culate the probability distribution of the shared cost. 1 Introduction. Multicast ... cost per node shown in the following is used in the sim- ulation. Let Cn denote the ...
Cost Sharing Methods for Multicast Services and Probability Distribution of Shared Cost Eiji TAKAHASHI††,††† , Takaaki OHARA †,††† , Takumi MIYOSHI††††,††,††† and Yoshiaki TANAKA††,††† †
Graduate School of Science and Engineering, Waseda University Global Information and Telecommunication Institute, Waseda University ††† Okinawa Research Center, Telecommunications Advancement Organization of Japan 5th Floor, Nishi–Waseda Bldg., 1–21–1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169–0051, JAPAN Tel : +81 3 3207 6373 †††† Faculty of Systems Engineering, Shibaura Institute of Technology 307 Fukasaku, Saitama–shi, Saitama, 330–8570, JAPAN Tel : +81 48 683 2020 ††
Abstract In multicast-type services, many users share a single data flow. The question is how to split the cost of such a flow among the users. There is a need to provide users with reasonable feedback on the cost that their network usage incurs. Therefore, cost sharing methods will play an important role in multicast-type services in the future. This paper discusses cost sharing methods where many users share multicast flows and describes how these costs are shared among the members of multicast groups. We consider the case where an Information Provider provides users with multicast-type video streaming services by using the network resources owned by an Internet Service Provider. First, we consider some cost sharing methods to determine the charge in a multicast-type services and then calculate the probability distribution of the shared cost.
1
Introduction
Multicast technology is expected to become widely used, since of its link sharing capability makes it well suited to broadband multimedia services. In multicast technology, a point-to-multipoint (multicast) connection is used to copy packets only at branching nodes, which ensures network efficiency [1] - [3]. In multicast-type services, many users share a single data flow. The question is how to split the cost of this flow among the users. There is a need to provide users with reasonable feedback on the cost that their network usage incurs. Therefore, cost sharing methods will play an important role in multicast-type services in the future. The case of multicast-type video streaming services is considered in this paper. A flat-rate pricing is unrealistic for pricing video streaming services in telecommunication networks as long as network resources aren’t plentiful, since there would be no incentive to limit wasteful traffic. Therefore, cost-based measured rate pricing is considered in this paper. To simplify the problem, information costs and fixed costs are not considered here; we only consider costs for network usage.
If the price is decided dynamically according to the shared cost, the final price cannot be informed to any user at the start of the service, as the price changes according to the change in the number of joining users. In this respect, dynamic pricing is not user-friendly. Thus, a time-based pricing where the unit price is fixed is considered in this paper. First we examine some cost sharing methods that determine charges in multicast-type services. Then, we calculate the probability distribution of the shared cost to analyze how the costs are shared among users in the proposed methods.
2 2.1
Cost in Multicast–type Services Definition of Cost
In this paper, we consider the case where the information provider provides users multicast–type video streaming services by using the network resources that an ISP (Internet Service Provider) owns. The ISP charges the information provider for its network usage, and this charge is the cost for the information provider to provide a multicast-type video streaming service. Here, information costs and fixed costs are not considered to make the problem simpler.
2.2
Cost–based Multicast Pricing
The accounting architecture among ISP, information provider, and end users is shown in Figure 1. The ISP charges the information provider for its network usage. The information provider charges the users for contents and network usage for delivering the contents. Hereafter, the charge for contents is not considered because it doesn’t depend on the state of network usage. charge for usage requests Information Provider
requests ISP
contents
Users contents
charge for information and network usage
Figure 1: Costs and profits for information provider.
Users Source Node Users
Users
Users
Users Users
Figure 2: Many users share a single data flow.
3
Cost Sharing Methods
In multicast-type services, many users share a single data flow as shown in Figure 2. This section defines three cost sharing methods among receivers for multicast-type service [4].
