3-Approximation Algorithm for Joint Routing and
Link Scheduling in Wireless Relay Networks
Chi-Yao Hong, Student Member, IEEE, and Ai-Chun Pang, Member, IEEE
Abstract—In emerging wireless relay networks (WRNs) such as IEEE 802.16j, efficient resource allocation is becoming a substantial issue for throughput optimization. In this paper, we propose an algorithm for joint routing and link scheduling in WRNs. The developed theoretical analysis indicates that the performance of the proposed algorithm is within a factor of three of that of any optimal algorithm in the worst case. Through simulation experiments, the numerical results show that our algorithm outperforms the previously proposed routing and link-scheduling algorithms. Furthermore, the proposed algorithm can effectively achieve near-optimal performance, and provide much better throughput than the theoretical worst-case bound in the average case.
Index Terms—Approximation algorithm, link scheduling, rout-ing, wireless relay networks.
I. Introduction
R
ECENTLY, IEEE 802.16j task group has been devoted to the development of multi-hop relaying technology as an enhancement for IEEE 802.16 point-to-multipoint networks. While IEEE 802.16j subscribers are expected to experience performance improvement through low-cost relay stations (RSs), the degradation of system throughput would occur instead if multi-hop radio resources can not be well-utilized. Consequently, the routing and link-scheduling algorithms, which are of the important resource control functions in multi-hop networks, shall be specifically re-designed to facilitate the effective use of network resources for WRNs.The link-scheduling issues have been extensively investi-gated in wireless multi-hop networks (WMNs) with or without quality of service (QoS) consideration. For instance, [1]–[3] target at QoS-sensitive services while the performance of best-effort traffic over WMNs is studied in [4]–[7]. For non-QoS traffic, system throughput is considered as an essential perfor-mance criterion. Unfortunately, the problem of link scheduling for system throughput maximization in WMNs was proven to beNP-hard [7]. Due to the intractability of the problem, the previous works have focused on developing approximation algorithms for WMNs [4]–[6]. Besides, it was shown that cross-layer optimization takes the advantages over strictly layered design in mobile ad hoc networks [8]. The joint design
Manuscript received October 26, 2007; revised March 3, 2008; accepted April 15, 2008. The associate editor coordinating the review of this paper and approving it for publication was S. Shen.
C.-Y. Hong is with the Department of Computer Science and Information Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]).
A.-C. Pang is with the Graduate Institute of Networking and Multimedia, Department of Computer Science and Information Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: [email protected]).
Digital Object Identifier 10.1109/TWC.2009.071193
of link scheduling and routing for WMN poses a challenging research problem. Among the research related to join routing and link scheduling, the solution in [9] requires exponential-time complexity with respect to the number of stations, and is intractable for large-scale networks. In [10], Wei et al. pro-posed an interference-aware scheme for joint routing and link scheduling to maximize the system throughput of WiMAX-based WMNs. However, the proposed heuristic scheme can not guarantee the worst-case performance.
The previous schemes for WMNs could not properly applied to WRNs because of some inherent differences of routing and link scheduling for throughput maximization between WMNs and WRNs. The key discrepancies are substantially two-folds and described as follows. 1) In WRNs, RSs do not generate data traffic. This restriction may make the design of routing for throughput maximization in WRNs easier than that in WMNs. On the other hand, the routing problem in WRNs could be more difficult due to its limitation of transmission path of traffic flows. In a WMN, each node can exchange data packets with any other node in the network while a WRN traffic flow is either destined to a base station (BS) or delivered from a BS through one or more RSs. Such a limitation raises the routing complexity and reduces the routing flexibility for throughput maximization. 2) It is generally considered that WMN nodes are equipped with omni-directional anten-nas to provide a greater coverage area [4], [5], [9], [11]. However, WRN RSs are expected to have high-technology smart antennas with switchable/steerable beam to increase the spectral efficiency [12], [13]. In this case, the inter-station interference can be significantly reduced, and a higher degree of spatial reuse can be achieved. The decreased interference and increased degree of spatial reuse lead to a considerably large set of links that can be activated concurrently in WRNs. Thus the link scheduling problem for throughput maximization in WRNs would be more complex compared with that in WMNs.
