Mini-slot scheduling for IEEE 802.16d chain and grid mesh networks

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Mini-slot scheduling for IEEE 802.16d chain and grid mesh networks

Jia-Ming Liang

*

_{, Ho-Cheng Wu, Jen-Jee Chen, Yu-Chee Tseng}

Department of Computer Science, National Chiao-Tung University, Hsin-Chu 30010, Taiwan

a r t i c l e

i n f o

Article history:

Available online 29 July 2010 Keywords:

IEEE 802.16 Routing tree Mini-slot scheduling WiMAX

Wireless mesh network

a b s t r a c t

This work considers the mini-slot scheduling problem in IEEE 802.16d wireless mesh networks (WMNs). An efficient mini-slot scheduling needs to take into account the transmission overhead, the scheduling complexity, and the signaling overhead to broadcast the scheduling results. We are interested in chain and grid WMNs, which are the basic topologies of many applications. We propose scheduling schemes that are featured by low complexity and low signaling overhead. Compared to existing works, this work contributes in developing low-cost schemes to find periodical and regular schedules that achieve near-optimal transmission latencies by balancing between transmission overhead and pipeline efficiency and that are more practical and easier to implement. To minimize the transmission latency, we model the transmission latency as a function of the transmission size and the subscriber stations’ traffic demands, and take the first-order derivative of the transmission size to find the minimum latency. Simulation results show that our schemes significantly improve over existing works in computational complexity while maintain similar or better transmission latencies.

1. Introduction

Recently, the IEEE 802.16 standard[1]has been proposed to support wide-range wireless broadband access. The standard is based on a common medium access control (MAC) protocol compli-ant with several physical layer speciﬁcations. The protocol sup-ports point-to-multipoint (PMP) and mesh modes. This work considers the IEEE 802.16d wireless mesh network (WMN), which can be used as a broadband wireless backbone[2,3]. A WMN has a base station (BS) connecting to multiple subscriber stations (SSs) via multi-hop transmissions. In an IEEE 802.16d WMN, transmis-sion follows a time divitransmis-sion multiple access (TDMA) mechanism over the underlying orthogonal frequency division multiplexing (OFDM) physical layer. In such WMNs, scheduling is a critical issue that may signiﬁcantly impact the system performance. It involves con-structing a scheduling tree from the network and planning wireless resources for SSs to send/receive their data to/from the BS.

The wireless resource on each link in an IEEE 802.16d WMN is a sequence of ﬁxed-length time slots, called mini-slots. However, be-fore actually transmitting on a mini-slot, a sender must wait for a ﬁxed number of mini-slots, called transmission overhead, to avoid collisions[4–6]. This is a guard time to synchronize and tolerate the air-propagation delay of the transmission occurring on the mini-slot right before the aforementioned overhead mini-slots. Once starting its actual transmission, a node may send on several consecutive mini-slots. References[6,7]observe that if the (actual)

transmission is too short, most of the time will be occupied by the transmission overhead. On the other hand, if the transmission is too long, it may hurt fairness and pipeline efﬁciency (i.e., there could be less concurrent transmissions) in the pipelines. So, a good scheduling should balance between the ratio of transmission over-head and the pipeline efﬁciency by adjusting the sizes of (actual) transmissions. In this work, we propose to use three metrics to evaluate a scheduling scheme (i) the total latency (i.e., the time to deliver all data to BS), (ii) the scheduling complexity, and (iii) the signaling overhead (i.e., the cost to notify all SSs their schedules).

In the literature, several works[8–15]have studied the sched-uling problem in WMNs. Reference[8]proposes an interference-aware, cross-layer scheduling scheme to increase the utilization of a WiMAX mesh network. Reference[9]suggests using concur-rent transmissions to improve overall end-to-end throughput of a WMN. Reference[10]shows how to increase spatial reuse to im-prove system throughput and provide fair access for subscriber stations. Reference[11]presents a flow-control scheduling to pro-vide quality of service guarantee for flows. Reference[12]shows how to maximize spatial reuse to achieve better overall network throughput and higher spectral efficiency. Reference[13]proposes four criteria to select conflict-free links to reduce the scheduling length. These criteria include random selection, min-interference selection, nearest-to-BS selection, and farthest-to-BS selection. Ref-erence [14] considers that each transmission can transmit one piece of data and tries to maximize spatial reuse to minimize the total transmission time. However, all above schemes do not con-sider the cost of transmission overhead (to be defined later on).

