preferences of the consumer. Let’s consider that ϵibst has a Weibull distribution. The value of hs gives the characteristic measures of the dispersion of tastes and preferences of the consumers, that is, hs allows us to estimate the substitutability telecommunication services s О {1,†,m} that provide companies Ti and Tj (i,j О {1,†,n},i№j). The services s О {1,†,m} of companies become total substitutes with hs ® 0, and it is total complementary with hs ® Ґ. Let’s assume that each consumer b (b О {1,†,Bt}) chooses the company Ti and rejects the company Tj (i,j О {1,†,t},i№j) at the period t then there is inequality Uibstehsϵibs і Ujbstehsϵjbs . Thus, the probability Pibst that the consumer b gives preference to the company Ti and rejects the company Tj (i,j О {1,†,n},i№j) equals to Pibst = P{U ibstehsϵibs > Ujbstehsϵjbs }. (11) Since the values ϵibs are independent and have a Weibull distribution we have that Pibst = 1 1 + Uibst Ujbst 1 hs = (rbstpist)st (rbstpist)st + (rbst - pjst)st, (12) where ts = 2∕hs. Similarly for the company Tj we have the same Pjbst = 1 1 + Ujbst Uibst 1 hs = rbstpjst st rbstpjst st + rbst - pist st. (13) Thus, each consumer chooses one service s in the company Ti with probability pibs and in the company Tj with probability pjbs. We can generalize this approach for the case when the consumer chooses one company Ti from the set of companies {T1,†,Tn} to obtain the service s, and we can get the probability in case the consumer gives preference to the company Ti: Pibst = (rbst - p ist) st е j=1n(rbst - pjst)st. (14) The probability that the consumer chooses one company Ti from a set of companies {T1,†,Tn} to receive service package SR has the form: Pibt = е s=1m(rbst - pist)st е s=1m е j=1n(rbst - pjst)st. (15) The expected value of consumers bi(t) who chooses a company Ti is determined by the probability Pibt, which can be considered as the market share mit of a company Ti, and has the form mit = P ibt = е s=1m rbst - pist st е s=1m е j=1n rbst - pjst st,е i=1nm it = 1. (16) The demand of consumers for services s О {1,†,m} of the company Ti (i О {1,†,n}) has the form: Dibst(p ist) = BtP ibt 2sbst rbst - p ist = Btm it 2sbst rbst - p ist . (17) Demand function of the consumers Dsiit who have plan to use the service SR of a company Ti, which may be implemented within network NWi, and demand function of the consumer Dijt who has plan to use the service SR implemented with resources of the networks NWi and NWj, have the form: Dsiit = Btm it2 2sbst rbst - p ist ,D ijst = Btm itm jt 2sbst rbst - p ist , (18) where the aggregated s-service demand Dist has the form: Dist = D siit + е j=1nD sijt = Btm it2 2sbst rbst - p ist + е j=1;i№jnBtm itm jt 2sbst rbst - p ist , (19) and the total network traffic volume demand Dit for company Ti has the form: Dit = е s=1m D siit + е j=1nD sijt = е s=1m Btm it2 2sbst (rbst - p ist) + е j =1;i№jnBtm itm jt 2sbst rbst - p ist , where Diit = е s=1mD siit,D ijt = е s=1mD sijt, and the total network traffic volume for a company Ti has the form: Y it = θD it = е s=1mθ sDist = е s=1mθ s Btmit2 2sbst rbst - p ist + е j=1; i№j nBtm itm jt 2sbst rbst - p ist , (20) where θ is an “average” linking parameter for function Y it and Dit. Revenue function TRit of companies Ti (i О 1,†,n) at the period t (t = 1,2,†,Tmax) has the form: TRit = е i,j=1; i№j n pМ„ itD iit pМ„ it + pМ„ it - d ijtpМ„ jt D ijt pМ„ i + dijtpМ„ itD jit pМ„ jt , (21) where dijt О 0,1 is a parameter to be defined during negotiations between companies Ti and Tj. 