Based on the above discussion, it is still an open and interesting question: whether there exists a time-efficient joint routing and link scheduling scheme that provides near-optimal system throughput for WRNs. In this study, we present an algorithm to support interference-aware routing and link scheduling to maximize the system throughput for WRNs. The algorithm accommodates a general workload model for data requests, and fits many other polling-based wireless systems as well as IEEE 802.16 [14]. The achieved throughput of our pro-posed algorithm is proven to be within a factor of three of that of any optimal solution in the worst case. To the best of our knowledge, the 3-approximation algorithm for joint routing
and link scheduling is one of the first works for WRNs that can guarantee the worst-case performance within a small-constant bound. Through our simulation experiments, the numerical results show that our proposed algorithm outperforms the previously proposed routing and link-scheduling algorithms. Furthermore, the algorithm effectively achieves near-optimal performance, and provides much better throughput than that of the theoretical worst-case bound in the average case.
The remainder of the paper is organized as follows. tion II formally presents our problem formulation. In Sec-tion III, we present our joint routing and link-scheduling algorithm. A time complexity analysis and a worst-case performance analysis for the proposed algorithm are also presented. Section IV presents some simulation experiments and numerical results of the proposed algorithm. Finally, the paper is concluded with a brief summary.
II. Problem Definition
This section presents the definition of our joint routing and link scheduling problem. We assume that the WRN under investigation adopts Time Division Multiple Access (TDMA) protocol [14], and accommodates best-effort services in IEEE 802.16j. The data transmission among stations in the WRN is synchronized on frame basis. A frame consists of several time slots which is defined as a basic time unit for transmission in the system. A centralized base station (BS) acts as a coordi-nator, and provides the contention resolution for data requests of best-effort service flows from serving subscriber stations (SSs). The BS also computes the corresponding transmission routes and schedules for these flows.
In each computation, the WRN is modeled as a simple directed graph G = (V, E), where the vertices represent the stations including a BS, several RSs and SSs, and the edges denote the wireless links between two stations. The vertex set V is divided into three disjoint subsets VB,VR,VS, and they
respectively include a BS, RSs, and SSs. Each SS ns ∈ VS
has a traffic demand ps. ps signifies the aggregate uplink
traffic request for ns. For simplicity, only uplink traffic is
considered in this study. The downlink traffic can be computed in a similar way, and the details are omitted. A wireless link (i, j) is an element of E if and only if station nj is within the
maximum transmission range of station ni. Each edge (i, j) in
E is associated with capacity ri, j.
Given the graph G, a conflict graph Gc = (Vc, Ec) is
defined as a simple undirected graph to indicate wireless interference among the stations. Any vertex of Gc bijectively
corresponds to an edge in G. Two vertices inVcare adjacent
if and only if the corresponding two edges inE are adjacent. This model reflects the primary interference, which captures the interference when a station 1) transmits packets to two different stations simultaneously, 2) receives packets from two different stations concurrently, or 3) transmits and receives packets at the same time. Another type of interference, namely, the secondary interference, occurs when a receiving station is interfered by other transmissions which are not intended for this station. We assume that the secondary interference can be avoided for the WRNs under investigation through the beam and null steering capability of smart antennas.
Based on the above definition of the problem, our proposed solution for joint routing and link scheduling is developed to produce a transmission schedule such that the schedule is interference-free and the system throughput is maximized. Note that the throughput maximization problem in our study is equivalent to the minimization of the schedule length in terms of time-slots because the achieved system throughput is inversely proportional to the schedule length [4], [5], [7], [11].
III. A Joint Routing and Link-Scheduling Algorithm This section elaborates on our routing and link-scheduling algorithm to achieve the maximal system throughput for WRNs.
