Role and channel assignments for wireless mesh networks
using hybrid approach
q
Andy An-Kai Jeng, Rong-Hong Jan
*Department of Computer Science, National Chiao Tung University, Hsinchu 30050, Taiwan, ROC
a r t i c l e
i n f o
Article history:
Received 7 September 2008
Received in revised form 20 January 2009 Accepted 29 March 2009
Available online 17 April 2009 Responsible Editor: L. Lenzini Keywords:
Wireless mesh network Multi-channel multi-interface Channel assignment Role assignment NP-hardness Greedy algorithm
a b s t r a c t
The wireless mesh network has been considered one of the most promising techniques to extend the broadband access to the last mile. To utilize multiple channels on more than one interface, a number of approaches have been proposed. In particular, the hybrid strat-egy that combines the benefits of high channel diversity and low coordination cost has seen growing interest in recent studies. In this paper, we define two optimization prob-lems, named the role assignment and the semi-fixed channel assignment, to characterize the unique feature in the hybrid strategy. We proved that the two problems are NP-hard even if the transmission ranges of interfaces are equal. In order to solve our problems in reasonable time, we design efficient algorithms. For the role assignment problem, we give an 1/2-approximate algorithm to find a nearly optimal solution. For the semi-fixed channel assignment problem, a heuristic algorithm, based on transferring from a coloring-based problem, is proposed. Experimental results show that optimizing the defined problems is indeed beneficial to improve the network throughput, and the proposed algorithms are sig-nificantly superior to existing methods.
Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
The demand on wireless broadband access has been continually burgeoning in the recent years. Particularly, the wireless mesh network (WMN) is considered one of the most promising techniques for extending broadband access to the last mile[1]. The WMN consists of a set of mesh access points (MAPs). A mobile station can access to the network by connecting to a nearby MAP. Each MAP acts as a wireless router to forward traffic hop-by-hop to destinations. Thus, by deploying in this fashion, a backhaul network can be easily built up without wired connection.
Unlike the mobile ad hoc network (MAENT), nodes in WMNs are usually static with continuous power supplies.
Hence, the issues about mobility and energy efficiency are less critical. Instead, the capacity of the backhaul is a major concern. The backhaul has to provide sufficient bandwidth to support traffic between MAPs and Internet gateways as well as communications between MAPs them-selves. To increase the capacity, one approach is to utilize multiple channels[2,3]. The IEEE 802.11 b/g and 802.11a standards provide up to 3 and 12 non-overlapped chan-nels, respectively, in 2.4 GHz and 5 GHz spectrums. Nodes within the transmission range of each other can turn their interfaces to different non-overlapped channels to avoid interference. Besides, the throughput can be further en-hanced by equipping nodes with multiple interfaces[4,5]. That is, a node with two or more interfaces can perform simultaneous transmissions and/or receptions on different channels to increase throughput in parallel.
Ideally, the capacity can be multiplied by H times if there are H interfaces and channels available to each node. However, in practice, it is too expensive to equip nodes with as many interfaces as the number of channels (recall that IEEE 802.11a has 12 channels). Therefore, how to
1389-1286/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.comnet.2009.03.020
q
This research was supported in part by the National Science Council, Taiwan, ROC, under Grants NSC2752-E-009 -005-PAE and NSC 97-2219-009-006.
*Corresponding author. Tel.: +886 3 5731637; fax: +886 3 5721490. E-mail address:[email protected](R.-H. Jan).
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Computer Networks
exploit multiples channels with a few interfaces (usually 1–4) by each node is a key issue. On the other hand, although using diverse channels is attractive, whenever two neighboring nodes want to communicate, they have to ensure that some of their interfaces are on a common channel; otherwise, they cannot detect signals from each other.
To address these issues, a number of studies have been conducted in the literature[6]. According to the classifica-tion in Skalli et al.[7]and Kyasanur and Vaidya[8], exist-ing approaches can be categorized into the static, dynamic, and hybrid strategies.
In the static strategy[5,9–11], each interface is fixed on a channel permanently or for a long period of time. If a node A wants to communicate with a neighboring node B, they must have some interfaces fixed on the same chan-nel. Then, A can send packets (e.g. RTS/CTS/DATA) directly to B, without an additional control process to find a com-mon channel. However, it also limits the ability of using di-verse channels. For example, as shown inFig. 1a, there are
four channels ch1;ch2;ch3and ch4, and A is relaying packets from S to B on ch2. If ch2is now interfered by other trans-missions, A cannot utilize other channel except ch1and ch2. In other words, the number of channels that can be utilized by a node is limited by the number of its interfaces. In addition, if there is no common channel shared by two adjacent nodes (e.g., A and C), their traffic has to be relayed through a longer path (e.g., A; B; T; C). Even worse, the net-work will be partitioned if there is no alternative path.
In contrast, the dynamic strategy allows each interface to switch its channel from time to time to exploit the max-imum channels diversity[12–15]. For example, inFig. 1b, node A can turn its interface to a less interfered channel (e.g., ch3), if it observes that the current signal-to-noise ra-tio (SINR) is below a threshold. Nonetheless, since those as-signed channels are not fixed, before a transmission, a coordination mechanism is required to find a common channel for the sender–receiver pair. In[13], all nodes have to periodically rendezvous on a default channel to negotiate data channels for the hereafter transmissions. Another way is to use pseudo-random channel hopping sequences[14]. These mechanisms may spend considerable bandwidth for exchanging control packets or depend on sophisticated time synchronization. Therefore, the implementation is harder.
The hybrid strategy [8,16–19] combines the benefits
from both the static and dynamic strategies. Herein, each interface is either fixed or switchable. Like the static strat-egy, a fixed interface will stay on a channel permanently or for a long period of time. On the other hand, a switch-able interface can switch among different channels. For a transmission, the sender turns one of its switchable inter-face to a channel that is fixed on some interinter-face of the re-ceiver, and then starts transmitting on that channel. In this way, a node can utilize diverse channels via its switchable interface(s). For example, assume that node S wants to send a message to node T in Fig. 1c. First, node S sends the message to node A. If A observes that ch2is interfered, it can just switch its switchable interface to ch3which is the channel of the relaying node C’s fixed interface. More importantly, since channels of fixed interfaces are rarely changed, switching can be made immediately without any coordination.
Overall, the hybrid strategy is attractive due to its prac-ticality (the implementation of channel switching is easier comparing with channel coordination) and the flexibility of using diverse channels. However, so far, only a few studies have been made for this strategy. There was no research thoroughly exploring the unique feature about coexisting the dynamic channel switch and fixed channel usage. For this reason, we attempt to further investigated into this strategy. There are three major contributions in this paper. First, we define two optimization problems for the hybrid strategy as follows.
(1) Role assignment problem: Recall that an interface can be either fixed or switchable. Given a set of inter-faces, the role assignment problem is to decide which one should be fixed and which one should be switchable. The problem was not discussed before, because the previous works usually assumed that
each node has only two interfaces with the same coverage. In this setting, there is no choice but assigning one to be fixed and another to be switch-able. But when there are more than two interfaces or having varied transmission ranges, a various assignment would result in a completely different topology. In order to preserve the original topology, we aim to maximize the total number of switching pairs among interfaces, which can help nodes to find more paths to avoid interference, balance traffic load, and increase throughput. In addition, the con-nectivity between nodes should be guaranteed. (2) Semi-fixed channel assignment problem: The channel
assignment problem has been extensively studied for the static and dynamic strategies [7]. But, the concern for the hybrid strategy is quite different. To explain it, let us see an example in Fig. 1c. No matter which neighbor of A (i.e. B; C; or S) wants to transmit to A, it has to switch its switchable inter-face to ch1that is fixed on A. Now assume that some nodes surrounding to C (not drawn here) are also contending for ch1. To avoid interference, intuitively, A should change its fixed interface to other channel before receiving from C. However, it is not appropri-ate to change the channel of a fixed interface fre-quently; otherwise, the node may need to spend more time and bandwidth to update its fixed nel to nearby nodes. Moreover, if the updated chan-nel was not correctly received by all nearby nodes, a miss matching channel switching may occur. As shown inFig. 1c, assume that A has changed its fixed interface from ch1to ch2and broadcasted ch2to its neighbors. If S did not detect this event due to inter-ference, S may still switch to ch1and send RTS on this channel. As a result, the RTS will not be received by A, which in turn incurs additional cost for retrans-missions. For these reasons, the channel assigned on each fixed interface should be rarely changed and meanwhile should satisfy the possible channel switching from nearby nodes. We called such prob-lem as the semi-fixed channel assignment. To charac-terize this problem, we define a cost metric that measures the average number of links interfered by any possible channel switching. The cost metric is not specific to any traffic pattern, which can help nodes to adapt to time-varying traffic demands without changing to the assigned result. This is the most desired feature in the hybrid strategy. The details of the two problems will be explained in Sec-tion3.2.
