^{1}

^{1}

^{1}

This paper investigates an incentive pricing problem for relaying services in multihop cellular networks. Providing incentives to encourage mobile nodes to relay data is a critical factor in building successful multihop cellular networks. Most existing approaches adopt fixed-rate or location-based pricing on rewarding packets forwarding. This study applies a mathematical programming model to determine an optimal incentive price for each intermediate node that provides relaying services. Under the obtained incentive price, the connection availability of the networks is maximized by using the same relaying costs as other pricing schemes. A signomial geometric programming problem is constructed, and a deterministic optimization approach is employed to solve the problem. Besides, quality-of-service constraints are added in the proposed model to mitigate the unfairness between connection availabilities of individual nodes. Computational results demonstrate that the proposed model obtains the optimal incentive price on relaying services to maximize connection availability of the networks.

Over the past few years, wireless networks and wireless devices have rapidly developed and undergone significant advances. More and more services that dramatically affect personal and business communications are provided by wireless access networks. How to build a seamless wireless network has received increasing attention from the practitioners and the researchers. Most wireless networks are based on cellular architecture, which means that a mobile host is handled by a central base station in a limited range. Cellular networks have inherent limitations on cell coverage and the dead spot problem. Traditionally, the network providers utilize more infrastructure equipments such as base stations, to solve these problems. However, this method is expensive. Therefore relaying technology has been developed to solve this problem. In the last decade, multihop cellular networks have been proposed to harness the benefits of conventional cellular networks and emerging multihop ad hoc networks. In cellular networks, a mobile device directly connects with the base station; in multihop networks, a mobile device communicates with others over peer-to-peer connections. Figure

Increases the speed of data transmission.

Reduces total transmission power.

Extends the service area.

Increases system capacity.

Balances traffic load.

Reduces the interference with other nodes.

Reduces the number of base station sites.

Scenario of general multihop cellular networks.

Cooperation among nodes is a critical factor for ensuring the success of the relaying ad hoc networks [

This paper constructs a mathematical programming model to the problem of optimal pricing on relaying services provided by the mobile nodes in the multihop cellular networks. The formulated model that maximizes connection availability of the networks under identical relaying costs used by the fixed-rate pricing scheme and the location-based pricing scheme [

The rest of the paper is organized as follows. Section

Opportunity-driven multiple access (ODMA) is an ad hoc multihop protocol where the transmissions from the mobile hosts to the base station are broken into multiple wireless hops, thereby reducing transmission power [

Since forwarding data for others consumes battery energy and delays its own data, providing incentives for mobile nodes to cooperate as relaying entries is necessary. The existing incentive schemes can be classified into detection-based and motivation-based approaches. The detection-based approach finds out the misbehaving nodes and reduces their impact in the networks. Marti et al. [

Instead of discouraging misbehavior by punishing misbehavior node, the motivation-based approach encourages positive cooperation by rewarding incentives for relaying packets. Buttyán and Hubaux [

Lo and Lin [

Pricing is an inducer for suppliers to provide services. Monetary incentives can affect the motivation of mobile nodes providing services and are usually characterized by a supply function that represents the reaction of mobile nodes to the change of the price [

In multihop cellular networks, data packets must be relayed hop by hop from a given mobile node to a base station; thus the connection availability of node

The connection availability maximization problem in the multihop cellular networks considered in this paper can be formulated as follows:

In the numerical examples, we find the connection availabilities of some mobile nodes are zero by using the proposed model described previously. In order to alleviate the unfairness situation between connection availabilities of individual nodes, this study employs QoS constraints in the original model to guarantee each mobile node with a minimum successful connection probability. The connection availability maximization problem with QoS requirements considered in this study can be formulated as follows:

Since the problem described in the previous section is an SGP problem, that is, a class of nonconvex programming problems. SGP problems generally possess multiple local optima and experience much more theoretical and computational difficulties. This study uses variable transformations and piecewise linearization techniques to reformulate the problem into a convex MINLP problem that can be globally solved by conventional MINLP methods. Much research has proposed variable transformation techniques to solve optimization problems including signomial functions to global optimality [

If

For convexifying negative signomial terms, we apply the power transformation to reformulate a negative signomial function

If

Herein the concept of special ordered set of type 2 (SOS-2) constraints can be utilized to formulate the piecewise linear function [

The original model has one nonconvex objective function, one constraint, and

The following example is used to illustrate how the proposed method discussed previously determines the incentive price on relaying services provided by each mobile node.