A. Linear Programming Based Routing
For any edge (i, j) ∈ E, let fi, j be the aggregate traffic transmitted from station ni to station nj. The total time of
station ni for transmitting and receiving the data is denoted
by ρi = (i, j)∈Erfii, j, j
+( j,i)∈Erfjj,i,i
. Due to the primary in-terference, any two transmissions that are originated/destined from/to a station can not be overlapped, and the schedule length would be limited by ρi. Consider a routing with a
large ρi. The schedule length is inevitably large, regardless
of which link scheduler is adopted. In order to minimize the schedule length, we shall find a routing such that maxni∈V{ρi}
is minimized. To achieve this, the linear programming (LP) technique is used, and its objective function is
minimize β (1a)
where
β ≥ ρi, ∀ni∈ V. (1b)
For any RS nj, the flow conservation constraint is
(r, j)∈E fr, j = ( j,r)∈E fj,r, ∀nr∈ Vr. (1c)
Each SS has no incoming traffic, i.e.,
fi,s= 0, ∀ns∈ Vs, ∀ni∈ V, (1d)
and each BS has no outgoing traffic, i.e.,
fb,i= 0, ∀nb∈ Vb, ∀ni∈ V. (1e)
Furthermore, the nonnegative flow constraint is
fi, j ≥ 0, ∀(i, j) ∈ E. (1f) The traffic demand constraint psfor each SS ns
(s,i)∈E
fs,i≥ ps, ∀ni∈ Vs (1g)
Note that we make no assumption on the number of routing paths for any subscriber. That is, the traffic load from each SS could be distributed to multiple routes to alleviate the network congestion.
B. Makespan Link Scheduling
Once the routing is determined, this section shows how we efficiently schedule the link traffic so that the schedule length can be minimized and the system throughput can be accordingly increased. Before presenting our scheduling algorithm, we define several terms used in this section. For each edge (i, j) ∈ E, ti, j represents the number of time-slots for transmitting fi, j, i.e., ti, j = fi, j/ri, j. Since each link in E corresponds to a vertex in Vc, t
i, j can be bijective-mapped to c(i) for any vertex nc
i in Vc. The definition of
c(i) corresponding to vertex nc
i inVc is equivalent to that of ti, j for edge (i, j) ∈ E.
For any vertex nc
i in Vc, we define that ξc(i) is the total
number of time-slots required by nc
i and its neighbors to fulfill
their traffic demands, i.e., ξc(i)= c(i)+ ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ (i, j)∈Ec c( j) ⎞ ⎟⎟⎟⎟⎟ ⎟⎠ . (2)
For any time-slot τ, Iτ denotes a set of the links that are scheduled to transmit the data in time-slotτ. In order to avoid the collision resulting from the primary interference,Iτ shall be an independent set of vertices in Gc.
Given a flow requirement fi, j of each link in G, we present an Interference-aware Makespan Scheduling (IMS) algorithm to minimize the schedule length for the traffic flows. As shown in Algorithm 1, the inputs include the conflict graph Gc, and
the corresponding c(i) and ξc(i) for each vertex nc i in Vc.
Initially, G is set to Gc for the operations of the following
loop. For theτth time slot (see the outer repeat loop), some proper vertices in Gc will be selected into I
τ. The selection is based on their up-to-date values of ξ, and the process is shown in the inner repeat loop of lines 6-15. G(τ)in line 4 will be used for the worst-case analysis in the following section, and is not discussed here. Let ℘( j) be a flag to indicate whether vertex nj ∈ V is able to be scheduled into Iτ due to the primary interference. Initially, all vertices inV could be scheduled, and their ℘ is set to 0. In each iteration of the inner repeat loop, a vertex in V with minimalξand its ℘ = 0 is selected into Iτ. Then the ℘ values of the vertex and its neighbors are set by 1 for collision avoidance. Also, if vertex n
i ∈ Vhas been satisfied with its scheduled time slots, i.e., (i) = 0, it will be removed from V. The inner loop will be performed until℘= 1 for all vertices in V. Then the link-scheduling operation for the following time slot will be activated. The outer repeat loop terminates when all link flows are properly scheduled (i.e.,|V| = 0). Through our Algorithm IMS, a collision-free makespan schedule I1, I2, . . . , Iκ will be produced, whereκ is the schedule length in the unit of time-slots.