*Corresponding author. Tel.: +886 3 5712121; fax: +886 3 5721490. E-mail address:jmliang@cs.nctu.edu.tw(J.-M. Liang).

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For example, the results in[13,14]are not optimal because they do not take this into account. Considering transmission overheads,

[15]proposes to always ﬁnd the maximal number of concurrent transmission links in each round. This problem has been shown to be NP-hard[16]. Although it performs close to optimum, its computational complexity is too high to be used by the BS. Also, the signaling overhead incurred by[15]is quite high because the scheduling patterns for SSs are not regular.

In this paper, given the uplink loads of all SSs in a WMN, we consider the problem of scheduling their traffics such that the total latency to transmit all data to the BS is minimized and the sched-uling complexity and signaling overhead are as low as possible. (The scheduling in the downlink direction is similar, so we focus in only one direction.) In our approach, we first try to find the opti-mal transmission size for the given loads to strike a balance be-tween the ratio of transmission overhead and the pipeline efficiency. We observe that when the actual transmission size is small, the pipeline could be full for the most of the time, but the transmission overhead could occupy too much time. On the other hand, when the actual transmission size is too large, the above problem may be fixed, but the pipelines may not be filled with suf-ficient concurrent transmissions, thus hurting spatial reuse. We then assign the transmissions of each link in a periodic and regular manner with a proper transmission size. Since our scheduling is periodical, the signaling overhead to inform each SS is also quite low. To the best of our knowledge, our work is the first one with these properties. Our scheme incurs low complexity and the result is applicable to most regular topologies, such as chain, grid, and tri-angle networks, which have been proved to have many applica-tions, such as the mesh networks deployed in rural areas in South Africa to provide Internet access [17], the VoIP testbed

[18], and the mobility testbed developed in[19]. In addition, these topologies outperform random topologies in terms of their achiev-able network capacity, connectivity maintenance capability, and coverage-to-node ratios (about two times that of random topolo-gies)[20,21]. We remark that the chain topology is a special case of grid topologies, which is the most suitable for long-thin areas, such as railways and highways[14]. Simulation results are pro-vided to verify our claims on these topologies.

The rest of this paper is organized as follows. Section2formally deﬁnes our mini-slot scheduling problem. Section3presents our schemes. Simulation results are given in Section4. Section5 con-cludes this paper.

2. Problem deﬁnition

We are given one BS and n SSs, SSi, i = 1, . . . , n. These BS and SSs

are deployed in a chain, grid or triangle topologies, as shown in

Fig. 1. The BS can be placed at any location in the topology. All nodes share the same communication channel. The amount of data that a node can transmit per mini-slot is d bytes. Since the topology is regular, two nodes are allowed to transmit concurrently if they are at least H hops away from each other. We consider the uplink

scheduling. So we abstract the uplink mini-slots of the system by concatenating them together into an inﬁnite sequence and ignore the downlink mini-slots. Each SSihas a trafﬁc demand of pibytes.

Our goal is to construct a scheduling tree T such that each SSi

re-ceives a collision-free schedule Tiand the total time to deliver all

SSs’ data to the BS is as less as possible. In our work, we impose that the schedule Tifor each SSishould be periodical as deﬁned

below.

