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International Journal of Distributed Sensor Networks
Volume 2012 (2012), Article ID 840391, 9 pages
http://dx.doi.org/10.1155/2012/840391
Research Article

A Dynamic Pricing Scheme for Congestion Game in Wireless Machine-to-Machine Networks

1Institute of System Engineering, Tianjin University, Tianjin 300072, China
2Department of Information Management and Management Science, Tianjin University, Tianjin 300072, China

Received 8 April 2012; Revised 9 July 2012; Accepted 9 July 2012

Academic Editor: Jianhua He

Copyright © 2012 Zhifei Mao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The problem of assigning a set of source nodes to a set of routes in wireless machine-to-machine (M2M) networks is addressed using a game theoretic approach. The objective is to minimize the maximum latency over all source nodes as far as possible while the game achieves a pure Nash Equilibrium (NE). To compute such an NE efficiently, we present a distributed dynamic pricing (DP) scheme, where each source node is assumed to pay for using any route so that the route has incentive to relay data for the source node. A loose upper bound is given for the convergence time of DP, and simulation results show that it performs much faster in practice. The price of anarchy in this game is also investigated by comparing DP with a cost-reducing path method; the results show that DP produces optimum assignment in more than 90% of the simulation runs.

1. Introduction

Features such as self-organization, ease of deployment, low cost, and infrastructurelessness bring to wireless M2M networks the advantages of robustness, easy maintenance, economy, and flexibility. Therefore, M2M networks are envisioned to be the key technology for next generation wireless networks [1]. An M2M network consists of a large number of self-organized and self-configured nodes, the behavior of which cannot be controlled easily without any available centralized scheduling command. Thus, the network can easily be congested when there are a great many nodes that want to transmit data via a common set of routes.

Extensive work has addressed the congestion problem by adjusting the source rate based on a feedback method [24], but prevention strategies for the congestion have been ignored. Nevertheless, by joining the congestion control in the routing layer, this problem can be solved via the coordination among nodes [5, 6]. However, in most wireless M2M network applications, each user is its own authority, and unconditional forwarding of packets to other users cannot be assumed directly. Therefore, a scheme that stimulates routes to relay the packets of other nodes is required. Motivated by the general network congestion game [79] and pricing scheme [10], our study jointly considers the problems of routing congestion and incentive provision.

In this paper, a distributed dynamic pricing (DP) scheme is developed for wireless M2M networks to minimize the maximum latency over all source nodes. First, routing congestion is formulated as a dynamic game in which pricing is employed in the definition of players’ utility function. Each route is encouraged to charge source nodes for delivering their information, and the amount is decided by its latency. Second, dynamics is introduced in the game. The price of each route is changeable rather than fixed so that its load can be adjusted dynamically. We assume that the number of source nodes a route can take has an upper limit, and while the number of source nodes assigned to this route equals its upper limit, we say that this route is supply-demand balanced. Once all routes achieve supply-demand balance, the game terminates and converges to a pure Nash Equilibrium (NE). Finally, simulations are conducted to evaluate the performance of the proposed scheme.

The rest of this paper is organized as follows. Section 2 provides related work in recent years. Section 3 illustrates the system model and Section 4 formulates the network congestion game. A DP scheme is proposed in Section 5. In Section 6, the simulation results are presented. And we conclude in Section 7.

2. Related Work

In [2], an algorithm jointly addressing congestion control and scheduling in wireless M2M networks is developed, the basis of which is rate allocation among flows in the network. By adjusting backoff min-slot and window size of each flow, the network can achieve high overall throughput and low per-flow delay. However, this algorithm cannot be applied for scenarios with dynamic routing, since route of each flow is assumed to be fixed. EWCCP, a protocol that can be added as a thin layer between IP and TCP, is proposed in [3]. It coordinates flows that compete for common channel according to explicit multibit congestion feedback from routers. Authors in [4] found that the main reason of congestion collapse in wireless M2M networks for streaming services is the severe contention among individual nodes at MAC layer, and thus presented a TCP-friendly congestion control scheme where a contention state estimator is defined as the difference between the number of arriving packets and that of leaving packets during each control interval. Similar to the aforementioned literature, congestion control schemes proposed in [1113] are all feedback-based. Although these existing feedback-based approaches have achieved some success in congestion control for M2M networks, most of them ignored prevention/avoidance strategies for congestion control.