3.1
Equal Total Cost Split
The equal total cost split method is the simplest method for equally sharing the total cost of all links among all receivers joining the service. Let C denote the cost per second per user. Let T (i) denote the continuation time of the ith state of the network. Here, the ith state means that the composition of the multicast receivers has changed i − 1 times since the beginning of the service. Let CT (i) denote the total cost per second in the ith state of the network. Let N (i) denote the number of receivers receiving the stream. Then, CT (i)T (i) C = i=1 . (1) i=1 N (i)T (i) The state of each node is on-state if at least one downstream receiver exists, and is off-state if no downstream receiver exists. In Chapters 4 and 5, not the behaviour of each user but the on-off pattern of each node is required in the approximative calculation. The cost per node shown in the following is used in the simulation. Let Cn denote the cost per second per node. Let t(j) denote the continuation time of the jth state of the network. Here, the jth state means that the multicast tree has changed j − 1 times since the beginning of the service. Let cT (j) denote the total cost per second in the jth state of the network. Let n(j) denote the number of nodes that contain at least one receiver. Then, j=1 cT (j)t(j) Cn = . (2) j=1 n(j)t(j) The equal total cost split method is a desirable method for its ability to realize simple accounting and billing. However, the method does not discriminate between those receivers far from the source and receivers close to the source. Therefore, the method does not attempt to hold receivers accountable for the costs their individual membership incurs. Moreover, sometimes one’s share of the cost in the multicast-type service becomes higher than that of a unicast cost, for
instance in the case of some receivers connecting directly to the source node when there exists at least one receiver in another node. Therefore, this method does not satisfy individual rationality.
3.2
Equal Link Cost Split
The cost of a particular link is incurred because there is at least one downstream receiver. Consequently, while all downstream receivers can be considered equally responsible for the cost, all of the other receivers are not responsible at all. Therefore, the approach where the cost of each link is split equally among only the downstream receivers is considered here. Let Cq denote the cost per second shared by each receiver connected to node q. Let T (i) denote the continuation time of the ith state of the network. Here, the ith state means that the composition of the multicast receivers has changed i − 1 times since the beginning of the service. Let cost(u, v) denote the cost per second of the link(u, v). Let N (i, u, v) denote the number of receivers sharing the link(u, v) during the ith state. Let Nq (i) and Nq (i, u, v) denote the number of receivers and the number of receivers sharing the link(u, v) connected to node q during the ith state. Then,
i=1
Cq =
cost(u,v) u,v N (i,u,v) Nq (i, u, v)T (i)
i=1
Nq (i)T (i)
.
(3)
In Chapters 4 and 5, not the behaviour of each user but the on-off pattern of each node is required in the approximative calculation model. The cost per node shown in the following is used in the simulation. Let Cn,q denote node q’s share of cost per second. Let Tq denote the total time that node q is on-node; if there is at least one user receiving the stream in a node, we call this node on-node here. Let t(j) denote the time between the jth state of the network. Here, the jth state means that the multicast tree has changed j − 1 times since the beginning of the service. Let link(u, v) denote the link between adjoining nodes u and v. Then, Cn,q =
1 cost(u, v) sq (j, u, v)t(j), Tq n(j, u, v) u,v
(4)
j=1
where sq (j, u, v) =
0 : if node q is not sharing link(u, v), 1 : if node q is sharing link(u, v),
during the jth state. In this method, multicast costs are always lower than unicast costs. Therefore, this method satisfies individual rationality. However, the cost shared by one user can be different from that of other users, and as a result accounting and billing will be complex.
In the equal group cost split method, all nodes are divided into m groups, and the total cost the group incurs is equally split among users in the group. For example, in the case where m = 2, a group close to the source node and a group far from the source node would be formed. The former group is in the area where hop-count from the source is less than the dividing hop-count, and the latter group is in the area where hop-count from the source is larger than the dividing hop-count. For instance, the dividing hop-count is [ Hmax 2 ], where m = 2 and Hmax is the maximum hopcount in the multicast tree.
4 4.1
Arrival rate
Equal Group Cost Split
(b) Regular Broadcast Model The regular broadcast model is a representation of traffic in multicast-type video stream services [5],[6]. In this model, the program is broadcasted from the beginning to end, and users are classified into four types as shown below. Type 1 users: Users who watch the program from beginning to end. Type 2 users: Users who arrive randomly and leave the service randomly. Type 3 users: Users who arrive randomly and leave the service at the end of the program. Type 4 users: Users who watch the program from its beginning and leave the service randomly.
