Wireless Communications and Mobile Computing
Volume 2018 (2018), Article ID 9297536, 15 pages
https://doi.org/10.1155/2018/9297536

Research Article

Congestion-Optimal WiFi Offloading with User Mobility Management in Smart Communications

Bin Liu,¹ Qi Zhu,¹ Weiqiang Tan,² and Hongbo Zhu¹

¹Jiangsu Key Laboratory of Wireless Communications, Nanjing University of Posts and Telecommunications, Nanjing 210003, China
²School of Computer Science and Educational Software, Guangzhou University, Guangzhou 510006, China

Correspondence should be addressed to Hongbo Zhu; zhuhb@njupt.edu.cn

Received 12 April 2018; Accepted 11 June 2018; Published 1 August 2018

Academic Editor: Mu Zhou

Copyright © 2018 Bin Liu 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.

We study the WiFi offloading problem in smart communications and adaptively seek for the optimal offloading strategies with the consideration of the mobility management and the dynamical nature of network state. With users mobility management, we formulate the offloading ratio optimization problem based on Markov process. Then, we propose a novel Congestion-Optimal WiFi Offloading (COWO) algorithm based on subgradient method, which aims to obtain the optimal offloading ratio for each access point (AP) to maximize the throughput and minimize the network congestion. Due to the computational complexity of subgradient method, we further improve the COWO algorithm by the equivalent transformation. By viewing all the APs as one virtual WiFi network, we try to optimize the identical offloading ratio for virtual WiFi network and develop a Virtualized Congestion-Optimal WiFi Offloading (VCOWO) algorithm with lower complexity. Under the equivalent conditions, the performance of the VCOWO algorithm could well approximate the optimal results obtained by the COWO algorithm. It is found that the VCOWO algorithm could obtain the upper bound of multiple APs WiFi offloading performance. Moreover, we investigate the impacts of user mobility on the WiFi offloading performance. Simulation results show that the proposed algorithm could achieve higher throughput with lower network congestion compared with other current offloading schemes.

1. Introduction

With the proliferation of smart devices such as smartphones and tablets, cellular networks are facing an exponential growth of mobile data traffic. According to Cisco’s forecast, global mobile data traffic is expected to grow to 49 megabytes per month by 2021, a sevenfold increase over 2016 [1–4]. With the limited licensed bandwidth, the cellular network capacity, however, can not keep up with the explosive data growth [5–9]. The mobile operators have been seeking for the cost-effective and timely solution to alleviate the cellular network. Thanks to the abundant unlicensed spectrum and large-scale WiFi deployment, Wireless Local Area Network (WLAN) has attracted much attention as a promising approach to offload data from the cellular network [10–13] and enhance network survivability and resilience in smart communications.

Previous works have demonstrated WiFi offloading prospects in leveraging traffic load [14–18]. The work in [18] proposed the on-the-spot offloading (OTSO) scheme and showed that the OTSO offloading could leverage more than 65% traffic from the cellular network through an experiment in Seoul. Gass and Diot [19] further verified that the WiFi offloading is favorable even if the connecting time is insufficient. Much effort has also been seen in WiFi offloading schemes and performance analysis under the integrated cellular and WLAN networks framework [20]. The authors in [21] demonstrated that larger portions of cellular traffic could be offloaded to WiFi if the delay to wait for WiFi network is allowed during user movement. It presented the offloading scheme called Wiffler, which schedules the network access based on the historical access knowledge. By extracting typical users’ mobility profiles, the work of [22] optimized the energy consumption in offloading. The authors in [23] studied the capacity of delayed offloading without prior knowledge of users’ mobility patterns and proposed online scheduling policy to maximize the amount of offloaded data. By taking downloading cost and delay into the offloading decision, the delayed offloading scheme was proposed in [24] to harvest maximum benefits from WiFi offloading. The optimal transmission deadline for delayed offloading was further derived in [25] and could achieve the maximum monetary incentive while maintaining the outage probability. The WiFi offloading and LTE WLAN Aggregation (LWA) was jointly considered in [26] to make the full use of the spectrum in the licensed and unlicensed carriers aggregation approach.

