A theoretical approach to optimal association control in vehicular WiFi networks
 Jaeryong Hwang^{1},
 Jaehyuk Choi^{2},
 Joon Yoo^{2} and
 Chongkwon Kim^{3}Email author
https://doi.org/10.1186/168714992014192
© Hwang et al.; licensee Springer. 2014
Received: 30 January 2014
Accepted: 3 November 2014
Published: 17 November 2014
Abstract
The recent ubiquitous deployment of WiFi access points (APs) has offered vehicles to use the highspeed and lowcost Internet service via the roadside APs. However, the high mobility of vehicles and the limited coverage of APs render some challenges. First, it results in frequent handoffs, thus leading to long delay and low service availability. Second, the available AP sets and their channel quality change dynamically, making the AP selection problem even harder. Therefore, there is a strong need to develop an efficient association control mechanism that provides efficient vehicular WiFi access. In this paper, we present a theoretical framework to formulate the optimal association problem through nonlinear integer programming, whose objective function is to maximize the throughput or to minimize the handoff overhead. We show that this problem holds the totally unimodular (TU) property and is thus solvable in polynomial time. Then, we study the optimality of association control by comparing existing online algorithms through real tracebased simulations. The results show that there exists a large performance gap between the performance of existing online algorithms and the optimal one. We also observe that the association control algorithm can be further improved if it has access to future knowledge. Particularly, the offline optimal with future AP information improves the performance of the local optimal by up to 10%.
Keywords
1 Introduction
Recently, WiFi access technology for moving vehicles has drawn considerable attention, since it has been shown that WiFi connectivity is feasible even at vehicular speed [1–9]. Moreover, the wide deployment of numerous WiFi access points (APs) in the urban areas [3] provides costeffective and broadband Internet connectivity to the passengers. However, there are still several challenges remaining for vehicular WiFi access. First, different from the traditional data access systems for stationary users, the connection time in vehicular WiFi access is typically short, since the vehicles move fast and the coverage of roadside APs is limited. Furthermore, the available AP sets and their wireless channel change dynamically due to the high vehicle mobility. To sustain the connectivity and maintain high link quality, the vehicles have to continuously associate with different APs and thus conduct frequent handoffs. The frequent handoffs, however, involve very high overhead such as association, authentication, and DHCP latency [3, 6, 9, 10]. Therefore, deciding when to perform the handoff and determining the best AP among the candidate APs are a very important, yet challenging, problem.
Several studies have been proposed to address the association and handoff problem. Giannoulis et al. [4] have suggested a handoff protocol which supports a vehicle’s mobility in an urban mesh topology. Kim et al. [11] presented an association control solution that minimizes the frequency of handoffs of mobile devices. A recent study of Xie et al. [12] aims to improve the overall throughput and fairness for all users in an enterprise network. However, the association scheduling problem considering handoff overhead from the vehicle’s point of view has not yet been addressed well despite its importance, while these existing solutions improve the association performance and provide useful insights in the design of online association algorithms.
In this paper, we present a theoretical framework for analyzing the association problem in vehicular WiFi access networks, with the aim of understanding and improving the performance of several association control schemes. We first formulate the association problem as a nonlinear integer program in which we incorporate the influence of a vehicle’s mobility, available effective bit rate from APs, and handoff cost. Our key contribution is to show that the nonlinear integer program can be cased into a linear programming problem thanks to the properties of the binary decision variable. Furthermore, we show that the linearized problem is solvable in polynomial time because the constraint matrix of the problem meets the totally unimodular (TU) property. The TU implies that all vertexes of the polyhedron representing the feasible set are integral; thus, the linear program obtained by relaxing the integrality constraints of the corresponding integer program can find an optimal integral solution. We prove that the optimal association problems satisfy the TU property, and we validate that these problems are solvable in polynomial time.
We conduct extensive simulations by using the metropolitan city of Seoul bus mobility trace gathered from the Seoul Transport Operation & Information Service (TOPIS) database [13]. For the simulations, we have collected the real movement traces for more than a month of approximately 7,000 buses. We compare three existing online algorithms  bandwidthbased association, durationbased association, and combined metricbased algorithms  with the optimal solution. We have observed that there exists a big performance gap between the performance of the three existing association algorithms and the optimal throughput; these algorithms achieve only 54%, 68%, and 81% throughput compared to the local optimal association control. The results also show that the association control algorithm can be further improved if it has access to future knowledge. Particularly, the offline optimal algorithm with future AP information improves the performance of the local optimal by up to 10%.
