End-to-End Delay Constrained Routing and Scheduling for Wireless ...

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

End-to-End Delay Constrained Routing and Scheduling for Wireless Sensor Networks Qing Wang∗ , Pingyi Fan∗ , Dapeng Oliver Wu† and Khaled Ben Letaief ∗

{qing-wang06@mails, fpy@mail}.thu.edu.cn; † [email protected];  [email protected] ∗ Tsinghua National Laboratory for Information Science and Technology (TNList), ∗ Department of Electronic Engineering, Tsinghua University, Beijing 100084, China. † Department of Electrical & Computer Engineering, University of Florida, Gainesville, Florida 32611, USA.  Department of E.C.E., Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, HK.

Abstract— In the paper, we consider the end-to-end routing and link scheduling problem for multi-hop wireless sensor networks. The efficient link scheduler under our consideration is intended to assign time slots to different users so as to minimize channel usage subject to constraints on data rate, delay bound, and delay bound violation probability. We also present a coupled robust multi-path routing structure satisfying the restriction of flows over fading channels based on an SINR-based interference model. Here the effective capacity (EC) model is used and then the joint routing and link scheduling can be formulated as a mixed integer optimization problem. Moreover, because the mixed integer optimization problem is NP-complete, we propose a computationally feasible EC-based Column-Generation-Algorithm (EC-CGA) to search for a sub-optimal solution. Simulation results are given to evaluate the performance of our proposed scheme.

I. I NTRODUCTION Wireless sensor networks attracted much attention in recent years due to their wide applications. For such networks, link scheduling and routing are two important issues. Typically, an efficient link scheduling scheme is intended to schedule time slots and possibly transmit powers for multiple users so that certain optimized criteria (such as throughput, spatial reuse or fairness index) are satisfied. One key issue in the design of link scheduling is how to model and mitigate/avoid interference. In the literature, there are two channel models to characterize interference, i.e. 1) the protocol interference model (PrIM) or disk model and 2) the physical model [1]. Though the physical model is more accurate, it is more difficult to solve the link scheduling problem under the SINR constrains, especially for dynamic power adaption. Thus, to achieve reliable communications, an efficient way is to find a carefully constructed link scheduling scheme. On the other hand, for wireless sensor networks, the traffic load of each node usually changes. Thus, another challenging issue is to route the flows cooperatively to guarantee the network throughput. Essentially, the routing problem is to decide a multi-path routing structure for each source node and an assignment of its flow to all links. Moreover, the flow assignment needs to satisfy certain practical restrictions. Most importantly, the assigned flow should be under maintained schedulability associated with the link scheduling. Now, we review some formulations of the joint routing and link scheduling problem and the solution space, which has been studied in both networking and theory fields. The

joint routing and link scheduling problem can be formulated by optimizing the total network throughput or spatial reuse, the uniform throughput [1] [2], max-min fairness [2] [3], minimum potential delay fairness, proportional fairness [4], possibly subject to some constraints on power, channel resources, and/or QoS. A linear programming formulation under the RTS/CTS and PrIM model was proposed in [2]. The solution under the disk model, is usually based on a linkcontention graph or a conflict graph. Its idea is to find the maximum independent sets so that the nodes in a maximum independent set can transmit simultaneously without causing collision while potential interfering users are allocated to disjoint time slots. Such a scheduling problem is equivalent to the well-known graph coloring problem [3]. Since the graph coloring problem is NP-complete, people seek for heuristic algorithms or polynomial-time approximation algorithms [5]. Different from the existing works that either use the disk model or the physical model, in this paper, we use the Effective Capacity (EC) technique [6] to quantify the effect of interference on system performance, which we call EC-based model. Since we consider fading channels, the received SINR is random variable, actually, a stochastic process. Hence, it is possible to use less transmit power to reach the same distance, resulting in resource efficiency. Note that both the physical model and the effective capacity model are based on SINR. In addition, we consider statistic delay performance, i.e., the triplet of data rate, delay bound, and delay bound violation probability. Since the effective capacity model captures the effect of time diversity in fading channels, we will use the EC-based model in the end-to-end routing and link scheduling problem. Our main focus in this paper is how to schedule links such that the channel usage is minimized while satisfies the restriction of the flow assignment, other than only considering the link scheduling in [7]. The rest of this paper is organized as follows: In Section II, to clarify the effective capacity model, we review some basic concepts and important results in the effective capacity theory. In Section III, we describe our network model and formulate the end-to-end routing and link scheduling problem into a mixed integer programming problem. In Section IV, we develop a feasible EC-based Column-Generation Algorithm (EC-CGA) to find a sub-optimal solution. Section V shows the simulation results. Section VI concludes the paper.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

