where wpen is a penalty weight. Note that for the packet loss probability and average delay, the utility function can be defined similar to that in (2) and (3). III.
Coalition Formation Games for Bandwidth Sharing in Vehicle-to-Roadside Communications ∗
Dusit Niyato∗ , Ping Wang∗ , Walid Saad† , and Are Hjørungnes† School of Computer Engineering, Nanyang Technological University (NTU), Singapore † UNIK - University Graduate Center, University of Oslo, Kjeller, Norway
Abstract—In vehicular-to-roadside (V2R) communications the bandwidth from roadside units (RSUs) can be shared among the vehicular users in order to improve the resource utilization and reduce the costs of bandwidth reservation. We formulate a coalitional game model to analyze the situation in which multiple vehicular users can cooperate for sharing the bandwidth from serving RSUs. First, we consider a rational coalition formation approach in which each vehicular user is self-interested, and, hence, decides to join the coalition which maximizes its individual utility. For this approach, we propose a dynamic model based on Markov chain which allows to obtain a stable coalitional structure. Further, for implementation of rational coalition formation, we propose a distributed algorithm based on well-defined merge and split mechanisms. Then, we consider the optimal coalition formation process in which the coalitions are formed so that the social welfare of all vehicular users is maximized. The performance evaluation shows that optimal coalition formation yields a higher utility than rational coalition formation due to the group-interest of all vehicular users. Also, both optimal and rational coalition formation achieve a significantly higher utility than the case without bandwidth sharing (non-cooperative case).
Keywords-Intelligent Transportation Systems (ITS), vehicleto-roadside (V2R) communications, coalitional games. I. I NTRODUCTION Vehicle-to-roadside (V2R) communications based on wireless technologies will be a key technique for supporting various intelligent transportation system (ITS) applications. In V2R communications, vehicular users reserve the bandwidth of wireless RSUs such as base stations or access points from the network service provider. This bandwidth is used to transfer the data (e.g., road traffic information and infotainment data) intended for the vehicular users [2], [3]. To efficiently utilize the reserved bandwidth from RSUs, multiple vehicular users can share this bandwidth in order to reduce the cost paid to the network service provider. However, depending on the mobility pattern and the wireless channel conditions, the quality-of-service (QoS) of the vehicular users may be degraded if the bandwidth is poorly shared (e.g., many users accessing the bandwidth of the same RSU). Therefore, rational vehicular users will seek the best cooperation strategy among each other which can minimize the cost of bandwidth reservation while the QoS performance is maintained at a desired target level. In this context, the problem of cooperation for bandwidth sharing among vehicular users in V2R-based vehicular networks can be formulated as a coalitional game [1] whereby the vehicular users can negotiate the formation of cooperative groups, i.e., coalitions, for improving their performance. We first consider the case where each vehicular user is selfinterested, and, hence, seeks to maximize its own individual
payoff (i.e., utility). We refer to this scheme as the rational coalition formation scheme for which a dynamic model based on Markov chain is developed in order to analyze the vehicular users’ cooperative behavior. Specifically, a stable coalitional structure (i.e., the network partition in which none of the vehicular users is able to improve its payoff by changing its coalition formation decision) can be obtained analytically. Also, a distributed learning algorithm based on merge and split mechanisms is presented as an implementation for the rational coalition formation process. In addition, we consider the optimal coalition formation scheme in which all vehicular users take the coalition formation decisions which maximize the social welfare (i.e., total network utility). Given the proposed analytical model and distributed algorithm, the performance evaluation shows that, by performing coalition formation, the vehicular users can improve the utilization of the bandwidth by increasing the total utility by 17% while meeting the QoS requirements. In [4], different wireless technologies were adopted in V2R communications to support safety and non-safety ITS applications. In [5], a cooperative protocol among the vehicular user for improving the throughput of data transmission was proposed. In [6], a proxy-based vehicles to RSU access (PVR) was proposed to support collaborative and opportunistic packet forwarding in V2R communications. An efficient protocol was also introduced. However, to the best of our knowledge none of the work in vehicular networks’ literature considered the cooperation among the vehicular users for bandwidth sharing of V2R communications, notably using the framework of coalitional game theory. II. S YSTEM M ODEL AND A SSUMPTIONS We consider a vehicular network with V2R communications. We denote by R the set of roadside units (RSUs) in the service area. Further, there are N vehicular users communicating with the RSUs in this service area. An example of a service area with 4 users (i.e., 4 vehicles) and 9 RSUs is shown in Fig. 1. To download1 data from the infotainment server (e.g., video streaming), the bandwidth of RSU k ∈ R can be reserved with the cost of Ck per unit of bandwidth per unit of time. The set of RSUs whose bandwidth is reserved by vehicular user i is denoted by Ri ⊆ R. To reduce the cost and to improve the utilization of the reserved bandwidth, multiple vehicular users can cooperate and form a coalition to share the bandwidth. In the same coalition, the cost of bandwidth reservation will be equally divided among vehicular 1 The model presented in this paper is also applicable to uplink communications.