The rest of the paper is organized as follows. We present the related work in Section 2, and in Section 3, we describe the system model and problem statement. From the given information of APs, we present the optimal association control problems in Section 4. We evaluate the performance of the online algorithms using simulation in Section 5. We finally conclude our paper in Section 6.
2 Related work
Related work falls under two categories: (i) techniques for improving the reliability and performance of vehicular WiFi Internet access, and (ii) studies addressing the association and handoff problems of mobile users.
Several recent studies have explored the Internet access via roadside access points for the moving vehicles. In [1], Ott and Kutscher identified the characteristics of the drivethru network access. They show that vehicles experience three different connectivity phases while driving by an access point: the entry phase, production phase, and exit phase. During the entry phase and exit phase, the vehicles suffer from poor connectivity, while in the production phase, it maintains good connection quality. In [2], Hadaller et al. studied the impact of the above three phases on the TCP goodput through detailed experiments in several drivethru network access scenarios. In [3], Bychkovsky et al. conducted experimental studies with numerous open WiFi access points for vehicular Internet access in urban environments. The experiments observe that the mean connection duration is fairly short (12 to 13 s), and it requires high latency to complete the IP address acquisition. This highly affects the application layer performance. Our goal is to improve the application layer performance through the association control techniques.
Recently, there are also efforts to support IP connectivity in vehicular Internet access by utilizing Network Mobility (NEMO) [14–16]. They consider IPlayer handover for vehicular network, but we are dealing with the AP association problem. Ad hoc communication based on relay access has been explored to extend the infrastructure coverage in drivethru network access [17]. Although we can utilize the ad hoc communication in this paper, for simplicity, we only consider the singlehop communication between vehicles and APs.
There are various approaches to improve the mobile user performance by association control. Association control is a decision process of (i) how to select an AP and (ii) when to conduct handoff to another AP. Typically, mobile users use the channel quality as the association metric for handoff initiation. In [4], Giannoulis et al. proposed a handoff technique that supports vehicular users in the urban mesh network deployed in the southeast part of Houston. They consider APquality scores as well as channel quality to associate with the APs. The APquality scores are determined by average load and backhaul connectivity. However, they do not consider the connection duration when controlling the handoff frequency. In [11], Kim et al. proposed an association control solution that minimizes the frequency of handoffs. The mobile devices always select the APs with the longest connection time due to the severe handoff overhead, such as association and DHCP latency. In [9], Deshpande et al. developed new handoff and data transfer strategies for moving vehicles in urban areas. They reduce the connection setup latency by using RF fingerprints of APs. The vehicles know the useful APs in the current location beforehand through the RF fingerprint, and thus, the vehicle conducts scripted handoff. In [12], Lei et al. designed an efficient association algorithm to achieve total throughput maximization and fairness of all users. Similar to our method, they also presented an optimization problem of association control for efficiency and fairness. However, they do not consider the handoff overhead. In this paper, we deal with the association scheduling problem considering handoff overhead from the vehicle’s point of view.
3 System model
This section describes the network model and assumptions and then summarizes the notations which are used throughout the paper.
3.1 System model and assumptions
We consider a vehicletoinfrastructure communication system in which vehicles move along roads and opportunistically connect to the roadside WiFi APs for wireless Internet access. Each vehicle adopts an association control mechanism to associate with the APs based on the predefined association criterion (e.g., the AP with the strongest signal strength or the least offered load). Due to the mobility of the vehicle and the limited coverage of APs, the number of available APs to a vehicle is finite and varies over time. The vehicle can associate with only one AP at a given time. A handoff is conducted if it is better to use a new AP compared to the current AP. This handoff may result in throughput gain but incur overhead in terms of delays^{a}.