The transmission schedule is formulated as follows. For each SSi

and each mini-slot, we use a character in {0, 1, h} to represent its state. A ‘0’ means that the mini-slot is idle for SSi. A ‘1’ means that

SSican transmit at most d bytes in this mini-slot. An ‘h’ means that

SSi is preparing to transmit (i.e., this mini-slot is considered a

transmission overhead). To start an actual transmission, a SS must wait for

a

mini-slots of state ‘h’ so as to synchronize and tolerate the air-propagation time of the transmission occurring right before the overhead mini-slots, where

a

is a system-dependent constant. For example, when

a

= 2, we can use a string ‘000hh111100’ to indicate that a SS is idle in the ﬁrst three mini-slots, waits for two overhead mini-slots, transmits for four mini-slots, and then stays idle in the last two mini-slots.

In this work, we enforce that all SSs’ transmission schedules are periodical and regular. Speciﬁcally, all SSs’ schedules have the same of period of

q

. Each SS’s transmission schedule has the format of (0a_ha₁b₀c_{)*, where a P 0, b > 0, and c P 0 are integers and}

a +

a

+ b + c =

q

. The 0s at the end of a schedule are necessary when we consider periodical schedule. Symbol ‘*’ means a number of repetitions of the string in parentheses until all necessary data is delivered. Different SSs may have different patterns. For example,

Fig. 2 shows a chain network with one BS and seven SSs. Only SS7 has a trafﬁc demand of p7= 4 bytes. Assuming

a

= 1, d = 1,

and H = 3, we show three schedules. In the first schedule, b = 1 mini-slot of data is transmitted in each cycle. The other parameters a = 0/2/4 and c = 4/2/0, respectively. So there are three types of schedule patterns: (h10000)*, (00h100)*, and (0000h1)*. In the sec-ond schedule, b = 2 mini-slots in each cycle; a = 0/3/6 and c = 6/3/0, respectively. In the third schedule, b = 4; a = 0/5/10/15/20/25/30 and c = 30/25/20/15/10/5/0, respectively (however, only one cycle is needed). The second schedule is the most efficient. Our goal is to find the most efficient regular schedules.

3. Scheduling tree construction schemes

Next, we present our scheduling schemes for chain and grid topologies. We first consider the chain topology with different locations of source SSs on the chain. Then we use these results as basic components to solve the scheduling problem for grid and tri-angle topologies. Given a grid/tritri-angle network, we first construct a fishbone-like tree. The fishbone-like tree is decomposed into indi-vidual chains, each of which can be scheduled using the previous chain solutions. Below, we present three solutions for a chain, from simpler to more complicated cases. Then, we combine these solu-tions for the grid/triangle topologies.

3.1. A chain with a single request

Since there are only one source and one destination, we can model the chain, without loss of generality, as a path SSn?SSn1

? ? SS1?BS such that only SSnhas a non-zero demand pn. To

increase parallelism, we partition these SSs into k concurrent transmission-able groups, where k = H if n P H and k = n otherwise (recall that H is the least spatial-reuse distance). Speciﬁcally, we deﬁne group Gj, j = 0, . . . , k 1, as follows:

Gj¼ fSSijðn iÞ mod k ¼ j; i ¼ 1; . . . ; ng: ð1Þ

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(c)

(a)

Fig. 1. (a) A 5-node chain topology, (b) a 4 4 grid topology, and (c) a 4 4 triangle topology.

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Nodes in the same group have the same schedule. We simply denote by Tjthe transmission schedule of Gj. Since we are

inter-ested in having regular schedules, we enforce each Gjto have a

schedule of the format ð0aj_ha₁b₀cj_Þ_{, where a}

jand cjare

group-spe-ciﬁc constants and b is a ﬁxed constant for all groups, such that the following conditions hold: (i) a0= 0, (ii) aj+

a

+ b + cj=

q

is a

con-stant and

q

is the period for all groups, and (iii) aj+

a

+ b = aj+1,

j = 0, . . . , k 2. Conditions (iii) means that each Gj+1is obtained

from Gjby shifting the latter to the right by (

a

+ b) positions. Given

any b, we can compute the total latency L1(n, pn, b) to deliver SSn’s

data to the BS: L1ðn; pn;bÞ ¼ pn bd H ð

a

þ bÞ þ ðn HÞ ð

a

þ bÞ; if n P H pn bd n ð

a

þ bÞ; otherwise: ( ð2Þ

When n P H, each cycle has a length of H (

a

+ b) mini-slots. It takes pn

bd

cycles for SSnto transmit its last piece of data. At the

end of the pn

bd

th cycle, the last piece of SSn’s data will arrive at node

SSnH. Then it takes another (n H) hops, each requiring (

a

+ b)

mini-slots, to travel to the BS. This gives the upper term in Eq.