Second, we analyze the complexity issue. It can be shown that the two defined problems are NP-hard even if the transmission ranges are equal. Since the problems are very difficult, the third contribution of this paper is to design efficient algorithms: For the role assignment prob-lem, we give an 1/2-approximate algorithm to find nearly optimal solution. On the other hand, a heuristic algorithm, based on transferring from a coloring-based problem, is presented for solving the semi-fixed channel assignment problem. Experimental results show that optimizing the
two problems is indeed beneficial to improve the perfor-mance and our algorithms are significantly superior to existing methods.
The rest of this paper is organized as follows. In Section
2, we review articles related to our study. Next, the net-work model and two concerned problems are formally de-fined in Section 3. In Sections 4 and 5, we present the complexity and algorithm, respectively, for the two de-fined problems. Section6conducts simulations to evaluate our designs. Conclusion remarks and future research are given in the last section. The proofs of theoretic properties are detailed in AppendixA.
2. Related works
The hybrid strategy first appears in[8]. It is similar to an approach in[3], where each node has one interface fixed on a common channel for exchanging control packets, and the other interfaces are switched among the remaining channels for data transmissions. The discrepancy is that the hybrid strategy allows fixed interfaces operating on different channels instead of a single one, which can avoid performance saturation from the common channel.
A hybrid multi-channel protocol (HMCP) was proposed in[16,17]. This protocol consists of two parts: The first part handles MAC issues, including queuing, switching, and broadcasting, in the hybrid strategy. Two timers are set to avoid the cost from frequent channel switching and hidden terminal nodes. The second part is a distributed channel assignment algorithm. Each node maintains a table to record the channels being used by its neighbors. Based on this table, a node periodically checks the number of other nodes using the same channel as itself. If the number is larger than average, the node will adjust its fixed interface to a less used channel with a probability p, and advertise this information through periodic ‘‘Hello” message. This approach can distribute channels equally on neighboring fixed interfaces. An abstraction module was implemented in[18]. It provides the requisite kernel support for this pro-tocol. In addition, a multi-channel routing protocol (MCR) for the hybrid strategy was proposed in[16]. This protocol incorporates switching costs of interfaces and expected transmission time of links to select routing path. It can reduce the latency from switch and link loss.
Nonetheless, the HMCP protocol has no guarantee on its convergency; the channels of fixed interfaces may con-stantly change if the numbers among nodes recursively de-pend on each other. Besides, since nodes are not on the same channel, a Hello message has to be broadcasted on every channel to ensure that all neighbors will receive it, which will incur considerable overhead and latency. The relationship between the required number of interfaces and the available channels was also discussed in[8]. But, they do not explicitly point out how to assign the role for each interface.
Consider other major channel assignment protocols under the static and dynamic strategies [5,11,24–26]. A flow-based approach, named the load-aware channel assignment (LACA) was proposed in[5]. This protocol can iteratively adjust channels and routing pathes among
nodes to allocate sufficient bandwidth to each wireless link for a given traffic profile. Similarly, a protocol, named the balanced-static channel assignment (BSCA), was designed to maximize the bandwidth allocated to each traffic aggre-gation point subject to fairness constraint [11]. Most recently, a flow-based approach in conjunction with rate control (FCRA) was presented in[24]. The FCRA jointly as-signs links with proper channels and transmission rates to make a given set of flow rates on links schedulable[24]. Although these flow-based approaches can optimize the performance for a given traffic profile, they may not be suitable for dynamic environments, where the traffic demand cannot be known a prior. The common channel
assignment (CCA) [25] and connected-low-interference
channel assignment (CLICA)[26] are two
flow-indepen-dent approaches. The CCA statically assigns nodes with a common set of channels. By doing so, the connectivity be-tween nodes can be easily preserved. However, it also lim-its the number of channels to be used by each node. For example, with 2 interfaces, only 2 channels can be utilized by all nodes. The CLICA improves the CCA by allowing each node to operate on a different set of channels and at the same time guarantees the node connectivity. But, since the CLICA is also a static approach, the number of channels that can be used by a node is still limited by the number of its interfaces. By contrast, the hybrid strategy allows each node to switch to diverse channels. To the best of our knowledge, there was no research exploring the concerned problems in this paper.
3. Network model and problems
In this section, we formally define the network model and the two concerned problems under study.
3.1. Network model
The network consists of a set VN¼ fu1;u2; . . . ;ung of n static mesh nodes. Each node ui has Ri interfaces, where RiP2. Besides, H orthogonal channels are available to each interface. Let uirstand for the rth interface of a node ui. The transmission power (and the corresponding trans-mission/interference ranges) of each uir is fixed, i.e. no power control. As shown inFig. 2a, the underlying topol-ogy under the same channel can be modeled as a digraph GT¼ VI;ET[ E0T
, named the transmission graph, in which VI¼ fuirji ¼ 1; 2; . . . ; n; r ¼ 1; 2; . . . ; Rig, representing the set of interfaces, a directed edge uirujt2 ET if and only if uir can transmit data to ujt, and a directed edge uirujt2 E0T if and only if uir can simply interfere with ujt but cannot transmit data to ujt.
We assume that the IEEE 802.11 DCF (or other conten-tion-based MAC) is employed to avoid hidden terminal nodes. To support it, the link between any sender–receiver pair of interfaces should be bidirectional; otherwise, the RTS–CTS–DATA–ACK sequence cannot be successfully ex-changed. Moreover, there is no need to transmit on the air between any two interfaces of the same node, since in this case the traffic can be bridged inside by hardware cir-cuits. With these considerations, given a GT, all
communi-cable links can be represented as a subgraph GC¼ ðVI;ECÞ of GT, named the communication graph, seeFig. 2b, where an edge uirujt2 EC if and only if uirujt2 ET;ujtuir2 ET, and i – j.
Throughout this paper, we denote NirðGÞ and EirðGÞ, respectively, as the sets of adjacent vertices and incident edges of some uirin a graph G. In addition, the sets of out-going and incoming edges (vertices) are distinguished by the superscripts of ‘‘” and ‘‘+”, correspondingly, i.e. NirðGÞ ¼ NþirðGÞ þ NirðGÞ, and EirðGÞ ¼ EþirðGÞ þ EirðGÞ. The terms ‘‘link” and ‘‘edge” will be used interchangeably. Be-sides, to differentiate from the graphic term ‘‘vertex”, the term ‘‘node” is referred to ‘‘mesh node”.
3.2. Role assignment problem
Given a set VI of interfaces, a role assignment
q
¼ ðVF;VSÞ is a disjointed partition of VI, where VF and VS correspond to the sets of fixed and switchable inter-faces, respectively. For any two interfaces uir and ujt;uir can switch to the channel of ujt only if uir is switchable and ujtis fixed. So, given a role assignmentq
, all possible switching pairs among interfaces in a communication graph GC can be represented as a digraph GS¼ ðVI;ESÞ,where a directed edge uirujt2 ES if and only if
uirujt2 EC;uir2 VS, and ujt2 VF. We named GS the switch-ability graph on interfaces and any edge in ESthe switchable link.
