# Strategic Interaction between Operators in the Context of Spectrum Sharing for 5G Networks

Authors: Erwin J. Sacoto-Cabrera; Angel Sanchis-Cano; Luis Guijarro; José Ramón Vidal; Vicent Pla
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

5G networks will make network sharing agreements between mobile operators technically possible. However, depending on the agreed and implemented quality-of-service isolation, the provision of services may lead to unsustainable business cases. In this paper, the economic feasibility of such arrangements is analyzed for the case of two operators. Concretely, while one network operator owns the spectrum, one virtual operator does not, and each one provides service to its subscriber base. Two sharing alternatives, namely, pooling and priority sharing, are studied regarding the profits that each operator gets. We conclude that the network operator is worse off under any circumstances under a pooling agreement, while a lump sum payment may leave the network operator better off under a priority sharing agreement.


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## Body

## 1. Introduction

The fifth generation (5G) wireless networks will appear in the market in 2020 and is expected to improve customers’ quality of service (QoS) significantly with the increasing of data volume in mobile networks [1]. Furthermore, 5G networks will have the capacity to provide different QoS to users with a wide range of needs, such as new applications, different traffic types, and a wide range of services [2, 3].The sharing of any sort of resources among network operators has been shown as an efficient way of spreading out the technology at a lower cost. The resources in this paper will be typically spectrum that is owned by the network operator, but similar analysis may be performed when the resources are, e.g., radio-access-network (RAN), network roaming, and core-network [4, 5]. We will refer to the sharing of resources with the general termnetwork sharing. Authors of [6] describe a scenario where infrastructure providers enable the leasing of resources, which allow service providers to share the cost of providing coverage with virtual operators, Over-The-Top providers and industry vertical market players. However, network sharing is based on the dynamic deployment and scaling of functions in the 5G networks and requires a careful orchestration of resources to preserve bilateral agreements among operators, as stated by [7].Regarding the costs involved in moving to 5G, they can be significantly reduced through network densification and network sharing. The business models analyzing these scenarios such as [8] are usually focused on the relation between the service provider and the network operator, where their physical and logical network infrastructures are tightly coupled. There are several ways to implement network sharing, as proposed in [9]. A possible new business model could be based on the sharing of resources, where the owner of the resources and the operator of the resources are different actors.In the literature, some researches are focused on the study of the economic feasibility of different network sharing scenarios based on elements of queuing theory, such as [10, 11], as well as microeconomics and game theory concepts [12, 13]. For instance, in the context of network sharing, in [14] the effects of infrastructure sharing and competition regulation on cellular network planning are analyzed. The authors of [15] analyze a network slicing scenario, based on game theory and a framework called “share-constrained proportional allocation”; the results obtained provide an effective and implementable scheme for dynamically sharing resources across slices. In [16], the authors investigate the generic instruments for addressing a spectrum sharing in 4G networks from a cooperative games perspective and conclude that, in most of the considered instances, the mobile network operators (MNOs) are better off building a unique shared RAN than creating subcoalitions or building individual RANs due to the combined gain from spectrum aggregation and cost reduction from sharing the network infrastructure. In [17], the authors investigate a particular class of congestion games and learning mechanisms to design a distributed solution to the wireless network slicing problem. In the same way, there are some studies that analyze the network sharing, based on game theory, such as [18–21]. In the framework of the queue theory, in [22] the effects of queueing delays and users’ related costs on the management and control computing resources are analyzed. The authors of [23] propose an analysis from a pricing perspective, based on priority queuing (PQ) and Generalized Processor Sharing, with the issue of maximizing networks operators revenue. In [24], the authors investigate the priority queuing as a way to establish service differentiation; to do that, they consider the Discriminatory Processor Sharing discipline for two models of service with different QoS and determine the prices that maximize the provider’s profit. In the same way, [25] studies pricing for heterogeneous services based on a priority queuing as a way to establish service differentiation. Additionally, in [26] the authors analyze the cooperation strategies among mobile network operators competitors, customers, and different types of partners based on network sharing. However, the mentioned studies do not analyze the incentives to mobile operators to engage in spectrum sharing schemes using priority sharing disciplines in a network sharing environment from an economic point of view.Our aim in this paper is to proceed one step further and model the strategic interaction in network sharing between a network operator and a virtual operator, providing service to their users. We analyze two alternatives: pooling and priority sharing. The analysis is conducted by means of game theory. Our main contribution is to prove that network sharing is economically viable and allows the network and virtual operators to coexist under a priority sharing agreement.The rest of this paper is organized as follows. In Section2, we describe in detail the model with the actors, the utility of each actor, and the pricing scheme. In Section 3, we analyze and solve the subscription and pricing strategies of the different models. In Section 4, we show and discuss the results. Finally, in Section 5, we present the conclusions.