(2). For the lower term, the derivation is similar.

Given ﬁxed n, pn, and

a

, we are interested in knowing the

opti-mal value of b, denoted by ^b, that gives the minimum latency L1. To

do so, we need to conﬁne that pnis divisible by b d in Eq.(2). To

minimize Eq.(2), we can let L1= 0 and take the ﬁrst-order

deriva-tive of b. This leads to

^ b ¼ ffiffiffiffiffiffiffiffiffiffiffi_a pnH dðnHÞ q ; if n P H; pn d; otherwise: 8 < : ð3Þ

The value of ^b in Eq.(3)is a real. The best value may appear in d^be or b^bc. Plugging this into Eq.(2), we can get the minimum ^L1.

3.2. A chain with multiple requests

Next, we consider a path SSn?SSn1? ? SS1?BS with

multiple non-zero-load nodes. Without loss of generality, we as-sume SSn’s load is non-zero. Similar to Section3.1, we divide SSs

into k groups Gj, j = 1, . . . , k 1. Again, we enforce Gj’s schedule

with the format ð0aj_ha₁b₀cj_Þ_{, where a}

j and cjare group-speciﬁc

constants and b is a ﬁxed constant for all groups, such that the fol-lowing conditions hold: (i) a0= 0, (ii) aj+

a

+ b + cj=

q

is a constant

and

q

is the period for all groups, and (iii) aj+

a

+ b = aj+1,

j = 0, . . . , k 2. Conditions (iii) means that each Gj+1 is obtained

from Gjby shifting the latter to the right by (

a

+ b) positions. To

ﬁnd an appropriate value of b, we imagine that all data are origi-nated from SSnby assuming that all SSs have zero loads except that

SSnhas a load p0n¼

Pn

i¼1pi. Then we plug p0ninto pnin Eq.(3)to ﬁnd

the best ^b.

With this ^b, we need to ﬁnd the latency L2ðn; p1;p2; . . . ;pn; ^bÞ to

deliver all SSs’ data on the original path. The transmission is similar to a pipeline delivery, but with some bubbles sometimes. To model the pipeline behavior, we do not take a ‘micro-view’ on the system. Instead, we take a ‘macro-view’ to partition the path into n0_¼ n

k

trains, by traversing from the end (i.e., SSn) toward the head (i.e.,

SS1) of the path by grouping, every consecutive k SSs are as one

train (when n is not divisible by k, the last few SSs are grouped into one train). We make two observations on these trains.

Observation 1. In each cycle, a train can deliver up to b d bytes of data to the next train, no matter where these data are located in which SSs of the train.

However, a bubble appears when a train does not have sufﬁ-cient data to be delivered to the next train. Below, we show when bubbles will not appear.

Observation 2. Except the ﬁrst n0_¼ n k

cycles, the BS will contin-uously receive b d bytes of data in every cycle until no more data exists in the path.

period

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Proof. Consider the ﬁrst cycle, the data delivered by n0_{th train is}

only its data. However, in the second cycle, the data delivered by the n0_{th train will be the n}0_{th chain’s data or both the n}0_{th and}

(n0_{1) th chains’ data. So, in the n}0_{th cycle, the data delivered by}

n0_{th train will be the n}0_{th chain’s data 1st chain’s data. If the n}0_th

chain’s has no sufﬁcient data (i.e., <b d), the delivery will combine the previous train’s data, i.e., (n0_{1)th chain’s data or even}

