Construction of the mathematical model of Pricing for telecommunication services with Allowance for congestion in networks Текст научной статьи по специальности «Компьютерные и информационные науки»

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Вестник РУДН. Серия МИФ

RUDN Journal of MIPh

http://journals.rudn.ru/miph

2018 Vol.26 No. 2 155-166

UDC 330.4, 519.83, 519.85, 621.39 DOI: 10.22363/2312-9735-2018-26-2-155-166

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

Networks S. A. Vasilyev, Haroun Hassan Salih

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.

Key words and phrases: queueing theory, game theory, optimization methods, probability theory, industrial market theory, economic and mathematical modeling

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

Department of Applied Probability and Informatics Peoples' Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Introduction

Received 2nd March, 2018.

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

n

Let's consider a network NW (NW = U NW.) consisting of n equivalent multiservice

i=1

m

network (numbered in a certain order multiservice network SR = (J SRs ) belonging to

__s=l

different telecommunication companies Ti (i = 1,n), and it is assumed that in between all the networking companies there are switching nodes.

Let t G {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.

si

Let's assume that the network NW consists of a set of nodes J1 = (J J* and a set

i=1

si

of channels Ll = \j Lj, and NW = J1 U ll.

i=1 _

In the time period t each network NWi of the company Ti (i = 1,n) is represented by the set of nodes J^ (j = 1,... ,sf) and channel set L\j (j = 1,..., s f), numbered

J L

Sj si

in a certain way, where J-; = (J Jfj, Lj = (J Ljk and NWi = J.f U L\, and the total

3 = 1 k=1

n n

number of nodes is (t) = Y1 si , and the total number of channels is Sfw (t) = Y1 sf

i=1 i=1

for network NW. _

Let Hfj be a capacity (bits/sec) of j-node (j = 1, Jsj ), and Sjk a throughput (bits/sec)

fc-link (k = 1,Lsl ) Ti of network NW. company T. 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 G {1,2,..., Tmax}. Let b (b G (1,2,..., Bl)) 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:

fy^i' _ fr-jt rr> t 1

Dts(pts) = ^snr^ = 4s - blps, ats = ^, bl = , (1)

Dbs(Pts) is a linear function of the price pi, and r\s > 0 and stbs > 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^s be an individual traffic volume of a consumer b, and let Y£s = Xlshtbs be the average value of Yls, where the parameter Xls is the average intensity of the flow of requests and the parameter hls 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

Yfs = ^Ishl = OsDlM) = 0S (at - b^pS) , (2)

where ds 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

Bt Bt

Yls = £ = £ 0S (at - btpl) = ^ - Blspt

b=l b=l

Bt Bt (3)

% = E abs, Ës = E ^

5=1 5=1

where as, Bt are parameters of the function Y*.

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:

B B

dUpD = E = E K - btpi),

b=i b=i

Bt v '

Bt(PD = K - blpl) , a* = ^ a^ bl = ^ Btb\s

where the parameters al ^ 0 and bl ^ 0 are determined from market research of services in the period t.

We can get a link between the network traffic volume Yg(pl) and the demand function Dt(Ps) of the service s during the period t:

yUpD = QtiplWsDlM) = 0S (al - blpl) = a\ - Blpl, (5)

where Y^(pts) is linear price functions and Als = dsal, Bls = dsbl are coefficients.

We can get the network traffic volume that is associated with the consumer b (b = 1, Bl)

m m

Yl = £ Yt = £ 9s (at - bi^s) < A\ - B\ft,

s=i s=i (6) m m m m ^ '

Al = £ Os als, Bl = Y, bt, ti = Etis, BlPl < E dsbisPss.

where Atb > 0, Bt > 0 are parameters load functions Yt associated with the consumer b, and a parameter pl is a tariff for services SR (service package) during the time period t.