## 2. Model Description

### 2.1. General Model

This section models the following two sharing agreements:(i)
Pooling: the operators share equally the resource.(ii)
Priority sharing: the operators share the resource under different priorities.In addition, we analyze in the Appendix the case where there is only one operator providing service; i.e., no priority is implemented and the operator behaves as a monopolist. This case will be used as a baseline for comparison in Section4. A summary of the notation used in this paper is given in Table 1.Table 1
Summary of notations.
Eq. General Model Mean packet system time T i (1) Mean service rate μ (1) Subscribers mean arrival rate λ d (1) Operator 1’s number of users n 1 (1) Operator 2’s number of users n2 (1) Virtual Operator’s relative priority γ (2) Quality perceived by the users Q i (4) Users sensitivity to delay α (4) Conversion factor c (4) Users utility U i (5) Price charged by operatori pi (5) Operator i’s profits Π i (6)  Game Model Operator 1’s best response B R 1 (11) Operator 2’s best response B R 2 (12) Operator 1’s equilibrium price p 1 ∗ (13) Operator 2’s equilibrium price p 2 ∗ (14) Operator 1’s equilibrium number of users n 1 ∗ (16) Operator 2’s equilibrium number of users n 2 ∗ (16) Arbitrarily small positive value ϵ Boundary cases I-II p ^ 1 γ (34) Boundary cases I-II p ^ 2 γ (35) Boundary cases III-IV p ^ 1 U (36) Boundary cases II-IV p ^ 2 U (37)  The Appendix- Monopoly Monopolistic operator’s profits Π 0 (A.1) Monopolistic operator’s number of users n 0 (A.1) Price charged by monopolistic operator p 0 (A.1) Users Utility U0 (A.3) Monopolistic operator’s optimal price p 0 ∗ (A.9)The network operator (aka. Operator1) is the spectrum owner, while the virtual operator (aka. Operator 2) leases the resource under the specific sharing agreement. As part of the sharing agreement, the operator undertakes to communicate the data of the number of users to the other operator. We assume that each operator has its own subscriber base.The network, i.e., the spectrum resource that is used for providing service, is modeled as an M/M/1/∞ queue, where packets are generated by the subscribers of the two operators and are served according to the scheduling discipline described below. Operator i has ni subscribers that independently generate packets following a Poisson process with a rate λd, so that for each operator we can define a Poisson packet arrival process with rate λi=λdni, and the sum of the two processes is also Poisson with rate λ=λ1+λ2. The service times of all packets are exponentially distributed with mean 1/μ. We assume for stability reasons that λ<μ.Under the pooling agreement, the scheduling discipline is First-Come-First-Served and there is no priority. The mean packet system timeTi can be computed as(1)T1=T2=1μ-n1+n2λd.Under the priority sharing agreement, we propose that the service provided by the network be modeled by a Discriminatory Processor Sharing (DPS) discipline, where each customer has a relative priority and receives service at an instantaneous rate proportional to the priority [11, 27, 28]. A DPS queue basically works as follows: if there are n customers with priorities x1,x2,…,xn (xi≤1 for i=1,…,n), then the customer i is served at a fraction xi/∑j=1nxj of the servers capacity [11]. In our analysis we use a DPS model with two relative priorities: x1=1-γ for the network operator and x2=γ for the virtual operator, where x1+x2=1 and 0≤γ≤1. DPS is a mechanism much more flexible than priority queuing in order to model the sharing of a common resource, thanks to the γ parameter. Ti can be computed [11] as(2)T1=1μ-λdn1-λdn21+λdn22γ-1μ-1-γλdn1-γλdn2,(3)T2=1μ-λdn1-λdn21-λdn12γ-1μ-1-γλdn1-γλdn2.Note that Preemptive Resume PQ, if the users were serviced in a processor sharing manner, is a special case of DPS discipline, sinceγ=0 gives strict priority to operator 1 users while γ=1 would give strict priority to operator 2 users [24].We quantify the quality perceived by the users of each operator borrowing a variation of the expression used in [23, 25, 29, 30]. In order to analyze the impact of the priority parameter γ, we assume that the users are homogeneous in their service time requirements for both operators.(4)Qi≡cTi-α,i=1,2,wherec>0 is a conversion factor and Ti has been defined and computed above. We define 0≤α≤1 as the user sensitivity to delay, since a greater α translates into a worse perceived quality, for a given delay. The user utility is proposed to be given by the difference between the quality perceived by the users in monetary units minus the price charged by the operator. Specifically, the user utility we model has the advantage of being interpretable in monetary-equivalent terms: since cTiα is the value that the outcome has to user i and pi (in monetary units per second (m.u./s.)) is the price that the user pays, Ui can be seen as the benefit (m.u./s.) of user i.(5)Ui≡cTi-α-pi,i=1,2.We set the utility when users do not subscribe to the service to zero.Note that a similar approach for modeling the user utility can be found at, e.g., [22, 31–33]. And this form of utility function can be related to the quasi-linear function widely used in microeconomic and telecommunications networks analysis as described in [34].The network operator charges a pricep1 to its users, while the virtual operator charges a price p2. We assume that no costs are incurred by the operators. The operators’ profits will then be given by(6)Πi=λdnipipi,i=1,2.Remark that the profits functions described in (6) should have been decreased by the operator’s costs. Important components of these costs are the operating costs and the investment costs. We do not consider the operating costs, as their inclusion does not provide additional insight since they do not depend on the price of the service, while it makes the expression of profits less explicit. Related to the investment costs, we can consider them constant, given that the time scale on which the provider can adapt his network capacity μ is relatively long compared to the time scale on which prices vary [25, 30].