C. Time Complexity Analysis
Then we analyze the time complexity of our proposed joint routing and link-scheduling algorithm. The proposed LP-based routing algorithm can be solved in polynomial time by the well-known projective method [15]. Let(Gc) be the maximal
degree of vertices in Gc. The time complexity of scheduling
each time-slot in AlgorithmIMS is O(|E|×max lg|E|, (Gc) ),
because choosing a vertex n
i∈ Vtakes O(max lg|E|, (Gc) )
1 ) 1 ( , () ) ( 1W "W n 3 ) 2 ( , () ) ( 2W "W n (), ()(3) 2 3W "W n (a) ) ( 1 ~W ˖ ) ( 2 ~W ˖ ) ( 3 ~W ˖ (b)
Fig. 1. An example of graph reduction from (a) G(τ)to (b) ˜G(τ)
time and at most |E| links could be scheduled to the slot. Thus Algorithm IMS requires O(κ|E| × max lg|E|, (Gc) )
computations. Since(Gc)≤ |E|, the worst-case running time
of Algorithm IMS will be O(κ|E|2), which is acceptable in a WMN/WRN network [4], [5], [10].
Algorithm 1 IMS Require: Gc= (Vc, Ec)
Ensure: I1, I2, . . .
1: G ← Gc (V ← Vc, E← Ec, (i) ← c(i), and ξ(i) ← ξc(i) for all vertex nc
i inVc)
2: τ ← 1 3: repeat
4: G(τ)← G
5: ℘( j)← 0 for all vertices njinV
6: repeat
7: ni← arg minnj∈V,℘( j)=0ξ( j)
8: Iτ← Iτ∪ {ni}
9: ℘( j)= 1 for all vertices nj∈ {ni} ∪ {nk|(i, k) ∈ E}
10: (i)← (i)− 1
11: if (i)= 0 then
12: V← V\ {ni}
13: end if
14: updateξ( j) for all vertices nj∈ {ni} ∪ {nk|(i, k) ∈ E}
according to (2)
15: until℘( j)= 1 for all vertices nj∈ V
16: τ ← τ + 1 17: until|V| = 0 18: κ ← τ
19: return I1, I2, . . . , Iκ
D. Worst-Case Performance Analysis
In this section, a worst-case analysis for system throughput of our routing and link scheduling algorithm against any of optimal solutions is presented. For any G(τ) (1≤ τ ≤ κ), we construct a simple undirected graph ˜G(τ) = ( ˜V(τ), ˜E(τ)) such that for each vertex n(iτ)inV(τ), a clique ˜C(τ)
i with(τ)(i)
fully-connected vertices is added into ˜G(τ). For any two different cliques, say ˜Ci(τ)and ˜C(jτ), ˜Ci(τ)is fully connected to ˜C(jτ)if (i, j) is inE(τ). If (i, j) is not in E(τ), ˜C(τ)
i and ˜C
(τ)
j are disconnected.
An example of the construction is shown in Figure 1, and the following facts are observed for the construction.
Fact 1: | ˜V(τ)| = n(iτ)∈V(τ) (τ)(i) = |V(τ)| × (τ), where (τ)= n(τ) i ∈V(τ)
Fact 2: ∀˜n(τ)i ∈ ˜C(τ)j ∈ ˜V(τ), ˜di(τ) = (τ)( j) − 1 +
( j,k)∈E(τ)(τ)(k), where ˜di(τ) represents the vertex degree
of ˜n(iτ). Fact 3: ˜δ(τ) = ˜n(τ) i ∈ ˜V(τ) ˜d (τ) i /| ˜V(τ)| = 2| ˜E(τ)|/| ˜V(τ)|, where ˜δ(τ)represents the average vertex degree in ˜V(τ).