Congestion avoidance can be achieved by jointly considering congestion and routing. The work in [11] provides a multipath routing algorithm, I2MR, which discovers zone disjoint paths using the concept of path correlation so as to avoid interference in multiple nodes and increase system throughput. Cross-layer design of joint congestion control, routing, and scheduling was presented in [12], which enable each source node to adjust its routes for data transmission in each period according to a local congestion price from its neighbors. Other works such as [1417] have also jointly considered the problems of congestion control and routing in multihop wireless networks or wireless sensor networks. The major distinction between our work and aforementioned work lies in the formulation of the congestion routing. Specifically, we propose a congestion game to model the route-selection behavior of each node who wants to transmit its data via a route with the lowest price. Moreover, we design a distributed algorithm with low complexity by making use of pricing to implement the congestion game framework.

3. Network Model

Consider a wireless M2M network with a set 𝑁 of 𝑛 parallel routes from a set 𝑀 of 𝑚 source nodes to a destination node, 𝑀={1,2,,𝑚} and 𝑁={1,2,,𝑛}. Figure 1 shows an illustration of the network. For each source node 𝑖, let the strategy set 𝑆𝑖𝑁 denote the set of routes to which source node 𝑖 can possibly be assigned. For each route 𝑗, let 𝐵𝑗𝑀 denote the set of source nodes that can possibly be served by this route. Each source node 𝑖 intends to send a particular amount of traffic along a fixed route to the destination; thus, their choices form an assignment 𝑠=(𝑠1,𝑠2,,𝑠𝑚), where 𝑠𝑖𝑆𝑖 is the chosen route of source 𝑖. Given an assignment, let 𝑞𝑗 be the number of the source nodes choosing route 𝑗. Let 𝑗 be the length of route 𝑗, which equals the total of nodes contained in this route.

840391.fig.001
Figure 1: An example of wireless M2M network.

In the network, each route 𝑗 has a latency 𝑑𝑗. Since the packet of each source node has been generated and processed before its transmission toward one route, the process time will not be included in routes’ latency. Furthermore, comparing to queuing time and transmission time, the propagation time can be neglected in most M2M networks except some like satellite networks. Thus, the latency 𝑑𝑗 of route 𝑗 is dependent on the number of its length and current load, which can be expressed as 𝑓(𝑞𝑗)𝑔(𝑗). We assume that all source nodes transmit their packets via a common wireless channel, which means that routes will receive the packets one by one when multiple transmissions arrive simultaneously. Thus, both 𝑓(𝑞𝑗) and 𝑔(𝑗) can be set as a linear function. Based on the above assumptions, we compute the latency 𝑑𝑗 as 𝑑𝑗=𝑞𝑗×𝐿𝑅𝑠+𝑗×𝐿𝑅𝑟,(1) where 𝐿 is the size of packet generated by source nodes, and 𝑅𝑠 is the rate of transmitting one packet from a source node to a route while 𝑅𝑟 is the rate of transmitting the same packet between two nodes within the same route. Moreover, we assume that 𝑅𝑠 equals 𝑅𝑟. Then, 𝑑𝑗 can be rewritten as 𝑑𝑗=𝑞𝑗+𝑗×𝐷,(2) where 𝐷=𝐿/𝑅𝑠=𝐿/𝑅𝑟.

Our study aims to minimize the maximum latency over all source nodes. Hence, the congestion control problem can be modeled as minmax𝑗𝑀𝑞𝑗+𝑗𝐷.(3)

4. Network Congestion Game

The network congestion game deals with the problem of congestion which is fundamental in networks and distributed systems [7]. Whenever a set of users intend to utilize a much smaller set of resources in a common period, one needs to schedule these users orderly so as to reduce the overall cost and exploit available resources efficiently.