### 2.2. Game Model

From the above model description, we can observe the following strategic interactions:(i)
Flowi’s subscription decision is influenced by operatori’s pricing decision.(ii)
Operatori’s profit depends on flowi’s subscription decision.(iii)
Flowi’s subscription decision depends on flowj’s decision with (j≠i), through the Qi factor.(iv)
Operatori’s profit is then influenced by operatorj’s pricing decision, indirectly through flowj’s subscription decision.We conclude then that the above strategic scenario is amenable to game theory. Specifically, each operator will choose its pricing strategypi aiming at maximizing its profit Πi.We will use a two-stage sequential game of the multileader follower kind, where, in the first stage, each operator chooses a pricepi in order to maximize its profit Πi. The game is solved using backward induction [35], which means that at Stage I players proceed strategically anticipating the solution of Stage II.In the second stage, for each operator, the following applies. Each user takes its own subscription decision, trying to maximize the utility it gets from either subscribing to the operator or not. The user will observe the price published by the operator and will take the decision of service subscription. Assuming that number of users is high enough, the individual subscription decision of one user does not affect the utility of the rest of the users. Under these conditions, the equilibrium reached is that postulated by Wardrop [36]. Applying this notion of equilibrium, known asWardrop equilibrium, to the user subscription problem, we postulate that(i)
eitherni≥0 and Ui=0(ii)
orni=0 and Ui<0.Combining the above alternatives, we can identify four outcomes:(i)
Case I:(7)U1=0,U2=0andn1≥0,n2≥0(ii)
Case II:(8)U1=0,U2<0andn1≥0,n2=0(iii)
Case III:(9)U1<0,U2=0andn1=0,n2≥0(iv)
Case IV:(10)U1<0,U2<0andn1=0,n2=0In the first stage, each operator chooses a price in order to maximize its profits in a simultaneous and independent way. Each operator is not only aware of the users subscription decision in the second stage, but also of the rational behavior of the other operator. Under such circumstances, the equilibrium reached is theNash equilibrium [37], where no player has an incentive to change its own strategy unilaterally.The general method to discover the set of Nash equilibria is to obtain the best-response (BR) function of each operator and identify the crossing points [34]. The BR functions are defined as follows:(11)BR1p2=argmaxp1⁡Π1p1,p2,(12)BR2p1=argmaxp2⁡Π2p1,p2.Once we have obtained the BR functions, we can obtain the set of Nash equilibria solving the following system of equations:(13)p1∗=argmaxp1⁡Π1p1,p2∗,(14)p2∗=argmaxp2⁡Π2p1∗,p2.


