Joint routing, scheduling, and power control for ... - Semantic Scholar

J Control Theory Appl 2011 9 (1) 93–105 DOI 10.1007/s11768-011-0227-8

Joint routing, scheduling, and power control for multichannel wireless sensor networks with physical interference Xiaoling ZHANG 1,2 , Haibin YU 1 , Wei LIANG 1 , Meng ZHENG 1,2 1.Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang Liaoning 110016, China; 2.Graduate School of the Chinese Academy of Sciences, Beijing 100039, China

Abstract: Reliability and real-time requirements bring new challenges to the energy-constrained wireless sensor networks, especially to the industrial wireless sensor networks. Meanwhile, the capacity of wireless sensor networks can be substantially increased by operating on multiple nonoverlapping channels. In this context, new routing, scheduling, and power control algorithms are required to achieve reliable and real-time communications and to fully utilize the increased bandwidth in multichannel wireless sensor networks. In this paper, we develop a distributed and online algorithm that jointly solves multipath routing, link scheduling, and power control problem, which can adapt automatically to the changes in the network topology and offered load. We particularly focus on finding the resource allocation that realizes trade-off among energy consumption, end-to-end delay, and network throughput for multichannel networks with physical interference model. Our algorithm jointly considers 1) delay and energy-aware power control for optimal transmission radius and rate with physical interference model, 2) throughput efficient multipath routing based on the given optimal transmission rate between the given source-destination pairs, and 3) reliable-aware and throughput efficient multichannel maximal link scheduling for time slots and channels based on the designated paths, and the new physical interference model that is updated by the optimal transmission radius. By proving and simulation, we show that our algorithm is provably efficient compared with the optimal centralized and offline algorithm and other comparable algorithms. Keywords: Multichannel wireless sensor networks; Optimization; Power control; Scheduling; Routing; Physical interference

1

Introduction ical layer, data-link layer, and network layer to the appliAdvances in low-power integrated circuit devices and cation layer. The above problem has an extremely active communication technologies have enabled the deployment area of research in recent years [1∼11]. Prior work can be of low-cost low-power sensors that can be integrated to broadly divided into two categories. The works in the first form wireless sensor networks (WSNs) [1]. The WSNs have category consider the practical and efficient protocol design vast important applications and have been identified as one for wireless networks [2, 3], while those in the second catof the most important technologies nowadays. The deploy- egory consider the achievable capacity region [4∼6] and ment of the low cost and energy limited sensors implies design some approximation algorithms [7∼11]. Our work that the energy efficient communication protocol is impera- falls into the second category and aims to find an online and tive to extend the lifetime of the network. Meanwhile, the distributed approximation algorithm but differs from other applications, such as industrial process monitoring and works. We consider a more general framework that includes control, give new challenges to the protocol design, such not only the scheduling and routing but also the power as reliability and real-time requirements. The problem of control with physical interference model of multichannel energy efficient, reliable, and real-time protocol can be ap- wireless networks compared with [4∼6]. The comparisons proached from different communication layers: from phys- between [7∼12] and our work is shown in Table 1. Table 1 Papers related to our work. Citation

Layers

Objective

Interference model

Channel allocation

Algorithm

[7] [8] [9] [10] [12] [11] Our work

RSP RSP RSP RS RS RS RSP

Min-P Min-P Max-T Max-T Max-T Max-D Tradeoff-D/T/P

Pairwise SINR Pairwise Pairwise SINR SINR SINR

S S S M (dynamic) M (static) S M (dynamic)

C+A C C D C C D+A

Received 15 October 2010. This work was supported by the Natural Science Foundation of China (No. 60704046, 60725312), the National High-Tech Research Development Plan (863 plan) of China (No. 2007AA041201), and the Natural Science Foundation of Liaoning Province (No. 20092083). c South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2011

94

X. ZHANG et al. / J Control Theory Appl 2011 9 (1) 93–105

Note that R denotes routing, S denotes scheduling, P denotes power control, C denotes centralized algorithm, D denotes distributed algorithm, and A denotes approximation algorithm. Min-P denotes minimizing energy, Max-T denotes maximization throughput, and Min-D denotes minimizing delay. The definitions of pairwise interference model and the signal-to-interference-noise-ratio (SINR) interference model in Table 1 are described in Section 2.1. Motivated by the above considerations, we will address the following important problem: Given a multichannel wireless sensor network, what is the joint assignment of routing, scheduling, and power control that realizes tradeoff among energy consumption, end-to-end delay, and network throughput? We solve the multichannel joint routing, scheduling, and power control problem (MJRSP) in two parts and propose an online and distributed solution that can adapt automatically to the changes in the network topology and offered load. First, considering the delay requirements given by applications, we choose optimal transmission radius and rate for all nodes to minimize the energy consumption and maximize the network lifetime. As aforementioned, our algorithms use the SINR model of interference. It has been shown in [13] that SINR models can obtain significant performance gain in theory and practice. Further, reference [14] demonstrates that using SINR models for medium access control (MAC) scheduling can increase overall performance and decrease delays. Second, after calculating the optimal transmission radius and rate, we solve the routing, link allocation, and channel assignment together. First, the optimal routing strategy is carried out based on the optimal transmission rate. Then, based on the optimal transmission radius, we bound new interference relationships among network nodes by using the physical interference model and schedule each link with time slot and channel. Because the capacity of wireless sensor networks can be substantially increased by equipping each network node with a transceiver that can operate on multiple nonoverlapping channels, we adopt the dynamic channel assignment method [15], which can choose different channels in different time slots to overcome burst interference. We also study how to generate alternative paths and how each source node should optimally route packets among these alternative paths. We make five contributions toward this end: 1) modeling the reliability requirement and transforming this constraint into the number of time slots for the first transmission and the subsequent retransmissions; 2) considering the multichannel WSNs with one sensor having one radio to reduce the hardware costs; 3) addressing the above joint problem with physical model and multipath routing; 4) generalizing single-channel distributed algorithm to multichannel joint optimization problem; 5) designing a dynamical channel allocation scheme rather than traditional static allocation. The remainder of the paper is organized as follows. Section 2 describes the system models. In Section 3, we derive the MJRSP algorithm for multichannel WSNs with physical interference under considerations of energy consumption, reliability, and end-to-end delay. Meanwhile, we analyze the efficiency ratio of our algorithm to the centralized greedy maximal scheduling algorithm [16]. The performance of our proposed algorithm is evaluated in Section

4. Finally, conclusions are drawn in Section 5.

2 System models 2.1 Network model We model the network as a directed graph G = (V, E). V is the set of network nodes, which includes one sink node V0 and several normal nodes Vi (i = 1, 2, · · · , n). E is the set of directed links each connecting a pair of nodes. A directed link l = (i, j) denotes that node i can transmit to node j directly in the situation of noninterfering. L is the total number of links in a wireless network. The nodes in the network will monitor the environment and report collected data to the sink periodically. Each node can work at C channels. We consume static network. Let the sink be the center, and the maximum distance between the sink and other node be the network radius R. All network nodes, namely, sink and nodes, will be bounded into this circle. We assume that the network density is ρ, and the transmission radius of node is ru , see Fig. 1 for a counterexample. In our model, the network operates synchronously in a time-slotted model. We assume that time is slotted into intervals of equal length. The duration of one time slot is equal to the time required to transmit a packet and return an acknowledgement (ACK). A collection of time slots repeating cyclically consist a superframe. In any time slot, a set of noninterfering links is active, and each link is assigned a channel.