In wireless M2M networks, users and resources are referred to as source nodes and routes, respectively. The network congestion game can be formally defined as a tuple Γ=𝑁,(𝑆𝑖)𝑖𝑀,(𝑢𝑖)𝑖𝑀, where 𝑀 is the set of players, that is, the source nodes. For the remainder of this paper, the terms “source node” and “player” are used interchangeably. 𝑆𝑖 is the strategy set of player 𝑖. Let 𝑠=(𝑠1,𝑠2,,𝑠𝑚) be the strategy profile when each player 𝑖 chooses 𝑠𝑖, which is equivalent to the assignment in Section 2. 𝑢𝑖Π=𝑆1×𝑆2××𝑆𝑚𝑅 is the utility function of player 𝑖. Since each player intends to connect to a route with the lowest latency, the utility function of player 𝑖 can be formulated as 𝑢𝑖=𝑑𝑠𝑖𝑠𝑖𝑆𝑖.(4)

The problem of incentive provision is managed here by transforming the latency to price which is charged by a route. The source node(s) that chooses this route is the price taker(s). This transformation provides the following benefits: (1) payment can effectively motivate a route to relay the data of a source node, and (2) the route can exploit its price as a lever to adjust its load. The instrument of pricing adopted here is not for maximizing the routes’ revenues, but for achieving a socially beneficial objective. The price of route 𝑗 is set as a function of its latency and denoted as 𝑝𝑗(𝑑𝑗). The utility function of each player 𝑖 can then be rewritten as 𝑢𝑖=𝑝𝑠𝑖𝑑𝑠𝑖𝑠𝑖𝑆𝑖.(5)

Suppose the game is played in a one-shot style, which indicates the game experiences only one iteration and then terminates. Since each player has to make a decision without the decisions of other players, the result may be extreme; that is, routes with low prices may share the entire load, whereas those with high prices may share no load at all. Furthermore, a single iteration does not allow routes to adjust their load. Therefore, we treat this game as a dynamic game where players interact with each other by playing the one-shot game numerous times until it converges to an NE. The next section proposes specific rules for the dynamic game.

5. Algorithm

In this section, a distributed DP scheme is presented to implement the dynamic game and enable it to converge in polynomial time. If a source node wants to deliver information via some routes, it first sends a request to all available routes in the strategy set. The game goes by the following specific rules.(1) Each route 𝑗 broadcasts an initial price to all source nodes that can communicate with 𝑗 by wireless connection, and then waits for their responses. The initial price is set as 𝑝𝑗=𝛼(|𝐵𝑗|+𝑗), where 𝛼 is a price factor.(2) After receiving the prices from all available routes, each source node 𝑖 sends a response to the one(s) with minimum price.(3) After obtaining the response from each source node in 𝐵𝑗, route 𝑗 decides which node should be accepted. The result is sent to all source nodes in 𝐵𝑗. Whether to accept a source node or not depends on the supply-demand degree of this route, which is formally defined as follows.

Definition 1. The supply-demand degree of a route 𝑗 is 𝜂𝑡𝑗=||𝐵𝑡𝑗||+𝑗𝑝𝑡𝑗=𝛼||𝐵/𝛼𝑡𝑗||+𝑗𝑝𝑡𝑗.(6)
The above equation means that at iteration 𝑡, the maximum number of connections (old and new) that each route can take in is 𝑝𝑡𝑗/𝛼𝑗. Thus, route 𝑗 balances its supply-demand when the actual number of taken-in source nodes equals the maximum. If newly requested connections plus old connections exceed the maximum, 𝑗 needs to randomly reject some of the newly requested connections to obtain its supply-demand balance. 𝑗 then broadcasts the decision.(4) After receiving the decisions of all routes in 𝑆𝑖, each source node 𝑖 responds with an acknowledgement to the chosen route. Node 𝑖 must choose one route randomly if multiple routes intend to accept its request.(5) Thus far, the interaction between routes and source nodes in a single iteration is finished. Afterwards, each route 𝑗 needs to determine the price of the next iteration. If the supply-demand is not balanced at the current iteration and there is no new connection accepted or no old connection broken, the price of the next iteration is set to 𝑝𝑗𝑝𝑗𝛼. Otherwise, the price remains unchanged. If the supply-demand is balanced at the current iteration, the price also remains unchanged and adds a “balanced” mark with the announced price.
The above five steps illustrate the process of each iteration in the dynamic game. The following step decides when the game would be over. First, a definition of the market level supply-demand degree is provided.