As shown inFig. 2c, after assigning the roles of inter-faces, some communicable links in GCare no longer in GS. In order to preserve the original topology of GC, our goal is to maximize the total number of switchable links ðTSLÞ in the resulting GS. There are three reasons:
1. In comparison with the static strategy, the hybrid strat-egy allows each node to transmit to different targets using diverse channels to avoid interference. For exam-ple, node u2can transmit to u3using the channel fixed on either u31or u32, depending on real-time conditions. Therefore, if channels are properly arranged for fixed interfaces, a larger TSL can increase the chance of switching to non-interfered channels.
2. Increasing the TSL can also help nodes to forward pack-ets on alternative routes to balance traffic load. For instance, with the link of u51u62, node u5can communi-cate with u1 by relaying through u6 whenever u3 is overloaded.
3. For two adjacent nodes, if there are multiple switchable links between them, such as links u51u31 and u52u32 between u5and u3, their throughput can be multiplied by transmitting on different channels in parallel. On the other hand, the network connectivity should be guaranteed. To be precise, consider an role assignment
q
, we define GS¼ ðVN;ESÞ as the switchability graph on mesh nodes, where a directed edge uiuj2 ESif and only if there is a switchable interface uir on ui and a fixed interface ujt on uj such that uirujt2 ES. In other words, an edge uiuj2 ES means that node ui can initiate a transmission with node uj, seeFig. 2d. We say that a role assignmentconnected, i.e. for any two mesh nodes ui and uj in VN, there is a path from uito uj, and vice versa. Accordingly, the role assignment problem can be now defined as follows.
Definition 1 (Max TSL). Given a communication graph GC¼ ðVI;ECÞ, find a feasible role assignment
q
¼ ðVF;VSÞ ofVIsuch that jESj is maximized.
Note that in the original design of the hybrid strategy
[8], if two fixed interfaces are on the same channel, their transmission is also allowed. However, our preliminary experiments show that the performance has little change when prohibiting nodes from transmitting in such case. There are two reasons: First, as channels are uniformly spread on fixed interfaces, the chance of finding such a communication pair is lower. Second, most of fixed inter-faces already spend a large portion of time to receive from other switchable interfaces so that they nearly have no time to do transmitting, especially when traffic load is hea-vy. On the other hand, if this case is considered, the chan-nels assigned on fixed interfaces will change the topology which however is further an influential factor of the assignment of channels. Therefore, the channel assignment problem, as defined later, would become very complicated and intractable. For these reasons, we do not consider this circumstance in our study.
3.3. Semi-fixed channel assignment problem
As a switchable link is active for some transmission, it may conflict to other links so that they cannot be active simultaneously. Furthermore, when there are many links conflicting to each other, the throughput will degrade
sig-nificantly. Therefore, our primary goal in this channel assignment problem is to minimize the possible conflict between switchable links. Now let us consider two switch-able links e and e0. As e is active, there are two circum-stances that e0cannot be active at the same time.
(1) Interface blocking: This happens when the required interface of e0 is occupied by e. See for example in
Fig. 3a. As u8u1 is active, u9u1;u12u1, and u13u1 should be silent, since they share a common fixed interface u1. On the other hand, u8u3 cannot be
active; otherwise, the channel of u8 has to be
switched from ch1to ch2. Notice that although u4is on the same channel of u8u1, a collision would occur when u8receives ACKs from both u4and u1(recall that the 802.11DCF is used). Hence, u8u4cannot be active. In these cases, we say e0 is blocked by e, or say, e0is a blocked link of e. Let e ¼ u
irujt. All blocked links of e can be defined by
BLðeÞ ¼ E
irðGSÞ [ EþjtðGSÞ feg:
An important feature of this circumstance is that any blocked link is determined as long as the assignment
q
is given, no matter what channels are assigned on VF.(2) Co-channel interference: As e is active, the RTS–CTS– DATA–ACK sequence has to be exchanged between uirand ujt. Therefore, all other interfaces within the interference range of uirand/or ujt, i.e.
PINðeÞ ¼ NirðGTÞ [ NjtðGTÞ fuir;ujtg;
are possibly interfered by e. In other words, e0can be interfered by e only if e0is adjacent to some interface
in PINðeÞ. With this fact, the set of links that are pos-sibly interfered by e can be defined as
PILðeÞ ¼ fe0je02 E
lsðGSÞ; uls2 PINðeÞg BLðeÞ:
Notice that BLðeÞ is subtracted from PILðeÞ, since for any blocked link e02 BLðeÞ, it can never be active simulta-neously with e even when not interfered by e. Let
v
ðeÞ de-note the channel of e, i.e. the channel assigned on ujt. The set of interfered links of e can be given byILðeÞ ¼ fe0je02 PILðeÞ;
v
ðe0Þ ¼v
ðeÞg:As shown in Fig. 3a, the interfered links of u8u1 are u10u4;u11u4;u12u4, and u13u7. Combining the two circum-stances, we say that e0is the conflicted link of e whenever it is either blocked or interfered by e. The set of conflicted links of e is defined by
CLðeÞ ¼ BLðeÞ [ ILðeÞ:
Our ultimate goal is to minimize the total number of con-flicted links ðTCLÞ, i.e.
TCL ¼X e2ES
jCLðeÞj:
Note that since the set of blocked links of any e 2 ES is independent to the assignment of channels on VF, the min-imization of the TCL is equivalent to minimizing the total number of interfered links ðTILÞ, i.e.
TIL ¼X e2ES
jILðeÞj: ð1Þ
In addition, for any given
q
, the cardinality of ESis a con-stant. Hence, it is also equivalent to minimizing the aver-age number of links that can be interfered as one of the switchable links is active. For example, inFig. 3b, the value indicated on each link e is jILðeÞj. By summing up these val-ues, we get TIL ¼ 66. On the other hand, there are totally 21 switchable links. So, on average a transmission in this graph interfere with at most 3.14 links ð66=21 ffi 3:14Þ. That is, the TIL can be treated as an average-case perfor-mance measurement at the link-layer. The problem is now defined as follows:Definition 2 (Min TIL). Given a transmission graph GT¼ ðVI;ET[ ET0Þ, a role assignment
q
¼ ðVF;VSÞ, and Hchannels, find a channel assignment
v
:VF! ½H such thatthe TIL is minimized.
In Section6.1, we will show that minimizing the TIL is beneficial to improve network throughput by simulations. 4. Role assignment
This section begins by showing the NP-hardness of the MINTSL. Next, we present a linear time 1/2-approximation
algorithm to find nearly optimal solutions. 4.1. NP-Hardness
Since the role assignment problem is first defined in this paper, we need to analyze its time complexity at the very beginning. Let us consider a special case of inputs, where the neighbor sets of all interfaces on the same node in GC are equal, called the identical GC. When each node has only two interfaces, the MAXTSL with identical GCis clearly triv-ial, since there is no choice but assigning one interface to be fixed and another to be switchable for each node. How-ever, if a lot of nodes have more than two interfaces, the problem will become much complicated.
In the following theorem, we show that the MAX TSL with more than two interfaces is NP-hard even if the gi-ven GCis identical. The proof is based on a polynomial time reduction from the maximum cut problem (MAXCUT). The
problem, as defined below, is one of the Karp’s original
NP-complete problems[20].
Definition 3 [20]MAXCUT. Given a graph G ¼ ðV; EÞ, find a
partition ðS; SÞ of V such that the cardinality of ðS; SÞ, i.e. the number of edges with one end point in S and another end point in S, is maximized.
Theorem 1. If the number of nodes having more than two interfaces is not a constant, the MAXTSL with identical GCis NP-hard.
The proof of Theorem1is given in AppendixA. The the-orem indicates that the MAXTSL cannot be optimally solved
in polynomial time unless NP ¼ P. Therefore, in the next subsection, we present an 1/2-approximate algorithm to find nearly optimal solutions.