We assume that vehicles have Global Positioning System (GPS)equipped devices, so that they know their location, speed, and direction information. When the vehicles get their mobility information and the traffic information of roads on runtime, they can predict their location after some period of time by dead reckoning [18]. We also assume that we have the location information of the roadside APs. In fact, several open or commercial service providers have provided a list of APs and their attributes such as the geographical coordinates, channel numbers, and estimated backhaul capacity [19, 20]. From the vehicle’s mobility prediction and roadside AP’s information, they can compute the estimated handoff overhead and effective bit rate. We study an optimal association scheduling method that determines the optimal AP selection sequence by considering the tradeoff between the throughput gain from conducting handoff and the resulting overhead.
Without loss of generality, we consider the association control of a target vehicle over the time interval [0,T]. Time is divided into discrete time slots that have variable durations [t_{ i },t_{i+1}], represented by a set of slot index I=1,2,…,m, where 0≤t_{ i }<t_{i+1}<…<t_{ m }≤T for time index i∈I. At each time slot i, the vehicle is given a set of available roadside APs, J_{ i }, which will be fixed during the given time interval.
Let r_{ ij } be the effective bit rate (bit/s) of the link between the vehicle and AP j at time slot i. The bit rate r_{ ij } may vary over time as the vehicle moves. For simplicity of analysis without losing the key insights of the association scheduling problem, however, we assume that at a given time slot i, the effective bit rate r_{ ij } is fixed. We also assume that the scheduling algorithms know the effective bit rate r_{ ij } at that time. Nevertheless, note that our analytical framework can capture the change of the bit rate by dividing a time slot into several nonoverlapping subintervals with different bit rates, at the cost of analytical complexity.
If the AP associated with the vehicle at slot i+1 is different from the AP of the duration of slot i, then the vehicle is said to have undergone a handoff at slot i+1. When the handoff takes place, the mobile users in the vehicle cannot access the Internet for some time duration. We assume that the handoff overhead is constant and c (second) denotes the overhead.
3.2 Notations
Prior to the problem formulation, we summarize the indices and notations which are used throughout the paper in the following:

i∈I index interval, where I = {1,2,…,m}

j∈J index APs, where J = {1,2,…,n} the set of all the APs

J_{ i }⊆J represents available APs in interval i

s_{ i } is the length of interval i, i.e., s_{ i } = t_{i+1}t_{ i }

r_{ ij } is the effective bit rate of AP j in interval i

c is the handoff overhead (second)
For instance, the available APs at time slot 3 are AP1, AP2, and AP3; thus, J_{3}={AP1,AP2,AP3}.
4 The offline optimal scheduling problem
Based on the above network model and assumptions, our objective is to derive an optimal association scheduling method that maximizes the throughput of a target vehicle. In this section, we show that computing the offline optimal schedule can be casted into a linear programming problem, and thus, we have the optimal solution in polynomial time. In addition to the optimal solution that maximizes the throughput, our framework can be extended to obtain the optimality of association that minimizes the frequency of handoffs occurring in vehicles (see Appendix).
4.1 Objective function
The objective of the association scheduling is to maximize the total transmitted bits during the time intervals I = {1,2,…,m} for the given available APs and effective bit rate information. Computing the optimal association scheduling is equivalent to finding the optimal values for x_{ ij },i∈I andj∈J_{ i } by optimizing the tradeoff between the throughput gain from a handoff and the resulting reassociation overhead c.
Also, Equation 2 is provided since the vehicle can associate with at most one AP among the available APs according to the system model given above.
4.2 Computing the optimal scheduling problem
The objective function given in Equation 1 is a nonlinear integer programming problem with binary integer decision variable x_{ ij }. Although the optimal solution of the problem is a subset of all combinatorial and thus is solvable by bruteforce enumerative algorithms [21], its calculation complexity is too high to be practical.
To address this challenging issue, we transform the above nonlinear function into a linear function by exploiting the properties of the binary decision variable.
However, it is still nontrivial to solve the above integer program (IP) because of the integrality constraints. We will discuss the property of the objective function and show that its solution can be obtained in polynomial time.
Let us consider a IP max $\{{c}^{T}x:\mathit{\text{Ax}}\le b,x\in {\mathbb{Z}}_{n}\}$ and its linear programming (LP) relaxation max $\{{c}^{T}x:\mathit{\text{Ax}}\le b,x\in {\mathbb{R}}_{n}\}$. A linear program in real variables is said to be integral if it has at least one optimal solution which is integral. Likewise, a polyhedron P={x:A x≤b} is said to be integral if the linear program has an optimum solution x^{∗} with integer coordinates. According to the LP theory [22], solving an IP over an integral polyhedron can be done in polynomial time through LP relaxation of integral constraints, and then the LP has an integral optimal solution.