(n0_{2)th chain’s data, etc. So, if the n}0_{th train deliver less than}

b d (so called bubble) to BS after n0_{th cycle, it means that the}

amount of data on all trains must be less than b d after n0_{th cycle.}

Or it must deliver b d data without bubbles. It’s proved. h

Observation 2implies that if we can derive the network state at the end of the n0_{th cycle, the latency can be easily derived. To}

de-rive the network state after each cycle, let Si¼ ðwðiÞ1; . . . ;w ðiÞ n0Þ be the

network state at the end of the ith cycle, i = 0, . . . , n0_{, where w}ðiÞ j is

the total load remaining in the jth train at the end of the ith cycle. Initially, wð0Þ

j is the total loads of those SSs in the jth train. Then we

enter a recursive process to ﬁnd Si+1from Si, i = 0, . . . , n0 1 as

follows: wðiþ1Þ j ¼ max wðiÞ j bd; 0 n o þ min wn ðiÞ_j1;bdo; j ¼ 2; . . . ; n0_; max wðiÞj bd; 0 n o ; j ¼ 1: 8 > < > : ð4Þ

Eq.(4)is derived based onObservation 1. In the upper equality, the ﬁrst term is the remaining load of the jth train after subtracting delivered data and the second term is the amount of data received from the previous train. The lower equality is delivered similarly.

According toObservation 2, after the n0_{th cycle, the BS will see}

no bubble until all data on the path is empty and it will take Pn0 j¼1w ðn0 Þ j bd & ’

more cycles to deliver all remaining data. This leads to

L2ðn; p1;p2; . . . ;pn; ^bÞ ¼ Pn0 j¼1 wðnj 0Þ b d 2 6 6 6 6 6 6 3 7 7 7 7 7 7 þ n0 0 B B B @ 1 C C C A

q

; ð5Þ

where

q

= k (

a

+ b) is the period of cycles. As has been clear from the context, previous ^b in Eq.(5)is just an estimation. The optimal b may appear at a point to the left of ^b.1_{One may repeatedly decrease} ^

b to ﬁnd a better value.

3.3. A chain with multiple requests and BS in the middle

Since the BS in the middle, we model the chain as a path SS‘! SS‘1! . . . ! SS1! BS SS01 SS

0

2 SS 0

r. Without

loss of generality, we assume ‘ P r and both SS‘ and SS0r have

non-zero loads. For simplicity, we call SS‘?SS‘1? ? SS1?BS

the left chain CL, and BS SS01 SS 0

2 SS 0

r the right chain

CR. The arrangement of concurrent transmission-able groups is

more difﬁcult because we intend to transmit sufﬁcient data to the BS from both chains without congestion in each cycle. First, we need to identify a new value for k (the number of groups):

k ¼ H; if 1 6 H 6 4; 2H 4; if H P 5:

ð6Þ

Eq.(6)means that when 1 6 H 6 4, we can still manage to have the most compact number of concurrent transmission-able groups. However, when H P 5, the number of groups will exceed H, which is not most compact. Fortunately, in practice, H 6 4 in most cases.

Now, for j = 0, . . . , k 1, we deﬁne group Gjas follows: Gj¼ fSSijð‘ iÞmod k ¼ j; i ¼ 1; . . . ; ‘g

[

SS0ijð‘ i þ

D

Þ mod k ¼ j; i ¼ 1; . . . ; r

ð7Þ

In Eq.(7), nodes in CLand CRare grouped sequentially similar to the

earlier cases. However, for CR, the grouping of nodes is shifted by a

value ofDto allow concurrent transmissions, whereD= 1 if H = 2 and D= H 2 if H P 3. Fig. 3 shows some examples with ‘ = 6 and r = 6. In the example of H = 2, we shift the grouping of nodes in CRby a value ofD= 1. In the examples of H = 3 and H = 4, we shift

the grouping of nodes in CRby a value ofD= H 2. Such shifting

avoids nodes nearby the BS from colliding with each other. When H P 5, the value of k is deﬁned differently. However, shifting by

D= H 2 still helps avoid collision. Note that when H = 5 and 6, there are 6 and 8 groups, respectively, where these numbers of groups are least for grouping on both chains to transmit without congestion. We will explain later on.