Definition 2. The market level supply-demand degree is the average of all the degrees of the routes’ supply-demand, that is, 𝑁𝑡=𝑗𝑁𝜂𝑡𝑗𝑛.(7) Once 𝑁𝑡=1, the whole market is supply-demand balanced, and the game is considered finished (Algorithms 1 and 2).
Two basic metrics for evaluating this algorithm are (1) its ability to converge to a pure NE and (2) the speed it needs to do so. Some significant features of DP are proven from these two aspects.

alg1
Algorithm 1:

alg2
Algorithm 2:

Theorem 3. The proposed algorithm converges to a pure NE.

Proof. Let 𝑡 be the number of iterations that the proposed algorithm needs to converge, and let 𝑝𝑡𝑗 be the final price of any route 𝑗.
First, by contradiction, we prove that for every source node 𝑖𝐵𝑡𝑗, 𝑝𝑡𝑗𝛼𝑝𝑡𝑘𝑝𝑡𝑗+𝛼 holds, where 𝑘𝑆𝑖 is any route available to source node 𝑖. Assume 𝑝𝑡𝑘𝑝𝑡𝑗2𝛼, and without losing generality let 𝑝𝑡𝑘=𝑝𝑡𝑗2𝛼. Then(1) suppose the price of route 𝑘 at iteration 𝑡=1 already equals 𝑝𝑡𝑗2𝛼, which means 𝑘 has balanced its supply-demand from the beginning. Clealy, 𝑝𝑡𝑗2𝛼<𝑝𝑡𝑗𝑝1𝑗. Thus, the chosen route of player 𝑖 would be 𝑘 rather than 𝑗, which contradicts the preassumption that 𝑖 decides to connect to route 𝑗;(2) suppose the price of route 𝑘 at iteration 𝑡=1 is not 𝑝𝑡𝑗2𝛼. There must be some iteration 𝑡 between 𝑡=1 and 𝑡=𝑡, at which the price of 𝑘 equals 𝑝𝑡𝑗𝛼 and at iteration 𝑡+1, the price equals 𝑝𝑡𝑗2𝛼. Since 𝑝𝑡𝑗𝛼<𝑝𝑡𝑗, the chosen route of player 𝑖 at iteration 𝑡 would be 𝑘 rather than 𝑗, which also contradicts the preassumption that 𝑖 decides to connect to route 𝑗.
Considering (1) and (2) jointly, 𝑝𝑡𝑘𝑝𝑡𝑗2𝛼 is invalid. Therefore, 𝑝𝑡𝑘𝑝𝑡𝑗𝛼. Similarly, we can prove that 𝑝𝑡𝑘𝑝𝑡𝑗+𝛼. Thus, 𝑝𝑡𝑗𝛼𝑝𝑡𝑘𝑝𝑡𝑗+𝛼 holds for every player 𝑖𝐵𝑡𝑗.
Second, we prove that each player 𝑖 has no incentive to change its choice unilaterally. Without loss of generality, let 𝑝𝑡𝑘=𝑝𝑡𝑗𝛼. If 𝑖 changes its choice from 𝑗 to 𝑘, then 𝑘 and 𝑗 need to adjust their prices to 𝑝𝑡𝑗 and 𝑝𝑡𝑗𝛼, respectively, to keep their supply-demand balanced. Note that, the price 𝑖 has to take is still unchanged. Hence, its choice will not change.

Lemma 4. If there is a source node that changes its choice or a route that lowers its price at an iteration 𝑡, then we have 𝑁𝑡>𝑁𝑡1 and Δ=𝑁𝑡𝑁𝑡1𝛼/𝑛Λ(Λ1) where Λ=max𝑗(|𝐵𝑗|+𝑗).