One common way of proving that a polyhedron is integral is to show that its constraint matrix is TU [21], where a matrix is said to be totally unimodular if the determinant of each square submatrix is 0,+1,or 1. This leads to the following proposition.
Proposition 1
If a matrix A is totally unimodular and b is an integral vector, then the polyhedron P = {x:A x≤b} is integral.
Proof
See [21].
We now show that the constraint matrix of the objective function given in Equation 7 satisfies TU. At this point, however, it is not easy to recognize that the constraint matrix in Equation 7 is TU by directly using the definition of TU. Thus, we consider a general sufficient condition with the following theorem.
Theorem 1
 (i)
a _{ pq }∈{0,+1,1} for all p, q, and
 (ii)for any subset M of the rows of A, there exists a partition (M _{1}, M _{2}) of M such that each column q satisfies$\left\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\right\le 1.$
Proof.
See [22].
Then, one can easily express the above constraint as the form of A x≤b, where A denotes the constraint matric, defined as $A\triangleq \left[{a}_{\mathit{\text{pq}}}\right]$, $\mathbf{\text{x}}\triangleq {(x,z)}^{T}\text{and}b\triangleq \left({b}_{q}\right)$ such that b_{ q }≥0. Then, we have the following theorem.
Theorem 2.
Let A be the constraint matrix of the linear integer problem given in Equation 7 (i.e., in the form of A x≤b), then matrix A is totally unimodular.
Proof.
We first introduce a partitioning method that obtains a partition (M_{1},M_{2}) for an arbitrary subset M of the rows of A. Let B_{ i } be a subset of A such that B_{ i } includes all the rows corresponding to the coefficients of x_{ ij } in the constraints given in Equation 8. Let ${Q}_{i}^{1}$, ${Q}_{i}^{2}$, and ${Q}_{i}^{3}$ denote the submatrices of A containing all the rows corresponding to each constraint in Equations 8a, 8b, and 8c for interval i, respectively. Then, B_{ i } is given as ${B}_{i}={Q}_{i}^{1}\cup {Q}_{i}^{2}\cup {Q}_{i+1}^{3}$. For an arbitrary subset M, Algorithm 1 generates a partition (M_{1},M_{2}) by iteratively testing every row in B_{ i }∩M and assigning the row into M_{1} or M_{2} for i∈I. It is straightforward to show that this portioning rule obtains a partition (M_{1},M_{2}) of M correctly.
For a given arbitrary subset M of rows of A=[a_{ pq }], we can have its partition (M_{1},M_{2}) based on Algorithm 1. Since condition (i) of Theorem 1 is already satisfied in Equation 8, we only need to show that the partition (M_{1},M_{2}) will satisfy condition (ii) of Theorem 1.
 (i)
N _{ q }=0,1: nothing to prove.
 (ii)N _{ q }= 2: There are three subcases for $({a}_{\mathit{\text{pq}}},{a}_{{p}^{\prime}q})$ where p≠p ^{′}:
 (ii1)
$({a}_{\mathit{\text{pq}}},{a}_{{p}^{\prime}q})=\{1,1\}$: In this case, $p\in {Q}_{i}^{2}$ and ${p}^{\prime}\in {Q}_{i+1}^{3}$ (or ${p}^{\prime}\in {Q}_{i}^{2}$ and $p\in {Q}_{i+1}^{3}$) for ∃i. This happens only if B_{ i }∩M≠∅ by the definition of B_{ i }. Thus, it is belonging to the case 12 in Algorithm 1; the two rows p and p^{′} were partitioned to different partitions. Thus, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
 (ii2)
(ii2) {1,1} (or {1,1}): Since $p\in {Q}_{i}^{1}$ for ∃i, and this is belonging to case 11 in Algorithm 1, the two rows p and p^{′} in B_{ i }∩M should be assigned to the same partition M_{1} or M_{2}. Thus, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
 (ii3)
{1,1}: This can only happen for a column with respect to z_{ ij } for ∃i(B_{i1}and B_{ i }). Then, the columns of M_{1}(or M_{2}) and B_{ i }∩M are belonging to the case 12 in Algorithm 1; thus, the two rows p and p^{′} are assigned to different partitions. Thus, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
 (ii1)
 (iii)
N _{ q } = 3: In this case, among three nonzero a _{ pq }, two of them should be 1 and the other one is +1 where p∈M. The partition (M _{1}, M _{2}) is performed according to case 11 in Algorithm 1, and the summation of 1,1,1 is 1. Thus, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
In all cases, nonzeros in a column for the partition (M_{1},M_{2}) obtained from Algorithm 1 satisfy condition (ii) of Theorem 1. Therefore, the constraint matrix A is totally unimodular.