Fig. 1 Network topology.

2.2 Physical interference model Due to the broadcast nature of the wireless links, a link may interfere with other links when they transmit through the same channel (links on different channels do not interfere). An interference model defines which set of links can be active simultaneously without interfering. As pointed in [12], there are two kinds of interference models, namely, pairwise interference and physical interference. A pairwise interference model is represented by a set of pairs of links that interfere with each other. In the physical interference model, successful transmission over a link (i, j) depends on the SINR at j. Because of the physical interference is less restrictive and entails more capacity than the pairwise interference model, we use the physical SINR model in this paper. Given that Pij is the transmission power from node i to node j, Gij is the channel gain between node i and node j, and ηj is the thermal noise at j, the SINR in the presence of other transmission is given by SINRij =

ηj +

Gij Pij k,m∈V \{i,j}

Pkm Gkj

γ0 .

(1)


In (1),

k,m∈V \{i,j}

Pkm Gkj is the accumulated interfer-

ence with respect to link (i, j); Gij is calculated by the wide used far-field model Gij = d−α ij ; dij is the Euclidean distance between i and j; and α ∈ [2, 4] is the path loss index; γ0 is a given threshold determined by some quality-ofservice (QoS) requirements, such as bit error rate (BER). If (1) is not satisfied, j will not correctly receive data from i. 2.3 Reliability model Reliability shall be guaranteed by allocating enough time slots for retransmissions. Bij , which represents the number of time slots needed by link (i, j), needs to be fixed under different channel conditions and reliability requirements. We assume unicast transmission and consider the situation that a node returns ACK after correctly receiving a packet from a sender in a time slot, and the sender will stop transmitting the packet after receiving ACK. We also assume that ACK will not be lost. Let the sender be h hops away from the final destination, and r is the end-to-end reliability requirement of packet delivery. Let the reliability h ri = r. We assume that at the ith hop be ri , such that i=1

1

ri = r h , and the channel error e is a constant at each hop. A sender will retransmit a packet the ith time only if previous (i − 1) copies of the packet are not be transmitted correctly. The number of unsuccessfully transmissions is denoted as NNACK . Thus, 1

r h = 1 − eNNACK .

(2)

The expected number of transmissions Bij at a hop is given by NNACK

Bij = 1+

ei−1 = 1+

i=1

1

1 − NNACK rh = 1+ . (3) 1−e 1−e

The detailed analysis of our reliability mode can be found in Section 4.

3

MJRSP algorithm

We now present the main technical contribution: the MJRSP algorithm for multichannel WSNs with physical SINR interference. Our MJRSP algorithm has two procedures: 1) Power control algorithm, PowerControl, assigns optimal transmission radius and transmission rate for each node, which are used as the inputs of the second procedure. 2) Joint routing and scheduling algorithm, SPCAlgo, allocates each link with a channel and a time slot with reliability constraint and searches multiple paths for source node. The architecture of MJRSP algorithm is shown in Fig. 2.

Fig. 2 Architecture of MJRSP algorithm.

95

The key variables in this algorithm are summarized in Table 2. Table 2 Notations. R, ru E(i), Einit Emax , Eelec δfs d2 , δmp d4 Ln Γ Dtw , Drw Ce , Cl b, , η

Network radius and transmission radius; Energy consumption of node i, initial energy; Maximum energy consumption, electronics energy; Amplifier energy under free space model and multipath fading model; Network lifetime; Delay requirement that depends on special applications; Overall data quantity transmitted and received by node with distance D = vru +w to the sink; Amplification factors of signal amplifiers that respectively are in the free space model and the multipath fading model; Number of bits per code that is relative with the modulation technique; code rate; transmission rate η = b × ;

l = (i, j), l = (b(l), e(l)) Link l from node i to node j or from node b(l) to node e(l); W Channel bandwidth; M c (t), M (t) Set of noninterfering links that are chosen to transmit information on channel c at time slot t; a scheduling; Transmission power of node i, i ∈ {1, 2, · · · , Pi N }, N = n + 1; Interfering superset, defined in Section 3.2; Il Θ Network interference degree, maximum cardinality of any noninterfering subset of Il ; xcl (t), ylc (t) Number of packets that link l can assign to channel c at time slot t, and number of packets assigned to each channel queue; Link queue; retransmission queue; channel ql , τl , δlc queues; Denote the number of packets that link l can Dl (t) serve at time slot t; Transmission rate at which link l can transfer ηlc data on channel c; S, C, L Total number of source nodes, channels and links in the network; J(s) The number of alternative paths for source node s; F Constant that decided by network parameters; l ] Routing matrix, which value is 1 or 0; [Hsj Data generation rate; λs Fraction of traffic from source node s that is Psj (t) routed to path j at time slot t, j ∈ [1, · · · , J(s)]; γ Efficiency ratio that presents how much percentage of the optimal-throughput; Mb(l) , Me(l) Number of available interfaces at node b(l) and e(l), which are 1 in this paper.

3.1 Power control algorithm The objectives of our power control algorithm PowerControl are twofold. The first one is to minimize the maximum consumed energy min(max E(i)) that maximizes the network lifetime max(Ln ). The network lifetime is the time until the first node dies: Ln = Einit /Emax . Moreover, the

96


second one is to minimize the end-to-end delay. The endto-end delay is the accumulation of time across one path, tj ) Γ . which is represented as max( path j