Proof. Λ=max𝑗(|𝐵𝑗|+𝑗) implies that 𝑝𝑡𝑗𝛼Λ for each 𝑗𝑁.(1) Suppose player 𝑖 changes its choice from 𝑘 to 𝑗, then 𝑝𝑡𝑘<𝑝𝑡𝑗. Considering that there may be no alternation in the sets of taken-in source nodes of 𝑗 and 𝑘 at iteration 𝑡1, their prices may be reduced at iteration 𝑡 accordingly. Thus, we have 𝑝𝑡𝑗𝑝𝑗𝑡1 and 𝑝𝑡𝑘𝑝𝑘𝑡1. Then, the following can be obtained: 1Δ=𝑛𝜂𝑡𝑗𝜂𝑗𝑡1+𝜂𝑡𝑘𝜂𝑘𝑡1=1𝑛𝛼||𝐵𝑡𝑗||+𝑗𝑝𝑡𝑗𝛼||𝐵𝑗𝑡1||+𝑗𝑝𝑗𝑡1+𝛼||𝐵𝑡𝑘||+𝑘𝑝𝑡𝑘𝛼||𝐵𝑘𝑡1||+𝑘𝑝𝑘𝑡1=𝛼𝑛||𝐵𝑡𝑗||1+𝑗𝑝𝑡𝑗||𝐵𝑗𝑡1||+𝑗𝑝𝑗𝑡1+||𝐵𝑡𝑘||+1+𝑘𝑝𝑡𝑘||𝐵𝑘𝑡1||+𝑘𝑝𝑘𝑡1𝛼𝑛||𝐵𝑡𝑗||1+𝑗𝑝𝑡𝑗||𝐵𝑗𝑡1||+𝑗𝑝𝑡𝑗+||𝐵𝑡𝑘||+1+𝑘𝑝𝑡𝑘||𝐵𝑘𝑡1||+𝑘𝑝𝑡𝑘=𝛼𝑛1𝑝𝑡𝑘1𝑝𝑡𝑗𝛼𝑛1𝑝𝑡𝑗1𝛼𝑝𝑡𝑗=𝛼2𝑛𝑝𝑡𝑗𝑝𝑡𝑗𝛼𝛼2=𝛼𝑛𝛼Λ(𝛼Λ𝛼)𝑛Λ(Λ1)>0.(8)This means that 𝑁𝑡+1>𝑁𝑡 holds if a source node changes its choice at iteration 𝑡.(2) Suppose route 𝑗 lowers its price at some iteration 𝑡, then 𝑝𝑡𝑗=𝑝𝑗𝑡1𝛼. Furthermore, a lower price may attract extra source nodes to choose 𝑗, thus |𝐵𝑡𝑗||𝐵𝑗𝑡1|. Hence, the following can be obtained: 1Δ=𝑛𝜂𝑡𝑗𝜂𝑗𝑡1=1𝑛𝛼||𝐵𝑡𝑗||+𝑗𝑝𝑡𝑗𝛼||𝐵𝑗𝑡1||+𝑗𝑝𝑗𝑡1𝛼𝑛||𝐵𝑗𝑡1||+𝑗𝑝𝑡𝑗||𝐵𝑗𝑡1||+𝑗𝑝𝑗𝑡1=𝛼𝑛||𝐵𝑗𝑡1||+𝑗1𝑝𝑗𝑡11𝛼𝑝𝑗𝑡1𝛼𝑛1𝑝𝑗𝑡11𝛼𝑝𝑗𝑡1=𝛼2𝑛𝑝𝑗𝑡1𝑝𝑗𝑡1𝛼𝛼2=𝛼𝑛𝛼Λ(𝛼Λ𝛼)𝑛Λ(Λ1)>0,(9)which implies that 𝑁𝑡+1>𝑁𝑡 holds if a route changes its price at iteration 𝑡.

Lemma 5. 𝑁𝑡𝑁𝑡1 holds and 𝑁𝑡 can remain unchanged during two successive iterations at most.

Proof. Lemma 4 states that 𝑁𝑡+1>𝑁𝑡 if there is a source node that changes its choice or if a route lowers its price at an iteration 𝑡. Consider the following case: there is neither any source node changes its choice nor any route lowers its price at an iteration 𝑡, and some routes are not supply-demand balanced. It can be computed that 𝜂𝑡𝑗=𝜂𝑗𝑡1 according to Definition 1. Thus we have 𝑁𝑡𝑁𝑡1=(1/𝑛)𝑗𝑁(𝜂𝑡𝑗𝜂𝑗𝑡1)=0. By jointly considering Lemma 4, 𝑁𝑡𝑁𝑡1 can be obtained. 𝑁𝑡 can remain unchanged during two successive iterations. Since some routes are still in an unbalanced state and no source node deviates from its initial decision, these routes will reduce their prices at iteration 𝑡+1 (according to Rule 3). 𝑁𝑡+1>𝑁𝑡 can then be obtained according to Lemma 4. This completes the proof.

