A Nash Stackelberg Game for Revenue Maximization and Vertical Handover Decision Making Mariem Zekri∗ , Makhlouf Hadji∗† , Badii Jouaber∗† and Djamal Zeghlache∗† ∗ INSTITUT Telecom - Telecom SudParis, Evry, France † CNRS, UMR SAMOVAR Emails: {mariem.zekri, makhlouf.hadji, badii.jouaber, djamal.zeghlache}@it-sudparis.eu

Abstract—Radio resource and mobility managements are becoming more and more complex within nowadays rich and heterogeneous wireless access networking systems. Multiple requirements, challenges and constraints, at both technical and economical perspectives have to be considered. While the main objective remains guaranteeing the best Quality of Service and optimal radio resource utilization, economical aspects have also to be considered including cost minimization for users and revenue maximization for network providers. In this paper, we propose a game theoretic scheme where each available network plays a Stackelberg game with a finite set of users, while users are playing a Nash game among themselves to share the limited radio resources. A Nash equilibrium point is found and used for vertical handover decision making and admission control. Index Terms—Heterogeneous wireless networks, Vertical HandoverDecision, Revenue Maximization, Stackelberg game, Noncooperative game, Nash equilibruim, Admission control

I. I NTRODUCTION Nowadays, service providers are relying on different wireless access technologies to handle the increasing amount of subscribers’ demands. These heterogeneous networks would be able to insure the ”Always best connected” paradigm by providing different service classes with their corresponding required QoS. In this paper, we consider multihomed mobile terminals able to connect simultaneously to more than one access network and to seamlessly switch from one access network to another. An illustration of a heterogeneous wireless network is represented in Fig.1 where the mobile terminal can connect to three different access networks, namely, 3G, wireless local area networks (WLAN) and wireless metropolitan area network (WMAN). The available wireless technologies have different characteristics including coverage, mobility management, security and capacity. To select the most appropriate acces network, new solutions are required to meet both users’ and networks’ objectives. On the one hand, users seek the most suitable access network regarding their needs and cost preferences. On the other hand, service providers may aim to maximize their revenues that are proportional to the resource utilization while remaining competitive to attract users. Most of existing vertical handover decision mechanisms are mainly based on technical network aspects like RSS and QoS parameters and do not consider interactions that may exist between the actors concerned by the decision making

3G Base Station

Wifi Access Point

Wimax Base Station

MultiHomed Mobile Terminal

Fig. 1: Heterogeneous network

(i.e. users, networks and service providers). In [1] different MADM (Multiple Attributes Decision Making) models are introduced and a comparison between the most popular, namely SAW, TOPSIS, GRA and AHP is established. It shows that the provided performances depend on the adopted decision scheme. An appropriate choice of the decision mechanism is then required. Authors in [2] presented a vertical handover decision mechanism that enables network selection using Fuzzy Logic (FL) concepts. They showed that this solution is able to determine when a handover is required, and to select the best access network according to quality of service requirements, network and mobile terminal conditions and user preferences. In [3] an intelligent context aware vertical handover decision mechanism that combines MADM methods and FL and considers networks’ capabilities and QoS requirements is proposed. This solution is based on a Fuzzy Logic Controller (FLC) that checks whether a serving network is still able to provide a good QoS and on an Analytic Hierarchy Processe (AHP) method that scores the available networks to indicate over which one to hand over. In [4] a user centric decision algorithm that gives the end user the control on the selection of the wireless access network that best fits his preferences is described. Authors consider that ”good” or ”best” connectivity is relative to the user preference. For instance, the user may prefer to ensure a good QoS for his ongoing applications as long as possible, no matter the costs. He may also opt for saving on the connection cost even if the session continuity is not guaranteed. Alternatively, the user may prefer to find some