In general, the objective of the PowerControl algorithm is

⎧ ⎨ max(Ln ), tj ) Γ. ⎩ max(

(4)

path j

We use a simplified energy consumption model shown in [17]. Both the free space (d2 power loss) and the multipath fading (d4 power loss) models are used, depending on the distance between the transmitter and receiver. The energy spent for transmission of a z-bit packet over distance d is zEelec +zδfs d2 , d < d0 , ETx (z, d) = zEelec +zδdα = zEelec +zδmp d4 , d d0 . (5) Eelec depends on factors, such as the digital coding and modulation; δfs d2 or δmp d4 depends on the transmission distance; and the acceptable BER; d0 is the threshold. Their values are listed in Section 4.2. The energy used to receive a message is (6) ERx (z) = zEelec . We can present the relationship between ru and R as R , g ru . R = aru + g, a = (7) ru From reference [18], we can conclude the data quantity and energy consumption from one node to sink when the distance between them is D = vru + w, v ∈ {0, · · · , a}, w ∈ [0, ru ). The results of data quantity and energy consumption are respectively listed in (8) and (9). ⎧ ru ⎪ (a−v−1)w+(a−v−1)(a+v) ⎪ ⎪ 2 , ⎪ Dtw (v) = 1+ ⎪ ⎪ ⎪ vru + w ⎪ ru ⎪ ⎪ ⎪ (a−v−1)w+(a−v−1)(a+v) ⎪ ⎪ 2 , w ⎪ Dr (v) = ⎪ ⎪ ⎪ vr + w u ⎪ ⎨ v ∈ [0, a], w ∈ [g, ru ); (8) ru ⎪ (a−v)w+(a+v+1)(a−v) ⎪ ⎪ 2 , ⎪ ⎪ Dtw (v) = 1+ ⎪ ⎪ vr + w ⎪ u ⎪ ru ⎪ ⎪ (a−v)w+(a+v+1)(a−v) ⎪ ⎪ ⎪ 2 , ⎪ Drw (v) = ⎪ ⎪ vru + w ⎪ ⎩ v ∈ [0, a], w ∈ [0, g], ⎧ F w2 F ⎪ ⎪ + ] + (Dtw (v) − 1) , ⎪ Dtw (v)[Ce (2b − 1) ⎪ b b b ⎪ ⎪ ⎪ ⎪ w d0 and v = 0, ⎪ ⎪ ⎪ ⎪ w4 F F ⎪ w b ⎪ (v)[C (2 − 1) D + ] + (Dtw (v) − 1) , l ⎪ t ⎪ b b b ⎪ ⎨ w > d0 and v = 0, (9) ru2 F F ⎪ w b w ⎪ (v)[C (2 − 1) (v) − 1) D + ] + (D , ⎪ e t t ⎪ ⎪ b b b ⎪ ⎪ ⎪ ru d0 and v = 0, ⎪ ⎪ ⎪ ⎪ w ru4 F F ⎪ b ⎪ (v)[C (2 − 1) D + ] + (Dtw (v) − 1) , ⎪ l t ⎪ b b b ⎪ ⎩ ru > d0 and v = 0. The fluctuations of energy consumption with differ-

ent transmission radii and different transmission rates are shown in Section 4.2. Each link l = (i, j) can be viewed as a single-user Gaussian channel with Shannon capacity cl = W log2 (1 + SINRl ). However, each node may choose among a finite number of transmission rates it can sustain. This leads to a finite number of achievable link-rate vectors for some rate levels (η0 , · · · , ηM ) without considering the modulation mechanism, which is represented as 0 M cl = (cηtgt,l , · · · , cηtgt,l ), l = 1, · · · , L, (10) 0 is the target l − η0 vector. We let c0tgt,l = 0. where cηtgt,l Meanwhile, each node can determine its transmission power according to the SINR level. We assume that η SINRηl k < SINRl k+1 , and SINR0l = 0, η ηk if SINRl SINRηl k SINRl k+1 . (11) Pi = Ptgt,l

If the transmission rate of one link is η = b × , we can deduce proposition [18]. Proposition 1 When the distance between one node and the sink is D = vru + w, the maximum delay is v Dtw (k) v k=0 τkw = Γvw = . (12) b× k=0 Proof From (8), we can conclude that the data quantity of the node with distance D = vr + w to sink is Dtx (i), and the transmission rate is η = b × . Therefore, the deDw (v) lay from this node to the sink is τiw = t . Meanwhile, b× the packet at position D = vru + w needs to be forwarded through (v + 1) times. Then, the sum of these (v + 1) delay is the end-to-end delay of one path. The fluctuations of end-to-end delay with different transmission radii and different transmission rates are shown in Section 4.2. Based on the aforementioned calculations, we propose our power control algorithm – PowerControl. PowerControl (Optimization of energy consumption and delay) Input Network radius R, code rate , allowed maximum delay Γ , node initial energy Einit or related parameters of energy model, and SINR constraint. Output Transmission radius ru , transmission rate η, and energy consumption below maximum delay Γ . Step 1 For each link originating from node i, obtain each transmission power Pk from the transmission power set {1, 2, · · · , Λ} and calculate the corresponding transmission radius ruk . Step 2 If Pk satisfies the SINR requirement, choose the first suitable transmission rate η0 ; note that {ηi |i ∈ {0, 1 · · · , M }} is the available set of transmission rates. Step 3 For ruk , calculate the energy consumption of each node by using equation (9). Step 4 For ruk , calculate the delay by using Proposition 1. Step 5 For ηi , calculate the maximum energy consumpEinit ηi ,k ηi ,k tion Emax , network lifetime ηi ,k , and delay Γmax . Emax Step 6 Obtain next available transmission power and return to Step 2; If there is no next transmission power, go


to Step 7. Step 7 Obtain the transmission radius ru , transmission rate η, and network lifetime that are corresponding to ηi ,k min(Emax ) and under delay requirement Γ . End. The PowerControl algorithm is a polynomial time heuristic algorithm. It has N ΛM iterations, and the total running time is bounded by O(N ΛM ). 3.2 Joint scheduling and path selection algorithm We start with a few definitions here. Definition 1 (Interfering superset Il ) If link l and other links in set Il are transmitting on the same channel at the same time, neither of the links can transfer any useful data. Definition 2 (Noninterfering subset) Noninterfering subset is a subset of Il such that any two links in this subset do not interfere with each other. Definition 3 (Network interference degree Θ) Interference degree is the maximum cardinality of any noninterfering subset of Il . In other words, it is the maximum number of links in Il that interfere with link l but not with each other. The network interference degree is the maximum possible interference degree over all links. Note that the interference degree of the network can often be determined directly from the physical SINR model and thus is independent of the network topology [10]. Definition 4 (Scheduling M (t)) A Scheduling is a specification of a certain number of time slots. For each time slot, we specify a multiset of active links with a channel assigned to each link. A valid and reliable scheduling must satisfy two constraints: 1) the links active in the same time slot do not interfere; 2) the transmission of a link must guarantee the reliability requirement. At time slot t, M (t) = [M c (t)], where c indicates one channel. Definition 5 (Optimal capacity region Ω and efficiency ¯ = [λ1 , · · · , λS ] denote the offered load ratio γ [10]) Let λ to the network. The capacity region under a particular channel assignment, link scheduling, and routing algorithm is ¯ such that the network remains stable. Under the set of λ possible routing constraints, we define the optimal capacity region Ω as the supremum of the capacity regions of all algorithms. An algorithm is said to achieve an efficiency ratio of γ if it can stabilize the network under any load such that ¯ lies strictly in γΩ. λ 3.2.1 A multichannel scheduling and routing algorithm based on maximal scheduling algorithm One will naturally hope that the generalization of the single-channel scheduling algorithms may lead to equally efficient and low-complexity scheduling algorithms for multichannel WSNs. In [10], the authors study the generalization of two typical single-channel scheduling algorithms, i.e., greedy maximal scheduling [16] and maximal scheduling [19], to multichannel networks. They show that the generalization of greedy maximal scheduling, which is a low-complexity but centralized algorithm, can still guarantee efficiency-ratios almost as tight as in single-channel networks, while the greedy maximal scheduling algorithm is not very easy to implement in a distributive fashion. Moreover, the straightforward extensions of maximal scheduling, a low-complexity, and distributed algorithm results in lower efficiency ratios in multichannel networks. Mathematically speaking, the efficiency ratios of the greedy maxi-