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 Pit of companies Ti (i О 1,†,n) at the period t (t = 1,2,†,Tmax) has the form: Pit = TR it - TCt w Jikt,H ikt,w Likt,c ikt,Ft , TCt = е k=1siJ wJiktH ikt + е k=1siL wLiktc ikt + Ft, (22) where TCt is a total costs function and Ft is a fix cost. 3. Profit Company Control Problem and Overloads in Networks We can formulate an optimization problem for each company Ti (i О 1,†,n) at any time t О 1,2,†,Tmax: ¶Pit∕¶pit = 0; ¶2Pit∕¶pit2< 0. (23) The following theorem holds true. Theorem 1. Provided that the parameters θs > 0, ДЃt > 0, bМ„t > 0, dijt О 0,1, wJijt і 0, wLijt і 0, Ft і 0, there is a unique solution of the problem (??) in the form of the equilibrium value of the tariff for the use of services SR of company i О 1,†,n during the period t: pМ„t* = m it + е j=1;i№jnd ijtm jt ДЃt 2bМ„t. Proof. Let’s write out the profit function of i company in the form of: Pit = е i,j; i№j n pМ„ itm it2 ДЃt -ДЃtpМ„ it + m itm jt pМ„ it - d ijtpМ„ jt ДЃt -bМ„tpМ„ it + + dijtm jtm itpМ„ it ДЃt -bМ„tpМ„ jt -е k=1siJ wJiktH ikt + е k=1siL wLiktc ikt - Ft, We can calculate the derivatives of pМ„it and equal them to zero, thus we obtain a system of algebraic equations of the form: mit ДЃt - 2bМ„tpМ„ it+е j=1;j№in m jt ДЃt - 2bМ„tpМ„ it + d ijtbМ„tpМ„ jt + d ijtm jt ДЃt -bМ„tpМ„ jt = 0, and the equilibrium value of the tariff has the form: pМ„t* = m it + е j=1;j№ind ijtm jt ДЃt 2bМ„t. We can obtain for ¶2Pit∕¶pМ„it2, ¶2Pit ¶pМ„it2 = е i,j;i№j -mit22bМ„t - m itm jt2bМ„t - d ijtm jtm itbМ„tpМ„ jt < 0. The theorem is proved. We can formulate an optimization problem for each company Ti (i О 1,†,n) at any time t О 1,2,†,Tmax for the tariff value pМ„t* : ¶Pit(pМ„t*,dijt)∕¶dijt = 0; ¶2Pit(pМ„t*,dijt)∕¶dijt2 < 0; which allows maximizing the profit of each company of Ti using the parameter dijt. After substituting the corresponding equilibrium tariffs pМ„t* in the profit function, we obtain the following equation Pit = е i,j; i№j nДЃt2m it m it + m jt 2bМ„t mit + е j=1; j№i nd ijtm jt 1 - 0.5 m it + е j=1; j№i nd ijtm jt - -е k=1siJ wJiktH ikt + е k=1siL wLiktC ikt - Ft, and differentiating by dijt and equaling to zero, we have a system of algebraic equations, by solving which, we obtain an equilibrium value of dt* = 0.5. The equilibrium tariff pМ„t* for the services of company Ti, taking into account the optimal value dt* = 0.5 during the period t, has the form: pМ„t* = m it + 1 ДЃt 4bМ„t, The equilibrium demand function for the company Ti (i О 1,†,n) services SR at any t can be represented as follows: Dit*pМ„ t* = m itD t pМ„t* = 0.25 Ч m itДЃ t 3 - mit , and the total network traffic volume for a company Ti with the equilibrium tariff has the form: Y it = θD it* = 0.25 Ч θm itДЃ t 3 - mit . The total equilibrium market demand function Dt* and the total equilibrium traffic volume Y t* for services SR at any t has the form: Dt* = ДЃt 3 -е i=1nm it2 ,Y t* = θДЃt 3 -е i=1nm it2 and we can show that with a uniform distribution of customers between all companies Ti (i О 1,†,n) the total equilibrium traffic volume for services SR 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.

About the authors

S A Vasilyev

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: vasilyev_sa@rudn.university

Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Hassan Salih Haroun

Peoples’ Friendship University of Russia (RUDN University)

Email: harounhassan198@yahoo.fr

PhD student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)

6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

References