II. R EVIEW OF E FFECTIVE C APACITY Effective Capacity (EC) [6] is a connection-layer model in which a wireless link is modeled by two EC functions, respectively, the probability of nonempty buffer γ(μ) and the QoS exponent of this connection θ(μ). Both of them are functions of the source traffic rate μ. Specifically, the key idea in the theory of effective capacity is that, if the source traffic has a communication delay bound of Dmax and can only tolerate a delay-bound violation probability of ε at most, then we need to limit the source data rate to a maximum of μ, where μ is the solution to ε = γ(μ)e−θ(μ)Dmax in which θ(μ) = μα−1 (μ). Here α(·) is exactly the originally defined function of effective capacity. We review its definition first. Let r(t) be the  t instantaneous channel capacity at time t. Define S(t) = 0 r(τ )dτ , which is the service provided by the channel. Suppose the channel is ergodic and stationary. Then the effective capacity function of r(t) is defined as [6] −Λ(−u) , ∀u > 0 (1) u 1 where Λ(−u) = lim log E[e−uS(t) ]. t→∞ t Thus, if we can derive the effective capacity function α(u) based on different kinds of fading channels, already discussed in [8], [9] then we can find the QoS exponent function θ(·), according to θ(μ) = μα−1 (μ). Finally, associated with the QoS requirement of the source traffic, respectively, the communication delay bound of Dmax and a delay-bound violation probability ε, we can estimate the probability of nonempty buffer γ(μ) and then tune the source rate μ to guarantee its QoS requirement. Now, we can see that the effective capacity model is actually a triplet of data rate, delay bound and delay bound violation probability, i.e., {μ, Dmax , ε} or another useful form, i.e. {μ, Dmax , Perr } also derived by the authors in [6], where Perr is the packet error probability and the relation between them is given by u = − log Perr /(μ · Dmax ). α(u) =

III. P ROBLEM F ORMULATION We model a sensor network by a set of N nodes, denoted by set N , and a set of directed links, denoted by set E. Then, the complete communication graph G is a directed graph G = (N , E). Moreover, for a node n, we use En+ to denote the set of incoming links (all directed links pointed to n). Similarly, we use En− to denote the set of outgoing links at node n. We assume two types of nodes, respectively, ordinary nodes denoted by the set NI and gateway nodes denoted by the set NII . Ordinary nodes will sense and generate traffic. (n) We use rs to denote the required data rate by ordinary node n for its generated traffic where n ∈ NI . Ordinary node can also play as relay to route the information to gateway nodes. The gateway nodes do not generate their own traffic and will not act as relay, either. They only provide the connectivity to an information process center or other network systems such as Internet, etc. Assume that there is only one radio frequency or say, channel and a node can not transmit and receive simultaneously. For each link {i, j} ∈ E, transmitting

node i can communicate successfully with receiving node j only if specified QoS (SINR or BER or delay) is satisfied. Let Pi (t) be the transmission power for node i at time t, Gij (t) the gain of the fading channel from node i to node j and N0j be the variance of the thermal noise at receiver j. The SINR at receiver j due to transmission from node i is given by SIN Rij (t) =

N0j +

P (t)Gij (t) i . l=i,j Pl (t)Glj (t)

(2)