2
users. When the vehicular users are at the different RSUs, each of the users can fully access the reserved bandwidth. In this way, the utilization of the bandwidth increases, while the cost per user decreases. However, if multiple users are at the same RSU, a congestion will occur and we assume that all users will have an equal chance to access the reserved bandwidth (e.g., using a contention protocol such as CSMA/CA). With this bandwidth sharing scheme, if the users have a high chance to be at the different RSUs, the reserved bandwidth will be efficiently utilized. As a result, these users tend to form a coalition. However, if the users have a high chance to be at the same RSU, then, the performance will be degraded due to congestion. Consequently, these users will not form a coalition. Hence, coalitional games techniques are applied to investigate this coalition formation among the vehicular users for RSUs’ bandwidth sharing. Packet arrival
Data queue
Primary user’s transmitter Credit queue
ary user n by prim nsmissio Direct tra by ion iss er sm y us n r tra da lay on Re sec Primary user’s receiver Data queue
1
Direct
2
Direct
Relay
3
Direct
Secondary user
Direct
Ck |Sˆi,k |
is the cost of reserving ˆ bandwidth at RSU k per user, Si,k ⊆ S is the subcoalition whose members share the bandwidth with user i at RSU k, and Pi (Q(S), qthr ) is the penalty cost function due to performance degradation below threshold qthr . This penalty cost function can be defined as follows: ½ wpen , Q(S) < qthr , Pi (Q(S), qthr ) = (3) 0, otherwise, where wpen is a penalty weight. Note that for the packet loss probability and average delay, the utility function can be defined similar to that in (2) and (3). III. C OALITIONAL G AME F ORMULATION OF V EHICULAR U SERS Given the proposed model in the previous section, we have a coalition formation game among the vehicular users, where the objective of the users is to form coalitions among each other to share the bandwidth and improve their utility in (2).
Three types of transmission burst
S ec
onda trans ry user’ s miss ion
Fig. 1. System model for the vehicular network with coalition formation among vehicular users to share the bandwidth.