Solving an IP over an integral polyhedron P can be done in polynomial time. Therefore, from the results of Theorems 1 and 2, we can obtain the solution of the integer program given in Equation 8 in polynomial time.
Remark
In addition to the optimal scheduling problem that maximizes throughput, our framework can be extended to obtain the optimality of association that minimizes the frequency of handoffs occurring in vehicles. In the Appendix, we formulate the optimal handoff minimization scheduling problem as a nonlinear integer programming and show that this problem also holds the TU property, and thus is solvable in polynomial time.
5 Performance comparison of online algorithms with perfect link information
The optimal association scheduling solution renders practical challenges since it requires perfect future knowledge, such as the set of available roadside APs, the available link duration (connection time), and the effective bit rate between the vehicle and the roadside APs. Nevertheless, the optimal scheduling solution still provides an important insight. Further, it can also be used as a baseline to compare the performance of various online association control algorithms.
In this section, we evaluate the performance of three existing online algorithms by comparing them with the optimal association algorithm given in Section 4 in a mobile wireless network via real bus tracedriven simulations. We first describe the evaluation methodology for the performance study, then discuss the evaluation results that compare the performance of the optimal association with conventional association algorithms using various handoff metrics.
5.1 Methodology
5.1.1 Association control algorithms under consideration
We compare three association control schemes with the optimal solution: (i) bandwidthbased association, (ii) durationbased association, and (iii) combined metricbased association. We give some explanations for these schemes as follows:

Bandwidthbased association (Ba): The most widely used association control method is to use some measurable metrics (e.g., signal strength or collision probability) that infer the achievable throughput. We study the online association control algorithm proposed in [23], in which the mobile nodes select the AP with the largest effective bit rate whenever it meets a new AP or the current associated AP is disconnected. This greedy decisionbased algorithm does not consider the handoff overhead, thus suffering from frequent handoffs.

Durationbased association (Du): There have been some research efforts to alleviate the handoff latency problem. We study the online greedy algorithm proposed in [11] in which the mobile nodes select the AP with the longest duration whenever the current AP becomes unavailable. In [11], this greedy online algorithm is optimal with an offline setting where the perfect future knowledge is given.

Combined metricbased association (BaDu): The association control scheme proposed in [24] exploits the tradeoff between the bandwidthbased association and durationbased association algorithms. The mobile node selects the AP with the highest bandwidth duration whenever the mobile node meets new APs or the current associated AP becomes unavailable.
Note that the performance of these online association control algorithms significantly hinges on how they can accurately estimate the required metrics, such as available bandwidth and connection duration. However, our focus in this section is not to study the accuracy of the link information measurement, but to compare various online scheduling algorithms with the optimal association control algorithm and extract the relative merits of these algorithms. Therefore, we assume that these online algorithms have perfect knowledge of the link between the vehicle and available APs, e.g., the effective bit rate and connection time information^{b}, so that we focus on characterizing their difference induced from their different handoff criteria.
5.1.2 Online optimum and offline optimum
We use the solution presented in Section 4 as the optimal association control algorithm. We categorize the solution into two algorithms  (i) offline optimal and (ii) local optimal  depending on the availability of future knowledge.

Local online optimal association (LO): The local optimal (LO) is an online algorithm where only J_{ i } is given at slot i without any knowledge of future sets J_{i+1},J_{i+2},⋯,J_{ m }. Therefore, with the local online optimal algorithm, a mobile user schedules the optimal association sequence only with a set of current available APs according to the presented method in Section 4. The mobile user reschedules the association sequence whenever the mobile node meets new APs or the current associated AP becomes unavailable.