However, previously proposed schemes mainly focus on using WiFi offloading merely as a capacity-augment solution and try to offload as much data traffic to WiFi as possible, without systematically considering the dynamic nature of the network, which may result in the network congestion and degrade user’s quality of experience (QoE). For instance, the OTSO scheme, which has been used as a default setting in most of the smartphones, enables users to handover and offload data through WiFi whenever users enter the network coverage. Rather than consider the load balance and network congestion in the WiFi network, this scheme simply decides the offloading by the availability of the network. It nevertheless increases the access conflicts and degrades the performance [27]. The network selection game was proposed in [28, 29] to analyze the offloading by jointly considering the network congestion and the cost of switching between different networks. The Pareto-efficiency of the equilibria in congestion game was proved in [30], and a client-centric network selection is further proposed to reduce the network congestion. Though the load balance and network congestion are considered, few of these works are involved in the optimal users offloading ratio during their movement or analyze the impacts of network congestion and user mobility on WiFi offloading.

In this paper, we investigate WiFi offloading with consideration of the network congestion and the user mobility management in smart communications and deduce the throughput and blocking probability based on Markov process. From the geometric deduction, the user flow equilibrium is observed, i.e., the rate of the users entering and leaving out of the network is equal, and thus we obtain the user flow rate between different networks. We optimize the ratio of users performing offloading when they move into the WiFi access points (APs), which aims to maximize the network throughput and minimize the network congestion. In this paper, the network congestion is characterized by the blocking probability. To obtain the optimal offloading ratio for each WiFi network, the Congestion-Optimal WiFi Offloading (COWO) algorithm is proposed based on the subgradient method. High computational complexity often occurs, especially in the dense APs deployed scenario. To this end, we make the nearly equivalent transformation and view all the WiFi networks as a virtual network. In this case, the Virtualized Congestion-Optimal WiFi Offloading (VCOWO) algorithm could well approximate to the optimal results under equivalent conditions. Moreover, the effects of users mobility on offloading effectiveness are demonstrated.

The rest of this paper is organized as follows. In Section 2, we describe the scenario of WiFi offloading in Section 2. We analyze the WiFi offloading performance and formulate the offloading optimization problem in Section 3. The Congestion-Optimal WiFi Offloading algorithm is proposed in Section 4. With equivalent conditions, we develop the Virtualized Congestion-Optimal WiFi offloading algorithm in Section 5. We present the numerical results in Section 6 and draw conclusions in Section 7. The symbols used throughout the paper are summarized in Table 1.

Table 1: Notations and definitions.

Notation
Definition

Coverage of the cellular network

Coverage of the th AP

The number of APs

Radius of cellular network coverage

Radius of th AP coverage

Total active users arriving rate

Active users arriving rate in the cellular network

Active users arriving rate in the th AP

Average user flow rate in the cellular network

Average user flow rate in the th AP

Average service completing rate in the cellular network

Average service completing rate in the th AP

Service arriving rate in the cellular network

Service arriving rate in the th AP

Service leaving rate in the cellular network

Service leaving rate in the th AP

Average number of users in the cellular network

Average number of users in the th AP

Blocking probability in the cellular network

Blocking probability in the th AP

Throughput in the cellular network

Throughput in the th AP

Total number of channels in the cellular network

Total number of channels in the th AP

Average velocity of user mobility

Offloading probability in th AP

2. System Model

We describe the WiFi offloading scenario under the integrated cellular and WLAN networks framework [31–34] in Figure 1, where the MO can tightly integrate the WiFi networks with the cellular networks through the recent IEEE and 3rd Generation Partnership Project (3GPP) standards. For example, the network discovery and selection functionality (ANDSF) reports the network information related to the access network type, roaming consortium, and venue information through management frames to the macro base station (MBS) [28]. Then, the MBS decides and schedules the user offloading based on the reported information. In this paper, we consider that the MBS covers the whole scenario, with WiFi APs randomly distributed in its coverage without overlapping with each other. The active users, defined as the users need network service at this moment, are assumed to have arriving rate with Poisson arrivals, and the active users follow the uniform distribution on the geometry coverage; i.e., the active user arriving rate for each network is proportional to their coverage, and the active user arriving rates for th AP and cellular networks are , , and , respectively,

where and are the coverage area of the scenario and th AP, respectively.