To facilitate the coalition formation among vehicular users, we assume that there is a coordinator at the application server. This coordinator provides information of active vehicular users. A. Utility of Vehicular User For downlink V2R communications, there is a buffer to store the data packets to be delivered to vehicular user i through the transmission of RSU k ∈ Ri . To obtain the performance measures (e.g., average number of packets in queue, packet loss probability, average packet delay, and throughput), a queueing model analogous to the one in [7] can be used given the mobility model of the users, the traffic arrival parameters, the channel quality, and the coalitions of the users. We consider a wireless transmission with adaptive modulation and coding (e.g., such as in DSRC). Let pr (k) denote the probability of using transmission rate r (r = 0, . . . , rmax with totally rmax modes) at RSU k. Given the channel quality (i.e., average SNR), this probability determines the departure process of the queueing model [8]. When there are n users from the same coalition at the same RSU k, this transmission probability becomes pr (k) , (1) pr (k, n) = n Prmax for r = 1, . . . , rmax and p0 (k, n) = 1 − r=1 pr (k, n). With bandwidth sharing in the coalition S, the utility of any vehicular user i, given the queueing performance (i.e., throughput) Qi (S), can be defined as follows: X φi (S) = Qi (S) − Ck (S) − Pi (Q(S), qthr ), (2) k∈Ri
for i ∈ S and k ∈ Ri . Ck (S) =
A. Rational Coalition Formulation The proposed model is formulated as a coalition formation game with the rational players being the vehicular users. The set of players is denoted by N = {1, . . . , N }. A coalition of users is denoted by S ⊆ N. The coalition of all players is denoted by N and referred to as the grand coalition. The strategy of each player is to decide on which coalition to join, and the payoff is the utility given in (2). The value of any coalition S ⊆ N is given by a characteristic function v(S) that determines the worth of the coalition and is defined as follows: X v(S) = φi (S), (4) i∈S
with v(∅) = 0. A partition or coalitional structure is a group of coalitions that spans all the users in N and is defined as ω =S{S1 , . . . , Sj , . . . , Ss } where Sj ∩ Sj 0 = ∅ for j 6= j 0 , s and j=1 Sj = N, and s is the total number of coalitions in partition ω. The total number of possible partitions for N users is given by the Bell number for DN where n−1 Xµ n−1 ¶ Dn = Dj , (5) j j=0
for n ≥ 1 and D0 = 1. The proposed model is formulated as a non-transferable utility coalitional game in which the value of a coalition of vehicular users, defined in (4) as a sum of payoffs with each being a function of the queueing performance of each user, cannot be arbitrarily apportioned among the users in a coalition. The solution of the proposed coalition formation game is the coalitional structure, i.e., partition ω, which can exhibit the following internal and external stability notions: • Internal stability: A coalition S ∈ ω is internally stable if no user can improve its payoff by leaving its coalition and acting alone, i.e., φi (S) ≥ v({i}) = φi ({i}) (this is related to the concept of individual rationality).
3
•
External stability: A coalition S ∈ ω is externally stable if no other coalition R ∈ ω can improve the payoff by joining coalition S, i.e., v(R) ≥ v(S ∪ R) − v(S) for R, S ∈ ω and R ∩ S = ∅.
Subsequently, a partition ω is stable if the conditions of internal and external stability are verified for all the coalitions in ω. The probabilities that a vehicular user joins or leaves any coalition are determined from the received utility given that decision. For instance, the coalition formation decision of any user i can be made using one of the following choices: Partitioning: Given the original coalition Sj , users in this coalition can collectively form (i.e., be partitioned into) mul(i)† tiple new coalitions Sj 0 whose set is denoted by M† , if the following condition is satisfied: [ (i)† (i)† (i) (i) φi (Sj 0 ) > φi (Sj ) ∀i ∈ Sj = Sj 0 . (6) j 0 ∈M†
Vehicular users agree to split into multiple coalitions if the payoffs of all users is higher than that in one coalition. Joining: Multiple coalitions Sj in a set M‡ can collectively (i)‡ agree to form a single new coalition Sj 0 , if the following condition is satisfied: (i)‡
(i)
(i)‡
(i)‡
φi (Sj 0 ) > φi (Sj ) and φi0 (Sj 0 ) > φi0 (Sj 0
(i)
− Sj ), (7)