compromise between sessions’ continuity and cost saving. These solutions are very interesting in the sense that different decision parameters related to different requirements are considered. However, other considerations related to the real interaction of all the actors involved in an heterogeneous environment (access networks, users, service providers,...) should be taken into account to make appropriate decisions. Indeed, interactions across actors are non-negligible because the choices of any one may influence the choices of the others. In this context,it is important to examine the economic concern by introducing the service provider and mobile users in a market like environment, allowing to jointly optimize both resource consumption and utilities of both users and providers. Game theory has shown to be a powerful tool for the analysis of interactive decision-making processes. It provides mathematical tools to predict what should happen when agents (or players) with conflicting interests interact. In this paper, the problem is modeled as an hierarchical game among heterogeneous available networks and multiple users running various services and having different requirements. We propose a scheme where each available network plays a Stackelberg game with users to maximize the service provider revenue, while these latter are playing a Nash game among them selves to maximize their utilities. The remainder of this work is organized as follows. In section II, a basic introduction of the tool of game theory and its known applications in telecommunications are given. Section III formulates the game and its resolution and analyses the service provider’s (the networks) revenue. In section IV, a vertical handover decision algorithm with a selection process based on the obtained Nash equilibrium is proposed. Section V provides the performances results and, finally, section VI concludes the proposed work.

one are blocked when there is no space. The objective is to minimize the delay for the first class and the blocking probability for the second. [7] presents a routing problem in which non-cooperating agents wish to establish paths from sources to destinations to transport a fixed amount of traffic. The authors study the equilibrium that arise in networks of general-topology under some polynomial cost functions and obtain conditions for the uniqueness of the equilibrium. A promising potential application of game theory is also the area of network security. In [8], authors resort to game theory to develop a network packet sampling strategy that detects network intrusions taking into consideration the constraint of not exceeding a given total sampling budget. They model the problem as a non-cooperative game between intruders and networks providers. The intruder injects malicious packets and picks paths to minimize chances of detection and the network operator chooses a sampling strategy to maximize the chances of detection. Another problem that is well studied using game theory is flow control. [9] presents a game theoretic framework in which each user aims to maximize its performance measure expressed by a standard utility function. It demonstrates the existence and the uniqueness of Nash equilibrium and gives a proposal on how non-cooperative users can distribute their flows among numerous links, by imposing a suitable pricing method that encourages load balancing. Authors in [10], propose a game theory based model for revenue maximization pricing and capacity expansion in a Many-Users regime. They consider a model where many users are accessing a single link and capacities are increased in proportion to the number of users. They show that, as the number of users increases, the service provider’s revenue-perunit-bandwidth increases for all values of the link capacity and the overall performance of each user improves.

II. G AME THEORY IN TELECOMMUNICATIONS

III. A T WO -L EVEL H IERARCHICAL G AME

Game theory is a powerful tool that has been widely used in different fields to study the behavior of interacting actors having conflicting interests. It has been applied in real games, economics, politics, commerce and recently in telecommunications and networking. For instance, intensive research effort has been devoted to game models in wireless networks. Some of the main studied issues are power control, pricing, security issues, access and flow control and auctions for resource reservation. In [5], Xiao et al. present a power control framework called utility-based power control (UBPC) cost. This framework ameliorates system convergence and satisfies QoS requirements in term of delay and bit error rate for different service classes in code division multiple access (CDMA) cellular systems. The UBPC is represented as a non-cooperative N-person game where each user aims to maximize its satisfaction by increasing its QoS and minimizing its power consumption. There is also an extensive literature on game theoretic models of routing problems. [6] presents an approach that formulated a multiple class routing problem based on game-theory as a Nash game and solved the routing problem for two classes of packets sharing two links. The first class may be queued at the link buffers and second

A. Game formulation In this paper, we assume that there is a single service provider that manages the available networks. Let’s denote by: - N j the available networks ; j = {1, ..., k}, and users by I = {1, ..., n}. Network N j has a total available bandwidth denoted by C j . - Bij the bandwidth provided by N j to a user i. - pji the charged price to user i by network N j ; - wi > 0 the user i ability to pay [11]. - rj the network N j reputation, it represents the network reliability in terms of good QoS providing and depends on QoS parameter including delay, jitter, bit error rate, etc [12]. rj varies between 1 and 10, 1 for very bad reputation and 10 for excellent reputation. - qi the user i requirement in term of QoS according to its running application. qi is between 1 and 5, 1 for low QoS requirements and 5 very high QoS requirements. The problem is modeled as a two-level hierarchical game [13]: • The upper level is a Stackelberg game with each of the