Table 1: Parameters of traffic model. Parameter λ µ λ1 λ2 µ1 τ1 d N
Value 3.01 ×10−6 (/sec) 3.70 ×10 −2 (/sec) 8.00 ×10 −6 (/sec) 9.00 ×10 −6 (/sec) 1.00 ×10 −2 (/sec) 500 (sec) 3600 (sec) 10,000
t
d End
2
Traffic Model
(a) Continuous Broadcast Model In this model, the program is broadcasted continuously, and their is neither beginning nor end of the program. To make the calculation simple, it is assumed that an arriving call follows a Poisson distribution and that the holding time follows an exponential distribution. Values of arrival rate λ and service rate µ of each user are shown in Table 1.
1
Figure 3: Arrival rate of calls in regular broadcast model.
Simulation Model
In this paper, we use two traffic models, a continuous broadcast model and a regular broadcast model, in order to determine whether the simulation results depend on traffic models or not.
1 2
0 Start
Service rate
3.3
2
1
0 Start
t
d End
Figure 4: Service rate of calls in regular broadcast model. Poisson arrival (arrival rate: 1 or Type 1 calls Holding time following a unit distribution of d - T
2 Type 2 calls
Holding time following an exponential distribution (service rate: 1)
d - T: remaining time of the program
Figure 5: Types of holding time in the regular broadcast model. The change in arrival rate and service rate of calls are shown in Figures 3 and 4. However, to make the problem simple, we set τ2 to zero and use a simple model where holding time is classified randomly into two types; the holding time of Type 1 calls follows a unit distribution of d−T , and the holding time of Type 2 calls follow an exponential distribution as shown in Figure 5. The ratio of Type 1 calls to Type 2 calls is 1 to 9. Values of arrival rate λ 1 , λ2 per user and service rate µ1 per user are shown in Table 1. Values of τ1 and d are also shown in Table 1.
4.2
Approximative Traffic Model
In multicast technology, a point-to-multipoint (multicast) connection is used to copy packets only at branching nodes, which ensures network efficiency. Each node has the ability to copy packets and to forward the copies to users or other nodes. The state of each node depends on whether at least one downstream receiver exists. The state of each node is on-state if at least one downstream receiver exists. The state of each node is off-state if one downstream receiver does not exist. Therefore, not the behaviour of each user but the on-off pattern of each node is used in the simulation to reduce the calculation time.
Let N and λ0 denote the the number of users connected to each node and the arrival rate of each user, respectively. Then, the holding time of the off-state follows the exponential distribution of mean value 1 N λ0 . The approximative model is used for calculating the holding time of the on-state [5], [6]. The average holding time of on-state E is given as N
E=
(a + 1) − 1 , λ0 N
(5)
where a = λµ00 and µ0 denotes the service rate of each user. Accordingly, the service rate of each node µ is given as µ = E1 . In each calculation, λ0 is equivalent to λ, λ1 or λ2 , and µ0 is equivalent to µ or µ1 . The network has many nodes and these nodes contain many users. The simulation time would be astronomical if the behaviours of all users were individually monitored. Therefore, we use an approximative model to calculate the cost (per second per user) as shown below. Cn , (6) RN Cn,q Cq = , (7) RN where R denotes the average audience rating, and C, Cn , Cq and Cn,q have values as mentioned in Chapter 3. Each node accommodates 10,000 users. C=
4.3
Network Model
A network was modeled as a random graph possessing some of the characteristics of an actual network [7]. The vertices representing nodes were randomly distributed on a rectangular coordinate grid, and each vertex had integer coordinates. For a pair of vertices, say u and v (0 ≤ u, v < 1), an edge was added according to the following probability: k¯ e d(u, v) β exp{− }, (8) n Lα where n is the number of vertices in the graph, e¯ is the mean number of degrees of a vertex, k is a scale factor related to the mean distance between two vertices, d(u, v) is Euclidean distance between vertices u and v, L is the maximum distance between any two vertices in the graph, and α and β are the parameters Pe (u, v) =
Table 2: Parameters used for generating random networks. Parameter n α β e¯ k
Network Model 1 50 0.25 0.20 3.00 25
Network Model 2 50 0.50 0.40 2.50 15
(real numbers between 0 and 1). In this paper, we use two types of random networks to confirm whether the simulation results depend on the network model. These parameters are shown in Table 2. A source node is selected randomly.