A consumer's b (b = 1,Bl) demand for S'.R-services in the considered time period has the form:

qKpD = Y.Dbs(A) = £ K - blsPl) < K - bip'),

s=i s= (7)

mm m V /

ai = £ai,, bl = ^bl^ bip* < ^bips.

s=l s=l s=l

Aggregating the network traffic volume Y^p*) from (5) for all services s = {1,...,m}, we can get the total network traffic volume Y(t) for the period t in the form:

m m m

y(t) = £ YM) = £ K - bsPs) = £ 0s {as - bsPs) = A* - W,

s=l s=l s=l

m m m

A* = £ esals, Btpt > £ esbtspts, B* = Y,°sK,

s=l s=l s=l

(8)

where A1 ^ 0 and Bl ^ 0 are aggregated parameters of function Y(t), and where function of aggregated demand for services SR (service package) has the form:

D(i) = £>s - ^s) = a* - Vp

s=l

m m

Y as, W > Y bsf ps, ¥ = !>:

s=1 (9)

m m

( = £as, ltPl >£ M^, bl = Y bs,

s =1 s =1 s =1

where the parameters a1 ^ 0 and bl ^ 0 are aggregated parameters of the demand function D(t).

We can assume that for each company T. (i = 1, n) there exists a function of consumer demand for services SR (service package) during the time period t. Let Dsii (i G

m

{1,..., n}) be a demand function of services SR = (J SRs provided by the company T

s=1

using its NW. network resource only, and let Dsij (i, j G {1,..., n}, i = j) be a demand function of services provided together with a network NW. of a company T. and a network NWj of a company Tj (i, j G {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 T. and Tj (i,j G {1,...,n}, i = j) agree on the charges and aj., where ajj is a charge, which company T. pays the company Tj (i,j G {1,..., n}, i = j) for the use of its network resources in connection with the service of s G {1,... ,m} (traffic from the network NW. to the network NWj or outgoing traffic for the company T. and incoming traffic for the company Tj), and aji is a corresponding charge at which the company Tj pays the company T. (i, j G {1,..., n}, i = j) for the use of network resources in connection with the provision of a similar service s G {1,... ,m} (traffic from the network NWj to the network NW. or outgoing traffic for the company Tj and incoming traffic for the company T.) during the time period t.

Suppose that any two companies Ti and Tj (i,j e {1,... ,n}, i = j) charges ci\- and aji depend on tariffs pi and pj, and atij = aj (pi, pj) for any (i,j e {1,... ,n}, i = j) and s e {1,..., m} at any time t e {1, 2,..., Tmax}.

We assume that there is the proportional dependence between ajj and pthen atj = atpt, where the proportionality factor is 0 < a\ < 1 for i e {1,..., n} and s e {1,..., m}.

2. Multiservice Demand Function

Suppose that each consumer can use telecommunication multiservice network of companies Ti (i e {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 e {1,..., B1}), which is ready to choose one service from the set s e {1,..., m} of the company Ti (i e {1,..., n}), has the following utility function:

utbs = Utbse^ = Uts(Qi(pts),pls)e^, (10)

uL = [rL - stbsQtb.M)] Qis(pS) - pSQÍs(pS),

where the random parameter e\bs characterizes individual tastes and preferences of the consumer. Let's consider that elbs has a Weibull distribution. The value of ^s gives the characteristic measures of the dispersion of tastes and preferences of the consumers, that is, 'qs allows us to estimate the substitutability telecommunication services s e {1,..., m} that provide companies Ti and Tj (i,j e {1,..., n}, i = j). The services s e {1,..., m} of companies become total substitutes with ^s ^ 0, and it is total complementary with ■qs ^ to.

Let's assume that each consumer b (b e {1,..., B1}) chooses the company Ti and rejects the company Tj (i,j e {1,... ,t}, i = j) at the period t then there is inequality

TJ1 p'sUbs > TJ1 p'stjbs its jts

Thus, the probability Pjtbs that the consumer b gives preference to the company Ti and rejects the company Tj (i,j e {1,... ,n}, i = j) equals to

PL = ?{Ulbse'^b* > Utbse'^ }. (11)

Since the values eibs are independent and have a Weibull distribution we have that

(^i1 ) ^

_ _ _v bs^is)s_

1 + I Ulb^\ ^ (rbsP\sYs + (rts - PjsYs ,

p t _ ___ _ _y bsi is)s__/in\

ribs = . = l„t W , f„t W , (12)

Ulb Ubu

jbs

where tS = 2/'qs. Similarly for the company Tj we have the same

p t =_1_=_(rbsPjs)s__( )

rjbs ^ l / t t \r , t t ,T . (13)

j 1 + (gbs) ^ (rbsPjs)s + (rbs - PDs

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 {T\,... ,Tn} to obtain the service s, and we can get the

probability in case the consumer gives preference to the company T.:

P

i b s

(f t _ ) S

V bs iJis)s n '

E (rbs — PjsYs t=1

(14)

The probability that the consumer chooses one company T. from a set of companies {T1,... ,Tn} to receive service package SR has the form:

E (rbs - Ptis)l

P

s=l

b

E E (TIs - PjsY

s=l =l b s s

(15)

The expected value of consumers b.(t) who chooses a company T. is determined by the probability P/t, which can be considered as the market share mf of a company T., and has the form

mi — Pib —

m

E (r

s=1

t t )S bs Pis)s

E E (ris- pjs)s

s=l =l

— I-

(16)

=l

The demand of consumers for services s G {1,..., m} of the company T. (i G {1, has the form:

Blpt Btmt

Dtits(Pis) = (rts — Pis) = 0t . (rts — Pis) .

2s bs

2s bs

,nj) (17)

Demand function of the consumers D\u who have plan to use the service SR of a company T., which may be implemented within network NW., and demand function of the consumer D\j who has plan to use the service SR implemented with resources of the networks NW. and NWj, have the form:

t Btmi-2 , t t s t Btmt.mj

Dsii 0„i Vbs — Pis) , Dijs 2gt ybs — Pis) ,

2 s 1 2 b s

b s

(18)

where the aggregated s-service demand Djs has the form:

n T?t/rrtt2 n Btmtmt

Ds — DL + ^DL — (rbs - PL) + E -mp (rL - tis) , (19)

s

=l

2 s 1 2 b s

t=i;i=t

b s

and the total network traffic volume demand Dj for company T. has the form:

D —

s=1

s s

t=1

m

L

s=1

-W^t t ,, A B'mlmj

Vbs - Pis)+ £ o„t 3 {Ks - Pis)

2S bs

=1; =

2

b s

where

d7- = Vd7-., d7- = VD7-■,

Ii / J sil? IJ /j si J ?

s=1 s=1

and the total network traffic volume for a company T has the form:

Yl = QD\ = Y °sDi = Y °s

s=1

=1

t

B7 m

n Bt mtmt

2m-fa - pd - pts)

2 0 bs J=1. 2 0 bs

=1

=

, (20)

where d is an "average" linking parameter for function Yit and Dl.

Revenue function TRI of companies T. (i e {1,...,n}) at the period t (t = 1,2,..., Tmax) has the form:

TRt = Y [pD (Pt) + {pt - ) D% (Pi) + ¿¡jPtDti (p7)]

(21)

i,)=i;

=

where 57j G [0,1] is a parameter to be defined during negotiations between companies T. 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 n* of companies T. (i G {1,..., n}) at the period t (t = 1,2,..., Tmax) has the form:

n- = TRti - TC* (wjtik, Hlk, wLtik, ctik, Fl) ,

( * '' \

TC = I £ ^jikHtk + £ I + F*,

\k=1 k=l J

where TC* is a total costs function and F* is a fix cost.

(22)

3. Profit Company Control Problem and Overloads in

Networks

We can formulate an optimization problem for each company T (i G {1,..., n}) at any time t G {1, 2,... ,Tmax}:

{dnt/dpt =0;

[d^/dp? < o. (23)

The following theorem holds true.

Theorem 1. Provided that the parameters ds > 0, a* > 0, b* > 0, 57j G [0,1], wjtij > 0, wL\j > 0, F* > 0, there is a unique solution of the problem (23) in the form of the equilibrium value of the tariff for the use of services SR of company i G {1,..., n} during the period t:

n \ -t ,t , xt ™t I a

P* = (mt + £ ^ ) ^.

. , J I 2bt"

=1; =

Proof. Let's write out the profit function of i company in the form of:

n = E b^2 (cjt - atPi) + miimtj № - 6ijpj) (at - btPi) +

i, j; i=j

+ Sljmjml^ (a* - Fpj)] - ¡YwjlkHfk + V I - Fl,

, k=1

k=1

We can calculate the derivatives of tii and equal them to zero, thus we obtain a system of algebraic equations of the form:

n

mt (a? - 2btptt) + £ imj - 2btPt + ^Pj) + S^mj (a? - tfpj)] = 0,

j=i;j=i

and the equilibrium value of the tariff has the form:

j=i;j=i

ft = H + £ 5hmj I 2F.