When H 6 4, the collision-free property of the above grouping can be proved by case-by-case validation. When H P 5,Fig. 4gives a general proof. There are k = 2H 4 groups. Consider nodes SSi

and SS0

i; i ¼ 1; . . . ; 2H 4 on CLand CR. Assume that the former i

(1 6 i 6 2H 4) SSs are grouped from G0 and the remaining

2H 4 i SSs are grouped from Gi. We prove it in two aspects.

First, since the indexes of groups on both two chains are increasing toward BS, if the pair of SSxand SSy, where they are in the same

group but on different chains, have no interference to each other that will make all SSi, i P x and SS0i; i P y be interference free. By

Eq.(7), all SSs on CRare shifted by H 2 of grouping sequence,

the critical pairs (i.e., SS1and SS0H1Þ can be in same groups while

keep a distance of H hops. That means those SSi, i P 1 and

SS0i; i P ðH 1Þ can be grouped successfully and will not cause

any interference when those SSs are in their groups. On the other hand, as we know that SS0

1 is grouped by Gi+H3. Since SS2H4is

grouped in Gi, the smallest index of SS grouped by Gi+H3will be

SSH1, which has a distance of H hops to SS01. By these two aspects,

all SSi, i P (H 1) and SS0i; i P 1 will not cause any interference

when those SSs are in their groups. Then, we can promise that each SSiand each SS0i; i P 1 can be grouped by Eqs.(6) and (7)to

trans-mit without any interference and congestion.

Theorem 1. By Eqs.(6) and (7), SSs in the same group can transmit concurrently without collision for any value of H.

WithTheorem 1, we can allow CLand CRto transmit

concur-rently without collision. So the results in Section3.2can be applied here. (The only exception is that the ﬁrst transmitting node in CR,

i.e., SS0

r, may not start from group G0. In this case, we can add some

virtual nodes to CRand make one start from G0.) For CL, there will

exist an optimal value of ^b‘such that L2ð‘; p1;p2; . . . ;p‘; ^b‘Þ is

small-est. Similarly, for CR, there will exist an optimal value of ^br such

that L2ðr; p01;p20; . . . ;p0r; ^brÞ is smallest. Plugging in any possible b,

we can formulate the latency as follow:

L3 ‘;p1;p2; . . . ;p‘;r; p 0 1;p 0 2; . . . ;p 0 r;b ¼ max L2ð‘; p1;p2; . . . ;p‘;bÞ; L2 r; p01;p02; . . . ;p0r;b ð8Þ

The best value of b, called ^b, may appear nearby or between ^b‘and

^ br.

3.4. A general grid/triangle topology

Here we show how to extend our scheduling scheme to a grid/ triangle topology. The scheduling is built on top of the previous chain scheduling results. First, we will construct a ﬁshbone-like tree from the grid/triangle network. The ﬁshbone-like tree is further decomposed into horizontal and vertical chains. For example, 1

The current ^b is the upper bound of the optimal value because we previously imagine that all data are originated from SSn.

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Fig. 5(a) shows how such trees are formed. One of the chain pass-ing the BS is called the trunk chain, and the others are called branch chains. Then, we schedule all branch chains to transmit their data to the trunk chain. Branch chains are divided into H groups and we schedule these groups to transmit sequentially. Finally, we schedule SSs in the trunk chain to transmit their data to the BS.

Details of the scheme are as follows. We consider a X Y grid/ triangle topology. Without loss of generality, we assume X 6 Y and we decompose the tree into Y vertical chains (branch chains) and one horizontal chain (trunk chain). Intuitively, the trunk chain is larger than the branch chains. There are two phases. In the ﬁrst phase, branch chains are scheduled to transmit. These chains are divided into H groups. Since two parallel branch chains with a dis-tance of H hops can transmit concurrently without interference, we assign a number between 1 to H to each branch chain in rotation from left to right. Chains marked by the same number are in the same group. Then we schedule each group of chains to transmit

sequentially. For example, when H = 3, in Fig. 5(b), the seven branch chains are numbered by 1, . . . , 3 in rotation. Then we let group 1 to transmit until all data are forwarded to the trunk chain, followed by group 2, and then group 3 in a similar way. Since chains in the same group can transmit individually without inter-ference, we can apply the optimal ^b for each chain as formulated above. The latency of phase one is the sum of all groups’ latencies. In the second phase, data are already all aggregated at the trunk chain. So, we can apply the easier result again to schedule nodes’ transmissions on the trunk chain.