4.2. 1/2-Approximate algorithm
The following design is based on an algorithm for the MAX CUT proposed in [21]. Our algorithm is initiated by Fig. 3. (a) Blocked and interfered links of u8u1and (b) jILðeÞj of each e 2 ES.
placing two arbitrary interfaces (here we take ui1and ui2) of each mesh node ui into VF and VS, respectively. It is to ensure the network connectivity. Next, for each unconsid-ered interface uir, if uir’s neighbors in GC that have been placed in VF are fewer than those being placed in VS, i.e. jNirðGCÞ \ VFj < jNirðGCÞ \ VSj; uir is assigned to VF; other-wise, it is assigned to VS. In other words, an interface will be assigned to the set where the assigned interfaces has fewer linkages to itself. For example, in Fig. 4, interface uir will be assigned to VF, since jNirðGCÞ \ VFj ¼ 2 < jNirðGCÞ \ VSj ¼ 3, where NirðGCÞ ¼ fa; b; . . . ; gg. On the other hand, the objective value TSL is increased by the number of links between uir’s and all uir’s neighbors that have been placed to different side. The algorithm is sum-marized below.
Algorithm 1. MAXTSL
Input : An identical communication graph GC¼ ðVI;ECÞ. Output : A role assignment
q
and the TSL.Step 1: TSL :¼ 0; VF:¼ ;; VS:¼ ;; Step 2 : For each node ui2 VN,
VF:¼ VFþ fui1g; VS:¼ VSþ fui2g;
TSL :¼ TSL þ jNi1ðGCÞ \ VSj þ jNi2ðGCÞ \ VFj;
Step 3: For each interface uir2 VI VF VS, if jNirðGCÞ \ VFj < jNirðGCÞ \ VSj, VF:¼ VFþ fuirg; TSL :¼ TSL þ jNirðGCÞ \ VSj; otherwise, VS:¼ VSþ fuirg; TSL :¼ TSL þ jNirðGCÞ \ VFj;
Step 4: Stop, and return
q
¼ fVF;VSg and TSL.The following theorem shows that any TSL resulted from this algorithm is no less than a half of the optimal value.
Theorem 2. Algorithm MAXTSL is a 1/2-approximate algorithm for the MAXTSL with identical GC.
The proof of Theorem 2 is given in AppendixA. The time complexity of the Algorithm MAXTSL is analyzed as follows.
Note that each interface is examined exactly once by either
Step 2 or Step 3. Moreover, for each uir, the calculation of jNirðGCÞ \ VFj and jNirðGCÞ \ VSj needs to check at most jNirðGCÞj links. Hence, the time complexity is linear to the total number of links in GC, i.e. OðjECjÞ.
For the general GC, the MAX TSL would become more
complicated because finding a feasible assignment is no longer trivial. Consider an example in Fig. 5, where the GCis not identical. The two assignments inFig. 5b and c have been specified a pair of fixed and switchable inter-faces to each node. But, the resulting graph inFig. 5c is dis-connected. It is reasonable to assert that the problem is NP-hard even if the goal is simply to find a feasible assignment. If our assertion is true, designing a branch-and-bound algorithm that enumerates a confined solution space could be more appropriated. The proof of this asser-tion is part of our ongoing work.
5. Semi-fixed channel assignment
In this section, we investigate into the channel
assign-ment problem defined in Section 3.3. The NP-hardness
of the MINTIL is obvious. Consider the case of an identical GC with two interfaces on each node. It is equivalent to the traditional receiver-based channel assignment prob-lem with single interface [12]. Therefore, we will focus here on designing an efficient algorithm for the MINTIL.
The main idea of this algorithm is based on problem transformation. It consists of three parts: First, we trans-form our problem into a coloring-based problem, called the minimum k-partition problem (MINK-PARTITION). Next, a
greedy algorithm is designed to find a sufficiently good solution
r
0for the MINK-PARTITION. Finally, the solutionr
0 will be converted into a channel assignmentv
with the same objective value for our problem. Now, the three parts are given in the following subsections.5.1. Problem transformation
The MINK-PARTITION[22]is a coloring-based problem. Its
purpose is to find a set of edges with minimum total weight whose removal makes a graph K-colorable. The for-mal definition is as follows.
Fig. 4. An example ofALGORITHM MAXTSL.
Fig. 5. (a) General GC; (b) feasible assignment and (c) infeasible
Definition 4[22]. Given a graph G ¼ ðV; EÞ, with a weight function w : E ! N, find a K-color assignment
r
:V ! ½K such that the total weight W of monochromatic edges is minimized, whereW ¼ X
vivj2E:rðviÞ¼rðvjÞ
wðvi;vjÞ: ð2Þ
In the following context, we transform any input X of the MINTIL into an input Y of the MINK-PARTITION. It can
be shown that the optimization of our problem to X is equivalent to solving the MINK-PARTITIONto Y. First of all, from Eq.(1), we have the following derivation for the TIL:
TIL ¼X e2ES jILðeÞj ¼ X uir2VF X e2Eþ irðGSÞ [ ujt2VF fe02 Eþ jtðGSÞje02 ILðeÞg ¼ X uir2VF X e2Eþ irðGSÞ [ ujt2VF fe02 PILðeÞ \ Eþ jtðGSÞj
v
ðe0Þ ¼v
ðeÞ ¼ X ðuir;ujtÞ2VFVF X e2Eþ irðGSÞ fe02 PILðeÞ \ Eþ jtðGSÞjv
ðe0Þ ¼v
ðeÞg : Lets
ðuir;ujtÞ ¼ X e2Eþ irðGSÞ fPILðeÞ \ Eþ jtðGSÞg : ð3ÞThe TIL can be represented as
TIL ¼ X
ðuir;ujtÞ2VFVF:vðuirÞ¼vðujtÞ
s
ðuir;ujtÞ; ð4Þwhere
v
ðuirÞ andv
ðujtÞ are channels assigned on uirand ujt. Eq.(4)indicates that for any fixed interface ujt, if there is an interface uirfixed on the same channel (i.e.v
ðuirÞ ¼v
ðujtÞ), the total number of links incident to ujt that will be inter-fered by any transmission to uiriss
ðuir;ujtÞ. In other words, the increment to the TIL caused by assigning uirand ujton the same channel is the sum ofs
ðuir;ujtÞ ands
ðujt;uirÞ.Based on this relation, a transformation from the MINTIL
to MINK-PARTITIONis defined as follows.
Definition 5 (Transformation). Given a transmission graph GT¼ VI;ET[ E0T
, a role assignment
q
¼ ðVF;VSÞ, and Hchannels for the MINTIL, we construct an instance,
includ-ing a graph G0
¼ ðV0;E0
Þ, a weight function w0, and a
constant K0, for the M
INK-PARTITIONsuch that
(i) V0¼ fvirjuir2 VFg; (ii) E0 ¼ fvirvjtjuir2 VF;ujt2 VFg; (iii) w0ðv ir;vjtÞ ¼
s
ðuir;ujtÞ þs
ðujt;uirÞ;8virvjt2 E0; (iv) K0¼ H.Now, we show their equivalence. Let optPðIÞ denote the optimal value of a problem P with an input instance I. We have the following result.