## 2.1. General Model

This section models the following two sharing agreements:(i)
Pooling: the operators share equally the resource.(ii)
Priority sharing: the operators share the resource under different priorities.In addition, we analyze in the Appendix the case where there is only one operator providing service; i.e., no priority is implemented and the operator behaves as a monopolist. This case will be used as a baseline for comparison in Section4. A summary of the notation used in this paper is given in Table 1.Table 1
Summary of notations.
Eq. General Model Mean packet system time T i (1) Mean service rate μ (1) Subscribers mean arrival rate λ d (1) Operator 1’s number of users n 1 (1) Operator 2’s number of users n2 (1) Virtual Operator’s relative priority γ (2) Quality perceived by the users Q i (4) Users sensitivity to delay α (4) Conversion factor c (4) Users utility U i (5) Price charged by operatori pi (5) Operator i’s profits Π i (6)  Game Model Operator 1’s best response B R 1 (11) Operator 2’s best response B R 2 (12) Operator 1’s equilibrium price p 1 ∗ (13) Operator 2’s equilibrium price p 2 ∗ (14) Operator 1’s equilibrium number of users n 1 ∗ (16) Operator 2’s equilibrium number of users n 2 ∗ (16) Arbitrarily small positive value ϵ Boundary cases I-II p ^ 1 γ (34) Boundary cases I-II p ^ 2 γ (35) Boundary cases III-IV p ^ 1 U (36) Boundary cases II-IV p ^ 2 U (37)  The Appendix- Monopoly Monopolistic operator’s profits Π 0 (A.1) Monopolistic operator’s number of users n 0 (A.1) Price charged by monopolistic operator p 0 (A.1) Users Utility U0 (A.3) Monopolistic operator’s optimal price p 0 ∗ (A.9)The network operator (aka. Operator1) is the spectrum owner, while the virtual operator (aka. Operator 2) leases the resource under the specific sharing agreement. As part of the sharing agreement, the operator undertakes to communicate the data of the number of users to the other operator. We assume that each operator has its own subscriber base.The network, i.e., the spectrum resource that is used for providing service, is modeled as an M/M/1/∞ queue, where packets are generated by the subscribers of the two operators and are served according to the scheduling discipline described below. Operator i has ni subscribers that independently generate packets following a Poisson process with a rate λd, so that for each operator we can define a Poisson packet arrival process with rate λi=λdni, and the sum of the two processes is also Poisson with rate λ=λ1+λ2. The service times of all packets are exponentially distributed with mean 1/μ. We assume for stability reasons that λ<μ.Under the pooling agreement, the scheduling discipline is First-Come-First-Served and there is no priority. The mean packet system timeTi can be computed as(1)T1=T2=1μ-n1+n2λd.Under the priority sharing agreement, we propose that the service provided by the network be modeled by a Discriminatory Processor Sharing (DPS) discipline, where each customer has a relative priority and receives service at an instantaneous rate proportional to the priority [11, 27, 28]. A DPS queue basically works as follows: if there are n customers with priorities x1,x2,…,xn (xi≤1 for i=1,…,n), then the customer i is served at a fraction xi/∑j=1nxj of the servers capacity [11]. In our analysis we use a DPS model with two relative priorities: x1=1-γ for the network operator and x2=γ for the virtual operator, where x1+x2=1 and 0≤γ≤1. DPS is a mechanism much more flexible than priority queuing in order to model the sharing of a common resource, thanks to the γ parameter. Ti can be computed [11] as(2)T1=1μ-λdn1-λdn21+λdn22γ-1μ-1-γλdn1-γλdn2,(3)T2=1μ-λdn1-λdn21-λdn12γ-1μ-1-γλdn1-γλdn2.Note that Preemptive Resume PQ, if the users were serviced in a processor sharing manner, is a special case of DPS discipline, sinceγ=0 gives strict priority to operator 1 users while γ=1 would give strict priority to operator 2 users [24].We quantify the quality perceived by the users of each operator borrowing a variation of the expression used in [23, 25, 29, 30]. In order to analyze the impact of the priority parameter γ, we assume that the users are homogeneous in their service time requirements for both operators.(4)Qi≡cTi-α,i=1,2,wherec>0 is a conversion factor and Ti has been defined and computed above. We define 0≤α≤1 as the user sensitivity to delay, since a greater α translates into a worse perceived quality, for a given delay. The user utility is proposed to be given by the difference between the quality perceived by the users in monetary units minus the price charged by the operator. Specifically, the user utility we model has the advantage of being interpretable in monetary-equivalent terms: since cTiα is the value that the outcome has to user i and pi (in monetary units per second (m.u./s.)) is the price that the user pays, Ui can be seen as the benefit (m.u./s.) of user i.(5)Ui≡cTi-α-pi,i=1,2.We set the utility when users do not subscribe to the service to zero.Note that a similar approach for modeling the user utility can be found at, e.g., [22, 31–33]. And this form of utility function can be related to the quasi-linear function widely used in microeconomic and telecommunications networks analysis as described in [34].The network operator charges a pricep1 to its users, while the virtual operator charges a price p2. We assume that no costs are incurred by the operators. The operators’ profits will then be given by(6)Πi=λdnipipi,i=1,2.Remark that the profits functions described in (6) should have been decreased by the operator’s costs. Important components of these costs are the operating costs and the investment costs. We do not consider the operating costs, as their inclusion does not provide additional insight since they do not depend on the price of the service, while it makes the expression of profits less explicit. Related to the investment costs, we can consider them constant, given that the time scale on which the provider can adapt his network capacity μ is relatively long compared to the time scale on which prices vary [25, 30].