Figure 1: WiFi offloading scenario.

The user mobility management is considered, and the offloading ratio in user mobility is further investigated in this paper. With circular coverage, the user flow rates under fluid flow model and geometric angle mobility model were separately deduced in [35–38], and it is proved that the rate of the users entering and leaving the network is equal, which is called user flow equilibrium. With user mobility probability density function , we have the user flow rate as

where is the number of users in the network, is the radius of the network coverage, and denotes the first moment of . We start the analysis of user flow rate with the assumptions as follows:

(1)All the networks have circular geometric coverage.(2)User moves within the coverage area with the random direction; i.e., the angle of direction is uniformly distributed in .(3)The user's mobility is in uniform linear motion; i.e., .

The user flow rate can be written as follows [37]:

To be specific, for the th AP with radius and MBS with radius , we have the user flow rate separately:

where and are the numbers of active user in th AP and MBS.

We formulate the WiFi offloading model based on the discrete Markov process [39–41]. The occupation of network channel is described as a discrete Markov state, the service arriving means that the user enters the network service sequence, and service leaving means that the user cuts off the network service. Each user could access one channel at the same time. The maximum number of users served by the network is constrained by the number of channels . Let denote the state transmission rate from the th state to the ( +1)th state , and is the state transition rate of th state to the (-1)th state, , as shown in Figure 2. And we have the steady distribution as

where is the steady-state probability distribution for the th state, and

Figure 2: Markov state transition.

We try to find the optimal offloading ratio , aiming to maximize the network throughput and minimize the network congestion at both networks when the users enter into the th AP’s coverage area. With offloading ratio , the serve arriving rates at th AP and MBS are defined as

where the first term denotes the newly active users in their own coverage, and the second term denotes the service increase by the users flow in their mobility. Note that only of the users will be offloaded to the WiFi network when entering, while all the users moving out of the AP’s coverage will be immediately handed over to the cellular network. Assume the average service completing rate in WiFi and cellular network is and , respectively; the service leaving rates for th AP and MBS are

where the first part is by the user finishing their transmission and turning to inactive, and the second part is by the user moving out of the network coverage, and handover to another network. Only of the users could be accepted by the th AP, i.e., only of the users flowing out of the cellular network could leave the MBS service sequence, and the rest of users are still kept in service in the cellular network.

In this paper, we try to optimize the offloading ratio for multiple APs, which aims to increase the throughput and reduce the network congestion. On the one hand, it is an incentive to offload more user to leverage the cellular traffic load and to increase the throughput. On the other hand, the increase of offloading will nevertheless result in the conflicts and degrade the performance. Therefore, the offloading ratio should be carefully designed with the consideration of the network congestion and the user mobility.

3. Problem Formulation

In this section, we analyze the WiFi offloading performance from both cellular and WiFi network sides based on the Markov process and model the network congestion, which is characterized by the blocking probability. Then, we define the system utility as the function of throughput and blocking probability and formulate the offloading ratio optimization problem.

3.1. WiFi Network

First, we analyze the th AP network with channels in total. From (8), the steady-state probability distribution of th AP is

where

Thus, we have the number of users served by the th AP as the average queue length, and

In [42], the blocking probability and nonblocking probability for the th AP are defined as

where denotes the fact that the blocking probability depends on the ratio of service arriving and leaving rate in (10). Then, the average user number can be rewritten as

With the average user number in (13), the user flow rate can be expressed as

After modification, it is observed that the user flow rate is the function of blocking probability, which is denoted as , and we have

With the user flow rate in (15), we have the blocking probability in its implicit function form as

where the second equation is denoted as . Clearly, is the root of that holds

and it could be solved by the bisection method.