5
Simulation Results
The relationships between the cumulative probability and the cost (per second per user) are shown in Figures 6, 7, 8, 9 and 10. Figure 6 shows the results in the case of the Continuous Broadcast Model, Network Model 1 and Equal Link Cost Split Method. Figure 7 shows the results in the case of the Regular Broadcast Model, Network Model 1 and Equal Link Cost Split Method. Figure 8 shows the results in the case of the Continuous Broadcast Model, Network Model 1 and Equal Link Cost Split Method. Comparing Figures 6 and 7, there is little difference between then in both form and value. This result implies that the simulation results do not depend on the traffic model. Therefore, it is efficient to use the continuous broadcast model, as it takes less time in the simulation because of its simplicity. The comparison of Figures 6 and 8 shows that there is little difference in form between them. However, there is a difference between the values of the two figures. This result is caused by the differences in average dimensions and the average distances between nodes of the two network models. Therefore, the simulation results depended on the network model. Figure 9 shows the result in the case of the Regular Broadcast Model, Network Model 1 and Equal Link Cost Split Method. Figure 10 shows the result in the case of the Continuous Broadcast Model, Network Model 1 and Equal Link Cost Split Method, where all nodes are divided into two groups. Group 1 is the group of nodes in the area where hop-count from the source node is less than [ Hmax ]. Here, Hmax is the 2 maximum hop-count from the source node in the multicast tree. Group 2 is the group of nodes in the area where hop-count from the source node is above [ Hmax 2 ]. For example in Figure 9, when the information provider set the probability of having a deficit at 10%, the cost shared by each user should be determined as 0.035.
6
Conclusions
In this paper, three cost sharing methods in multicast-type services were defined. By using these methods, the charge for multicast-type video streaming service can be determined in such a way to reflect the cost that each user’s network usage incurs. Cumulative probability distributions of the shared costs in the above three methods were shown. We found that the result of our simulation does not depend on the traffic model or network model. However, values of
Acknowledgements This study was partly supported by a Waseda University Grant for Special Research Projects (Individual Research: 2000A-926). Part of this study is from the results of the project “Research on Pricing Controlled Highly Efficient Network” at Advanced Research Institute for Science and Engineering, Waseda University.
1.2 Probability that cost will be above x
costs depend on the network model. Though we mainly focused on algorithmic aspects, we are aware that a number of issues still remain unresolved, including users’ behaviour in responding to tariffs and a traffic forecasting method.
1.0
hop: 1 hop: 2
hop: 3 hop: 4
hop: 5 hop: 6
0.8 0.6 0.4 0.2 0.1 0.0
0.01 0.02 0.03 0.04 0.05 0.06 Cost x (per second per user) Figure 7: Distribution of cumulative probability (Regular broadcast model, Network model 1, Equal link cost split method).
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1.0
hop: 3 hop: 4
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0.8 0.6 0.4 0.2 0.1 0.0
1.2 1.0 0.8 0.6 0.4 0.2 0.1 0.0
1.2 hop: 1 hop: 2
hop: 1 hop: 2
0.01 0.02 0.03 0.04 0.05 0.06 Cost x (per second per user) Figure 9: Distribution of cumulative probability (Continuous broadcast model, Network model 1, Equal total cost split method). hop: 5 hop: 6
0.8 0.6 0.4 0.2 0.1 0.0
1.0
0.01 0.02 0.03 0.04 0.05 0.06 Cost x (per second per user) Figure 8: Distribution of cumulative probability (Continuous broadcast model, Network model 2, Equal link cost split method).
0.01 0.02 0.03 0.04 0.05 0.06 Cost x (per second per user) Figure 6: Distribution of cumulative probability (Continuous broadcast model, Network model 1, Equal link cost split method).
Probability that cost will be above x
Probability that cost will be above x
1.2
Probability that cost will be above x
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1.2
Probability that cost will be above x
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1.0
Group 1 (Close to source node) Group 2 (Far from source node)
0.8 0.6 0.4 0.2 0.1 0.0
0.01 0.02 0.03 0.04 0.05 0.06 Cost x (per second per user) Figure 10: Distribution of cumulative probability (Continuous broadcast model, Network model 1, Equal group cost split method m = 2).