Based on the facts, the upper bound of ˜δ(τ)in terms of(τ)( j) and(τ) will be derived in the following theorem.
Theorem 1: ˜δ(τ)≤ max n(τ)i ∈V(τ) (i,k)∈E(τ)(τ)(k) + (τ)− 1. Proof: ˜δ(τ) = ˜n(iτ)∈Cj∈ ˜V(τ) ˜d (τ) i | ˜V(τ)| (by Fact 3) = ⎡ ⎢⎢⎢⎢⎢ ⎢⎣ ˜n(iτ)∈Cj∈ ˜V(τ) ( j,k)∈E(τ)(τ)(k) | ˜V(τ)| ⎤ ⎥⎥⎥⎥⎥ ⎥⎦ + ⎡ ⎢⎢⎢⎢⎢ ⎢⎣ ˜n(iτ)∈Cj∈ ˜V(τ)[ (τ)( j)− 1 | ˜V(τ)| ⎤ ⎥⎥⎥⎥⎥ ⎥⎦ (by Fact 2) = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪ ⎩ n(τ)j ∈V(τ)(τ)( j)×( j,k)∈E(τ)(τ)(k) (τ)|V(τ)| ⎫⎪⎪⎪ ⎪⎪⎬ ⎪⎪⎪⎪⎪ ⎭+ (τ)− 1 (by Fact 1) ≤ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪ ⎩ n(jτ)∈V(τ)(τ)( j)× maxni(τ)∈V(τ)(i,k)∈E(τ)(k) (τ)|V(τ)| ⎫⎪⎪⎪ ⎪⎪⎬ ⎪⎪⎪⎪⎪ ⎭+ (τ)− 1 = max n(iτ)∈V(τ) ⎧⎪⎪⎪ ⎨ ⎪⎪⎪⎩(i,k)∈E(τ) (τ)(k)⎫⎪⎪⎪⎬ ⎪⎪⎪⎭ + (τ)− 1 (by Fact 1)
In addition to the observations from the ˜G(τ) construction, the following facts are observed for our AlgorithmIMS.
Fact 4: ˜d(τ)i ≤ ˜d(τ−1)i − 1 for τ = 2, . . . , κ, where di(τ) represents the vertex degree of n(τ)i inV(τ).
Corollary 1: ˜δ(τ) ≤ ˜δ(τ−1)− 1 for τ = 2, . . . , κ. (by Fact 3 and Fact 4) Fact 5: n(τ) i ∈V(τ) (τ)(i) = n(τ+1) i ∈V(τ+1)
(τ+1)(i)+ |Iτ| for τ = 1, . . . , κ − 1.
Corollary 2: | ˜V(τ)| = | ˜V(τ+1)| + |I
τ| for τ = 1, . . . , κ − 1. (by Fact 1 and Fact 5)
Since Algorithm IMS selects an independent set Iτ of vertices in G(τ)for each time slot τ, by [16], (2) and Fact 2, the cardinality|Iτ| satisfies
|Iτ| ≥ ˜n(τ) i ∈ ˜V(τ) 1 1+ ˜d(iτ) (3) For a sequence of positive numbers, their arithmetic mean can not be less than the harmonic mean. Thus we have
˜n(τ) i ∈ ˜V(τ)1+ ˜d (τ) i | ˜V(τ)| ≥ | ˜V(τ)| ˜n(τ) i ∈ ˜V(τ) 1 1+ ˜d(τ) i . (4)
Substituting (4) into (3), we have |Iτ| ≥ | ˜V(τ)|2 ˜n(iτ)∈ ˜V(τ)1+ ˜d (τ) i = | ˜V(τ)| 1+ ˜δ(τ) (by Fact 3) (5)
LetΨτ=1≤i≤τ|Ii| be the scheduled traffic demand (in the
unit of time-slots) in the firstτ slots for 1 ≤ τ ≤ κ and Ψ0= 0. Then we obtain Ψτ = Ψτ−1+ |Iτ| ≥ Ψτ−1+ | ˜V(τ)| 1+ ˜δ(τ) (by (5)) ≥ Ψτ−1+ | ˜V(τ)| ˜δ(1)− τ + 2 (by Corollary 1) = Ψτ−1+| ˜V(1)| − Ψτ−1 ˜δ(1)− τ + 2 (by Corollary 2) (6) From (6), we observe thatΨ1≥ | ˜V(1)|/(˜δ(1)+1). We will prove thatΨτ≥ τ | ˜V(1)| (˜δ(1)+1) by induction onτ. Ψτ ≥ Ψτ−1+| ˜V (1)| − Ψ τ−1 ˜δ(1)− τ + 2 ≥ | ˜V(1)| × (τ − 1) ˜δ(1)+ 1 + | ˜V(1)| −| ˜V(1)|×(τ−1) ˜δ(1)+1 ˜δ(1)− τ + 2 = | ˜V(1)| × τ ˜δ(1)+ 1 , forτ ≥ 1 (7) Substituting Ψκ = | ˜V(1)| into (7), we find that the schedule lengthκ of Algorithm IMS can be
κ ≤ ˜δ(1)+ 1 (8) ≤ max n(1) i ∈V(1) ⎧⎪⎪ ⎪⎨ ⎪⎪⎪⎩(i, j)∈E(1) (1)( j)⎫⎪⎪⎪⎬ ⎪⎪⎪⎭ + (1) (By Theorem 1) = max nc i∈Vc ⎧⎪⎪ ⎪⎨ ⎪⎪⎪⎩(i, j)∈Ec c( j)⎫⎪⎪⎪⎬ ⎪⎪⎪⎭ + c (9) slots.