Proof. See Appendix A.

From (16), it is also found that the depends on the offloading ratio , i.e., , and we thus have the user flow rate in the form of a function of

Then, the throughput achieved under the offloading probability in the th AP is for simplicity; we normalize the achievable data rate of per channel in the th AP network as , and the throughput can be expressed as

3.2. Cellular Network

The analysis of the cellular network follows the similar approaches in the WiFi network. Consider the cellular network with total channels, and the steady-state probability distribution of the cellular network is expressed as

where

The length of service queue is equal to the average number of users served by the cellular network,

where and are defined the blocking probability and nonblocking probability in the cellular network as (13):

where denotes that the blocking probability depending on the ratio of service arriving and leaving rate .

Fixing the average user number in (22), the user flow rate can be expressed as the function of blocking probability, which is denoted as ,

With the user flow rate in (24), we have the blocking probability in its implicit function form,

Similar to (16), could be obtained through solving (25) by the bisection method. From (25), it is also found that the depends on the offloading ratio for each AP , ; i.e., is the function of vector composed by , , . Thus, the user flow rate also has the formation of the function of offloading ratio vector as

The throughput of the cellular network is similarly given:

where is the normalized channel rate of the cellular network.

3.3. Congestion Optimization Problem

In this paper, we seek for the optimal offloading ratio for multiple APs in user mobility, which aims to maximize the throughput and minimize the network congestion. The throughput is the sum of all networks, i.e., , and the network congestion is characterized by the blocking probability, i.e., . Thus, we set the system utility function so that it increases with the throughput and decreases with the network congestion, where the blocking probability acts as a penalty for network congestion. The constant is defined as weight over the throughput and blocking probability, which embodies the s sensitivity to network congestion. The system utility function could be formulated as

4. Congestion-Optimal WiFi Offloading

In this section, we proposed the COWO algorithm to obtain the optimal offloading ratio for each AP, which is based on subgradient method [43] and further reveal the impacts of user mobility on WiFi offloading performance.

4.1. Congestion-Optimal WiFi Offloading Algorithm

The optimization problem in (28) is convex function with the convex feasible region. Thus, the subgradient method can be used to solve the optimization problem.

Proof. See Appendix B and Appendix C.

The augmented Lagrangian function of optimization problem (28) is written as

where is the Lagrange multiplier. The Lagrange problem in (29) could be solved in subgradient method, and the optimal satisfies

where the th element in the partial derivatives of can be calculated as (31), as shown below.

Equation (30) is nonlinear. Thus, we make the transformation to simplify the calculation. The partial derivative of is written as

where

To achieve the identity , for . Thus, we have

where , and the augmented Lagrangian function in (28) can be written as

Then, could be updated with pace , and we have

The following parts give the update process in detail. For th AP, the service arriving rate is , and . It is observed from (37) that is dependent on , and thus we define the function

It can be proved that the is a decreasing function of by the partial derivative deduction. Thus, if the zero point of exists, it can be obtained by the bisection method. Let us denote the zero point of as . If does not have zero points, we modify it as

If , , the offloading ratio is obtained by

When , , the offloading ratio is modified as

Based on the subgradient method, the implementation of the proposed COWO algorithm is provided as in Algorithm 1.

1: Initialize: , ,

2: while  

3:Calculate for and under the initial in the bisection method

4:Get initial updating factor

5:Solve equation (37) and get

6:Update by (38)

7:Solve out

8:Set and update according to (36)

9:Iterate into

10:  end while

11: Set the optimal offload factor vector

Algorithm 1: The proposed COWO algorithm.