97

mal scheduling and the maximal scheduling are respectively ¯ where I¯ is the maximum number of links 1/(Θ+2) and 1/I, that interfere with any link l [10]. Given the results in [10], we will consider the following question: Can we design a distributed scheduling algorithm for multichannel WSNs that can guarantee the same efficiency ratio as the centralized greedy maximal scheduling, but with lower complexity as maximal scheduling? In this section, we will develop such a distributed joint scheduling and routing algorithm that can not only consider reliability but also assign links to use channels having good quality. Our algorithm is based on the idea in [10] and realizes the reliability and packet retransmissions. In our algorithm, packets arriving to each link l are served in three steps. In the first step, we design the local optimization objective as maximum the routing fractions. In the second step, the arriving packets are assigned to queues that correspond to each channel c. In the third step, the links are assigned to time slot and channels according to the maximal scheduling. The characteristics of this algorithm are to prevent link to be assigned to weak channel and can be implemented online. Besides using the length information of queue at both the perlink level and the perchannel level as [10], we specify a retransmission queue that is used to guarantee the reliability. We suppose that each link has three kinds of queues: one link queue ql , one retransmission queue τl , and c channel queues δlc . The packets in the link queue and the retransmission queue are assigned to the channel queues, respectively. As discussed in Section 2.3, the reliability requirement is met by allocating certain number of time slots that are used for first transmission and subsequent retransmissions. If the first transmission of one packet is failed, this packet will be inserted to the retransmission queue τl . Moreover, the related information including the link index, the channel index used for the last transmission, and the maximum transmission number of the packet is also inserted into the retransmission queue. The failed packet is retried in the retransmission time slots, and the number of transmissions is decreased by one. If the number of transmissions is decreased to 0, the correspondent packet shall be discarded. Note that the maximum transmission number is the maximum value between (maximum transmission number +1) and the number of time slots used for guaranteing reliability. The channel index in the retransmission queue is used to extend our algorithm for dynamic channel allocation and adaptive channel hopping, which is our future work. According to the definitions of Dl (t) and ηlc , we can ob c ηl . We design a binary variable tain that Dl (t) = c:l∈M c (t)

l [Hsj ] that satisfies 1, if path j of source node s traverses link l, l Hsj = 0, else. (13) For simplicity, we assume that source node s injects packets into the network periodically, and the duration of cycle is λs . We assume that P¯s = [Ps1 , · · · , PsJ(s) ] and P¯ = [P¯1 , · · · , P¯S ] for all source nodes, where s ∈ {1, · · · , S}, P¯s indicates the set of paths of source node s, and P¯ indicates the set of paths of all source nodes in the network. J(s) Psj (t) = 1. Psj (t) 0 and j=1

98


Let xcl (t) denote the number of packets on link l that can be assigned to channel c at time slot t; ylc (t) denotes the actual number of packets assigned to each channel queue δlc , and ylc (t) ∈ [0, ηlc (t)]. If a packet on link l is assigned to channel c and time slot t, ylc (t) > 0; else, ylc (t) = 0. Motivated by the above considerations, the link queue ql includes not only new packets from other source node that traverses link l but also packets from the retransmission queue. Let ql (t) denote the number of packets queued on link l and τl (t) denote the number of packets needed to be retried at the beginning of time slot t. The evolution of ql (t) [10] may be written as ql (t + 1) = ql (t) + τl (t) +

S s=1

l Hsj Psj (t)λs −

C c=1

ylc (t). (14)

Next, we will solve the joint routing, link scheduling, and channel assignment problem by generalizing the maximal scheduling algorithm. We will study how each source node should optimally route packets among the alternate paths and how each link is assigned by some time slots and a channel by the following steps. SPCAlgo At the beginning of each time slot t Step 1 Design objective and constraints. β J(s)

J(s) L s l max − [Psj (t)]2 − Psj (t) Hsj ql (t)+esj 2 j=1 P¯s j=1 l=1 (15) subject to s ∈ {1, · · · , S}, Psj (t) 0 and

J(s)

Psj (t) = 1, (16)

j=1

where βs is a positive number chosen for each source node s, and esj is a constant chosen for each path j of source node s. Each source node then routes each arriving packet or retransmitted packet independently to path j with probability Psj (t). The optimization problem of equations (15) and (16) are formulated that the routing fraction Psj (t) is larger for a path j with a smaller congestion cost

L l Hsj ql (t) + esj . l=1

esj is used to embody the preference to choose route. That is, if we want to choose a route with certain hops, we can choose esj be a positive constant multiplied by the number of wanted hops. βs is an adjustable constant that is used to prevent the potential routing oscillation. If βs = 0, only paths that have the smallest congestion cost will have positive Psj (t). When the packet queues are being updated, this property will lead to oscillation of the routing fractions Psj (t) [20]. On the other hand, the objective function (15) is strictly concave with a quadratic term, which makes the optimal routing fraction become a continuous function of the queue length. Thus, βs J(s) [Psj (t)]2 2 j=1 is used to eliminate the routing oscillation. Step 2 Let each link l satisfy the following choices in