available networks as a leader and mobile users as followers. In this level each network adjusts its prices in order to maximize its total revenue when users respond with their bandwidth requests corresponding to their requirements. The network revenue is given by: Rj =

n ∑

pji Bij

under the constraints given by n ∑

Note all Bij , i ∈ {1, ..., n} and j ∈ {1, ..., k} such ∑nthat for j that i=1 Bi ≤ C j ∂Uij ∂Bij

and the service provider total revenue is: R=

wi r j qi

=

− pji

1 + rj qi Bij

and R

j

∂ 2 Uij

j=1

• The lower level is an I-players non cooperative game where each user i objective is to maximize the following utility function: Uij = wi ∗ log(1 + rj qi Bij ) − pji Bij subject to the constraint n ∑

(2)

i=1

i=1

k ∑

Bij ≤ C j

Blj 6 C j

l=1

Remark: The utility function chosen for user i is wi ∗ log(1 + rj qi Bij ). It is close to the utility function wi logxi used in [14] that leads to proportional fair resource allocation. However, in our case, if we use wi logrj qi Bij , a user will be obliged to ask for a nonzero Bij to avoid the case where his utility becomes equal to −∞ if his demand is equal to zero. In addition, if a user is obliged to ask for a nonzero bandwidth, the service provider may get profit of this situation by imposing high prices. Our utility function wi log(1 + rj qi Bij ) allows users to decide whether to join a network or not which ensures a nontrivial solution to the Stackelberg game. In pursuing a solution to the Stackelberg game, our intention is to find the Nash Equilibrium (NE) point where neither networks nor users have any incentive to deviate unilaterally from that point. This (NE) point is formally defined as follows: Definition: (Nash Equilibrium) Let pj∗ be the network i solution for the stackelberg problem and Bij∗ be a solution j∗ for the it h user’s Nash problem. The point (pj∗ i , Bi ) is a NE j j for the Stackelberg game if for any (pi , Bi ): j∗ j j∗ j Uij (pj∗ i , Bi ) ≥ Ui (pi , Bi )∀i, j j∗ j∗ j j Rj (pj∗ i , Bi ) ≥ R (pi , Bi )

Theorem I: (Existence of Unique Nash Equilibrium) For each price pji the n-player non cooperative game admits a unique Nash equilibrium solution. Proof : Uij (B, pj ) = wi log(1 + rj qi Bij ) − pji Bij

(1)

(1 + rj qi Bij )2

0)

(8)

If λ > 0, equations (6) and (7) lead to n ∑

Bij =

i=1 n ∑ k̸=i

B. Solution

wi (rj qi )2

It is now easy to notice that Uij is a concave function of Bij and the second derivative given in (3) is negative. This leads to conclude the uniqueness of the Nash equilibrium point.

⇔

and

=−

2 ∂Bij

(9)

n ∑ 1 wi + Cj − j jq r pk + λ i i=1

(10)

wi

i=1

pji + λ

wk pjk + λ ∑n

The expression k̸=i form: n ∑ wk pj k̸=i k

n ∑ 1 = Cj jq r i i=1

n ∑

+λ

=

wk pjk +λ

can be written in this equivalent

∑n

=

−

∏n

j l̸=k,i (pl + j m̸=i (pm + λ)

wk ∏n

k̸=i

λ)

(11)

Equations (10) and (11) lead to: ∑n ∏n j n ∑ wi 1 k̸=i wk l̸=k,i (pl + λ) + Cj − j (12) = ∏n j j r qi pk + λ m̸=i (pm + λ) i=1

Considering γ = to γ

n ∏

∑n

1 i=1 r j qi

(pjm + λ) −

i + C j − pjw+λ , equation (12) leads k

n ∑

n ∏

wk

(13)

l̸=k,i

k̸=i

m̸=i

(pjl + λ) = 0

⇔

C. Revenue analyses Considering the optimal prices given by (24) and the optimal bandwidth demands given by (23) we can calculate the optimal revenue of a network N j : n ∑ Rij∗ (25) Rj∗ = i=1

n ∏

γ(pjt + λ)