Assume that each link {i, j} ∈ E has a traffic demand of (ij) rs bits/sec, which needs to be transmitted across the link (ij) (ij) with delay bound Dmax and packet error probability Perr . (n) ra Moreover, a fairness exponent, denoted by η = (n) , is defined (n)

rs

as the ratio of the achieved data rate ra of node n over the (n) required data rate rs for its realtime generated traffic. In   (n) particular, ra = {i,j}∈En− rij − {i,j}∈En+ rij . In majority of applications, we have to guarantee certain fairness of the achieved flows for nodes, assuming η0 in our discussion. Now, we give a mixed integer programming formulation of the joint routing and link scheduling problem. First, we assume that the time is slotted where K time slots of lengths {ωk } (k = 1, · · · , K and ωk ∈ [0, 1]) are used to schedule links (ij) (ij) (ij) in E with QoS requirements {rs , Dmax , Perr }. If there is (ij) no traffic demand for a specific link {i, j}, then rs = 0. Denote the transmission power of node i in slot k by Pi (k). Assume that Pi (k) (∀i ∈ N , ∀k ∈ {1, · · · , S}) can only take two values, i.e., 0 and P0 . Pi (k) = 0 means that node i does not transmit in slot k, while Pi (k) = P0 means that node i transmits in slot k. Note that the complex case in which the power control, namely,Pi (k) ∈ [0, +∞), is considered can be easily extended from our current formulation of the end-to-end routing and link scheduling problem which is given by K 

min

(k)

{ωk }{Pi

s.t.

}

ωk

k=1 (k) Pi

(3)

∈ {0, P0 }, ∀i ∈ N , ∀k ∈ {1, · · · , K}

(4)

ωk ∈ [0, 1], ∀k ∈ {1, · · · , K} (5)   ∗ ∗ (n) αij (uij ) − αij (uij ) ≥ η0 · rs , − {i,j}∈En

αij (u∗ij ) K 

+ {i,j}∈En

≥

∀n ∈ NI , ∀{i, j} ∈ E, ∈ E,

rs(ij) , ∀{i, j}

ωk ≤ 1,

(6) (7) (8)

k=1

where u∗ij = αij (u) =

K  k=1

αij,P (k) (u) = lim i

t→∞

(ij)

− log Perr (ij)

rs

(ij)

× Dmax

,

ωk × αij,P (k) (ωk × u), i

(9)

(10)

t −1 log E[e−u 0 W log(1+SIN Rij,k (τ ))dτ ]. ut (11)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Note that Eqn.(6) can be written into a matrix form C· αij ≥ (n) η · rs where C is a NI × |E| dimensional matrix with all its elements equal to 1 or −1. α ij is an |E|×1 dimensional vector (n) and rs is a NI × 1 dimensional vector. Then, by combining (6) and (7), we can get a unified form of the constraint, namely, e, where A·α  ij ≥  the (NI + |E|) × |E| dimensional matrix (n) 1 C./ r s A = η 0 (ij) and the (NI +|E|)×1 dimensional vector I./ rs e = [1, 1, · · · , 1]T where I is an identity matrix. Note that the (n) notation C./ rs represents a new matrix and the operator ./ means that the elements in the k’th row of C are all divided (n) by the corresponding k’th element in the vector rs . The (ij) explanation of I./ rs is similar.  If Problem (3) results in a solution with ωk ≤ 1, then this solution is feasible; otherwise it is not feasible since we assume there is only one radio frequency or say, channel. The S time slots can form a super-frame and the same slot pattern repeats in each super-frame. We assume admission control is employed and its work procedure is explained as below: for a new link requesting for admission, if the corresponding problem (3) results in a solution with ωk ≤ 1, then the new link can be accepted by the admission control; otherwise, it will be rejected. Consequently, the achieved network throughput Q is defined as the amount of information collected by the gateway nodes and is given by NII 

where

ra(n)

ra(n) , ∀n ∈ NII , ∀{i, j} ∈ E Q=   (12) = αij (u∗ij ) − αij (u∗ij ). − {i,j}∈En

+ {i,j}∈En

Unfortunately, the optimization problem specified by (3) to (8) is NP-complete. In Section IV, we present a column generation based algorithm to find its suboptimal solution. IV. A C OLUMN -G ENERATION -BASED S OLUTION First, we explain the basic idea of column generation algorithm. Column generation is an iterative algorithm for solving huge linear or integer programming problems. More specifically, the original problem is decomposed into a master problem and a subproblem. The master problem and subproblem could be either linear or integer program depending on the problem formulation. The strategy of this decomposition procedure is to operate iteratively on two separate but easierto-solve problems. During each iteration, the algorithm tries to determine whether any variables exist that have a negative reduced cost (in the case of minimization problem) and adds the variable with the most negative reduced cost to the master problem. Thus, the key idea of this solution is to sequentially improve the current solution by solving the subproblem that identifies a single new variable (a column) during each iteration and adding it to the master problem until the algorithm terminates in a reasonable solution. Next we will present our algorithm described as follows: denote M the power set of E, i.e., M contains all possible