## 2.2. Game Model

From the above model description, we can observe the following strategic interactions:(i)
Flowi’s subscription decision is influenced by operatori’s pricing decision.(ii)
Operatori’s profit depends on flowi’s subscription decision.(iii)
Flowi’s subscription decision depends on flowj’s decision with (j≠i), through the Qi factor.(iv)
Operatori’s profit is then influenced by operatorj’s pricing decision, indirectly through flowj’s subscription decision.We conclude then that the above strategic scenario is amenable to game theory. Specifically, each operator will choose its pricing strategypi aiming at maximizing its profit Πi.We will use a two-stage sequential game of the multileader follower kind, where, in the first stage, each operator chooses a pricepi in order to maximize its profit Πi. The game is solved using backward induction [35], which means that at Stage I players proceed strategically anticipating the solution of Stage II.In the second stage, for each operator, the following applies. Each user takes its own subscription decision, trying to maximize the utility it gets from either subscribing to the operator or not. The user will observe the price published by the operator and will take the decision of service subscription. Assuming that number of users is high enough, the individual subscription decision of one user does not affect the utility of the rest of the users. Under these conditions, the equilibrium reached is that postulated by Wardrop [36]. Applying this notion of equilibrium, known asWardrop equilibrium, to the user subscription problem, we postulate that(i)
eitherni≥0 and Ui=0(ii)
orni=0 and Ui<0.Combining the above alternatives, we can identify four outcomes:(i)
Case I:(7)U1=0,U2=0andn1≥0,n2≥0(ii)
Case II:(8)U1=0,U2<0andn1≥0,n2=0(iii)
Case III:(9)U1<0,U2=0andn1=0,n2≥0(iv)
Case IV:(10)U1<0,U2<0andn1=0,n2=0In the first stage, each operator chooses a price in order to maximize its profits in a simultaneous and independent way. Each operator is not only aware of the users subscription decision in the second stage, but also of the rational behavior of the other operator. Under such circumstances, the equilibrium reached is theNash equilibrium [37], where no player has an incentive to change its own strategy unilaterally.The general method to discover the set of Nash equilibria is to obtain the best-response (BR) function of each operator and identify the crossing points [34]. The BR functions are defined as follows:(11)BR1p2=argmaxp1⁡Π1p1,p2,(12)BR2p1=argmaxp2⁡Π2p1,p2.Once we have obtained the BR functions, we can obtain the set of Nash equilibria solving the following system of equations:(13)p1∗=argmaxp1⁡Π1p1,p2∗,(14)p2∗=argmaxp2⁡Π2p1∗,p2.


## 3. Models Analysis

In this section, we analyze the models described in Section2.


### 3.1. Pooling

In this subsection, operator 1 and operator 2 are sharing the spectrum resource. The utility of the users can be obtained replacing (1) in (5).(15)Ui=c·μ-n1+n2·λdα-pii=1,2We obtain the equilibrium values forn1 and n2, given p1 and p2, as follows.(i) Case I: from (7) and (15) we obtain(16)n1∗+n2∗=μ-p1/c1/αλd=μ-p2/c1/αλd,which is only feasible ifp1=p2. Without loss of generality, we can assume that n1∗=n2∗, so that(17)n1∗=μ-p1/c1/α2λd,(18)n2∗=μ-p2/c1/α2λd.(ii) Case II: from (8) and (15) we obtain(19)n1∗=μ-p1/c1/αλd,(20)n2∗=0,(21)p1≤cuα.(22)p2>p1.(iii) Case III: from (9) and (15) we obtain(23)n1∗=0,(24)n2∗=μ-p2/c1/αλd,(25)p2≤cuα,(26)p1>p2.(iv) Case IV: finally, from (10) and (15)(27)n1∗=0(28)n2∗=0,(29)p1>cμα,(30)p2>cμα.As a corollary, price values for Case I comply withp1≤cμα.Table2 summarizes the results of the users’ subscription decision under a pooling agreement. Figure 1 shows a graphical representation of the regions for a specific configuration of the parameters.Table 2
Users subscription decision-pooling.
Case n 1 ∗ n 2 ∗ p 1 p 2 I μ - p 1 / c 1 / α 2 λ d μ - p 2 / c 1 / α 2 λ d p 1 ≤ c u α p2=p1 II μ - p 1 / c 1 / α λ d 0 0 ≤ p 1 ≤ c μ α p 2 > p 1 III 0 μ - p 2 / c 1 / α λ d p 1 > p 2 0 ≤ p 2 ≤ c μ α IV 0 0 p 1 > c μ α p 2 > c μ αFigure 1
Wardrop equilibrium regions for pooling withc=1, α=0.8, μ=1 packets per second (p/s), λd=0.5 p/s.At this point we proceed to analyze the equilibrium pricespi∗, given the values for n1∗ and n2∗. For this purpose we use (6) and we obtain the BRs as defined by (11) and (12):(31)BR1p2=p1≥0ifp2=0p2-ϵif0<p2≤p0∗cμαα1+ααifp2>p0∗BR2p1=p2≥0ifp1=0p1-ϵif0<p1≤p0∗cμαα1+ααifp1>p0∗where p0∗ is the profit maximizing price for a monopolistic operator, which is given by (A.9), and ϵ is an arbitrarily small positive value. The Nash equilibrium is the intersection of the BRs, which is easily derived as p1∗=0 and p2∗=0, with zero profits Π1∗=0,Π2∗=0. We conclude that a pooling agreement between a network operator and a virtual operator generates strategic interaction that drives prices and profits to zero.