(pjm + λ) −

n ∑

m̸=i,t

wk

k̸=i

n ∏

(pjl + λ) = 0 (14)

l̸=k,i

Where,

Rij∗ R

=

j∗

Bij∗ pj∗ i . n ∑

n ∑

γ(pjt +λ)

t̸=i

n ∏

i=1

(pjm +λ)−(n−1)

m̸=i,t

n ∑

wk

k̸=i

n ∏

(pjl +λ)

=0

l̸=k,i

(15) Summing up terms with the same indices and taking the product as a common factor give: γ(pjt ⇔

[

+ λ) = (n − 1)wt

] n ∑ 1 wi j +C − j (pjt + λ) = (n − 1)wt jq r p + λ i k i=1

⇔

(16)

(17)

n ∑ wt wi 1 = (n − 1) j + Cj − j jq r p + λ (p i t + λ) k i=1

(18)

n ∑ 1 wt + C j − (n − 1) j jq r (p i t + λ) i=1

(19)

⇔ wi pjk + λ n ∑

wi

i=1

pji + λ

=

= nC j + n

Equations (9) and (20) lead to n n ∑ ∑ 1 1 wt j j (21) + C = nC + n − n(n − 1) j jq jq r r (p i i t + λ) i=1 i=1

Cj 1∑ 1 + n n i=1 rj qi n

wt pjt + λ

=

(22)

Equations (22) and (6) give: Cj 1∑ 1 1 + − j j n n i=1 r qi r qi n

Bij∗ =

Cj +

(23)

1 i=1 r j qi

n ∑ wi jq r i i=1

(26)

It is interesting to study the behavior of Rj∗ when a user ability to pay and its requirement in terms of QoS change. We note that: nwi Rij∗ = wi −

rj q

Cj

+

∑ni

(27)

1 i=1 r j qi

1) Behavior of Rj∗ with respect to qi :

∑n n n ∑ ∑ nwi [rj (C j + l=1 rj1qi ) − q1i ] ∂Rj∗ ∂Rij∗ ∑n = = ∂qi ∂qi [rj qi (C j + l=1 rj1qi )]2 i=1 i=1 (28) We notice that: n n ∑ ∑ 1 1 1 rj C j + − = rj C j + > 0 ∀i, j (29) qi qi qi l=1

n ∑ 1 wt (20) − n(n − 1) j jq r (p i t + λ) i=1

⇔

wi −

=

Simple manipulations then lead to

n ∑n

l̸=i

∂Rj∗ ∂qi

It is then clear that is strictly positive ∀i ∈ {1, ..., n} and ∀j ∈ {1, ..., k}. This means that the revenue of a network N j increases when users’ requirements in terms of QoS increases. In fact, when a user is more exigent in terms of QoS, the network will charge him with a higher price which is logical (24). 2) Behavior of Rj∗ with respect to wi : ∑ ∂Rj∗ ∑ ∂B j∗ j∗ ∂pj∗ j∗ ∑ ∂pj∗ j∗ ∂Rj∗ i i i = = p + i Bi = B . ∂wi ∂wi ∂wi i ∂wi ∂wi i i=1 i=1 i=1 (30) ∂pj∗ j∗ ∂Rj∗ i It is clear that ∂wi is positive as ∂wi > 0 and Bi is strictly positive for all n > 1. Thus, Rj∗ increases when the user ability to pay increases. This means that the total revenue of a network N j increases when the users are more willing to pay. In the following, we propose a handover decision algorithm with a selection process based on the obtained Nash/Stackelberg equilibrium. n

n

n

IV. V ERTICAL HANDOVER DECISION MAKING AND ADMISSION CONTROL

and so pj∗ i =

Cj

+

nwi ∑n

1 i=1 r j qi

A. Proposed vertical handover decision making based on NE (24)

We notice that, when a user requirements in terms of QoS increase, its demand in terms of bandwidth at the NE point ∂B j∗ increases( ∂qii is positive). The same is for optimal prices,(24) suggests charging more the users that are more exigent in terms of QoS, i.e. higher qi , and who are more willing to pay for their utilities, i.e. higher wi .