4.2. Impacts of User Mobility on WiFi Offloading

In this section, we study the impacts of user mobility, mainly characterized by average velocity, on the WiFi offloading. First, we show its impacts on WiFi offloading throughput. The derivatives of throughput with respect to are

From (10) and (11), can be expressed as

and we take the derivatives of , and we have

Thus, the derivatives of throughput with respect to could be rewritten as

where the details about the computation and the associated analysis can be found in the literature [44–46]. With the fact that and the deduction in Appendix A and B, we have

Therefore, , which draws the conclusion that the average throughput for each network is decreasing with the velocity . It is also observed that decreases with . Furthermore, the blocking probability has

Thus, also decreases with the values of . Similarly, the similar conclusions can be drawn in the cellular network. The blocking ratio and throughput all decrease with velocity. Therefore, the higher speed will degrade the capacity, but could lower the congestion. It could be explained by the fact that the increase of mobility rate in one hand enhances the handover opportunity, as well as the fairness for each user, since they have the more chance to access network; i.e., the increase of user mobility gives more opportunities to enter another network and enables some user who is always failing to connect to the AP to reconsider the network access. It could also lower the blocking probability in the congested situation but can not increase the system capacity.

5. Virtualized Congestion WiFi Offloading Algorithm

The COWO algorithm could obtain the optimal offloading ratio for each AP. However, it is based on the subgradient method, whose computational complexity will drastically increase with the number of networks [47, 48]. In this section, rather than optimize the independent offloading ratio for each AP, we view all the APs as one virtual WiFi network, and the channels and other resources in the WiFi networks will be jointly scheduled. Then, we proposed VCOWO algorithm, which could obtain the approximately optimal results under the equivalent conditions with much lower complexity. Though we did not get the separate offloading ratio, we can obtain some general results and useful insights from it.

Through observation, it is found that the offloading decision mainly depends on the service arriving and leaving rate for given total channels in each network. As all the WiFi networks act as the capacity-augment utilities, it is reasonable to view all the AP as one virtual WiFi network, where all the channels are centrally controlled. For simplicity, we consider the AP to have the same coverage, i.e., , , and same serving completing rate . The virtual WiFi network satisfies the equivalent conditions as follows.

(1)The user flow rate of virtual WiFi network is the sum of user flow rate for each AP; i.e.,

 where .(2)The active users arriving rate of virtual WiFi network is equal to total active users arriving rate of all APs,

(3)All the channels in each AP are scheduled by the virtual WiFi network, i.e., the virtual WiFi network has .(4)The user serving rate in virtual WiFi network is .

We try to find the optimal offloading ratio for the virtual WiFi network. Thus, the steady-state distribution of virtual WiFi network can be expressed as

where

Therefore, the number of users served by the virtual WiFi network is the average queue length, and

Similarly, the blocking probability and unblocking probability in virtual WiFi network are defined as

Then the users flow rate in virtual network is written as

After modification, it is observed that the user flow rate is the function of blocking probability, which is denoted as ; i.e.,

Under the equivalent conditions, the analysis of offloading performance in the cellular network could be simplified. The users flowing rate in the cellular network is transformed as ,

where

The user flow rate can also be written as the function of blocking probability,

Similar to (17), the blocking probability could be solved in bisection method [49, 50]. Therefore, the optimization problem in (28) for all multiple APs could be transformed as follows:

From (58), it is observed that the optimization is aimed at finding the optimal scalar to maximize the throughput and minimize the blocking probability. The optimal scalar could be solved by the golden section method, and we proposed the VCOWO algorithm as illustrated in Algorithm 2. The computational complexity in COWO is , while the complexity is reduced to in VCOWO algorithm, where is the number of MBS, and is assumed in this paper. It means the complexity of COWO algorithm increases with the number of APs , while that of VCOWO algorithm is independent of . The VCOWO algorithm could approach the optimal results under the equivalent conditions. To be more specific, the VCOWO could achieve the upper bound of the WiFi offloading performance achieved by COWO algorithm. Under the equivalent conditions, all WiFi networks are assumed to serve the users in a cooperation approach and could achieve higher throughput with lower blocking probability with the centralized control of WiFi channels allocation.

1: Calculate parameters in equivalent WiFi network

2: Initialize: ,

3:  while    do

4:Calculate steady-state probability distribution under given

5:Calculate utility function

6:Update according to golden section method

7: end while

8: Set the optimal offload factor vector .