combinations of members in E. The master problem is a restriction of the original problem (3) to (8). In the first step of the algorithm, the master problem uses only a subset of columns indexed by s ∈ {1, · · · , |M|}, where |M| is the cardinality of M. The master problem is randomly initialized with any S ⊂ M that satisfies (4) to (8). For each (k) transmitter i ∈ {i, j} ∈ S, the transmit power Pi , (∀k) is equal to P0 . Thus, the master problem is given by min

{ωk }

s.t.

|S| 

ωk

(13)

k=1

ωk ∈ [0, 1], ∀k ∈ {1, · · · , |S|} (14)   ∗ ∗ (n) αij (uij ) − αij (uij ) ≥ η0 · rs ,

− {i,j}∈En

+ {i,j}∈En

∀n ∈ NI , ∀{i, j} ∈ E αij (u∗ij )

≥

rs(ij) , ∀{i, j}

(15)

∈ E.

(16)

Since this formulation optimizes over a subset S of all feasible solutions, the optimal solution to (13) to (16) provides an upper bound for the original problem (3) to (8). In the second step, we run the iteration. During every iteration, after the master problem (13) to (16) is solved, if the solution to the master problem also provides the solution to the original problem (3) to (8), the procedure terminates; otherwise, we need to solve a sub-problem, which identifies a new column that can improve the current solution. The subproblem for generating a new column is formulated as below. For each member Sm ∈ M\S, which refers to the set of all columns that is in M but are not in S,  (m) ATm ( πij . × α ij )}, (17) ζm = min {1 − {Eij }

{i,j}∈Sm

where πij is the vector of dual variables, Am the m’th column of matrix A, ATm the transpose of Am and the notation π.× α represents a new vector in which each element is the product of the two corresponding elements of π and α . Moreover, (m)

t −1 log E[e−u 0 W log(1+SIN Rij,m (τ ))dτ ], t→∞ ut (18)

αij (u) = lim where

SIN Rij,m (t) =

N0j +



P0 Gij (t) l=i,j,&{l,j}∈Sm

P0 Glj (t)

.

(19)

If ζm < 0, then add the column induced by Sm to S as a new member (column). Since there are exponential number of members in M, in practice, we need to randomly select Sm from M\S; the iteration process stops when the solution to the master problem provides an -approximation sub-optimal solution [5] to the original problem (3) to (8). indicates how far the obtained solution is away from the optimal solution. The literature [5] further gave the number of iterations, i.e. the complexity, required to achieve a certain approximation ratio. To implement the proposed algorithm, we try to give a suboptimal distributed MAC protocol for the link scheduling and meanwhile consider flow rate restrictions. First, we use

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Transmitter

Fading channel

Transmitted data

Data source

x Q(n)

+

FDWCA without Dmax EC−CGA FDWCA with Dmax AODV DSR

8 Interference+Noise

Queueing system model used in our simulation.

certain existing schemes to form clusters of nodes. Cluster heads coordinate the transmission initiation by periodically transmitting a beacon signal so that all other nodes can set up their networking parameters and synchronize to the cluster head. It is a TDMA-like protocol based on a well-defined super frame consisting of a beacon, a contention access period (CAP), management channel time allocation (MCTAs), and channel time allocations (CTAs). Second, we search for a feasible solution to the optimal joint flow routing and link scheduling problem in a cluster by using the CGA iterative algorithm. Third, we determine the length of slot CTAk in the super frame based on the obtained optimal wk values. V. N UMERICAL R ESULTS A. Simulation Setting To estimate the effective capacity of one link, we simulate the discrete-time system depicted in Fig. 1. In this system, the data source generates packets at a constant rate μ. Generated packets are first sent to the (infinite) buffer at the transmitter, whose queue length is Q(n), where n refers to the nth sample-interval. The head-of-line packet in the queue is transmitted over the fading channel at data rate r(n). The fading channel has a random power gain g(n). Thus, the transmission rate r(n) is equal to the instantaneous (timevarying) capacity of the fading channel, as below, r(n) = Bc log2 (1 + g(n) × P0 /(I + σn2 )),