Theorem 6. The number of iterations required to complete the proposed algorithm is 𝑂(𝑛Λ2), where Λ=max𝑗(|𝐵𝑗|+𝑗).

Proof. Evidently, 𝑁𝑡=1 and 𝑁00. Furthermore, Lemma 5 states that 𝑁𝑡 can remain unchanged during two successive iterations at most. The following can then be obtained: 𝑡𝑁2𝑡𝑁01𝛼/𝑛Λ(Λ1)2=𝛼/𝑛Λ(Λ1)2𝑛Λ(Λ1)𝛼=2𝛼𝑛Λ2.Λ(10) Equation (10) indicates that the proposed algorithm terminates in polynomial time, and the upper bound is 𝑂(𝑛Λ2).

6. Simulation Results

We conduct simulations to assess the performance of the proposed algorithm in terms of executed time and to ascertain the optimality of the NE and provide the results in this section. To obtain reliable results, several parameters are investigated, including the number of source nodes, the ratio of source nodes to routes denoted by 𝑟, and the degree of source nodes, which is defined as the number of routes that each source node can connect to. The values of these parameters are given in detail in Table 1. Since having assumed that the transmission rate and packet length are the same for all nodes, these two measurements are not included in the variable table. And note that our simulations are dedicated for quite generalized wireless M2M networks and thereby do not rely on specific protocols in MAC layer; it is unnecessary to specify which wireless channel is in use and how to set the window size. A large number of runs are conducted for each simulation and the results are averaged.

tab1
Table 1: Simulation parameters.
6.1. Executed Time

Section 4 shows that the proposed algorithm can be completed in polynomial time and gives an upper bound of the executed time. However, the real speed is essential in practice. Here, we evaluate the practical speed by comparing it with the theoretical time bound 𝑇 given by (10). 𝜆=𝑃/𝑇 is denoted as the ratio of practical time to the theoretical time bound (P-T ratio), where 𝑃 is the practical time that the proposed algorithm takes.

First, we study the effect of the number of source nodes on the proposed algorithm’s speed, and the other parameters are set as constants. Figure 2 shows how many iterations the proposed algorithm needs to be completed versus the different number of source nodes when the degree of source nodes is set to an average of 5, and Figure 3 illustrates the value of 𝜆 under each corresponding case. It can be seen that with the increase of the number of source nodes, the proposed algorithm requires more iterations to be completed, but the increasing trend is not rapid and it needs only 200 iterations with 1000 source nodes and 250 routes (𝑟=4). Almost in contrast to the trend of absolute iterations, the P-T ratio 𝜆 decreases sharply from the beginning with the number of source nodes increasing until around 200, where 𝜆<0.01, followed by a gradual decline. Moreover, the value of 𝜆 is fairly close to 0 with 1000 source nodes. The reason for this phenomenon may be that the given upper time bound is loose.

840391.fig.002
Figure 2: Number of iterations the proposed algorithm takes versus the number of source nodes.
840391.fig.003
Figure 3: P-T ratio 𝜆 versus the number of source nodes.

Second, the degree of source nodes is considered as variable, and 10 groups of the maximum degree of the source nodes are set to 1,2,,10, respectively. Figure 4 shows that the required average iterations by the proposed algorithm increases almost linearly with the increase in the number of source nodes. In particular, when the maximum degree of each source node is 1, only one iteration is needed because each has only one choice. When there are 400 source nodes with the maximum degree of 10, the proposed algorithm takes nearly 400 iterations to be completed. Figure 5 shows that 𝜆 reaches its maximum (slightly below 0.02) when each source node has two available routes on average. Except in a special case where each source node has only one choice, the P-T ratio 𝜆 decreases slowly as each source node has more choices because each has multiple routes to choose from.

840391.fig.004
Figure 4: Number of iterations the proposed algorithm takes versus the degree of source nodes.
840391.fig.005
Figure 5: P-T ratio 𝜆 versus the degree of source nodes.

On one hand, the given upper time bound is rather loose. On the other hand, the time bound is polynomial and the P-T ratio 𝜆 is extremely small in the given simulation. Therefore, the proposed algorithm has good speed performance.