4. Performance evaluation

In this section, we present our simulation results to verify the effectiveness of the proposed scheme. The simulator is written in JAVA language. Unless otherwise stated, the default parameters

Fig. 3. The arrangement of transmission-able groups for H = 2, 3, 4, 5, and 6 when BS is in the middle.

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used in our simulation are d = 1 byte,

a

= 3 mini-slots[4,5], and H = 3 hops.

We compare our scheme against four schemes, named the basic IEEE 802.16d mesh operation[1], the BGreedy scheme [14], the Max-transmission scheme[15], and the Priority-based scheme

[15]. The reason for selecting the BGreedy scheme for comparison is that it considers the pipeline efficiency, while that for selecting the Max-transmission scheme and the Priority-based scheme for comparison is that they consider the transmission overhead. The basic IEEE 802.16d mesh operation assigns the cumulated data plus a transmission overhead as the transmission for each SS with-out any spatial reuse. BGreedy scheme makes each transmission as short as possible to maximize pipeline efficiency. Max-transmis-sion scheme always finds the maximal number of concurrent transmission links in the network round by round and assigns those links to transmit the minimal buffered data plus a transmis-sion overhead. Priority-based scheme first finds all available links sets, which can transmit without interference, and chooses one with the maximal predefined priority. Then, it assigns those links to transmit the minimal buffered data plus a transmission over-head. Here, we adopt LQR priority, which performs the best perfor-mance in[15]. Such priority consults some network information, such as the hop counts, queue lengths, and transmission rates of the links. Then, except BGreedy scheme, we construct our fish-bone-like tree for all other schemes because they do not discuss the routing tree construction in their works.

In the following results, we use the total latency to compare dif-ferent schemes. We simulate three scenarios: a chain with a single request (SN1), a chain with multiple requests (SN2), and a grid with multiple requests (SN3). Unless otherwise stated, we use a 15-node chain and a 7 7 grid for the last two scenarios with BS in the middle. We remark that since the results of the triangle topology are almost the same as those in grid topology, we will only discuss the grid case.

4.1. Impact of network size

First, we investigate the effect of network size on the total la-tency (in mini-slots). Fig. 6(a) considers scenario SN1 with pn= 30 bytes by varying n. Clearly, the total latencies of all schemes

increase as n increases. BGreedy, Max-transmission, and Priority-based schemes perform the same because when the traffic demand is from only one SS, they schedule one transmission each time without any spatial reuse. Although BGreedy tries to maximize spatial reuse, its latency is still higher than ours because it disre-gards the transmission overhead. Ours has the best performance. This indicates the necessity of balancing between transmission overhead and pipeline efficiency. This effect is more significant when n is larger.Fig. 6(b) and (c) shows our results for SN2 and SN3, respectively. Each SS has a randomly traffic demand from 0 to 20 bytes. Similar to SN1, the total latencies of all schemes in-crease as the network size inin-creases. Although all schemes will ex-ploit spatial reuse when there are multiple traffic demands, our scheme still outperforms all the other schemes. In addition, it is to be noted that the schedules generated by our scheme are regular and periodical, which is not so for other schemes.

(a)

(b)

Fig. 5. (a) A ﬁshbone-like tree on the 5 7 grid/triangle topology. (b) The grouping of branch chains when H = 3.