time slot t. ⎧ c x (t) = ηlc (t), ⎪ ⎪ ⎪ l C δ d (t) ⎪ ql (t) 1 δkc (t) ⎪ k ⎪ + ⎪ ⎨ if α η c (t) [ c d l k∈Il ηk (t) k∈M (b(i))d=1 ηk (t) l d C δk (t) ⎪ ⎪ ⎪ ]; + ⎪ d ⎪ ⎪ k∈M (e(l))d=1 ηk (t) ⎪ ⎩ c xl (t) = 0, otherwise, (17) where αl is an arbitrary positive constant chosen for link l. Equation (17) is used to ensure that links will only be scheduled on good channels that have big capacity. That is, packets will be more possible to be assigned to a channel if the corresponding rate ηlc (t) is larger, because the ηlc (t) is in the denominator of equation (17). We use equation (14) to represent the evolution of ql . Moreover, link l will drain min{ql (t), δlc (t)} packets from queues ql and τl , and the packet priority can be designed according to the packets’ deadlines or other factors, such as packet length. In rule (17), ql (t) and δlc (t) can be interpreted as price δkc (t) functions: ql (t) is the congestion cost; is the conc k∈Il ηk (t) C δ d (t) C δ d (t) k k and tention cost; d d k∈M (b(i)) d=1 ηk (t) k∈M (e(l)) d=1 ηk (t) are the radio costs at the transmitter and the receivers; and ηlc (t) is the weight value. Hence, (17) can be interpreted as that each link will assign packets to channel c at the maximum rate only if the contention cost of the channel plus the radio costs, weighted by the channel transmission rate ηlc (t), is smaller than the congestion level. Step 3 Maximal scheduling is carried out to determine the channel assignment and link schedules. Mathematically, according to the maximal scheduling in [19], we will define our multichannel maximal scheduling algorithm as follows. (Multichannel maximal scheduling) Our algorithm gives high priority to backlogged links that satisfy ql (t) + τl (t) ηlc (t). Furthermore, for any link-channel pair (l, c), at least one of the following is true: Either link l is scheduled in channel c, i.e., l ∈ M c (t). One of the interfering links k to link l is backlogged and scheduled in channel c, i.e., k ∈ Il and k ∈ M c (t). The evolution of each channel queue is given by δlc (t + 1) = δlc (t) + ylc (t) − ηlc (t)1{l∈M c (t)} . (18) 3.2.2 Choosing candidate paths l In SPCAlgo, we can obtain the routing matrix [Hsj ], which includes all possible paths. In practice, we only need several alternate paths. Now, we will propose a method to l ] and find one choose paths online. First, we search [Hsj path. This path will be chosen as one alternate path if it satisfies the rule given by [10]: βs Psj +

L l=1

=

l Hsj ql (t) + esj

min k=1,··· ,J(s)

qs,min (t).

βs Psk +

L l=1

l Hsk ql (t) + esj

(19)

From (19), we can conclude that find path with conges-

99


tion costs

L l=1

l Hsj ql (t) + esj larger than qs,min (t) will have

no benefit. Hence, we can run a minimal cost routing algorithm using the queue length as the cost-metric for each link. If the minimal cost is smaller than qs,min (t), we choose this path as one of the candidate paths. 3.2.3 Efficiency ratio evaluation of MJRSP algorithm We shall study the efficiency ratio of our algorithm to the optimal capacity region. By proving, we can obtain the following proposition. Proposition 2 Assume that the set of alternate paths are given. If αl = α for all links l, then the efficiency ratio of MJRSP algorithm is γ = 1/(Θ + 2). αl is an arbitrary positive constant chosen for link l in (17). Proof We will show that for any data generation rate ¯ = (λ1 , · · · , λS ), the MJRSP algorithm can stabivector λ ¯ when the efficiency lize the network at the offered load λ ratio of the maximal scheduling algorithm is 1/(Θ + 2). We construct the following Lyapunov function to prove the network stability. ¯ ¯ q (t)) + Vδ (δ(t)), (20) V (¯ q (t), δ(t)) = Vq (¯

+

L (q (t))2 l , 2αl l=l C δ c (t) δ c (t) L C δ d (t) k k k ¯ = [ + Vδ (δ(t)) c c d l=1 c=1 2ηl k∈Il ηk k∈M (b(i)) d=1 ηk

C δ d (t) k ] d k∈M (e(l)) d=1 ηk

C δ c (t) y c (t) L C y d (t) k k k [ + c c d l=1 c=1 ηl k∈Il ηk k∈M (b(i)) d=1 ηk

+

C y d (t) k − μcl (t)] + C2 . d k∈M (e(l)) d=1 ηk

As aforementioned, if ql (t) C c=1

We can easily prove that these two functions guarantee that the Lyapunov function is local nonnegative. Next, we ¯ + 1)) and will calculate the drift between V (¯ q (t + 1), δ(t ¯ V (¯ q (t), δ(t)).

+

C δ c (t + 1) δ c (t + 1) L k k [ = 2ηlc ηkc l=1 c=1 k∈Il

C δ d (t + 1) C δ d (t + 1) k k + + ] ηkd ηkd k∈M (b(i)) d=1 k∈M (e(l)) d=1 C δ c (t) δ c (t) L C δ d (t) k k k [ + − c c d 2η η l=1 c=1 k∈M (b(i)) d=1 ηk l k∈Il k

c=1

ylc (t) =

C xd (t) k − μcl (t)] + C2 , d k∈M (e(l)) d=1 ηk

+

C

k∈M (b(i)) d=1

k∈Il

C

k∈M (e(l)) d=1

(25)

1{k∈M d (t)}

1{k∈M d (t)} .

(26)

The Lyapunov drift can be bounded as ¯ E(ΔV (t)|¯ q (t), δ(t)) L q (t) S J(s) C l l [τl (t) + Hsj Psj (t)λs − xcl (t)] s=1 j=1 c=1 l=l αl +C1 + +

where

C1 is the upper bound of (ql (t + 1)) − ql (t))2 . ¯ + 1)) − Vδ (δ(t)) ¯ Vδ (δ(t

C

where C2 is a positive constant and 1{k∈M c (t)} + μcl (t) =

¯ + 1)) − V (¯ ¯ ΔV = V (¯ q (t + 1), δ(t q (t), δ(t)) ¯ + 1)) q (t + 1)) − Vq (¯ q (t))] + [Vδ (δ(t = [Vq (¯ ¯ −Vδ (δ(t))], (21) Vq (¯ q (t + 1)) − Vq (¯ q (t)) L (q (t + 1))2 L (q (t + 1))2 l l = − 2α 2αl l l=l l=l 2 L 1 (ql (t + 1)) − (ql (t))2 = [ ] 2 l=l αl L q (t) S C l l [τl (t) + Hsj Psj (t)λs − ylc (t)] s=1 c=1 l=l αl + C1 . (22)

c=1

δlc (t), then

q (t + 1)) − Vq (¯ q (t)) Vq (¯ L S C ql (t) l [τl (t)+ Hsj Psj (t)λs − xcl (t)]+C1 , (24) s=1 c=1 l=l αl ¯ + 1)) − Vδ (δ(t)) ¯ Vδ (δ(t L C δ c (t) xc (t) C xd (t) k k k [ + c c ηkd l=1 c=1 ηl k∈Il ηk d=1 k∈M (b(i))