In this section, we mainly focus on the handover decision making step and we propose a VHO decision mechanism based on NE as presented in figure 2. The vertical handover decision making consists in two steps which are handover initiation and network selection. Our proposed solution considers the network and terminal context (especially for handover initiation) as well as users preferences (for network selection) in terms of cost and QoS.

for a user i the class Cl(1) is preferable to the class Cl(2) which is also preferable to Cl(3) with respect to the utility function Uij (Bij∗ , pj∗ i ) . In the following, let V be the number of available classes(V ∈ {1, 2, 3}) and xji be the variable of decision making. xji = 1 if user i decides to connect to network j, and xji = 0 otherwise. Bandi is the total value of allocated bandwidth to a user i. Algorithm 1 VHO decision making algorithm

Fig. 2: Vertical handover process

1) Handover initiation: A Fuzzy Logic Controller (FLC), based on Fuzzification/Defuzzification mechanisms [2], checks whether the current network is still able to handle a user connection. It uses contextual information (RSS, load and velocity) to detect whether a VHO is required or not. If so, the FLC initiates the handover. For instance, the input parameters are fed into a fuzzifier where they are transformed into fuzzy sets. A fuzzy set may have different membership degrees that are obtained by mapping the real values of a given variable into a membership function. For example the membership function of the load input parameter has three fuzzy sets which are low, medium and high. The fuzzy sets are then fed into an inference engine, where a set of fuzzy IF-THEN rules are applied to indicate whether a handover is required. (e.g. IF load = high AND velocity = low AND RSS = low THEN Handover=YES). Finally, the overall obtained fuzzy sets are defuzzified to make

Bandi = 0, index = 1; while (Bandi < Bi ) and (index ≤ V ) do j1∗ = ArgM axj {Uij , j ∈ Cl(index) }; xji 1 ∗ = 1; △Band = Bi − Band i j∗ Band+ = min{Bi 1 , △Band} index + +; end while if Bandi < Bi then Connection not admitted; end if

B. Admission Control When a new connection or a VHO is initiated by a user i, the required bandwidth is compared to the total bandwidth Bandi that user could be offered by the available networks. Cl Cl Bandi = (Bi (1) )∗ + . . . + (Bi (V ) )∗ ; We consider Bij∗ = 0 if network j is not available in a service area. If a connection required bandwidth is smaller than Bandi , the connection is admitted. Otherwise, it is rejected. V. N UMERICAL RESULTS A. simulation model We consider an heterogeneous wireless environment consisting of two IEEE 802.11 WLANs, one UMTS cellular network (3G) and one IEEE 802.16 WMAN as shown in Fig.4. We consider different areas where a multihomed mobile

Fig. 3: Fuzzification/Defuzzification mechanism a final precise decision (i.e. VHO is required or not). If a handover is required the network selection process is activated. 2) Network selection: First, we classify the finite set of networks into three classes : Cl1 for WLAN Networks, Cl2 for WMAN Networks and Cl3 for cellular Networks. Then, we order the three classes of networks according to the utility function Uij (Bij∗ , pj∗ i ) . If we suppose that all the three classes are available, let this order be as follows: Cl(1) ≽ Cl(2) ≽ Cl(3) . This means that

Fig. 4: Simulation Model terminal may connect to different access technologies. In area A, only the WMAN is available. In area B, 3G and WMAN are available. In area C, a mobile terminal is able to connect to WMAN, WiFi and 3G. Finally, in area D, WMAN and WiFi are available. The transmission rate is 2Mbps in the 3G cell, 10Mbps in the WMAN, and 11Mbps in the WLAN.

In this paragraph we numerically verify the results in section III-C and we discuss the user’s utility evolution when the network parameter vary (rj and C j ). To show the effect of the users’ parameter (qi and wi ) on the optimal prices and the j∗ network revenue, we calculate pj∗ i and Ri while varying qi . j∗ The case where wi increases is trivial as pj∗ i and Ri increase linearly with respect to wi . Figure 5 shows the variation of pji when a user i requirement in terms of QoS increases from 1 to 5 for different network capacities and reputations values. We set the number of users to 30 and their requirements in terms of quality of service are randomly generated in the range of 1 to 5. The user i ’s ability to pay is set to 3 (wi = 3). One can remark in figure 5 that the charged prices increase when users requirements in QoS increase. It is also shown that for the same amount of available capacity and for different reputation values of a network, the prices charged to user i are higher for networks with better reputation. For the same value of reputation, the charged prices are lower for higher network capacity.