(a)

(b)

(c)

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4.2. Impact of trafﬁc load

Next, we investigate the effect of average traffic load on total la-tency.Fig. 7(a) shows the results for SN1 when n = 15.Fig. 7(b) and (c) shows the results for SN2 and SN3, respectively, where each SS has a random traffic of 0 to 10, 0 to 20, and 0 to 40 bytes (thus the averages are 5, 10, and 20 bytes, respectively). The trends are sim-ilar to the previous cases; our scheme outperforms the other schemes significantly in SN1 and slightly in SN2 and SN3.

4.3. Impact of transmission overhead

Next, we investigate the impact of transmission overhead (

a

) on total latency.Fig. 8shows our results. The simulation environment

is similar to the previous cases except that we vary the value of

a

. Naturally, the total latencies of all schemes increase as

a

increases. In SN1, our signiﬁcantly outperforms the other schemes in all val-ues of

a

. In SN2 and SN3, the average trafﬁc load of each station is 10 bytes. We see that a larger

a

will favor our scheme as compared to Max-transmission and Priority-based schemes. This is because our scheme enforces a regular schedule for each SS, thus losing some degree of pipeline efﬁciency. The impact is higher when

a

= 1 and 2. When

a

P3, the transmission overhead is too high to be neglected. Thus, balancing between transmission overhead and pipeline efﬁciency becomes quite important. In practice,

a

is greater than 2[5].

(a)

(c)

(b)

Fig. 7. The impact of trafﬁc load on total latency in scenarios SN1, SN2, and SN3.

(a)

(b)

(c)

Fig. 8. The impact of transmission overhead (a) on total latency in scenarios SN1, SN2, and SN3.

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4.4. Impact of BS location

Next, we move the location of the BS around, as shown inFig. 9. In SN2, we place the BS in the ﬁrst, 4th, and 8th position of the chain (n = 15). In SN3, we place the BS at (1, 1), (3, 3), and (4, 4) of the 7 7 grid network. In both cases, the total latencies of all schemes reduce as the BS is moved toward the center of the net-work because SSs are closer to the BS.

4.5. Computational complexity

Finally, we investigate the computational complexities of differ-ent schemes. We mainly compare our scheme against Max-trans-mission and Priority-based schemes. Note that it has been proved in [16] that the problem that Max-transmission and Priority-based schemes intend to solve is NP-hard. So we are interested in seeing the total CPU time incurred by these schemes as com-pared to ours. (The computation time is measured by the platform of IBM R61 with Intel Core 2 Duo T7300 2.0 GHz and DDR2-800 SDRAM 2 GB). FromFig. 10, we see that the computational com-plexities of all schemes increase as the network size increases. Since both Max-transmission and Priority-based schemes try to ﬁnd out the maximal concurrent transmission set of SSs round by round until all data are delivered to BS, the processing time in-creases exponentially as n grows (note that the y-axis is drawn with exponential scales). For example, the computation costs of Priority-based are 4.03, 3.22, and 16.7 times of ours when n = 4, 7, and 15, respectively, in SN2 and 96, 1039, and 6070 times of ours in the 5 5, 7 7, and 9 9 grid topologies, respectively. Because our scheme simpliﬁes the grouping of SSs in both chain and grid

topologies, and schedules transmissions in quite a regular way, it achieves a much lower cost. This further veriﬁes that our scheme is more practical and is easier to implement, even when the net-work scales up.

5. Conclusions

This paper addresses the scheduling problem in chain- and grid-based WMNs. While most existing solutions try to address this NP-hard scheduling problem by searching for the sequence of concur-rent transmission-able sets to maximize the spatial reuse factor, our approach tries to identify regular patterns that SSs can follow and repeatedly transmit easily. One special feature of our scheme is that it tries to balance transmission overhead and pipeline effi-ciency. In particular, our scheme tries to fill up the pipeline as full as possible to improve the pipeline efficiency. With these designs, our scheme does achieve better or equal total latency as compared to existing schemes, incurs much low computational cost as com-pared to existing schemes, and allows an easy implementation of the scheduler.

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Fig. 10. The impact of network size on computational complexity in scenarios SN2 and SN3.

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