Offline optimal association (Optimal): The offline optimal algorithm is given J_{ i } information for ∀i∈I, i.e., all sets of available J_{ i } are given in advance.
5.1.3 Simulation setup
5.2 Comparison results and discussion
Execution time for solving optimal association
Number of APs  Time (s)  Number of variables 

50  0.016  98 ×50×2 
100  0.063  193 ×100×2 
200  0.296  574 ×300×2 
300  0.843  976 ×500×2 
To evaluate the throughput maximization problem, we selected a road segment as shown in Figure 2 and utilized the 1h movement traces of 434 buses passing this area. Then, based on the association metric of each association control algorithm, we scheduled the association of 434 buses. We set the effective bit rate r according to the geographical distance between the buses and the WiFi APs, i.e., for the short link distance, we use high bit rate and vice versa. In the network model explained in Section 3, the bit rate is known to each bus and fixed for a given time slot. The reassociation overhead is set to 2 s (c = 2).
Figure 3 shows the normalized transmitted bits of five association control algorithms: bandwidthbased association (Ba), durationbased association (Du), combined metricbased association (BaDu), local optimum association (LO), and offline optimal association (Optimal) algorithms. We observe that the durationbased association (Du) shows the worst performance compared to the other online algorithms. The main reason is that the durationbased algorithm connects to the APs with longer available link durations for less frequent reassociation, without considering the effective bit rate. As a result, the durationbased algorithm suffers from low throughput once the users associate with an AP with low bandwidth. The bandwidthbased association is shown to achieve a relatively high bit rate compared to the durationbased association. However, we clearly observe that it experiences frequent handoffs, because the connection duration is not considered, and thus, it cannot achieve the best performance. The combined metricbased association control algorithm (BaDu), is shown to achieve the highest performance among the three online association control algorithms. This is mainly because it considers the link bandwidth as well as the available link duration and exploits their tradeoff. However, we observe that it sometimes selects the AP that has low bit rates but with a very long connection duration, and suffers from low throughput. From Figure 4, we can see that these three algorithms achieve reasonable performance, but there is a large performance gap between them and the optimal association control. The bandwidthbased association (Ba), durationbased association (Du), and combined metricbased association (BaDu) achieve only 54%, 68%, and 81% throughput compared to the local optimal association control (LO). We also observe that the association control algorithm can be further improved if it has access to future knowledge. Particularly, the offline optimal (Optimal) improves the performance of local optimal (LO) by up to 10%.
6 Conclusions
In this paper, we studied the optimal association control techniques for the vehicular WiFi networks. The tradeoff between using the effective bit rate and available link duration of APs as an association metric motivates the need for an efficient association control scheme. We have introduced a theoretical framework for optimal association control and prove that the optimization problem is solved in polynomial time. We performed extensive simulation studies to compare the existing online association control schemes with the obtained optimal control. We observed that there exists a large performance gap between the performance of the three existing association algorithms and the optimal throughput. We have shown that the association control algorithm can be further improved if it has access to future knowledge. Particularly, the offline optimal with future AP information improves the performance of local optimal by up to 10%.
Endnotes
^{a} The handoff overhead usually consists of association delay, IP acquisition delay (DHCP delay), and ARP lookup delay [9].
^{b} Note that several advanced estimation mechanisms have been proposed [24], so that the above online algorithms can acquire the link information in practice. For example, the IEEE 802.11 beacon message of roadside APs can be used to help moving vehicles obtain the link information. The beacon message has reserved fields which may include vendorspecific information. Thus, the APs can deliver information, such as the current location and backhaul capacity, through the reserved fields in the beacon message. The client vehicle that receives the beacon message estimates the duration through the above equations.
Appendix
Handoff minimization
In this section, we tackle the problem of minimizing handoffs. As vehicles move along their route, they continuously keep associating with different APs to maintain the connectivity to the Internet. However, such frequent handoffs may cause unacceptable service interruption. Therefore, it is desirable to take an approach to derive the optimal association scheduling that minimizes the handoff frequency. The definitions of the notations used in this section can be found in the previous sections.
where z_{ ij }=x_{ ij }·x_{i1j} which is equivalent to the settings in the throughput maximization problem given in Equation 7.