6.2. Optimality

Section 4 indicates that a one-shot style may render the NE of the game undesirable. This part shows how the dynamic game can improve NE by simulation. Figure 6 provides the comparison results when the degree of source nodes is set to an average of 5 and 𝑟 to 4. Evidently, the latency under OneShotHop is nearly triple of DynamicPricing. Moreover, OneShotPricing is merely better by one point compared with OneShotHop.

840391.fig.006
Figure 6: Latency under DynamicPricing, OneShotHop, and OneShotPricing versus the number of source nodes.

Since our proposed algorithm is distributed, there is no guarantee that it can compute the optimal NE every time as a centralized approach can. Therefore, learning the optimality of the NE generated by DP is significant. Here, several experiments are conducted to examine the optimality of the proposed algorithm by comparing it with a well-known centralized approach based on the cost-reducing path and to determine whether it can terminate in polynomial time [18]. Before showing the results, price of anarchy (PoA) [19], which measures the optimality of a game, is introduced as a metric and formally given as follows.

Definition 7 (PoA). For game G, let cos𝑡(𝐺) denote the social cost of G’s NE, and let opt(𝐺) denote the minimum social cost over all assignments. The price of anarchy is then defined by PoA(𝐺)=cos𝑡(𝐺).opt(𝐺)(11)PoA is constantly not smaller than 1 (1 means optimal). The smaller the PoA is, the better the NE becomes.

First, the effect of the number of source nodes on the proposed algorithm’s optimality is studied. The degree of source nodes is set to an average of 5. Figure 7 shows that as the number of source nodes increases, the NE’s PoA tends to become larger despite small fluctuations. The maximum value of PoA is below 1.40 with 1000 source nodes and 500 routes. The corresponding ratio of the number of runs that generates an optimal NE to that of all runs (optimal-to-all for short) is given in Figure 8. Most of the runs can generate an optimal assignment.

840391.fig.007
Figure 7: PoA versus the number of source nodes.
840391.fig.008
Figure 8: Ratio of optimal-to-all versus the number of source nodes.

Second, the degree of source nodes is considered to be variable. Ten groups of the maximum degree of the source nodes are set to 1,2,,10, respectively. When each source node has only one route to connect to, the assignment is unique, and thus PoA=1, which is in accordance with that shown in Figure 9. If each source node has multiple choices, the optimality of the proposed algorithm becomes slightly poor. However, the maximum PoA does not exceed 1.16, which is close to 1. Figure 10 provides the information on the ratio of optimal-to-all in each case corresponding to Figure 9. Over 90% of runs can generate an optimal assignment.

840391.fig.009
Figure 9: PoA versus the degree of source nodes.
840391.fig.0010
Figure 10: Ratio of optimal-to-all versus the degree of source nodes.

7. Conclusion

The routing congestion problem in wireless M2M networks has been investigated using a game theoretic approach. A network congestion game has been formulated, in which many selfish source nodes compete for connecting to a smaller set of routes. Since full cooperation among nodes cannot be assumed in wireless M2M networks, pricing has been applied in the game to motivate cooperative data relay. By setting the pricing scheme dynamic, routes altering their prices and source nodes accordingly altering their connected routes, we have proved that the network congestion game converges to a pure NE in polynomial time. The optimality of the proposed algorithm was evaluated by simulation. The results indicate good practical performance.

Acknowledgments

This research is partially supported by research grants from the National Science Foundation of China under Grant nos. 70701025 and 71071105, the Program for New Century Excellent Talents in Universities of China under Grant no. NCET-08-0396, and a National Science Fund for Distinguished Young Scholars of China under Grant no. 70925005, and the Program for Changjiang Scholars and Innovative Research Team in the university (IRT1028).