Consider an example taken from Lo and Lin [

Relaying topology of Example

Assume each mobile node has identical traffic load

Comparison between the fixed-rate pricing scheme, the location-based pricing scheme, and the proposed pricing scheme of Example

Fixed-rate pricing scheme | Location-based pricing scheme | Proposed pricing scheme | |

Incentive price on relaying services | |||

Connection availability of each | |||

Connection availability of the networks | |||

Relaying costs |

From Table

Comparison between the location-based pricing scheme and the proposed pricing scheme with QoS requirements of Example

Location-based pricing scheme | Proposed pricing scheme with QoS requirements | |

Incentive price on relaying services | ||

Connection availability of each node | ||

Connection availability of the networks | ||

Relaying costs |

This section describes the simulation results for verifying the advantages of the proposed pricing scheme. We design our simulation tests by C++ language. All simulations are run on a Notebook with an Intel CPU P8700 and 4 GB RAM. The simulation environment is a rectangular region of 100 units width and 100 units height with a single base station of 30 units radius located in the central point. The radius of each mobile node is 20 units. In this study, a shortest path tree is built such that each mobile node connects to the base station with a minimum number of hops.

In the experiments 32, 64, and 128, mobile nodes, respectively, are randomly distributed in the rectangular region. 10 simulations are run for each set of parameter settings. Table

Comparison of connection availability of the networks by three methods in the simulation space of (width, height) = (100 units, 100 units).

Number of mobile nodes |
CA_{FR} |
CA_{LB} | Average path length | |||
---|---|---|---|---|---|---|

32 | 0.47898643 | 0.48730129 | 0.53038439 | 10.82% | 8.88% | 1.0840542 |

64 | 0.48652063 | 0.49130662 | 0.51966696 | 6.87% | 5.81% | 1.0539176 |

128 | 0.49012412 | 0.49131695 | 0.51875024 | 5.88% | 5.62% | 1.0395036 |

CA_{FR}: connection availability of the networks from the fixed-rate pricing scheme.

CA_{LB}: connection availability of the networks from the location-based pricing scheme.

Comparison of connection availability of the networks by three methods in the simulation space of (width, height) = (100 units, 100 units).

To investigate the advantages of the proposed pricing scheme under a longer path, in this section we change the simulation space to a rectangular region of 200 units width and 200 units height. If the simulation area becomes larger, the path of the hop-by-hop connection to the base station required by a mobile node will be longer. Then the impact of the path length on the performance of the proposed pricing method can be observed. Table

Comparison of connection availability of the networks by three methods in the simulation space of (width, height) = (200 units, 200 units).

Number of mobile nodes |
CA_{FR} |
CA_{LB} | Average path length | |||
---|---|---|---|---|---|---|

32 | 0.22764841 | 0.23610142 | 0.53722796 | 138.12% | 130.04% | 2.6502985 |

64 | 0.23798218 | 0.24306587 | 0.53669912 | 125.83% | 121.06% | 2.5031434 |

128 | 0.24686002 | 0.24986204 | 0.51335212 | 108.03% | 105.54% | 2.3962481 |

CA_{FR}: connection availability of the networks from the fixed-rate pricing scheme.

CA_{LB}: connection availability of the networks from the location-based pricing scheme.

Comparison of connection availability of the networks by three methods in the simulation space of (width, height) = (200 units, 200 units).

Section

Tables

Comparison of connection availability by two methods in the simulation space of (width, height) = (100 units, 100 units).

Number of mobile nodes |
CA_{LB} | ||
---|---|---|---|

32 | 0.48730129 | 0.49575548 | 1.74% |

64 | 0.49130662 | 0.49617013 | 1.00% |

128 | 0.49131695 | 0.49246111 | 0.24% |

CA_{LB}: connection availability of the networks from the location-based pricing scheme.

Comparison of connection availability of the networks by two methods in the simulation space of (width, height) = (200 units, 200 units).

Number of mobile nodes |
CA_{LB} | ||
---|---|---|---|

32 | 0.23610142 | 0.3183623 | 34.84% |

64 | 0.24306587 | 0.31280795 | 28.69% |

128 | 0.24986204 | 0.29797671 | 19.26% |

CA_{LB}: connection availability of the networks from the location-based pricing scheme.

Comparison of connection availability of the networks by two methods in the simulation space of (width, height) = (100 units, 100 units).

Comparison of connection availability of the networks by two methods in simulation space of (width, height) = (200 units, 200 units).

Cost savings and connection availability are two crucial issues of a network provider adopting multihop cellular networking technology. This paper determines the optimal incentive price on relaying services for each mobile node by constructing a mathematical programming model that maximizes connection availability without extra relaying costs. A deterministic optimization approach based on variable transformations and piecewise linearization techniques is utilized to solve the formulated problem. Simulation results demonstrate that the proposed pricing model results in higher connection availability than the fixed-rate pricing scheme and the location-based pricing scheme. In addition, a mathematical programming model involving QoS requirements in connection availability of each individual mobile node is developed to eliminate the unfairness situation in the original model.

The authors thank the anonymous referees for contributing their valuable comments regarding this paper and thus significantly improving the quality of this paper. The research is supported by Taiwan NSC Grant NSC 99-2410-H-158-010-MY2.