We can obtain for d2nt/dp^2,

^ = £ -

dp i

mimj.

1 2 5 1 - d^mjmt^pj] < 0.

i,j;i=j

The theorem is proved.

□

We can formulate an optimization problem for each company Ti (i e {1,..., n}) at any time t e {1, 2,..., Tmax} for the tariff value p* :

idnt(p*, Stj)/dStj = 0; \d2nt(pt, Stj)/d5¡2 < 0;

which allows maximizing the profit of each company of Ti using the parameter 5\j.

After substituting the corresponding equilibrium tariffs p* in the profit function, we obtain the following equation

n =t2™t L^t i ™t

n = £

a^mt [ml + mj]

/

i,j;

i=j

2 t

\

m.

+

ijmj

j=i; j=i

(

(

1 — 0.5

m.

+ E*

tjmj

j=i; j=i

\\ //

- (E wjttkHik+E wLttkcik) - F 1 ,

, k=1

k=1

and differentiating by djj and equaling to zero, we have a system of algebraic equations, by solving which, we obtain an equilibrium value of St = 0.5.

The equilibrium tariff p't for the services of company Ti, taking into account the optimal value 5't =0.5 during the period t, has the form:

ft = N + ,

The equilibrium demand function for the company Ti (i e {1,..., n}) services SR at any can be represented as follows:

D*t (pt) = m^Dt (^) = 0.25 ■ m\at (3 - m]t) ,

and the total network traffic volume for a company Ti with the equilibrium tariff has the form:

Y] = 9D*it = 0.25 ■ Qm^t (3 - m^) .

The total equilibrium market demand function Dt and the total equilibrium traffic volume Yt* for services SR at any t has the form:

n

t t2 t

D*t = -a - £ mf^J, y; = ^ - Y

and we can show that with a uniform distribution of customers between all companies Ti (i e {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.

References

1. M. Armstrong, Network Interconnection, The Economic Journal 108 (1998) 545-564.

2. M. Carter, J. Wright, Interconnection in Network Industries, Review of Industrial Organization 14 (1999) 1-25.

3. M. Carter, J. Wright, Asymmetric Network Interconnection, Review of Industrial Organization 22 (2003) 27-46.

4. W. Dessein, Network Competition in Nonlinear Pricing, Rand Journal of Economics 34 (2003) 593-611.

5. W. Dessein, Network Competition with Heterogeneous Customers and Calling Patterns, Information Economics and Policy 16 (2004) 323-345.

6. T. Doganoglu, Y. Tauman, Network Competition and Access ChargeRules, The Manchester School 70 (2002) 16-35.

7. J.-H. Hahn, Network Competition and Interconnection with Heterogeneous Subscribers, International Journal of Industrial Organization 22 (2004) 611-631.

8. J.-J. Laffont, J. Tirole, Access Pricing and Competition, European Economic Review 38 (1994) 1673.

9. J.-J. Laffont, P. Rey, J. Tirole, Network Competition I: Overview and Nondiscriminatory Pricing, The Rand Journal of Economics 29 (1998) 1-37.

10. J.-J. Laffont, P. Rey, J. Tirole, Network Competition II: Price Discrimination, The Rand Journal of Economics 29 (1998) 38-56.

11. J.-J. Laffont, J. Tirole, Internet Interconnection and the Off-Net-Cost Pricing Principle, Rand Journal of Economics 34 (2003) 73-95.

12. J.-J. Laffont, J. Tirole, Receiver-Pays Principle, Rand Journal of Economics 35 (2004) 85-110.

13. S.-H. Chuna, Network Capacity and Access Pricing for Cloud Services, Procedia -Social and Behavioral Sciences 109 (2014) 1348-1352.

14. S. A. Vasilyev, D. G. Vasilyeva, M. E. Kostenko, L. A. Sevastianov, D. A. Urusova, Economics and Mathematical Modeling of Duopoly Telecommunication Market, Bulletin of Peoples' Friendship University of Russia. Series: Mathematics. Information Sciences. Physics (3) (2009) 57-67, in Russian.

15. S. A. Vasilyev, L. A. Sevastianov, D. A. Urusova, Economics and Mathematical Modeling of Oligopoly Telecommunication Market, Bulletin of Peoples' Friendship University of Russia. Series: Mathematics. Information Sciences. Physics (2) (2011) 59-69, in Russian.