Then, we have the following theorem.
Theorem 3.
Let A be the constraint matrix given in Equation 10, then A is totally unimodular.
Proof.
 (i)
N _{ q } = 0,1: nothing to prove.
 (ii)N _{ q } = 2: There are three cases for (${a}_{\mathit{\text{pq}}},{a}_{{p}^{\prime}q}$) where p≠p ^{′}
 (ii1)
(${a}_{\mathit{\text{pq}}},{a}_{{p}^{\prime}q}$) = {1,1}: There are two possible subcases: (1) $p\in {Q}_{i}^{2}$ and ${p}^{\prime}\in {Q}_{i+1}^{3}$ for ∃i, (2) $p\in {Q}_{i}^{12}$ and ${p}^{\prime}\in {Q}_{i+1}^{3}\left(\text{or}{Q}_{i}^{2}\right)$ for ∃i. By the definition of B_{ i }, this is treated in M∩B_{ i }≠∅, and by case 22 and case 23 of Algorithm 2, the two rows p and p^{′} are partitioned to different partitions. Thus, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
 (ii2)
{1,1}: In this case, it is given that $p\in {Q}_{i}^{11}$ and ${p}^{\prime}\in {Q}_{i}^{12},{Q}_{i}^{2},$ or ${Q}_{i+1}^{3}$. Since $p\in {Q}_{i}^{11}$ for ∃i, and this is dealt with case 21 and case 22, all rows in M∩B_{ i } are assigned to the same partition. Thus, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
 (ii3)
{1,1}: This can happen for a column with respect to z_{ ij } for B_{i1} and B_{ i }. By checking the columns of M_{1}(or M_{2}) and M∩B_{ i } in each case of Algorithm 1, they are assigned to a different partition M_{1} or M_{2}. Thus, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
 (ii1)
 (iii)N _{ q } = 3: In this case, we have two possible cases of ${a}_{\mathit{\text{pq}}},{a}_{{p}^{\prime}q},{a}_{{p}^{\mathrm{\prime \prime}}q}$ for three rows p, p ^{′}, and p ^{′′}.
 (iii1)
{1,1,1}: There are two subcases: (1) $p\in {Q}_{i}^{11}$, ${p}^{\prime}\in {Q}_{i}^{12}$ and ${p}^{\mathrm{\prime \prime}}\in {Q}_{i+1}^{3}\left(\text{or}{Q}_{i}^{2}\right)$ or (2) $p\in {Q}_{i}^{11}$, ${p}^{\prime}\in {Q}_{i}^{2}$ and ${p}^{\mathrm{\prime \prime}}\in {Q}_{i+1}^{3}$. In the case of (1), case 22 of Algorithm 2 assigns rows {p,p^{′},p^{′′}} into {p,p^{′}}∈M_{1}(or M_{2}) and {p^{′′}}∈M_{1}(or M_{2}) or {p,p^{′},p^{′′}}∈M_{1}(or M_{2}). On the other hand, in case of the (2), all rows are assigned to the same partition M_{1} or M_{2} by case 21 of Algorithm 2. Since the difference of their summations in the above cases cannot exceed 1, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
 (iii2)
{1,1,1}: By case 23 of Algorithm 2, all rows are partitioned to different partitions {p,p^{′}(or p^{″})} and {p^{″}(or p^{′})}. Thus, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
 (iii1)
 (iv)
N _{ q }=4: Note that we have only one +1 and three 1 among a _{ pq } for p∈M. By case 22, {1,1} is first assigned to the same partition, then {1,1} is divided into M _{1} or M _{2}. Since its summation is zero, $\sum _{p\in {M}_{1}}{a}_{\mathit{\text{pq}}}\sum _{p\in {M}_{2}}{a}_{\mathit{\text{pq}}}\le 1$ holds.
For all possible cases, we prove that the partition satisfies condition (ii) of Theorem 1. Thus, the constraint matrix A is totally unimodular.
Therefore, we have the solution of the original integer problem given in Equation 9 in polynomial time.
Declarations
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (No. 2012R1A1A1014755, NRF2013R1A1A1006823) and Korea Naval Institute for Ocean Research.
Authors’ Affiliations
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