Theorem 3. Given a transmission graph GT¼ ðVI;ET[ E0TÞ, a
role assignment
q
¼ ðVF;VSÞ, and H channels, optMin TILðGT;q
;HÞ ¼ optMin KPartitionðG0 ;w0;K0
Þ:
The proof of Theorem3 is also given in Appendix A. Theorem3indicates that with the transformation in Def-inition 5, any existing algorithm for the MIN K-PARTITION
can be applied to solve our problem and obtain the same objective value. An example is shown in Fig. 6a, where the weighted graph is transformed from the transmission graph inFig. 3a. The weight wðvir;vjtÞ on each edge virvjtis calculated from Table 1. The table shows the
s
ðuir;ujtÞ (marked in boldface) for any pair of fixed interfaces uir and ujt in Fig. 3a. For instance, wðv1;v4Þ ¼s
ðu1;u4Þþs
ðu4;u1Þ ¼ 10 þ 14 ¼ 24. Note that any edge of zero weight (e.g. v3v6) can be removed without loss of gener-ality. A 2-color assignment of the weighted graph is given in Fig. 6b. We can see that the total weight W ¼ 66, which is exactly equal to the TIL of Fig. 3b, obtained before.A number of studies have been conducted for theMIN
K-PARTITION. For K ¼ 2, Garg et al. [22] presented an
Oðlog jVjÞ-approximation based on the multi-commodity flow technique. For K ¼ 3, it can be approximated in
e
jVj2 for anye
>0[23]. However, for K > 3, it has been shown that the problem cannot be approximated in OðjVj2eÞ for any
>0[23]. In other words, there is no constant-ratio approximation algorithm for the MIN K-PARTITION unless NP ¼ P. Therefore, in the next subsection, we intend to design a greedy algorithm to find ‘‘sufficiently good” solu-tion for the MINK-PARTITION.5.2. Algorithm for the MINK-PARTITION
The algorithm is presented as follows. At the beginning, the total weigh W is set as 0. Besides, we initiate an empty set S to keep track of any vertex that has been assigned a color. Before describing the following processes, let us first make some observations on the relation between W and S. Assume that a subset S of vertices in V have been assigned colors and the W has been updated to the total weight of monochromatic edges among vertices in S. Now, if we want to assign a color k to an unassigned vertex vi2 V S, the W can be increased by
Wþðvi;S; kÞ ¼
X vj2fNiðGÞ\Sg:rðvjÞ¼k
wðvi;vjÞ; ð5Þ
since Wþðv
i;S; kÞ is the total weight of edges connecting vi and its neighbors that have been assigned the same color. On the other had, for any vi’s neighbor vjin S, if
r
ðvjÞ – k, the weight wðvi;vjÞ will not be counted in W. Hence, the upper bound of the W can be decreased byWðv i;S; kÞ ¼
X vj2fNiðGÞ\Sg:rðvjÞ–k
wðvi;vjÞ: ð6Þ
Combining Eqs.(5) and (6), we can observe that the final value of the W can be reduced by finding a smaller Wþ
ðvi;S; kÞ and/or a larger Wðvi;S; kÞ. Hence, in our algo-rithm, we prefer to give higher priority to assigning a color k to an unassigned vertex visuch that
Wþðvi;S; kÞ Wðvi;S; kÞ ð7Þ
is minimal. After assigning vi, the S and W should be up-dated so that S ¼ S þ fvig and W ¼ W þ Wþðvi;S; kÞ. The above processes will continue until all vertices are as-signed. The algorithm is summarized below.
Algorithm 2. MINKPT
Input: A graph G ¼ ðV; EÞ, a weight function w : E ! N, and K colors.
Output: A K-color assignment
r
and the total weight W. Step 1: W :¼ 0; S :¼ ;;Step 2: Choice a vertex vj2 V S and a color k 2 ½1; . . . ; K such that Wþðv
i;S; kÞ Wðvi;S; kÞ is minimal; Step 3: Assign
r
ðvjÞ ¼ k;Step 4: W :¼ W þ Wðvi;kÞ; S :¼ S þ fvig; Step 5: If S – V, go back to Step 2; Step 6: Stop, and return
r
and W.5.3. Algorithm for the MINTIL
Based on the previous two subsections, a channel assignment algorithm for the MINTIL is now presented
be-low. The input is first transformed into an instance of the MINK-PARTITION, according to Definition5. Next, the MIN KPT
is applied to solve the transformed instance and obtain a color assignment
r
0 and the corresponding total weightTable 1
Derivation of Eq.(3)fromFig. 3. uir e 2 EþirðGSÞ ujt u1 u2 u3 u4 u5 u6 u7 jPILðeÞj u1 u8u1 0 3 3 3 1 2 1 13 u9u1 2 3 2 0 2 2 11 u12u1 3 2 3 1 1 1 11 u13u1 2 2 2 0 1 1 8 sðu1;ujtÞ 10 10 10 2 6 5 u2 u9u2 3 0 3 0 0 1 2 9 u13u2 3 2 0 0 1 1 7 u14u2 4 3 0 0 1 1 9 sðu2;ujtÞ 10 8 0 0 3 4 u3 u8u3 3 2 0 3 1 0 1 10 u9u3 3 2 2 1 1 9 u10u3 2 2 3 1 0 1 9 u14u3 2 2 2 1 0 1 8 sðu3;ujtÞ 10 8 10 4 0 4 u4 u8u4 3 1 3 0 2 1 0 10 u10u4 4 0 3 1 1 0 9 u11u4 4 0 2 1 1 0 8 u12u4 3 1 2 2 1 1 10 sðu4;ujtÞ 14 2 10 6 4 1 u5 u10u5 0 0 3 3 0 0 0 6 u11u5 1 0 1 3 1 0 6 sðu5;ujtÞ 1 0 4 6 1 0 u6 u12u6 3 1 0 3 1 0 1 9 u13u6 3 2 0 1 0 1 7 sðu6;ujtÞ 6 3 0 4 1 2 u7 u13u7 3 2 1 0 0 1 0 7 u14u7 1 2 3 0 0 1 7 sðu7;ujtÞ 4 4 4 0 0 2
W0. The
r
0 and W0 are then converted into a channel assignmentv
and the objective value TIL, according to Eqs.(8) and (9)in the proof of Theorem 3 at AppendixA. Algorithm 3. MINTILInput: A transmission graph GT¼ VI;ET[ E0T
, a role assignment
q
¼ fVF;VSg, and H channels.Output: A channel assignment
v
and the TIL.Step 1: Transform GT;
q
, and H into G0;w0and K0according to Definition5.Step 2: Apply the MINKPT to G0 ¼ ðV0;E0
Þ with w0 to find a K0-color assignment
r
0and the total weight W0. Step 3: For any uir2 VF, assignv
ðuirÞ ¼r
0ðvirÞ. Step 4: Stop, and returnv
and TIL ¼ W0.The time complexity of the MINTIL is analyzed in the following: About the transformation in Step 1, the input consists of jVFj2pairs of fixed interfaces. For each pair of uir and ujt, the calculation of
s
ðvir;vjtÞ ands
ðvjt;virÞ needs OðjVSj2Þ time, since there are at most jVSj edges in EþirðGSÞ and EþirðGSÞ. Hence, Step 1 can be done in OðjVFj2jVSj2Þ. About the MINKPT in Step 2, for each iteration of assigning a color to an unassigned vertex, there are OðK0
jV0jÞ choices. For a choice, the calculation of Eq.(7)takes OðjV0jÞ time. Moreover, there are totally jV0j iterations. Besides, we know that jV0
j ¼ jVFj and K0¼ H. Thus, Step 2 can be done in OðHjVFj3Þ. Combining together, the time complexity of the MINTIL is O maxfjVFj2jVSj2;HjVFj3g
.
6. Experiment results
In this section, we conduct simulations using the ns2 simulator. Our experiment consists of three parts: In the first part, we provide statistical results to show that opti-mizing MINTIL and MAXTSL indeed help to improve the performance. Next, we compare the single-hop and mul-ti-hop performances of our algorithm with other ap-proaches in the last two parts.
The network environment and test scenarios are mostly adapted from those in[26], in which a flow-independent protocol (CLICA) similar to our approach was proposed. For any network under test, 50 static mesh nodes are ran-domly deployed on a 1000 m by 1000 m area. Each mesh node has an equal number of radio interfaces ðRÞ with an identical transmission range of 250 m and interference range of 550 m. The number of channels ðHÞ will be varied from 3 to 12. The data rate is fixed on 2 Mbps. Any channel switch will incur a hardware delay of 1 ms. Note that although higher data rates are specified in IEEE 802.11 standards (e.g. 802.11 b/g) we are more interested in rela-tive performance behavior.