Theorem 2: Our joint routing and link-scheduling
algo-rithm is a 3-approximation algoalgo-rithm.
Proof: Assume that an “optimum” algorithm for routing
and link scheduling outputs the smallest schedule length Φ, andΦ can be derived as
Φ ≥ max ni∈V ⎧⎪⎪ ⎪⎨ ⎪⎪⎪⎩ ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ (i, j)∈E f∗ i, j ri, j ⎞ ⎟⎟⎟⎟⎟ ⎟⎠ + ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ ( j,i)∈E f∗ j,i rj,i ⎞ ⎟⎟⎟⎟⎟ ⎟⎠⎫⎪⎪⎪⎬⎪⎪⎪⎭ (10) where f∗
i, j is the aggregate traffic from ni to nj
produced by the optimum algorithm. Since our LP-based routing provides an optimal solution to minimize maxni∈V ! (i, j)∈Erfii, j, j +( j,i)∈E rfjj,i,i " , (10) can be rewritten as Φ ≥ maxn i∈V ⎧⎪⎪⎨ ⎪⎪⎩ ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ (i, j)∈E fi, j ri, j ⎞ ⎟⎟⎟⎟⎟ ⎟⎠ + ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ ( j,i)∈E fj,i rj,i ⎞ ⎟⎟⎟⎟⎟ ⎟⎠⎫⎪⎪⎬⎪⎪⎭ = max ni∈V ⎧⎪⎪⎨ ⎪⎪⎩ ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ (i, j)∈E ti, j ⎞ ⎟⎟⎟⎟⎟ ⎟⎠ + ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ ( j,i)∈E tj,i ⎞ ⎟⎟⎟⎟⎟ ⎟⎠⎫⎪⎪⎬⎪⎪⎭ ≥ 12max (i, j)∈E ⎧⎪⎪⎪ ⎨ ⎪⎪⎪⎩ ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ (i,i)∈E ti,i ⎞ ⎟⎟⎟⎟⎟ ⎟⎠ + ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ (i,i)∈E ti,i ⎞ ⎟⎟⎟⎟⎟ ⎟⎠ + ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ ( j, j)∈E tj, j ⎞ ⎟⎟⎟⎟⎟ ⎟⎠ + ⎛ ⎜⎜⎜⎜⎜ ⎜⎝ ( j, j)∈E tj, j ⎞ ⎟⎟⎟⎟⎟ ⎟⎠⎫⎪⎪⎪⎬⎪⎪⎪⎭ ≥ 12max nc i∈Vc ⎧⎪⎪ ⎨ ⎪⎪⎩2c(i)+ (i, j)∈Ec c( j)⎫⎪⎪⎬ ⎪⎪⎭ (11)
Fig. 2. Topology used in the experiment
By (9) and (11), the approximation factor of our proposed algorithm can be derived as
≤ 2 × maxnci∈Vc (i, j)∈Ecc( j) + c maxnc i∈Vc 2c(i)+ (i, j)∈Ecc( j) (12) ≤ 2 × maxnci∈Vc (i, j)∈Ecc( j) + c maxnc i∈Vc (i, j)∈Ecc( j) + 2c min (13) where c min = minnci∈Vc{
c(i)}. Note that when c ≤ 2c
min, our algorithm has an approximation factor of 2. In a general case, (12) can be rewritten as
≤ 2 × # 1+2c c max+ cmin $ = 2 × % 1.5 − c max+ (cmin/2) − c 2c max+ minc & = 3 − % 2× c max+ (minc /2) − c 2c max+ cmin & (14) wherec max = maxnc i∈Vc{
c(i)}. Observe that cmax+(cmin/2)−c
2c
max+cmin > 0
completes the proof.
IV. Performance Evaluation