C δ d (t) k + ]. d k∈M (e(l)) d=1 ηk

C

(23)

xcl (t). We can write (22) and (23) as

where Vq (¯ q (t)) =

L C xd (t) C δ c (t) xc (t) k k k [ + c c d l=1 c=1 ηl k∈Il ηk k∈M (b(i)) d=1 ηk

C xd (t) k − μcl (t)] + C2 d k∈M (e(l)) d=1 ηk

L q (t) S J(s) C l l [τl (t) + Hsj Psj (t)λs − xcl (t)] α l s=1 j=1 c=1 l=l

+ +

C δ c (t) xc (t) L C xd (t) k k k [ + c c d l=1 c=1 ηl k∈Il ηk k∈M (b(i)) d=1 ηk

C xd (t) k − μcl (t)] + C3 , d η k∈M (e(l)) d=1 k

(27)

where C3 is a positive constant. Since our algorithm can stabilize the network when the ¯ there must exist data generation rate vector is (Θ + 2)λ, J(S) ˜ x ˜cl ∈ [0, ηlc ], P˜sj ∈ [0, 1], and Psj = 1 for each linkj=1

channel pair (l, c) such that for all link l, (1 + ε)2 (Θ + 2)

S J(S) s=1 j=1

for all l and channel c,

l Hsj Psj (t)λs

x ˜cl c Θ; k∈Il ηk

C c=1

x ˜cl ;

(28)

(29)

100


and for all node i, C x ˜cl c k∈M (b(i)) c=1 ηk C x ˜cl c k∈M (e(l)) c=1 ηk

1,

(30)

1,

(31)

where x ˜cl can be interpreted as the long-term average quantity of packets that link l processed at channel c. ε is a small positive number. Inequality (28) is due to the rate balance at link l; inequality (29) is due to the interference constraint; and inequalities (30) and (31) are due to the radio interface constraints. x ˜cl (1 + ε) and P¯sj = P˜sj , then we can have Let x ¯cl = Θ+2 the following results: J(S) ¯ Psj = 1 and 0 x 0 P¯sj 1, ¯cl ηlc . (32)

L q (t) S J(S) S J(s) λs βs (P¯sj )2 l ¯ 1 l − [ Hsj Psj λs − αl s=1 j=1 2 α l s=1 j=1 l=1 S J(s) S J(s) λs βs (Psj (t))2 λs βs (P¯sj )2 1 1 − + . αl s=1 j=1 2 αl s=1 j=1 2 (36) According to (15) and (32), L q (t) S J(S) S J(s) λs βs (Psj (t))2 l 1 l [ Hsj Psj (t)λs ] + αl s=1 j=1 2 s=1 j=1 l=1 αl S J(s) 1 λs Psj (t)esj αl s=1 j=1 L q (t) S J(S) S J(s) λs βs (P¯sj )2 l ¯ 1 l [ Hsj Psj λs ] + αl s=1 j=1 2 s=1 j=1 l=1 αl

+

+

j=1

All links l satisfy (1 + ε)[τl (t) +

S J(S) s=1 j=1

l Hsj Psj (t)λs ]

C c=1

x ¯cl . (33)

All link-channel pairs (l, c) satisfy C x C x x ¯cl ¯cl ¯cl (1 + ε)[ + + c c c] k∈Il ηk k∈M (b(i)) c=1 ηk k∈M (e(l)) c=1 ηk 1. Hence, the drift can be written as ¯ E(ΔV (t)|¯ q (t), δ(t)) L q (t) S J(S) l l [τl (t) + Hsj Psj (t)λs s=1 j=1 l=l αl −

S J(S) s=1 j=1 S J(s)

+

L C x C δ c (t) x ¯ck ¯dk k [ + c c d l=1 c=1 ηl k∈Il ηk k∈M (b(i)) d=1 ηk

(35)

where C4 is a constant. C x C x x¯ck ¯dk ¯dk + + Let mcl = c d d k∈Il ηk k∈M (b(i)) d=1 ηk k∈M (e(l)) d=1 ηk in order to reduce inequality (35). The first term of (35) can be written as L q (t) S J(S) S J(S) l l ¯ l [τl (t)+ Hsj Psj (t)λs − Hsj Psj λs ] s=1 j=1 s=1 j=1 l=l αl L q (t) S J(s) λs βs (Psj (t))2 1 l τl (t) + αl s=1 j=1 2 l=1 αl L q (t) S J(S) l l [ Hsj Psj (t)λs ] + α l s=1 j=1 l=1

1 αl

s=1 j=1

S J(s) s=1 j=1

−ε

C xd (t) − x xck (t) − x ¯dk ¯dk k ×[ + c d ηk ηk k∈Il k∈M (b(i)) d=1

=

αl

2

2λs |esj |+

L q (t) S 1 l λs βs J(s) τl (t), αl s=1 l=1 αl (38)

L q (t) S J(s) C l ¯ l [τl (t) + Hsj Psj λs − x ¯cl ] s=1 j=1 c=1 l=l αl

C δ c (t) L C x ¯dk k c − μ (t)] + l c d l=1 c=1 ηl k∈M (e(l)) d=1 ηk

C xd (t) − x ¯dk k ] + C4 , ηkd k∈M (e(l)) d=1

S J(s) S J(s) λs βs (Psj (t))2 1 1 λs P¯sj esj + αl s=1 j=1 2 αl s=1 j=1 S J(s) λs βs (P¯sj )2 1

which is nonnegative. According to (33), the second term of (35) satisfies the following relationship:

S J(s) L q (t) 1 l τl (t) − λs Psj (t)esj αl s=1 j=1 l=1 αl

−

C L q (t) C l l ¯ Psj λs − Hsj x ¯cl ] + [ x ¯cl −xcl ] s=1 j=1 c=1 l=l αl c=1

+

+

L q (t) l l ¯ Psj λs ] + Hsj [τl (t) l=l αl

(37)

Hence, (36) can be bounded as L q (t) S J(S) S J(S) l l ¯ l [τl (t)+ Hsj Psj (t)λs − Hsj Psj λs ] s=1 j=1 s=1 j=1 l=l αl

(34)

+

+

S J(s) 1 λs P¯sj esj . αl s=1 j=1

L q (t) S J(s) l ¯ l [τl (t) + Hsj Psj λs ]. α l s=1 j=1 l=1

(39)

According to (34) and μcl (t) 1, the forth term of (35) satisfies the following relationship: L C x C δ c (t) x ¯ck ¯dk k [ + c c d l=1 c=1 ηl k∈Il ηk k∈M (b(i)) d=1 ηk +

C x ¯dk − μcl (t)] d η k∈M (e(l)) d=1 k

C δ c (t) L k c (40) c ml . l=1 c=1 ηl When considering (17), the third and the fifth terms of (35) satisfy the following relationship: L q (t) C L C δ c (t) xc (t) − x ¯ck l k k [ x ¯cl − xcl ] + [ c c ηk l=l αl c=1 l=1 c=1 ηl k∈Il

−ε

+

C xd (t) − x C xd (t) − x ¯dk ¯dk k k + ] ηkd ηkd k∈M (b(i)) d=1 k∈M (e(l)) d=1

101


C x L C δ d (t) δkc (t) ¯cl −xcl ηlc ql (t) k [ −( + c c d η α η l l=1c=1 k∈Il k∈M (b(i))d=1 ηk l k

+

C δ d (t) k )], d k∈M (e(l)) d=1 ηk

(41)

which is also nonnegative. According to (38)∼(41), the Lyapunov drift can be bounded as ¯ E(ΔV (t)|¯ q (t), δ(t)) L q (t) S J(s) l ¯ l [τl (t) + Hsj Psj λs ] −ε α l s=1 j=1 l=1 −ε

L C δ c (t) k c c ml + C4 , l=1 c=1 ηl

which is negative and indicates that the network is stable. The network stability is been proven when the efficiency 1 ratio is . Θ+2

4

Performance analysis and simulation

4.1 Analysis result of reliability model From equation (3), we can observe the relationships among path hops h, end-to-end reliability r, channel error e, and number of time slots per hop Bij , which are shown in Figs. 3∼5.