In order to spread traffic load equally onto different channels, each node maintains a separate queue for any channel it can use. When some packet arrived at a sender, the packet will be dispatched to the queue having least packets among all channels which are communicable to the intended receiver. For example, if node A received a packet destined for node B and found that it can communi-cate with B using ch1and ch2, the packet will be dispatched
to the queue for ch1 as long as existing packets in this queue is less than that for ch2. Besides, if a node can com-municate with its neighbor via multiple interfaces, the traffic (packets) will be randomly striped across interfaces for parallel transmissions. These two mechanisms will be applied to all protocols in our simulation.
6.1. Problem evaluations
First, we evaluate our channel assignment problem, i.e. MINTIL, by computing the correlation coefficient between
the TIL and the aggregated throughput over H ¼ 3; 7; 12 and R ¼ 2; 3; 4. For each combination of H and R, we gener-ate 20 random networks, i.e. the transmission graph GT. The role assignment of each GTis specified using a simple random algorithm (RAN), where each mesh node is desig-nated one pair of switchable and fixed interfaces to ensure the network connectivity and the remaining interfaces are assigned at random. Next, for each network, we systemat-ically generate 100 various channel assignments from the solution space, such that the ith assignment has at least i% of fixed interfaces on the same channel and the others are arbitrary. This is to ensure that results are obtained from a wide variety of channel assignments with different quality. On top of each network, we establish unicast flows (back and forth) with identical poisson packet arrivals be-tween every pair of neighboring nodes in the network. The mean packet arrival rate is 0.5 Mbps, which is large enough to achieve the saturated throughput. The packet size is 1024 bytes. Each simulation will last for 50 s.
The correlation coefficients averaged from the 20 net-works for each pair of ðR; HÞ are summarized inTable 2. From Table 2, we learn that the correlation coefficient shows a strong negative association between the TIL and the aggregated throughput. That is, the smaller TIL the net-work has, the higher throughput it will achieve. In partic-ular, when the numbers of channels and interfaces are large, the value is very close to 1. For example, when R ¼ 4 and H ¼ 12, the correlation coefficient is 0.963. This means that the TIL and the aggregated throughput are almost linearly correlated. There are two reasons: First, for a given H, the co-channel interference among interfaces will become more serious if there are more interfaces on nodes. According to Eq.(4), the TIL is the total number of links incident to any pair of fixed interfaces which are on the same channel. Thus, the TIL can faithfully reflect the increasing interference. Second, a larger value of R and/or H corresponds to a larger solution space of possible chan-nel assignments, which will enlarge the gap between the worst and optimal results of the TIL.
Next, we evaluate the role assignment problem, i.e. MAX
TSL. As discussed in Section3.2, a larger TSL implies more possible switching pairs between interfaces, but it does
Table 2
Correlation coefficients between TIL and throughput.
H ¼ 3 H ¼ 7 H ¼ 12
R ¼ 2 0.884 0.889 0.908
R ¼ 3 0.914 0.928 0.937
not imply that nodes can switch their switchable interfaces to non-interfered channel more easily, unless channels are properly arranged among fixed interfaces. The TIL has been shown a suitable cost metric to find proper channel assign-ment. Therefore, instead of showing the correlation coeffi-cient between the TSL and the aggregated throughput with
random channel assignments, we evaluate the MAX TSL
problem by directly comparing the aggregated through-puts resulted from the two combinations of RAN + MINTIL and MAXTSL + MINTIL (Recall that RAN is the random role assignment algorithm described above).Fig. 7a and b re-port the results under varied numbers of channels and interfaces. In the two subfigures, we can see that MAXTSL shows clear gain over the random algorithm under the same channel assignment. The improvement can be more significant as R and H are large, since the more channels (interfaces) the network has, the more possible channel switching nodes can do. In summary, the above results have confirmed that minimizing the TIL and maximizing the TSL are indeed beneficial to improve the performance. 6.2. Comparisons: Single-hop performance
Now, we compare the MAXTSL + MINTIL with the fol-lowing channel assignment protocols (algorithms) to study the relative performance:
1. Hybrid multi-channel protocol (HMCP) [16,17]: The
HMCP was the only channel assignment approach designed for the hybrid strategy so far. Recall that this protocol requires each node to periodically check its channel statuses and broadcast a Hello message on every channel whenever some fixed channel was chan-ged. We set the checking period as 5 s, which is the default value in [16]. In addition, to compare HMCP with our channel assignment algorithm on the same base, the roles of interfaces for HMCP are also deter-mined by MAXTSL (i.e. MAXTSL + HMCP).
2. Common channel assignment (CCA)[25]: Akin to our
design, the CCA is a traffic-independent approach. It assigns a common set of channels to all nodes. More precisely, the rth interface at each node i, i.e. uir, is assigned the rth channel.
3. Connected-low-interference channel assignment (CLI-CA)[26]: The CLICA is also independent to any specific traffic pattern. Besides, similar to our approach, it allows each node to operate on a different set of chan-nels and is based on the greedy method. A phase-2 pro-cedure in[26]that assigns channels to uncolored radio interfaces has been implemented here.
Other details about these protocols have been reviewed in Section2. In addition, the performance in single channel case serves as a baseline in our comparison.
Between every pair of neighboring nodes in the net-work, we establish two poisson flows (back and forth) with mean packet arrival rate of x=2L Mbps, where L is the total number of neighboring node pairs and x is the expected load offered to nodes. We will compare the link-layer per-formance under different x started from 2 to 30 Mbps with an increment of 2 Mbps. Any result point is averaged from 20 networks and the simulation of each network lasts for 50 s.
Fig. 8reports the aggregated throughput for four combi-nations of R and H. The cases for 3 and 12 channels are rep-resentative of 802.11b and 802.11a networks, respectively.
As shown in Fig. 8a, when H ¼ 3, the CLICA,
MAX-TSL + HMCP and MAXMAX-TSL + MINTIL achieve very similar performance, because it is very easy to fully exploit the channel bandwidth if providing only a small number of channels. Even so, the MAXTSL + MINTIL obtains the larg-est throughput in this case.
When H is large, as shown inFig. 8b–d, our approach can perform significantly better than other approaches, especially when nodes are loaded with higher packet rates. Under 30 Mbps offered load, the MAXTSL + MINTIL
3 4 5 6 7 8 9 10 11 12 0 2 4 6 8 10 12 Number of Channels (H) Aggregrated Throughput (Mbps) MAXTSL+MINTIL RAN+MINTIL 2 3 4 5 0 2 4 6 8 10 12 14 Number of Interfaces (R) Aggregrated Throughput (Mbps) MAXTSL+MINTIL RAN+MINITL
improves the throughput in single channel case up to a factor of 5.58 with 2 interfaces, a factor of 6.83 with 3 interfaces, and a factor of 8.51 with 5 interfaces. This result indicates that MAXTSL + MINTIL can effectively exploit multiple channels by using only a small number of inter-faces. There are two main sources of such impressive per-formance. The first stems from the genius of our concerned strategy. The hybrid strategy allows each node to utilize di-verse channels via its switchable interface(s). Thus, nodes are more likely to find non-interfered channels. The second is due to the superiority of our designed algorithms. The MAXTSL and MINTIL can produce more non-interfered switching pairs so as to further reduce interference.
By contrast, the performance of CCA is strictly limited by the number of interfaces (R) on each node so that only R channels can be utilized at anytime. The CLICA allows nodes to operate on different sets of channels. Hence, it can provide much more throughput than CCA. However,
since CLICA is a static approach, the number of channels that can be used by each node is still limited by R. As shown inFig. 8b, with 2 interfaces and 12 channels, it only achieves at most 2.4 times throughput compared with the single channel case (Recall that ours can achieve 5.58 times in this case). For a comparable improvement (seeFig. 8d), the CLICA requires at least 5 interfaces on each node, which is very cost-inefficient in practice.