This section investigates the performance of the proposed algorithm through our developed Monte Carlo simulation. We use LINGO to solve the LP-based routing problem [17], and Dijkstra’s shortest-path algorithm with respect to the number of hop counts is adopted as the compared routing algorithm in the experiments. The centralized scheduling algorithm (Cent) proposed in [5] is used for comparison against ourIMS. For the joint design of routing and link scheduling, the heuristic (HEU) in [10] is used for comparison with our joint algorithm. We employ the Gamma distribution to approximate the traffic demand ps for each subscriber ns ∈ VS in G, where
the mean and variance of ps are denoted by pmean (Mbit)
and pvar (Mbit2), respectively. As shown in Figure 2, a 5× 5
grid topology is adopted to simulate a Manhattan-like urban environment [18], where the filled, blank and “⊗” circles represent the SSs, RSs and BS, respectively. The data rates of relay links (RS-to-BS or RS-to-RS) and access links (SS-to-BS or SS-to-RS) are set by 18.36 Mbps and 6 Mbps, which are the raw data rates of adopting the modulation scheme of quadrature phase-shift keying (QPSK) and 64-quadrature amplitude modulation (64-QAM) in IEEE 802.16 orthogonal frequency-division multiplexing (OFDM) [19].
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 4 8 16 32 64 128 256 512 Normalized throughput pvar (Mbit2) LP & IMS HEU LP & Cent Shortest Path & IMS Shortest Path & Cent
Fig. 3. Effects of pshapeon the normalized throughput
By setting pmean = 1 Mbit, Figure 3 shows the system
throughput for our joint routing and link-scheduling algo-rithm as well as the compared algoalgo-rithms. The throughput is normalized to an upper bound, and the upper bound can be obtained as follows. Due to the primary interference, any two links incident to the base station can not be scheduled simultaneously. Thus the system throughput is bounded to the data rate of relay links, i.e., 18.36 Mbps. From Figure 3, we observe that the normalized throughput of our joint routing and link-scheduling algorithm (LP & IMS) is much better than those compared algorithms for all pvar under investigation.
Specifically, the curves indicate that the normalized throughput of Algorithm Cent (with LP) is less than about 10% of that of our IMS (with LP). They also indicate that our LP-based routing provides at least a double of the throughput of “Short-est Path" (withIMS). For the joint design of routing and link scheduling, Figure 3 shows that ourLP&IMS outperforms HEU especially for small pvar. As pvar is small, another important
phenomenon is observed that the throughput of our algorithm is very close to the theoretical upper bound.