S (i) (i) (i)‡ (i) (i)‡ for i ∈ Sj and i0 ∈ Sj 0 − Sj where Sj 0 = j∈M‡ Sj . In particular, multiple coalitions will join and act as a single coalition if all users in all candidate coalitions receive a higher payoff through this joining. Depending on the current partition at time t denoted by (i) ω(t) = {S1 (t), . . . , Sj (t), . . . , Ss (t)}, the new coalition of user i is formed in a manner that maximizes the user’s immediate payoff (i.e., myopic policy) as follows: (i)† (i)† (i)‡ Sj 0 , φi (Sj 0 ) > φi (Sj 0 ) and (i)† (i) φi (Sj 0 ) > φi (Sj (t)), (i) (i)‡ (i)‡ (i)† Sj (t + 1) = Sj 0 , φi (Sj 0 ) > φi (Sj 0 ) and (8) (i)‡ (i) φi (Sj 0 ) > φi (Sj (t)), (i) Sj (t), otherwise. To obtain the stable coalitional structure, a dynamic coalition formation model based on discrete-time Markov chain can be formulated [9]. The state space of this Markov chain is defined as follows: ∆ = {(ωx ) such that x = {1, . . . , DN }},
(9)
where ωx represents a coalitional structure (spanning all vehicular users) and DN can be obtained from (5). The probability transition matrix of this Markov chain is denoted by P whose elements are Pω,ω0 . Each element Pω,ω0 represents the probability that the partition (i.e., state) of all users changes from ω to ω 0 during a certain time interval. This time interval can correspond to the transmission slot of the RSUs. Let Cω,ω0 denote the set of candidate users who are bound to make a coalition formation decision which will result in the change of
the network partition from ω to ω 0 . This transition probability can be obtained as follows: ( Q N −|Cω,ω0 | 0 , ω . ω0 , i∈Cω,ω0 γβi (ω |ω)(1 − γ) Pω,ω0 = 0, otherwise, (10) where |Cω,ω0 | is the cardinality of the set Cω,ω0 , and ω . ω 0 is a feasibility condition. In particular, if partition ω 0 is reachable from ω given the decision of all users defined in (8), then the condition ω .ω 0 is true and false otherwise. γ is the probability that a user makes a decision in a time interval. βi (ω 0 |ω) is the best-reply rule of user i. That is, βi (ω 0 |ω) is the probability that the decision of user i changes the current coalition from ω to ω 0 . This best-reply rule can be defined as follows: ½ ˆ φi (S (i) ⊂ ω 0 ) > φi (S (i) ⊂ ω), β, j j βi (ω 0 |ω) = (11) ², otherwise, where 0 < βˆ ≤ 1 is a constant, ² is a small number that corresponds to the probability that a user makes an irrational decision, e.g., ² = 0.01, and βˆ À ². Further, we consider that a user can make an irrational coalition formation decision due to either: (i)- a lack of information or, (ii)- a need for “exploration” in the learning process. In this case, the state transition probability Pω,ω0 is determined from the product of the transition probabilities of users who do and do not make decisions in a time interval. With a non-zero probability for irrational decisions (i.e., ² > 0), the stationary probability of the Markov chain can be obtained by ~TP = π ~ T and π ~ T ~1 = 1 where ~1 is a vector of solving π £ ¤T ~ = πω1 · · · πωx · · · πωDN ones. π is a vector of stationary probabilities where πω is the probability that the partition ω will be formed. The average individual utility of user i can be obtained from Ui =
DN X
(i)
(i)
πωx φi (Sj ) for Sj ∈ ωx .
(12)
x=1
Then, we let the probability of irrational decisions approach zero (i.e., ² → 0). In the Markov chain defined by the state space in (9) and the transition probability in (10), there could exist an ergodic set E ∈ ∆ of states if Pω,ω0 = 0 for ω ∈ E and ω 0 ∈ / E, and no nonempty proper subset of E has this property. In this regard, singleton ergodic sets are referred to as the absorbing states. Specifically, ω ∈ ∆ is absorbing if Pω,ω = 1. The concept of an ergodic set is important for the coalition formation process since once all vehicular users reach the state (i.e., coalitional structure) in an ergodic set, they will remain in this ergodic set forever. In addition, users will stop evolving once the absorbing state is reached. The absorbing state is referred to as the stable partition ω ? . In particular, no user has an incentive to change its decision given the prevailing partition. P ROPERTY 1: The proposed coalition formation game (N, v) for bandwidth sharing among vehicular users has at least one absorbing state. Proof: We use Theorem 1 in [9] to prove this property. Specifically, the Markov chain of a coalition formation process
4
with best-reply rule possesses at least P on absorbing state if for all coalitions S ⊆ N the condition i∈S v({i}) ≤ v(S) holds. Based on the value of the coalition P for bandwidth sharing defined in (4), it can be shown that i∈S v({i}) = P i∈S φi ({i}) = v(S). Therefore, there exists at least one absorbing state or equivalently the coalitional game of bandwidth sharing has at least one stable partition ω ? . B. Distributed Algorithm To obtain the stable partition for vehicular users sharing the bandwidth from RSUs, a distributed algorithm based on the mechanisms of merge and split [?, Section IV-C] is presented in Algorithm 1. This algorithm is based on reinforcement learning in which the users gradually learn the benefit of different coalitions. The decision is made by each vehicular user based on the knowledge κi (·) in order to maximize its payoff. Note that 0 ≤ rand ≤ 1 is a random number generator.