(20)

where I is the total average interference power  at receiver j from other simultaneous transmitters k, i.e., I = k P0 · d−α kj . Then, we collect the following measurements from the queueing system at the n-th sampling epoch (n = 1, 2, · · · , NT ): S(n) the indicator of whether a packets is in service (S(n) ∈ {0, 1}), Q(n) the number of bits in the queue (excluding the packet in service), and τ (n) the remaining service time of the packet in service (if there is one in service). We calculate the measured effective capacity function αs (u) by the following procedure. γˆ =

NT 1  S(n) NT t=1

,

qˆ =

NT 1  Q(n) NT t=1

(21)

τˆs =

NT 1  τ (n) NT t=1

,

θˆ =

γˆ × μ μ × τˆs + qˆ

(22)

αs (u) = μ,

x 10

Receiver

r(n) Gain

Fig. 1.

10

ˆ for u = θ/μ.

(23)

In our simulations, the sampling interval δ is set to 1ms. Each simulation run is 104 s long for all the scenarios, in order

network throughput (bits)

Rate = μ

4

Received signal

6

4

2

0 20

30

Fig. 2.

40 number of nodes

50

60

network throughput vs. number of nodes.

to obtain good estimate by Monte Carlo method. Since the sampling interval is 1ms, we have 107 samples for estimation. Next, we introduce the network scenario: we randomly generate N wireless nodes uniformly located in a 500×500m2 square area. The number N of node varies from 20 to 60. We assume there is a information collection center or gateway node in the network which is placed in the center of the region and all traffics generated by sensor nodes will forward to it. In our simulation, the power of thermal noise at the receiver is set to 3.34 × 10−9 W and the maximum transmit power P0 is set to 1mW which means the transmission range covers tens (n) of meters. We simulate the required source traffic rate rs for each node to the gateway node by using Constant Bit Rate (CBR) model, with a rate randomly selected from the range 20 ∼ 30kb/s for each node. In particular, the demand of traffic (ij) bit-rate rs of each link is randomly selected from the range 80 ∼ 120kb/s. The corresponding maximum delay bound Dmax is randomly selected from the range 100 ∼ 200ms and the delay bound violation probability ε is randomly selected from the range 1% ∼ 5%. Such kind of QoS requirements is similar to that for voice or compressed video traffic. The ¯ ij is proportional to d−α (α ≥ 2), average channel gain G ij where dij denotes the transmitter-receiver separation distance of Link ij. The fairness exponent η0 is 50%. The bandwidth W of each link is 100kHz. B. Simulation Results We evaluate different approaches, respectively the AODV, DSR, another joint routing and link scheduling scheme FDWCA (Fast Distributed Weighed Coloring Algorithm) [2] and our EC-CGA scheme, by comparing network throughput, length of required scheduling time and scheduling efficiency. Note that the network throughput is the amount of information (measured in bits) successfully collected by the gateway node during an observed time which has been given in (12). We first evaluate the network throughput, by fixing the running time as 5 seconds and varying the network size from

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

4

7

40 EC−CGA FDWCA with Dmax DSR AODV

scheduling time (sec)

30

EC−CGA FDWCA with Dmax AODV DSR

6

network throughput (bits)

35

x 10

25 20 15

5

4

3

2

10 1

5 0

0 20

30

40 number of nodes

50

0

1

60

Fig. 4. Fig. 3.