Algorithm 2: The proposed VCOWO.

6. Numerical Results

In this section, we evaluate the proposed algorithms with the OTSO scheme [18] in system utility as formulated in (28), blocking probability and throughput. The OTSO algorithm enables offloading user to a WiFi network whenever users enter into its coverage, which means the offloading ratio . Further, we show the approximation of VCOWO to COWO under equivalent conditions. Moreover, we verify the impacts of user mobility on WiFi offloading performance. In the simulation, we consider the scenario that contains one MBS and multiple APs. Each AP is assumed to have no overlap with the others, and other parameters are shown in Table 2.

Table 2: Simulation parameters.

Parameters
MBS
AP1
AP2
AP3
AP4

/(m)
1000
200
150
100
100

Channels
30
8
7
6
5

(Mbps)
9.6
13.6
13.6
13.6
13.6

(links/s)
1/120

(m/s)
8

6.1. Performance Comparison

In this section, we compare the WiFi offloading performance with the multiple APs between COWO and OTSO under user arriving rate ranging from 0 to 0.5. First, Figure 3 illustrates that the system utility varies with the total active user arriving rate . When , it is found that the system utility grows with , and COWO algorithm has similar performance as the OTSO. It is because when the network is unsaturated and the congestion is small, the system utility increases by the throughput bought by the user arriving. The throughput increases since more users flow to the network and make the use of channels when unsaturated. However, it can be observed from Figure 3 that the utility in OTSO decreases with . Intuitively, when traffic load in the WiFi network is heavy, offloading data to WiFi network whenever possible will nevertheless incur higher blocking probability, which may degrade the utility function.

Figure 3: System utility varies with the users arriving rate.

To further demonstrate the network congestion in WiFi offloading, we compare the blocking probability in Figure 4. It is obviously observed that the blocking probability in the proposed algorithm is much lower than the OTSO, especially when is larger. Without consideration of network congestion, the blocking probability grows rapidly with the active user arriving rate , which is also indicated in the decrease of system utility as shown in Figure 3.

Figure 4: Blocking probability varies with the users arriving rate.

Moreover, Figure 5 illustrates that the throughput varies with the user arriving rate . It can be observed that the throughput increases with . It is because the throughput increases with user arriving when the channels are not fully utilized. When is larger, the throughput does not change with the . Intuitively, when all the channels have been occupied, the increase of users will not bring higher throughput. Thus, it can be observed from Figure 5, constrained by the number of channels, that the throughput will not increase with the user arriving rate. The throughput is larger with more WiFi APs, e.g., more available channels. To conclude, the proposed algorithm could maximize the throughput with lower blocking probability than OTSO algorithm.

Figure 5: Throughput varies with the users arriving rate.

6.2. Approximation Comparison

In this section, we evaluate the VCOWO algorithm under the equivalent conditions. We consider the 4 or 5 APs separately with equal coverage (m), and each with 5 channels, and it correspondingly has 20 or 25 channels for the virtual WiFi network. Assume the maximum user arriving rate range is 0.7. Other parameters all satisfy the equivalent conditions in Section 5. In Figures 6–8, we compare the performance with OTSO under different channel numbers in WiFi network. Clearly, more channels offer service to more users and nevertheless could contribute to the larger system utility. It can be observed that the VCOWO achieves similar system utility, blocking probability and throughput to COWO algorithm. To be more specific, the performance in VCOWO is the upper bound of COWO under the equivalent situation. Recall the equivalent assumption in Section 5; it is found that all the channels of different APs could be jointly scheduled in the virtual WiFi network. That is to say, all the APs serve the users with the association, rather than separately schedule the users’ network access in COWO algorithm. Thus, VCOWO could give the tight upper bound of COWO algorithm performance with much lower complexity under the equivalent conditions.

Figure 6: System utility comparison varies with the users arriving rate.

Figure 7: Blocking probability comparison varies with the active users arriving rate.

Figure 8: Throughput comparison varies with the user arriving rate.