УДК 330.4, 519.83, 519.85, 621.39 Б01: 10.22363/2312-9735-2018-26-2-155-166

Построение математической модели ценообразования на телекоммуникационные услуги с учётом перегрузок в сетях

С. А. Васильев, Харун Хасан Салех

Кафедра прикладной информатики и теории вероятностей Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198

В работе строится модель динамического ценообразования на рынке телекоммуникаций при условии ограниченной конкуренции и с учётом перегрузок в мультисервисных сетях. Для построения и исследования модели был применён комплексный подход, заключающийся в использовании методов экономико-математического моделирования и теории массового обслуживания. В предлагаемой модели предполагается, что телекоммуникационные компании попарно договариваются о правилах тарификации входящего и исходящего трафика, причём эта тарификация строится как функция от тарифов, которые компании предлагают своим абонентам за обслуживание. Таким образом, эти компании ограничиваются на первом шаге договорённостями по обоюдным правилам пропорциональной тарификации за доступ трафика (ОППТДТ), которые впоследствии позволяют определить тарифы для пользователей услуг мультисервисных сетей, которыми владеют компании. Обоюдность правил означает, что компании подчиняются одним и тем же правилам на всем интервале времени, в течение которого действует договорённость. С учётом несовершенной конкуренции на рынке

телекоммуникаций и при условии максимизации прибыли каждой компанией, которая является поставщиком услуг, в рамках построенной модели были найдены равновесные тарифы на эти услуги с учётом перегрузок в мультисервисных сетях, а также объёмы этих услуг.

Ключевые слова: теория массового обслуживания, теория игр, методы оптимизации, теория вероятностей, теория отраслевых рынков, экономико-математическое моделирование

Литература

1. Armstrong M. Network Interconnection // The Economic Journal. — 1998. — Vol. 108. — Pp. 545-564.

2. Carter M., Wright J. Interconnection in Network Industries // Review of Industrial Organization. — 1999. — Vol. 14. — Pp. 1-25.

3. Carter M., Wright J. Asymmetric Network Interconnection // Review of Industrial Organization. — 2003. — Vol. 22. — Pp. 27-46.

4. Dessein W. Network Competition in Nonlinear Pricing // Rand Journal of Economics. — 2003. — Vol. 34. — Pp. 593-611.

5. Dessein W. Network Competition with Heterogeneous Customers and Calling Patterns // Information Economics and Policy. — 2004. — Vol. 16. — Pp. 323-345.

6. Doganoglu T., Tauman Y. Network Competition and Access ChargeRules // The Manchester School. — 2002. — Vol. 70. — Pp. 16-35.

7. Hahn J.-H. Network Competition and Interconnection with Heterogeneous Subscribers // International Journal of Industrial Organization. — 2004. — Vol. 22. — Pp. 611-631.

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© Vasilyev S. A., Haroun Hassan Salih, 2018

Для цитирования:

Vasilyev S.A., Haroun Hassan Salih Construction of the Mathematical Model of Pricing for Telecommunication Services with Allowance for Congestion in Networks // RUDN Journal of Mathematics, Information Sciences and Physics. — 2018. — Vol. 26, No 2. — Pp. 155-166. — DOI: 10.22363/2312-9735-2018-26-2-155-166.

For citation:

Vasilyev S. A., Haroun Hassan Salih Construction of the Mathematical Model of Pricing for Telecommunication Services with Allowance for Congestion in Networks, RUDN Journal of Mathematics, Information Sciences and Physics 26 (2) (2018) 155-166. DOI: 10.22363/2312-9735-2018-26-2-155-166.

Сведения об авторах:

Васильев Сергей Анатольевич — кандидат физико-математических наук, доцент кафедры прикладной информатики и теории вероятностей РУДН (e-mail: vasilyev_ sa@rudn.university, тел.: +7 (495)7287911)

Харун Хасан Салех — аспирант кафедры прикладной информатики и теории вероятностей РУДН (e-mail: harounhassan198@yahoo.fr, тел.: +7 (968)3376303)

Information about the authors:

Vasilyev S. A. — Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: vasilyev_sa@rudn.university, phone: +7 (495)7287911)

Haroun Hassan Salih — PhD student of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: harounhassan198@yahoo.fr, phone: +7 (968)3376303)