Fig. 3 Relationship between hops and time slots per hop.

Fig. 4 Relationship between reliability and time slots per hop.

Fig. 5 Relationship between channel errors and time slots per hop.

In Figs. 3∼5, we can conclude that the number of hops and the end-to-end reliability have little influence on the number of time slots needed to transmit a packet; however, the channel error has big effect on the number of time slots needed to transmit a packet. Serious channel condition will need more time slots to guarantee certain reliability. The total time slot overhead Or (total number of time slots) incurred over h hops when trying to achieve a relia1 bility r h at each hop is Or = Bij

h−1 i=0

1

i

r h = (1 +

rh 1−r . ) 1 − e 1 − r h1

(42)

The relationships among Or , hops h, channel error e, and end-to-end reliability r are studied in Figs. 6∼8.

Fig. 6 Relationship between hops and time slots per path.

Fig. 7 Relationship between reliability and time slots per path.

102


energy consumption decreases with the increase of distance. This is because if one node is within the scope of (0, ru ) the energy consumption decreases with the small transmission radius, and if one node is without the scope of (0, ru ), it transfers data with radius ru , and the data quality decreases with the large transmission radius. Hence, the energy consumption is high when a node is ru -distance away from the sink. Then, we formalize function (12) in Figs. 11 and 12 with different transmission radii and rates.

Fig. 8 Relationship between channel errors and time slots per path.

We can conclude that the number of hops has the greatest influence on the time slot overhead when the channel error is smaller than 0.7. Moreover, when the channel error is bigger than 0.7, its influence is exponential. 4.2 Analysis result of power control In this section, we will analyze the relationships among energy consumption, end-to-end delay, transmission radius, and transmission rate. We choose the same parameters as in [18], in which d0 = 87 m, Ce = 333 pJ/(bit×m2 )−1 , Cl = 0.04329 pJ/(bit×m4 )−1 , F = 50 nJ×bit−1 , and Einit = 0.5 J. We formalize function (9) in Figs. 9 and 10 with different transmission radii and rates.

Fig. 11 Relationship between end-to-end delay and transmission radius.

Fig. 12 Relationship between end-to-end delay and transmission rate.

Fig. 9 Relationship between energy consumption and transmission radius.

Fig. 10 Relationship between energy consumption and transmission rate.

In Figs. 9 and 10, the overall trend of the energy consumption is 1) if one node is within the scope of (0, ru ), the energy consumption increases with the growing ru ; 2) if the distance between one node and sink is larger than ru , the

In Figs. 11 and 12, we can deduce that the delay decreases with the increase of transmission radius and transmission rate. Because the data quantity in (0, ru ) is the biggest, the delay is determined mostly by the part of delay within scope (0, ru ). Meanwhile, in (0, ru ), the data quantity decreases when the distance between a node and the sink increases. The delay follows the same trend as that of the data quantity according to Proposition 1. When the distance between a node and the sink is out of the scope of (0, ru ), the delay follows the same trend as in (0, ru ). Therefore, the delay increases at first and then decreases, which causes a zigzagging trend. 4.3 Simulation results of MJRSP algorithm We utilize OPNET 10.0 to evaluate the performance of our algorithm. Our simulations are based on the network topology shown in Fig. 1. We compare the performance of our algorithm with those in [12,21]. Moreover, the performance metrics are designed as network throughput, end-to-end delay, and energy consumption. We consider five settings, physical interference with multipath routing, physical interference with singlepath routing, pairwise interference with single-path routing, fix rate/fix power, and variable rate/variable power. In each


103

setting, we vary the number of channels and the number of nodes. The performance comparisons among different algorithms and different settings are shown in Figs.13 ∼18. Based on the results in Figs. 13 and 14, we make the following observations: Increase in number of channels results in almost a proportional increase in the network throughput for all approaches, and our algorithm performs better than most of approximation algorithms. The variety of network throughput is minimal with increase of network scale if we average the network throughput by the number of nodes. For the given network setup, the degradation of performance from multipath to single-path routing is minimal. For pairwise interference with single-path routing, the randomized algorithm outperforms the Naive approach especially for large number of channels where interference awareness becomes more important. For physical interference model, the weight-based ap-

proach outperforms the length-class-based approach for both multipath and single-path routing. We assume that the end-to-end delay is unified and set as 150 ms. Then, we will compare the end-to-end delay among algorithms in [11, 12] and our algorithm. The comparisons are shown in Fig. 15. Based on the results in Figs. 15 and 16, we make the following observations: Increase in number of channels results in decrease in the end-to-end delay for all approaches. Our algorithm is closed to the latency minimization algorithm and more stable with the increase of channels and nodes. Based on the results in Figs. 17 and 18, we can conclude that the energy consumption of the MJRSP algorithm is moderate, compared with the famous scheduling algorithms in [12, 21]. In all, the MJRSP algorithm can perform better in aspects of network throughput, end-to-end delay, and energy consumption.

Fig. 13 Comparisons of network throughput among different algorithms with variable channels.

Fig. 14 Comparisons of network throughput among different algorithms with variable nodes.

Fig. 15 Comparisons of end-to-end delay among different algorithms with variable channels.

Fig. 16 Comparisons of end-to-end delay among different algorithms with variable nodes.

104


Fig. 17 Comparisons of energy consumption among different algorithms with variable channels.

Fig. 18 Comparisons of energy consumption among different algorithms with variable nodes.