Similar to our approach, the MAXTSL + HMCP also fol-lows the hybrid strategy. Nonetheless, its throughput is not comparable to ours in particular when H is large. As shown in Fig. 8b–d, it only achieves less than 80% of our throughput when H ¼ 12 and x ¼ 30 Mbps. There are three possible reasons: (1) Although HMCP can uniformly assign channels to fixed interfaces according to the channel’s usage on neighboring nodes, it does not take the possible channel switching into consideration. As a result, two potentially interfering fixed interfaces could be assigned
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 3 6 9 12 15 Offered Load (Mbps) Aggregrated Throughput (Mbps) Singe Channel CCA CLICA MAXTSL+HMCP MAXTSL+MINTIL 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 3 6 9 12 15 Offered Load (Mbps) Aggregrated Throughput (Mbps) Single Channel CCA CLICA MAXTSL+HMCP MAXTSL+MINTIL 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 3 6 9 12 15 Offered Load (Mbps) Aggregrated Throughput (Mbps) Single Channel CCA CLICA MAXTSL+HMCP MAXTSL+MINTIL 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 3 6 9 12 15 Offered Load (Mbps) Aggregrated Throughput (Mbps) Single Channel CCA CLICA MAXTSL+HMCP MAXTSL+MINTIL
the same channel even if there are many switchable inter-faces possibly switching to them; (2) Each node in this ap-proach has to broadcast a Hello message on every channel whenever some channel is changed at the periodic check point. The broadcasting process may consume consider-able bandwidth; (3) A miss matching channel switch may occur when some updated channel was not received, incurring additional cost for retransmissions (see Section
1for more detail explanation).
Fig. 9shows the average delay with 3 interfaces and 12 channels. (Note that we concentrate on the case of R ¼ 3 and H ¼ 12 for our hereafter comparisons). We can see that MAXTSL + MINTIL has the lowest delay compared with other approaches. Interestingly, although MAXTSL + HMCP and CLICA has almost the same performance under this setting (seeFig. 8c), the delay of MAXTSL + HMCP is much lower than that of CLICA. This phenomenon is explained as follows: In the hybrid strategy, each node can forward
packets using diverse channels to its neighboring nodes. Therefore, a node has to maintain more packet queues for various channels. This means that the average queue length of each channel is shorter than that of CLICA. As a result, each packet will stay in queue for a shorter period of time before being sent out. This is also the reason that MAXTSL + MINTIL has lower delay. Another reason is that MAXTSL + MINTIL can greatly reduce interference, in turn leading to lower channel access delays from the MAC layer
(include backoffs and retransmission). Notice that
although CCA has larger throughput than the single chan-nel case its delay rises drastically as the offered load over a certain limit. It is possibly due to the fact that the number of interfaces assigned to each channel is still equal to that of the single channel case, resulting in the same level of interference within each channel.
In Fig. 10a, we compare the performance under 50 nodes and 100 nodes for each protocol. The performance ratio ðPR1Þ is obtained by dividing the aggregated through-put of 100 nodes by that of 50 nodes. Observe that the per-formance ratio ðPR1Þ of any static approach (include CLICA, CCA, and single channel case) degrades as long as the of-fered load increases and below 1 when the load is over a certain limit. This phenomenon is possibly explained by the reason that interference in 100-node network is much severer than that of 50-node network but each node still utilizes a fixed number of channels. By contrast, the perfor-mance ratio of any hybrid approach (include MAX-TSL + MINTIL and MAXMAX-TSL + HMCP) increases with an increment in the offered load. Therefore, the hybrid strat-egy is more suitable for a large scale network.
Fig. 10b reports a comparison accessing the distance between the protocol and the physical interference mod-els. The channel access is resolved by the 802.11 DCF in the protocol model, while it only depends on the signal-to-noise ratio (SINR) in the physical model. The perfor-mance ratio ðPR2Þ is equal to the aggregated throughput of the physical model divided by that of the protocol
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 1 2 3 4 5 6 7 8 9 Offered Load (Mbps) Average Delay (s) Single Channel CCA CLICA MAXTSL+HMCP MAXTSL+MINTIL
Fig. 9. Average delay with R ¼ 3 and H ¼ 12.
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Offered Load (Mbps) Performance Ratio (PR 1 ) Single Channel CCA CLICA MAXTSL+HMCP MAXTSL+MINTIL 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 Offered Load (Mbps) Performance Ratio (PR 2 ) Single Channel CCA CLICA MAXTSL+HMCP MAXTSL+MINTIL
Fig. 10. Performance ratios vs. offered load with R ¼ 3 and H ¼ 12 (a) PR1= throughput under 100 nodes/throughput under 50 nodes and (b) PR2=
model. We can see that the performance ratios ðPR2Þ of CCA and single channel case degrade over 35% in the physical model. The degradation is mainly caused by hid-den terminal nodes in each channel. The CLICA, MAX-TSL + HMCP, and MAXMAX-TSL + MINTIL can spread nodes (interfaces) on different channels. Thus, they can greatly mitigate the hidden terminal problem. As shown in
Fig. 10b, there are relatively lower degradations in their performance ratios. Noticeably, although our approach is designed under the protocol model, it suffers only 12% decrement in throughput when applying to the physical model.
6.3. Comparisons: multi-hop performance
Lastly, we test the multi-hop performance. Following the settings in[26], we apply 50 s one-way bulk transfer
with FTP application in our test. Two different traffic pat-terns are considered. For the Internet access pattern, four Internet gateway nodes are randomly chosen from the 50 nodes. Each non-gateway node has a data transfer to the nearest gateway node determined by the shortest path length in terms of hops. For the peer-to-peer traffic pat-tern, 100 source-destination pairs (chosen at random) sep-arately start up a data transfer.
Fig. 11a and b show the average end-to-end TCP throughput for the two experiments. Normalized results (divided by the throughput in single channel case) are drawn inFig. 11c and d. We can see that MAXTSL + MIN-TIL substantially improves single channel performance especially for large-hop flows. The normalized throughput of our approach is also greater than the others in both cases. These results reveal that MAXTSL + MINTIL is not only superior in reducing inter-flow interference but also
1 2 3 0 200 400 600 800 1000 1200
Path Length (Hops)
Average TCP Throughput (Mbps) Single Channel CLICA MAXTSL+HMCP MAXTSL+MINTIL 1 2 3 4 5 6 0 200 400 600 800 1000 1200 Offered Load (Mb/s) Avg. TCP Throughput (Mb/s) Single Channel CLICA MAXTSL+HMCP MAXTSL+MINTIL 1 1.5 2 2.5 3 1 1.2 1.4 1.6 1.8 2 2.2 2.4
Path Length (Hops)
Normalized TCP Throughput CLICA MAXTSL+HMCP MAXTSL+MINTIL 1 2 3 4 5 6 1 1.5 2 2.5 3
Path Length (Hops)
Normalized TCP Throughput
CLICA MAXTSL+HMCP MAXTSL+MINTIL
Fig. 11. TCP throughput vs. path length with R ¼ 3 and H ¼ 12 (a) Aggregated TCP throughput under Internet access pattern; (b) aggregated TCP throughput under peer-to-peer pattern; (c) normalized TCP throughput under Internet access pattern and (d) normalized TCP throughput under peer-to-peer pattern.
superior in mitigating inter-hop interference to each flow. Note that since the tested flows are started separately, there is neither inter-flow interference nor inter-hop interference during any 1-hop flow. Besides, any channel switching requires a hardware delay of 1 ms. Therefore, our approach is slightly inferior to CLICA in 1-hop performance.