5 4 3 2 1 0 −1 −2 10 8

10 6

8 6

4

4

2

2 0

Price

0

Reputation

Fig. 7: User utility vs Reputation and prices

C=7, r=6.2 C=7, r=6 C=6.8, r=6.2

4.8

4.75

4.7 6 5

4.65

4.6

1

1.5

2 2.5 3 3.5 4 User i requirement in terms of QoS

4.5

5

Fig. 5: Optimum prices vs QoS, Capacity and Reputation

User utility

Price charged to user i from network j

6

a network capacity decreases, the network prices increase to improve the network revenue which strongly affects the user utility as depicted in figure 8. In this case, a network with scarce resource should expand his capacity to stay competitive with other networks and to keep attracting users.

4.9

4.85

If we look to this problem from the user side, it is important to notice that the utility of users also increases when the network reputation is improved. Thus, even if the network price is increased, users will still be attracted by this network because this prices rise is compensated by the reputation enhancement. This is illustrated in figure 7. However, when

User Utility

B. Revenue maximization

4 3 2 1 0 −1 −2 10 8

Figure 6 depicts the revenue variation when a user i requirements in QoS increase. When a network reputation (respectively capacity) increases, the network revenue increases. In other words, our results show that to enhance networks revenue for a given available capacity, it is interesting to improve the reputation by providing good QoS parameters (delay, jitter, Bit error rate...).

Revenue of network j from user i

10 C=7,r=8 C=7,r=6 C=5, r=8

9

8

7

6

5

4

3 1

1.5

2

2.5

3

3.5

4

4.5

5

User i requirement in terms of QoS

Fig. 6: Optimum revenue vs QoS, Capacity and Reputation

10 6

8 6

4

4

2

Price

2 0

0

Capacity

Fig. 8: User utility vs Capacities and prices

C. VHO decision making In this section, we consider the system model presented in paragraph V-A and we consider uniform demands in terms of QoS for all arriving users. Figure 9 illustrates the VHO dropping probability in the areas A, B and C. When the arrival rate of VHO connections is low, the VHO blocking rate in our scheme is almost equal to zero. However, when the number of simultaneous VHO connection requests increases, the VHO blocking rate increases to reach about 37 percent in area A, for a high amount of arrivals (40 simultaneous VHO arriving connection). Under the same conditions (same VHO connections arrival rate and the same bandwidth requests), fig.9 shows that the blocking rate in area C is less important than in area B which in turn is less important in area A. This is expected as users in area C may connect to three different

networks and get higher bandwidth than users in the two other areas. 0.4

VHO blocking rate

0.35

area A area B area C

0.3

compensate the bandwidth scarcity of some networks and to decrease the VHO blocking rate. Our future research will also address the non cooperative case in which different service providers are competing for resource charing while dealing with a finite set of users. This will be modeled as a game with multiple leaders and multiple followers.

0.25

R EFERENCES

0.2

0.15

0.1

0.05 10

15

20

25

30

35

40

Simultaneous VHO arriving connections

Fig. 9: VHO blocking rate

VI. C ONCLUDING REMARKS In this paper we proposed a tool to study the revenue of a service provider managing heterogeneous wireless access networks and dealing with a finite set of users that aim to maximize their utilities. This tool is then used for vertical handover decision making. We formulated and modeled mathematically the problem as a Stackelberg/Nash game and presented an optimal bandwidth/pricing policy for different players. Then we proposed a handover decision algorithm with a selection process based on the obtained Nash/Stackelberg equilibrium. The analyses of the optimal bandwidth/prices and the revenue at the equilibrium point show that these latter increase when user’s requirements increase in terms of QoS. We pointed out that networks having same capacities and different reputation values will charge users with different prices. Obviously, the one who has the best reputation is the most expensive. Nevertheless, users will still be attracted by good reputed networks as they provide them with better QoS which improve their utilities. It is important to mention that network reputation should be efficiently managed to avoid its falsification. We also show that our model implies that network having scarce resources should expand their capacities to stay competitive to other networks and to keep attracting users. Our ongoing work aims to propose a better scenario that considers interaction between different networks and gives higher priority to VHO connection to be fair to each area, to

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