6.3. User Mobility

In this section, we verify the impacts of user mobility on WiFi offloading as illustrated in Section 5. In Figures 9 and 10, the COWO and OTSO are evaluated under the different velocities with user arriving rate ranging from 0 to 0.5. It can be observed from Figure 9 that the throughput decreases with the velocity. Recall the fact that it was indicated in (41) that throughput is the monotonically decreasing function about the velocity . Intuitively, we deduce it in another way. At one-moment , suppose that there are users in th AP; the users move with the average velocity , and the . Then, for the next moment, , with the increase of , we have , and . Therefore, the increase of reduces the average number of users in the network and thus lowers the throughput and the blocking probability. The user mobility increases the opportunities to access, and could, to some extent, reduce the access blocking in the congested situation. The mobility only gives more users the chances to access the network. However, it can not guarantee the access success or improve the throughput. Actually, constrained by the number of total channels, the capacity will not increase. Even worse, the mobility impedes sustainable connection and nevertheless lowers the throughput in general.

Figure 9: Throughput evaluation under different mobility velocity .

Figure 10: Blocking probability evaluation under different user mobility velocity .

7. Conclusion

In this paper, we studied WiFi offloading with user mobility management in smart communications. With consideration of the dynamic network nature, including the network congestion and the user mobility, we proposed the COWO algorithm to optimize the offloading ratio for each AP, which aims to maximize the throughput with lower blocking probability. By viewing all the APs as one virtual WiFi network, the VCOWO algorithm is proposed with much lower complexity. Under equivalent conditions, the VCOWO algorithm could approximate optimal results by COWO algorithm and give an upper bound of offloading performance of COWO algorithm. Moreover, we investigated the impacts of user mobility on WiFi offloading. It is found that the increase of velocity in user mobility degrades the throughput performance, but could reduce the blocking in some way. For future work, it is an interesting direction for future research to consider other mobility models, especially the heavy-tailed distribution model, which was shown to be more accurate for modeling human mobility. It is promising to deal with the network congestion and load balance in smart communications through artificial intelligence technologies and wireless big data support, which could well predict the user random mobility patterns with statistical information of the possible user’s trajectories and schedule the network access in a smarter way.

Appendix

A. Proof of the Bisection Method for Solving

Due to being the root of , the derivation of is

where

and

The maximum number of users served by the network will not be more than the channel number; i.e., , and thus we have . Overall, it is found that , and thus is an increasing function of . Moreover, , , and . Therefore, , , and the bisection method can be used for the calculation of .

B. Proof of Convex Optimization

The utility function can be divided into two parts:

where the denotes the utility of th WiFi network and represents the cellular system utility.

The proof of convex optimization in (28) is equal to proving that the utility matrix is a negative definite matrix. Hessian matrix is

where

Since , we have

Then, the proof that the Hessian matrix is negative definite is equal to proving that its eigenvalue is nonnegative. That is,

We assume that and hold, which has been proved in Appendix B with the bound of . Thus, the convex optimization question is converted to prove . The first-order and second-order partial derivative of are, respectively, given by

and

The lower and upper bound of give the conditions [42]

and, thus, we have

which completes the proof.

C. Proof of

We prove the and with the bound constraint in Appendix B. From (10), the derivative of with is

Then, the fist-order derivative of with is calculated as

With (12), the derivative of with is

With (C.1) and (C.3), the derivative of with could be rewritten as

Since the channel used in is less than the total channel in -th AP, i.e., , thus, we have . Similarly, can be proved.

Data Availability

The authors confirm that the data supporting the findings of this study are available within the article. All the simulation parameters and other experimental details are clearly shown in this article, which could be used to reproduce the experiment related to our findings.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work of B. Liu, Q. Zhu, and H. Zhu was supported in part by the National Natural Science Foundation of China (61427801, 61571234, and 61631020) and in part by the Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX18_0893). The work of W. Tan was supported in part by the Project of Educational Commission of Guangdong Province of China under Grant 2017KQNCX155.

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