### 3.2. Priority Sharing

In this section, we analyze the case where operator 1 and operator 2 share the spectrum resource according to a distribution of relative priority. As justified in Section2.1, this distribution is modeled by means of DPS and Ti are now given by (2) and (3), so that the user utility will be(32)U1=cμ+γ-1λdn1+λdn2μ-λdn1-λdn2μ-γλdn2+γ-1λdn1-α-p1,(33)U2=cμ-γλdn1+λdn2μ-λdn1-λdn2μ-γλdn2+γ-1λdn1-α-p2.Following a similar reasoning as in Section3.1, we obtain the equilibrium values for n1∗ and n2∗ and the constraints on p1 and p2, which are shown in Table 3. Figure 2 shows a graphical representation of the cases for a specific configuration of the parameters.Table 3
Users subscription decision-priority sharing.
Case n 1 ∗ n 2 ∗ p 1 p 2 I (γ≠1/2) 1 λ d 1 ( γ - 1 ) c / p 1 1 / α + γ c / p 2 1 / α - 1 λ d μ γ c / p 2 - 1 / α c / p 1 1 / α + γ - 1 1 λ d μ ( γ - 1 ) c / p 2 1 / α c / p 1 - 1 / α + γ - 1 λ d 1 ( γ - 1 ) c / p 1 1 / α + γ c / p 2 1 / α p 1 ≤ p ^ 1 γ p 2 ≤ p ^ 2 γ I (γ=1/2) μ - p 1 / c 1 / α 2 λ d μ - p 2 / c 1 / α 2 λ d p 1 ≤ p ^ 1 γ p 2 = p 1 II μ - p 1 c 1 / α  1 λ d 0 0 ≤ p 1 ≤ p ^ 1 U p 2 > p ^ 2 γ III 0 μ - p 2 c 1 / α 1 λ d p 1 > p ^ 1 γ 0 ≤ p 2 ≤ p ^ 2 U IV 0 0 p 1 > p ^ 1 U p 2 > p ^ 2 UFigure 2
Wardrop equilibrium regions for priority sharing withγ=1/10, c=1, α=0.8, μ=1 packets per second (p/s), λd=0.5 p/s.The following expressions for the cases boundaries apply:(34)p^1γ=cγμp2/c-1/α-γ+1-γμ+γp2/c1/α+μ-α,(35)p^2γ=cp1/c-1/αγ-1μ-γp1/c1/αγ-1p1/c1/α-γμ-α,(36)p^1U=cμα,(37)p^2U=cμα.Note that whenγ=1/2, the priority sharing agreement reduces to the pooling agreement, since FCFS and DPS result in the same value for Ti.Unlike Section3.1, we have proceeded numerically in the analysis of Stage I, as regards the BRs and the Nash equilibrium.


## 3.1. Pooling

In this subsection, operator 1 and operator 2 are sharing the spectrum resource. The utility of the users can be obtained replacing (1) in (5).(15)Ui=c·μ-n1+n2·λdα-pii=1,2We obtain the equilibrium values forn1 and n2, given p1 and p2, as follows.(i) Case I: from (7) and (15) we obtain(16)n1∗+n2∗=μ-p1/c1/αλd=μ-p2/c1/αλd,which is only feasible ifp1=p2. Without loss of generality, we can assume that n1∗=n2∗, so that(17)n1∗=μ-p1/c1/α2λd,(18)n2∗=μ-p2/c1/α2λd.(ii) Case II: from (8) and (15) we obtain(19)n1∗=μ-p1/c1/αλd,(20)n2∗=0,(21)p1≤cuα.(22)p2>p1.(iii) Case III: from (9) and (15) we obtain(23)n1∗=0,(24)n2∗=μ-p2/c1/αλd,(25)p2≤cuα,(26)p1>p2.(iv) Case IV: finally, from (10) and (15)(27)n1∗=0(28)n2∗=0,(29)p1>cμα,(30)p2>cμα.As a corollary, price values for Case I comply withp1≤cμα.Table2 summarizes the results of the users’ subscription decision under a pooling agreement. Figure 1 shows a graphical representation of the regions for a specific configuration of the parameters.Table 2
Users subscription decision-pooling.
Case n 1 ∗ n 2 ∗ p 1 p 2 I μ - p 1 / c 1 / α 2 λ d μ - p 2 / c 1 / α 2 λ d p 1 ≤ c u α p2=p1 II μ - p 1 / c 1 / α λ d 0 0 ≤ p 1 ≤ c μ α p 2 > p 1 III 0 μ - p 2 / c 1 / α λ d p 1 > p 2 0 ≤ p 2 ≤ c μ α IV 0 0 p 1 > c μ α p 2 > c μ αFigure 1
Wardrop equilibrium regions for pooling withc=1, α=0.8, μ=1 packets per second (p/s), λd=0.5 p/s.At this point we proceed to analyze the equilibrium pricespi∗, given the values for n1∗ and n2∗. For this purpose we use (6) and we obtain the BRs as defined by (11) and (12):(31)BR1p2=p1≥0ifp2=0p2-ϵif0<p2≤p0∗cμαα1+ααifp2>p0∗BR2p1=p2≥0ifp1=0p1-ϵif0<p1≤p0∗cμαα1+ααifp1>p0∗where p0∗ is the profit maximizing price for a monopolistic operator, which is given by (A.9), and ϵ is an arbitrarily small positive value. The Nash equilibrium is the intersection of the BRs, which is easily derived as p1∗=0 and p2∗=0, with zero profits Π1∗=0,Π2∗=0. We conclude that a pooling agreement between a network operator and a virtual operator generates strategic interaction that drives prices and profits to zero.