References

  1. Q. Zhang and Y. Q. Zheng, “Cross-layer design for qos support in multihop wireless networks,” Proceedings of the IEEE, vol. 96, no. 1, pp. 64–76, 2008. View at Publisher · View at Google Scholar · View at Scopus
  2. P. K. Huang, X. Lin, and C. C. Wang, “A low-complexity congestion control and scheduling algorithm for multihop wireless networks with order-optimal per-flow delay,” in Proceedings of the IEEE International Conference on Computer Communications (INFOCOM '11), pp. 2588–2596, April 2011. View at Publisher · View at Google Scholar · View at Scopus
  3. K. Tan, F. Jiang, Q. Zhang, and X. Shen, “Congestion control in multihop wireless networks,” IEEE Transactions on Vehicular Technology, vol. 56, no. 2, pp. 863–873, 2007. View at Publisher · View at Google Scholar · View at Scopus
  4. H. J. Lee and J. T. Lim, “Congestion control for streaming service in IEEE 802.11 multihop networks,” IET Communications, vol. 4, no. 12, pp. 1415–1422, 2010. View at Publisher · View at Google Scholar · View at Scopus
  5. J. Y. Teo, Y. Ha, and C. K. Tham, “Interference-minimized multipath routing with congestion control in wireless sensor network for high-rate streaming,” IEEE Transactions on Mobile Computing, vol. 7, no. 9, pp. 1124–1137, 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. L. Chen, S. H. Low, M. Chiangs, and J. C. Doyle, “Cross-layer congestion control, routing and scheduling design in ad hoc wireless networks,” in Proceedings of the 25th IEEE International Conference on Computer Communications (INFOCOM '06), April 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. B. Vöcking and R. Aachen, “Congestion games: optimization in competition,” in Proceedings of the Proceedings of the 2nd Algorithms and Complexity in Durham Workshop, H. Broersma, S. Dantchev, M. Johnson, and S. Szeider, Eds., Kings College Publications, London, UK, 2006.
  8. H. Sperber, “How to find Nash equilibria with extreme total latency in network congestion games?” in Proceedings of the International Conference on Game Theory for Networks (GameNets '09), pp. 158–163, May 2009. View at Publisher · View at Google Scholar · View at Scopus
  9. D. Fotakis, S. Kontogiannis, and P. Spirakis, “Symmetry in network congestion games: pure equilibria and anarchy cost,” Approximation and Online Algorithms, vol. 3879, pp. 161–175, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. C. A. Gizelis and D. D. Vergados, “A survey of pricing schemes in wireless networks,” IEEE Communications Surveys and Tutorials, vol. 13, no. 1, pp. 126–145, 2011. View at Publisher · View at Google Scholar · View at Scopus
  11. X. Zhu, S. Han, and B. Girod, “Congestion-aware rate allocation for multipath video streaming over ad hoc wireless networks,” in Proceedings of the International Conference on Image Processing (ICIP '04), pp. 2547–2550, October 2004. View at Scopus
  12. L. Chen, S. H. Low, and J. C. Doyle, “Joint congestion control and media access control design for ad hoc wireless networks,” in Proceedings of the IEEE International Conference on Computer Communications (INFOCOM '05), pp. 2212–2222, March 2005. View at Scopus
  13. M. Chiang, “Balancing transport and physical layers in wireless multihop networks: jointly optimal congestion control and power control,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 1, pp. 104–116, 2005. View at Publisher · View at Google Scholar · View at Scopus
  14. C. Wang, B. Li, K. Sohraby, M. Daneshmand, and Y. Hu, “Upstream congestion control in wireless sensor networks through cross-layer optimization,” IEEE Journal on Selected Areas in Communications, vol. 25, no. 4, pp. 786–795, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. P. Wang and S. Bohacek, “Practical computation of optimal schedules in multihop wireless networks,” IEEE/ACM Transactions on Networking, vol. 19, no. 2, pp. 305–318, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. C. Jiang, Y. Shi, Y. T. Hou, and S. Kompella, “On optimal throughput-energy curve for multi-hop wireless networks,” in Proceedings of the IEEE International Conference on Computer Communications (INFOCOM '11), pp. 1341–1349, April 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. Y. P. Hsu and K. T. Feng, “Cross-layer routing for congestion control in wireless sensor networks,” in Proceedings of the IEEE Radio and Wireless Symposium (RWS '08), pp. 783–786, January 2008. View at Publisher · View at Google Scholar · View at Scopus
  18. N. J. A. Harvey, R. E. Ladner, L. Lovász, and T. Tamir, “Semi-matchings for bipartite graphs and load balancing,” Journal of Algorithms, vol. 59, no. 1, pp. 53–78, 2006. View at Publisher · View at Google Scholar · View at Scopus
  19. C. H. Papadimitriou, “Algorithms, games, and the internet,” in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 749–753, July 2001. View at Scopus