5

[5] M. Alicherry, R. Bhatia, L. Li. Joint channel assignment and routing for throughput optimization in multi-radio wireless mesh networks[C]//International Conference on Mobile Computer and Network (MobiCom). New York: ACM, 2005: 58 – 72. [6] J. Zhang, H. Wu, Q. Zhang, et al. Joint routing and scheduling in multi-radio multi-channel multi-hop wireless networks[C]//IEEE International Conference on Broadband Network Commuication System (BroadNETS). New York: IEEE, 2005: 631 – 640. [7] R. Bhatia, M. Kodialam. On power efficient communication over multi-hop wireless networks: joint routing, scheduling and power control[C]//IEEE INFORCOM. New York: IEEE, 2004: 1457 – 1466. [8] R. L. Cruz, A. V. Santhanam. Optimal routing, link scheduling and power control in multi-hop wireless networks[C]//IEEE INFORCOM. New York: IEEE, 2003: 702 – 711. [9] B. S. A. Kumar, M. Marathe, S. Parthasarathy, et al. Algorithmic aspects of capacity in wireless networks[C]//Proceedings of the 2005 ACM Sigmetrics International Conference on Measurement and Modeling of Computer System (ACM SIGMETRICS). Banff, Alberta, Canada, 2005: 133 – 144. [10] X. Lin, S. Rasool. A distributed joint channel-assignment, scheduling and routing algorithm for multi-channel ad hoc wireless networks[C]//IEEE INFORCOM. New York: IEEE, 2007: 1118 – 1126. [11] D. Chafekar, V. S. A. Kumar, M. V. Marathe. Cross-layer latency minimization in wireless networks with SINR constraint[C]// Proceedings of the 8th ACM International Symposium Mobile Ad-Hoc Networking and Computing (MOBIHOC). New York: ACM, 2007: 110 – 119. [12] M. A. Ayyoub, H. Gupta. Joint routing, channel assignment and scheduling for throughput maximization in general interference models[J]. IEEE Transactions on Mobile Computing, 2010, 9(4): 553 – 565. [13] T. Moscibroda, R. Wattenhofer, Y. Weber. Protocol design beyond graph-based models[C/OL]//Proceedings of the 5th Workshop on Hot Topics in Networks (HotNets). Irvine, CA, 2006: http://www.distcomp.ethz.ch/publications/hotnets06. [14] T. Moscibroda, R. Wattenhofer, Y. Weber. Topology control meets SINR: The scheduling complexity of arbitrary topologies[C]// Proceedings of the 7th ACM International Symposium Mobile Ad-Hoc Networking and Computing (MOBIHOC). New York: ACM, 2006: 310 – 321.

Conclusions

In this paper, we considered the joint routing, scheduling, and power control for multichannel WSNs with SINR model. Our objective is to achieve reliable and real-time communications while minimizing energy consumption and maximizing network throughput. In the context of dynamic channel assignment and multiple paths, our contributions include designing of approximation algorithm for the above joint problem. Meanwhile, our algorithm is distributed, and it can be implemented online. A number of questions remain open. First, we plan to extend our algorithm to handle nonorthogonal channels. Second, we should further improve the algorithm to efficiently handle actual network environment. Third, it would be interesting to investigate if our approach can be adapted to obtain provable algorithms for optimizing other performance metrics, such as fairness. Last but not least, our algorithm neglects the protocol overhead, and we should carefully design the protocols in real implementations. References [1] C. Pandana, W. P. Siriwongpairat, T. Himsoon, et al. Distributed cooperative routing algorithms for maximizing network lifetime[C]// Wireless Communication and Networking Conference (WCNC). Las Vegas, NV, 2006: 451 – 456. [2] M. Bahl, R. Chandra, J. Dunagan. SSCH: Slotted seeded channel hopping for capacity improvement in IEEE 802.11 ad hoc wireless networks[C]//International Conference on Mobile Computer and Network (MobiCom). New York: ACM, 2004: 216 – 230. [3] S. Wu, C. Lin, Y. Tseng, et al. A new multi-channel MAC protocol with on-demand channel assignment for multi-hop mobile ad hoc networks[C]//International Symposium on Parallel Architecture Algorithm and Netowrking (I SPAN). Los Alamitos, CA: IEEE Computer Society, 2000: 232 – 237. [4] M. Kodialam, T. Nandagopal. Characterizing the capacity region in multi-radio multi-channel wireless mesh networks[C]//International Conference on Mobile Computer and Network (MobiCom). New York: ACM, 2005: 73 – 87.

X. ZHANG et al. / J Control Theory Appl 2011 9 (1) 93–105 [15] P. Kyasanur, J. So, C. Chereddi, et al. Multi-channel mesh networks: Challenges and protocols[J]. IEEE Wireless Communications, 2006, 13(2): 30 – 36. [16] X. Lin, N. B. Shroff. The impact of imperfect scheduling on cross-layer rate control in multihop wireless networks[C]//IEEE INFORCOM. Piscataway: IEEE, 2005: 1804 – 1814. [17] G. Chen, C. Li, M. Ye, et al. An unequal cluster-based routing strategy in wireless sensor networks[J]. Wireless Networks, 2009, 15(2): 193 – 207. [18] Z. Zeng, Z. Chen, A. Liu. Energy-hole avoidance for WSN based on adjust transmission power[J]. Chinese Journal of Computers, 2010, 33(1): 12 – 22. [19] M. Hanckowiak, M. K. aronski, A. Panconesi. On the distributed complwxity of computing maximal matching[J]. SIAM Journal of Discrete Mathematics, 2001, 15(1): 41 – 57. [20] X. Lin, N. B. Shroff. Utility maximization for communications networks with multi-path routing[J]. IEEE Transactions on Automation Control, 2006, 51(5): 766 – 781. [21] M. Johansson, X. Lin. Corss-layer optimization of wireless networks using nonlinear column generation[J]. IEEE Transactions on Communications, 2006, 5(2): 435 – 445. Xiaoling ZHANG received her B.S. degree in Taiyuan University of Technology, Taiyuan, Shanxi, China, in 2005. She is currently working towards her M.S. and Ph.D. degrees in the area of wireless industrial sensor networks in Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, Liaoning, China, where her current research focus is on the industrial wireless standards and optimal scheduling algorithms for increasing the reliability and timeliness in wireless networks. E-mail: [email protected].

105

Haibin YU was born in Heilongjiang Province, China, in 1964. He received his Ph.D. degree in Automatic Control at Northeastern University, Shenyang, China. He is currently a professor of Key Laboratory of Industrial Informatics at Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China. His current interests include wireless sensor networks and networked manufacturing. E-mail: [email protected]. Wei LIANG received her Ph.D. degree in Mechatronic Engineering from Shenyang Institute of Automation, Chinese Academy of Sciences, in 2002. She is currently serving as an associate professor of Shenyang Institute of Automation. Her research interests are in the areas of wireless sensor network, industry communication and system simulation. Email: [email protected].

Meng ZHENG was born in Liaoning Province, China, in 1983. He received his B.S. degree in Applied Mathematics, and M.S. degree in Operational Research and Cybernetics at Northeastern University, Shenyang, China, in 2005 and 2008, respectively. He is working on his Ph.D. degree at the Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, China. His current interests include wireless sensor networks, industry wireless networks and networked control systems. E-mail: zhengmeng [email protected].