7. Conclusion
In this paper, we have defined two optimization prob-lems to characterize the unique feature of the hybrid strat-egy. The NP-hardness of the two defined problems have been proved. In order to solve our problems in reasonable time, we have designed efficient algorithms. Simulated re-sults have shown that optimizing our defined problems in-deed help to improve the performance. Besides, our algorithms are significantly superior to existing hybrid channel assignment approach and flow-independent approaches.
For the future research, it is worthwhile to extend our centralized algorithms to distributed protocols. We believe that the extensions are possible. Besides, we can see that the TIL is partially a function of the topology in GSwhich is further determined by the roles of interfaces. Therefore, to obtain the global optimum of the TIL, the two problems should be jointly considered. This is one of our ongoing works.
Acknowledgement
The authors would like to thank the anonymous refer-ees for their valuable comments, as well as Chi-Yu Li and Chin-Ching Chen for their helpful assistance in conducting the experimental results.
Appendix A. Proofs
Proof of Theorem 1. Given two integers p and q, we
denote TSLðqÞ and CUTðpÞ as two decision problems that determine whether jESj P p and jðS; SÞj P q, respectively.
We have the following argument.
Given a graph G ¼ ðV; EÞ, we constitute an instance GC¼ ðVI;ECÞ for the TSLðqÞ: For any vi2 V, there is a node ui
with three interfaces ui1;ui2 and ui3 in VI. In addition, a
node u0with two interfaces u01 and u02is in VI. For any
vivj2 E, a link is between uir and ujt in EC;8r; t ¼ 1; 2; 3.
Besides, for any vi2 V, a link is between uir and u0t in
EC;8r ¼ 1; 2; 3 and t ¼ 1; 2. We will show that G has a cut of
cardinality no less than p if and only if there is a feasible assignment of GCsuch that jESj P q ¼ 4jEj þ 3jVj þ p.
Consider a cut ðS; SÞ of G. A partition
q
¼ ðVF;VSÞ isconstructed by: (a) ui12 VS and ui22 VF;8vi2 V; (b)
ui32 VS;8vi2 S; (c) ui32 VF;8vi2 S. By (a), each node ui
has a switchable interface ui1 and a fixed interface ui2.
Besides, node u0has two links u02ui1 and u01ui2,
respec-tively, from and to any ui, i.e. u0 is a center point
connecting all nodes. Thus, the corresponding GS is
strongly connected. Now consider jESj. We decompose ES
into the following disjointed sets:
ES;1¼ fui1uj2;ui2uj12 ESjvivj2 Eg; ES;2¼ fui3uj22 ESjvivj2 E ^ vi2 Sg; ES;3¼ fuj1ui32 ESjvivj2 E ^ vi2 Sg; ES;4¼ fui1u02;ui2u012 ESjvi2 Vg; ES;5¼ fui3u012 ESjvi2 V ^ vi2 Sg; ES;6¼ fu01ui3;2 ESjvi2 V ^ vj2 Sg; ES;7¼ fui3ui3;2 ESjvivj2 E ^ vi2 S ^ vj2 Sg;
Since the partition of all ui1’s and ui2’s is unrelated to the cut ðS; SÞ, we get jES;1j ¼ 2jEj and jES;4j ¼ 2jVj. Besides, for any uj1;uj2and ui3, where vivj2 E, no matter what the par-tition of ui3 belongs to, links uj1ui3and uj2ui3 exist. Thus, jES;2j þ jES;3j ¼ 2jEj. Similarly, jES;5j þ jES;6j ¼ jVj. Moreover, for any pair of ui3 and uj3;ui3uj32 ES;7 if and only if vivj2 ðS; SÞ. As a sequel, if jðS; SÞj P p, we get jESj P q ¼ 4jEj þ 3jVj þ p.
Conversely, for any
q
¼ ðVF;VSÞ of GC, if it is feasible,there must be a pair of interfaces with different roles for each ui. In addition, since GCis identical, the order within a
node can be arbitrary. Thus, we can restrict our concern to the case, where ui12 VS and ui22 VF8vi2 V. In this case,
from the above observation, there must be 4jEj þ 3jVj links between uj1’s and uj2’s, uj1’s and ui3’s, and uj2’s and ui3’s. So,
if jESj P q ¼ 4jEj þ 3jVj þ p, we can obtain a cut ðS; SÞ such
that jðS; SÞj by choosing vi2 S;8ui32 VS and vi2 S; 8ui32 VF.
Clearly, the constructions can be carried out in polyno-mial time, and SLðqÞ is non-deterministic polynopolyno-mial. Thus the NP-completeness was established. h
Proof of Theorem 2. Firstly, we show that any resulting
q
is feasible. Without loss of generality, we assume that there is some feasible role assignment in the given GC. For any two mesh nodes uiand uj, connected by some link uirujt in GC, because GC is identical, there must be a link between any pair of interfaces on uiand uj in GC. Besides, with Step 2, at least one pair of fixed and switchable inter-faces has been assigned to both ui and uj. Hence, the two nodes can transmit to and receive from each other in the resulting GS.
On the other hand, for any partial solution ðVF;VSÞ, we
denote TNL as the total number of links in GC which are
non-switchable in GS, i.e. TNL ¼ X uir2VF NirðGCÞ \ VFþ X uir2VS NirðGCÞ \ VS !, 2:
In Step 2 and Step 3, if an interface viris assigned to VF, it means that jNirðGCÞ \ VFj < jNirðGCÞ \ VSj and the TSL is in-creased by jNirðGCÞ \ VSj; otherwise, it must be that jNirðGCÞ \ VFj P jNirðGCÞ \ VSj and the increment to the TSL is jNirðGCÞ \ VSj. In other words, no matter vir is assigned to VF or VS, the TNL and TSL are increased, respectively by minfjNirðGCÞ \ VSj; jNirðGCÞ \ VFjg and maxfjNirðGCÞ \ VSj; jNirðGCÞ \ VFjg. Consequently, at termi-nation, we get TNL < TSL. Let TSLbe the optimal. It is obvi-ous that TSLPTNL þ TSL ¼ jECj. In addition, due to the fact that TNL < TSL, the worst case occurs when TNL ap-proaches TSL. Hence, we get TSLPTNL þ TSL < 2TSL, i.e. TSL=TSL>1=2. h
Proof of Theorem 3. For any K0-color assignment
r
0of G0, if we assign the channels on VFsuch thatv
ðuirÞ ¼r
0ðvirÞ;8
uir2 VF; ð8Þit clearly satisfies that
X ðuir;ujtÞ2VFVF:vðuirÞ¼vðujtÞ
s
ðuir;ujtÞ ¼ X virvjt2E0:r0ðvirÞ¼r0ðvjtÞ w0ðv ir;vjtÞ: ð9ÞCombining Eq.(4)with Eq.(9), we have
TIL ¼ X virvjt2E0:rðvirÞ¼rðvjtÞ
wðvir;vjtÞ:
On the contrary, for any channel assignment
v
of VF, if we setr
ðvirÞ ¼v
ðuirÞ;8vir2 V0, Eq.(9)still holds, which is alsoequal to the W in Eq. (2). Hence, the statement is
proved. h References
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Andy An-Kai Jeng received the BS degree in statistics from Tamkang University, Taiwan, in 2001, the MS degree in management infor-mation systems from National Chi Nan Uni-versity, Taiwan,in 2003, and Ph.D. degree in the computer science from National Chiao Tung University, Taiwan, in 2007, where is currently a post-doctoral researcher. His research interests include wireless networks, distributed algorithm design and analysis, scheduling theory, and operations research.
Rong-Hong Jan received the B.S. and M.S. degrees in Industrial Engineering, and the Ph.D. degree in Computer Science from National Tsing Hua University, Taiwan, in 1979, 1983, and 1987, respectively. He joined the Department of Computer and Information Science, National Chiao Tung University, in 1987, where he is currently a Prof. During 1991–1992, he was a Visiting Associate Pro-fessor in the Department of Computer Sci-ence, University of Maryland, College Park, MD. His research interests include wireless networks, mobile computing, distributed systems, network reliability, and operations research.