2

3 running time

4

5

6

network throughput vs. running time

required schedule time vs. number of nodes

20 to 60. For each specific network size, the performance is shown in Fig. 2. It is observed that FDWCA and EC-CGA achieve larger network throughput than AODV and DSR. In particular, when we get a requirement of the delay bound (ij) of traffic, namely Dmax , which means that those packets arriving a gateway node after the maximum durable delay threshold will be dropped though they can also be received by a gateway node, the performance of our EC-CGA overweighs the FDWCA, since FDWCA does not consider the delay bound criteria in the end-to-end routing and link scheduling problem. Fig. 3 depicts that EC-CGA and FDWCA cost much shorter scheduling time than AODV and DSR. Here, the required scheduling time refers to the total run time when the network throughput needs to achieve a preset value, for example 5 × 104 bits in our simulation. Moreover, as the network size grows linearly, the required scheduling time in DSR and AODV increases very quickly, much faster than ECCGA and FDWCA. In particular, for the DSR, AODV and FDWCA, we use the 802.11b as the MAC layer protocol with RTS/CTS mechanism. In DSR and AODV, the large delay in the scheduling is mainly caused by the random resource competition of nodes, since every node is trying to send data to its parent node and eventually to a gateway node. However, the communication with EC-CGA is well coordinated. To evaluate the scheduling efficiency, we fix the network size at 40 and the running time from 0 to 6 second. The comparison of four schemes is shown in Fig. 4. As the time increases, the network throughput of our EC-CGA and FDWCA becomes much larger than that of DSR and AODV, especially after time increases to a relatively stable state, i.e., around 5 second. In other words, it is shown that our EC-CGA can enable finer and faster data collection than FDWCA, DSR and AODV. Overall, the simulation shows the overwhelming advantage of end-to-end routing and link scheduling and also the nice performance of the EC-CGA scheme.

VI. C ONCLUSIONS In this paper, we studied the end-to-end routing and link scheduling problem under fading channels in wireless sensor networks. We used the effective capacity model to formulate it as a mixed integer optimization problem to minimize channel usage subject to constraints on data rate, delay bound, and delay bound violation probability and the restriction of network flow assignment. Moreover, we proposed an iterative EC-based Column-Generation Algorithm (EC-CGA) to search for a sub-optimal solution to the problem. Simulation results confirmed the advantage of the EC-CGA scheme compared with some known methods. ACKNOWLEDGMENT This work is supported in part by the CN National Science and Technology Major Project No.2010ZX03003-003, the US NSF grant CNS-064373 and ONR grant N000140810873. R EFERENCES [1] P. Gupta and P. Kumar, “The capacity of wireless networks,” IEEE Transactions on information theory, vol. 46, no. 2, pp. 388–404, 2000. [2] Y. Wang, W. Wang, X. Li, and W. Song, “Interference-aware joint routing and tdma link scheduling for static wireless networks,” IEEE Transactions on Parallel and Distributed Systems, vol. 19, no. 12, pp. 1709–1726, 2008. [3] X. Huang and B. Bensaou, “On max-min fairness and scheduling in wireless ad-hoc networks: Analytical framework and implementation,” in ACM MobiHoc, 2001, pp. 221–231. [4] X. Lin, N. Shroff, and R. Srikant, “A tutorial on cross-layer optimization in wireless networks,” IEEE Journal on Selected Areas in Communications, vol. 24, no. 8, pp. 1452–1463, 2006. [5] K. Jansen and L. Porkolab, “On preemptive resource constrained scheduling: polynomial-time approximation schemes,” SIAM Journal on Discrete Mathematics, vol. 20, no. 3, pp. 545–563, 2007. [6] D. Wu and R. Negi, “Effective capacity: a wireless link model for support of quality of service,” IEEE Transactions on Wireless Communications, vol. 2, no. 4, pp. 630–643, 2003. [7] Q. Wang, D. Wu, and P. Fan, “Delay-Constrained Optimal Link Scheduling in Wireless Sensor Networks,” IEEE Transactions on Vehicular Technology, vol. 59, no. 9, pp. 4564–4577, 2010. [8] ——, “Effective capacity of a correlated Rayleigh fading channel,” online published by Wireless Communications and Mobile Computing. [9] ——, “Effective capacity of a correlated Nakagami-m fading channel,” online published by Wireless Communications and Mobile Computing.