## 3.2. Priority Sharing

In this section, we analyze the case where operator 1 and operator 2 share the spectrum resource according to a distribution of relative priority. As justified in Section2.1, this distribution is modeled by means of DPS and Ti are now given by (2) and (3), so that the user utility will be(32)U1=cμ+γ-1λdn1+λdn2μ-λdn1-λdn2μ-γλdn2+γ-1λdn1-α-p1,(33)U2=cμ-γλdn1+λdn2μ-λdn1-λdn2μ-γλdn2+γ-1λdn1-α-p2.Following a similar reasoning as in Section3.1, we obtain the equilibrium values for n1∗ and n2∗ and the constraints on p1 and p2, which are shown in Table 3. Figure 2 shows a graphical representation of the cases for a specific configuration of the parameters.Table 3
Users subscription decision-priority sharing.
Case n 1 ∗ n 2 ∗ p 1 p 2 I (γ≠1/2) 1 λ d 1 ( γ - 1 ) c / p 1 1 / α + γ c / p 2 1 / α - 1 λ d μ γ c / p 2 - 1 / α c / p 1 1 / α + γ - 1 1 λ d μ ( γ - 1 ) c / p 2 1 / α c / p 1 - 1 / α + γ - 1 λ d 1 ( γ - 1 ) c / p 1 1 / α + γ c / p 2 1 / α p 1 ≤ p ^ 1 γ p 2 ≤ p ^ 2 γ I (γ=1/2) μ - p 1 / c 1 / α 2 λ d μ - p 2 / c 1 / α 2 λ d p 1 ≤ p ^ 1 γ p 2 = p 1 II μ - p 1 c 1 / α  1 λ d 0 0 ≤ p 1 ≤ p ^ 1 U p 2 > p ^ 2 γ III 0 μ - p 2 c 1 / α 1 λ d p 1 > p ^ 1 γ 0 ≤ p 2 ≤ p ^ 2 U IV 0 0 p 1 > p ^ 1 U p 2 > p ^ 2 UFigure 2
Wardrop equilibrium regions for priority sharing withγ=1/10, c=1, α=0.8, μ=1 packets per second (p/s), λd=0.5 p/s.The following expressions for the cases boundaries apply:(34)p^1γ=cγμp2/c-1/α-γ+1-γμ+γp2/c1/α+μ-α,(35)p^2γ=cp1/c-1/αγ-1μ-γp1/c1/αγ-1p1/c1/α-γμ-α,(36)p^1U=cμα,(37)p^2U=cμα.Note that whenγ=1/2, the priority sharing agreement reduces to the pooling agreement, since FCFS and DPS result in the same value for Ti.Unlike Section3.1, we have proceeded numerically in the analysis of Stage I, as regards the BRs and the Nash equilibrium.