Algorithm 1 Distributed coalition formation algorithm for bandwidth sharing in vehicular networks. 1: t = 0 and the vehicular users are partitioned into Ss {S1 (t), . . . , Ss (t)} for j=1 Sj = N 2: loop 3: Vehicular user i observes the queueing performance P (i) (i) Qi (Sj (t)) and the cost k∈Ri Ck (Sj (t)) given its (i) current coalition Sj (t) (i) 4: Compute utility φi (Sj (t)) (i) 5: Update the knowledge of the utility, i.e., κi (Sj (t)) = (i) (i) αi φi (Sj (t)) + (1 − αi )κi (Sj (t)) where 0 < αi < 1 is the learning rate Merge mechanism S S (i) 6: if κi ( j∈M‡ Sj (t)) > κi (Sj (t)) for i ∈ j∈M‡ Sj (t), t
7: 8: 9: 10: 11: 12: 13:
14: 15: 16: 17: 18: 19: 20: 21:
t
where M‡t is a set of coalitions to be merged at time t then Merge coalitions Sj (t) for j ∈ M‡t else if rand ≤ ² then Merge coalitions Sj (t) for j ∈ M‡t end if end if Split mechanism (i) (i) if κi (Sj 0 ) > κi (Sj (t)) for i ∈ Sj (t), j 0 ∈ Mt† (i) where M†t is a set of coalitions split from Sj (t) at time t then (i) Split coalition Sj (t) into Sj 0 for j 0 ∈ M† else if rand ≤ ² then (i) Split coalition Sj (t) into Sj 0 for j 0 ∈ M† end if end if t=t+1 end loop
C. Optimal Coalition Formation In contrast to the previously devised rational coalition formation where the users are self-interested, the vehicular users can cooperate to form an optimal coalitional structure (partition) which maximizes the social welfare (i.e., total utility) instead of their individual utilities. For example, when all users are belong to the same public transportation service provider it is of their joint benefit to maximize the overall utility, i.e., the social welfare. The optimal partition is defined as follows: X (i) ω ∗ = {S1∗ , . . . , Sj∗ , . . . , Ss∗ } = arg max φi (Sj ). (13) (i)
Sj ∈ω i∈N
Again, in order to obtain this optimal partition, a Markov chain model can be formulated. The state space is the same as that defined in (9). The state transition Pω,ω0 can be defined similarly to that in (10). However, the best-reply rule βi (·|·) becomes ½ ˆ V (ω 0 ) > V (ω), β, (14) βi (ω 0 |ω) = ², otherwise, where V (ω) =
X (i) Sj ∈ω
(
X
(i)
φi (Sj ),
(15)
(i) i∈Sj
is the total utility of all vehicular users. Again, the stationary probability πωx can be obtained. The total utility of all users can be obtained from Utotal =
DN X x=1
πωx
X
v(Sj ).