Note that the derived upper bound could be much higher than the optimum solution since the bound might not be tight. It is suffice to show that the proposed algorithm yields the throughput that would be pretty close to the optimal solution for most of pvar values under investigation. Although our
LP&IMS is proven to be a 3-approximation algorithm in the worst case, the simulation results show that the throughput in the average case is much higher than that derived in the worst-case analysis.
As to the effects of pvar, we observe that the normalized
throughput of all algorithms under investigation decreases as pvar increases. The reason is that as the variance of
traffic demand of each SS increases, the load of each link is difficult to balance. To avoid collision, the unbalanced links cannot be scheduled simultaneously, and thus the lower spatial reuse leads to a lower throughput. However, when
pvar becomes large, our IMS provides more effective
link-scheduling capability compared to Cent. For example, the normalized throughput of AlgorithmCent (with LP) is about 90% and 68% of that of ourIMS (with LP) as pvar= 1 and 128,
respectively. Furthermore, we observe that the performance of HEU is relatively insensitive to pvar, and its throughput
becomes close to that of our proposed algorithm for large
pvar.
V. Conclusion
This paper proposed a joint routing and link-scheduling algorithm for TDMA-based wireless relay networks. The per-formance of the proposed algorithm was investigated through our developed theoretical analysis and simulation experiments. The theoretical analysis indicated that the performance of the proposed algorithm is within a factor of three of that of any optimal algorithm even in the worst case. Through simula-tion experiments, the numerical results demonstrated that our algorithm outperforms the previously proposed routing and link-scheduling algorithms under investigation. Furthermore, we observed that our algorithm can effectively achieve near-optimal performance, and provide much better throughput than that of the theoretical worst-case bound in the average case.
Acknowledgement
Our work was sponsored in part by Excellent Re-search Projects of National Taiwan University under contract 95R0062-AE00-07, National Science Council under contracts NSC96-2219-E-002-004 and NSC95-2221-E-002-096-MY3, Institute of Information Industry, Information and Commu-nications Research Lab/Industrial Technology Research Insti-tute,Intel, and Chunghwa Telecom.
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Chi-Yao Hong received his junior B.S. degree in
Electronic Engineering from National Ilan Institute of Technology, Taiwan, in 2004, and B.S. degree in Electronic Engineering from National Taiwan Uni-versity of Science and Technology, Taiwan, in 2006. He is currently working toward the M.S. degree in Computer Science, National Taiwan University, Taiwan. His current research interests are focused on wireless multihop networks, especially in QoS provisioning issues.
Ai-Chun Pang received the B.S., M.S. and Ph.D.
degrees in Computer Science and Information En-gineering from National Chiao Tung University, Taiwan, in 1996, 1998 and 2002, respectively. She joined the Department of Computer Science and Information Engineering (CSIE), National Taiwan University (NTU), Taipei, Taiwan, as an Assistant Professor in 2002. From August 2004 to July 2005, Dr. Pang served as an Assistant Professor in Gradu-ate Institute of Networking and Multimedia (INM),
and an adjunct Assistant Professor in CSIE/NTU,
Taipei, Taiwan. Currently, she is an Associate Professor in INM and CSIE of NTU, Taipei, Taiwan. Her research interests include design and analysis of personal communications services network, mobile computing, voice over IP and performance modeling.
Dr. Pang has served as the program co-chairs and committees of many inter-national conferences/workshops. She was the guest editor of IEEE Wireless Communications, and is currently an associate editor of International Journal
ofSensor Networks and ACM Wireless Networks. Also, Dr. Pang was a
recipient of the Teaching Award at NTU in 2005, 2006 and 2007, received the Investigative Research Award from the Pan Wen Yuan Foundation in 2006, received Wu Ta You Memorial Award from National Science Council (NSC) in 2007, received Excellent Young Engineer Award from the Chinese Institute of Electrical Engineering, and received K. T. Li Award for Young Researchers from ACM Taipei/Taiwan Chapter in 2007.