## 4. Results and Discussion

In this section, numerical values are computed for the users, prices, and profits for the network and virtual operators under a priority sharing agreement. Note that the prices and the profits are zero for pooling, as concluded in Section3.1.We set the parametersα=0.5,0.8,1, c=1, μ=1 packet/s, λd=0.01 packet/s and we vary γ∈[0,1]. Figures 3, 4, 5, 6, 7, and 8 show the prices, the number of users, and profits, respectively. The values p0∗, n0∗, and Π0∗ refer to the monopoly (the Appendix), while the values p1∗, n1∗, and Π1∗ refer to the network operator, and the values p2∗, n2∗, and Π2∗ refer to the virtual operator.Figure 3
Prices of users for the network operator and the monopolistic operator as a function ofγ for different values of α.Figure 4
Prices of users of the virtual operator as a function ofγ for different values of α.Figure 5
Number of users for the network operator and the monopolistic operator as a function ofγ for different values of α.Figure 6
Number of users for the virtual operator as a function ofγ for different values of α.Figure 7
Profits for the network operator and the virtual operator as a function ofγ for different values of α.Figure 8
Π 1 ∗ + Π 2 ∗ and profits for the monopolistic operator as a function of γ for different values of α.Figure3 shows the effect of α and γ (the virtual operator’s relative priority) on the equilibrium prices of the monopolistic operator and operator 1. The price p1 decreases as α and γ increase, since the QoS and priority (1-γ) of the service that the network operator can provide to its subscribers are lower. However, when γ approaches 1 the price p1 increases. In this situation the network operator is giving strict priority to the virtual operator; i.e., the DPS discipline behaves as a PQ. This specific setting is analyzed in [24, 29, 30] and the results match those shown in Figure 3. The monopolistic price p0∗ is a constant value because it does not depend on γ (Table 4). When γ∈0,1, p0∗ is greater than the equilibrium prices, whereas pi∗=p0∗ when operator i is given strict priority, since it receives a preemptive priority service; i.e., it is not affected by the nonpriority users.Table 4
Users, prices, and profit of one operator case.
Case n 0 ∗ p 0 ∗ Π 0 ∗ ( p 0 ) I μ - p o c 1 / α  1 λ 0 p 0 ∗ = c μ α α 1 + α α Π 0 ( p 0 ∗ ) = c μ α + 1 α 1 + α α + 1 1 α II 0 p0>cμα Π 0 ( p 0 ∗ ) = 0Figure4 shows the effect of α and γ on the equilibrium prices of operator 2. The prices (p1,p2) behavior is symmetrical; i.e., p2 exhibits the same behavior as the one described above but when α increases and γ decreases from 1 to 0.We can conclude that the entry of the virtual operator causes the price charged both by the network operator and by the virtual operator to decrease; therefore, it is beneficial for the users. And we have numerically investigated that PQ is the limit case of DPS always providing the highest prices.Figure5 shows the effect of α and γ on the number of users of operator 1. n1 increases when α increases from 0 to 1 and γ increases from 0 to 0.5, which can be explained by the decrease in price p1. At the same time, operator 2 loses all its users, and n2 gets to zero. When γ=0.5, the system behaves as in the pooling agreement, i.e., zero prices and equal level of subscription. When γ increases from 0.5 to 1, the number of users switches and operator 1 loses its users. Figure 6 shows the effect of α and γ on the number of users of operator 2. The behavior of n2 and n1 is symmetrical; i.e., n2 exhibits the same behavior as the one described above but when α increases from 0 to 1 and γ decreases from 1 to 0.The results show that the operator with the higher priority (e.g., operator 1 in the left hand side of Figure5) is able to provide a satisfactory service to a maximum number of users, although the equilibrium price needs to be very low but non-null (Figure 3). For γ near the values of 0 and 1, the number of users that the operator with the higher priority can serve is lower than the maximum; however, the equilibrium price is not so low as before, and the profits are greater (Figure 7). This specific setting is analyzed in [24, 29, 30], where the discipline was Preemptive Resume Priority Queueing (which, if users are served in a processor-sharing manner, is equivalent to DPS for γ=0 or 1). The explanation for these results is related to the fact that only when γ is near 0 or 1, the QoS received by each user group is sufficiently different from each other so that the competition between the two operators is not too fierce and the prices need not be too low.Figure7 shows the equilibrium profit of each operator as α and γ vary. We observe that the network operator suffers a significant reduction in its profit Π1 when α and γ increase between 0<γ<1. The profits only recover when γ approaches 0 or 1; i.e., the DPS discipline behaves as a PQ. The behavior described is consistent with the combined behavior of the prices and the number of users, described above. The virtual operator behavior is symmetrical; i.e., Π2 exibits the same behavior as the one described above but when α increases and γ decreases from 1 to 0.When compared to the profit that the network operator would get in a monopoly,Π0 (the Appendix), we conclude that the network operator is always worse off under a priority sharing agreement. Obviously, it is also worse off under a pooling agreement. However, we can determine a lump sum payment m that the virtual operator would make to the network operator in order to provide an incentive to the latter. Indeed, this payment should not only improve the situation of the network operator, that is, Π1+m≥Π0, but also allow the virtual operator to obtain nonnegative profits Π2-m≥0. Joining the two conditions we obtain Π0-Π1≤m≤Π2. Thus, a necessary condition for the existence of a possible payment is Π1+Π2≥Π0. This condition obviously does not hold for the pooling agreement. As regards the priority sharing agreement, Figure 8 shows that this condition holds only for sufficiently small values of γ or 1-γ.We can conclude that network sharing is incentive compatible under the following conditions: first, that a priority sharing agreement is reached and second, that either a very low or a very high relative priority is granted to the virtual operator willing to access the network operator resources.


## 5. Conclusions

In this paper, we have evaluated the issue of network sharing between a network operator and a virtual operator for profit-maximization purposes. We have studied the economic viability of the sharing under pooling and priority sharing agreements, and these are compared with a monopoly. Our main results suggest that the network operator is worse off under any circumstances under a pooling agreement. The entry of a virtual operator is desirable from the point of view of the users’ prices. Additionally, for each parameter configuration there exist a range of values ofγ, for which a lump sum payment can be designed so that the network operator has an incentive to let the virtual operator enter.



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*Source: 4308913-2018-10-11.xml*