(16)
Sj ∈ωx
IV. P ERFORMANCE E VALUATION We consider the V2R-based vehicular network shown in Fig. 1. There are 9 RSUs, and 4 vehicular users. Vehicular users visit the RSUs randomly as follows R1 = {RSU 1, . . . , RSU 6}, R2 = {RSU 4, . . . , RSU 9}, R3 = R4 = {RSU 1, . . . , RSU 4, RSU 6, . . . , RSU 9}. The channel quality (i.e., average SNR) of the connection between any vehicular user and any RSU is 14 dB. The utility function of the user is defined based on the queue throughput Qi (S). The threshold of minimum throughput is qthr = 1.2 packets/time slot, and wpen = 5 for the penalty cost function. The probability of an irrational decision is ² = 0.01. The parameters of the best-reply rule are βˆ = γ = 0.9. The distance between RSUs is 500 meters, and the average speed of vehicular user is 50km/h. We assume that¯ at ¯most two coalitions can join ¯ ¯ each other at one time, i.e., ¯M‡t ¯ = 2. Similarly, one coalition ¯can be ¯ partitioned into at most two coalitions at one time, i.e., ¯ †¯ ¯Mt ¯ = 2. A. Numerical Results Fig. 2 shows the state transition diagram of the coalition formation process among 4 vehicular users. There exists a total of 15 states with different coalitional structures. These states can be divided into four groups with the total number of coalitions (in a partition) ranging from one to four.
5
{1,2},{3},{4}
ω12
{1,3},{2},{4}
ω14
{1,2,3},{4} {1,2},{3,4} {1,2,4},{3}
{1,4},{2},{3}
{1},{2},{3},{4}
{1,3},{2,4}
ω13
{2,3},{1},{4}
ω11
{2,4},{1},{3}
ω9
{1,3,4},{2} {1,4},{2,3}
ω8 ω2 ω7 ω3
9.6
ω15
9.4 9.2
{1,2,3,4} Total utility
ω1
ω10
ω6 ω4
9
Optimal formation Rational formation Distributed algorithm Without coalition
8.8 8.6 8.4
{3,4},{1},{2}
{1},{2,3,4}
ω5
8.2 8
Fig. 2. State transition of the coalition formation process among 4 vehicular users.
7.8
Fig. 4.
8
9
10 11 12 13 Channel quality at RSU5 (dB)
14
15
Total utility under different channel quality at RSU 5.
9.8 9.6
maximize their individual utility. A dynamic model based on Markov chain for rational coalition formation is formulated. The stable coalitional structure is obtained based on the concept of absorbing states. For implementation, a distributed algorithm based on the merge and split mechanisms has been presented. In addition, we have considered the optimal coalition formation approach in which the vehicular users seek to maximize social welfare (i.e., total network utility).
Total utility
9.4 9.2
Optimal formation Rational formation Distributed algorithm Without coalition
9 8.8 8.6 8.4 8.2 20
Fig. 3.
25
30 35 40 Speed of vehicular users at RSU5 (km/h)
45
50
Total utility under different speeds of the vehicular users at RSU 5.
Fig. 3 shows the total utility as the speed of the vehicular users at RSU 5 varies. As expected, the optimal coalition formation achieves the highest total utility compared to the rational coalition formation and to the non-cooperative case. Due to the routes of vehicular users 1-4, the total utility increases, if the speed at RSU 5 decreases. Since only users 1 and 2 will be at RSU 5, when the speed is low users 3 and 4 will have higher chance to receive the full bandwidth from RSU 1-RSU 4 and RSU 6-RSU 9. Therefore, the total utility increases. Fig. 4 shows the total utility under different channel quality at RSU 5. Again, optimal coalition formation achieves the highest total utility. For both cases (Fig. 3 and Fig. 4 ), the proposed distributed algorithm based on the merge and split mechanisms achieves a total utility close to that of the Markov chain based rational coalition formation. V. S UMMARY In vehicle-to-roadside communications, the vehicular user can share the bandwidth for the connection to different roadside units. In this paper, a coalitional game model has been formulated to analyze the behavior of the vehicular users that seek to share the network’s bandwidth. The vehicular users can form different coalitions according to their mobility patterns. Coalition formation allows the users to reduce the cost of bandwidth reservation while meeting their QoS requirements. First, we have considered a rational coalition formation approach in which the vehicular users are selfinterested and take the coalition formation decisions that
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ACKNOWLEDGMENT This work was done in the Centre for Multimedia and Network Technology (CeMNet) of the School of Computer Engineering, Nanyang Technological University, Singapore. This work was supported by the Research Council of Norway through projects 183311/S10 and 176773/S10.
