Access granularity control of multichannel random access in next-generation wireless LANs

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Access granularity control of multichannel random access in next-generation wireless LANs Jiechen Yin a, Yuming Mao a, Supeng Leng a,∗, Yuming Jiang b, Muhammad Asad Khan a

Q1

a b

University of Electronic Science and Technology of China, Sichuan 611731, China Norwegian University of Science and Technology, Norway

a r t i c l e

i n f o

Article history: Received 11 December 2014 Revised 10 August 2015 Accepted 21 August 2015 Available online xxx Keywords: Fine-grained channel access Frequency-domain backoff Multichannel OFDMA Random access

a b s t r a c t As the next-generation Wireless LANs (WLANs) will provide the ubiquitous high-data-rate network coverage, the traditional contention-based Medium Access Control (MAC) scheme may be unable to fulfill the requirement of efficient channel access. To address this problem, several research works have proposed the random access systems combined with Orthogonal Frequency Division Multiplex Access (OFDMA) technology. Access granularity control is a crucial issue in this combination. Specifically, this issue focuses on how to tune the subchannel bandwidth and the number of accessible subchannels towards the maximum channel utilization. This paper analyzes access granularity control in an OFDMA system that adopts a multichannel Carrier Sensing Multiple Access (CSMA) MAC and resolves contention by the frequency-domain backoff. The theoretical analysis verifies the significance of access granularity control. In addition, the simulation experiments demonstrate that the proposed dynamic access granularity control algorithms notably outperform the traditional ones that divide channel band statically and adjust the number of accessible subchannels empirically. © 2015 Published by Elsevier B.V.

1

1. Introduction

2

Random access protocol is widely applied in Wireless Local Networks (WLANs) as a fundamental Medium Access Control (MAC) scheme. At the same time, it plays an important role in the emerging heterogeneous networks depending on its flexibility. The next-generation WLANs are expected to achieve more than 1 Gb/s network coverage. Moreover, the number of short packets grows due to the spread of Machine-to-Machine (M2M) communications. In these recency scenarios, the conventional random access MAC, e.g. 802.11 Distributed Coordination Function (DCF), may cause a drastic drop of channel utilization [1,2]. The root of this

3 4 5 6 7 8 9 10 11 12

∗

Corresponding author. Tel.: +86 28 61830520. E-mail address: [email protected] (S. Leng).

inefficiency is the fact that the impact of protocol overhead (such as backoff, inter-frame space, preamble and signaling) becomes increasingly notable when the high data rate causes a short data transmission time. For example, a backoff time slot in a 150 Mb/s 802.11n system is equal to 0.11 times the transmission duration of a 1500 byte packet, whereas this ratio rises to 1.05 in a 1.4 Gb/s 802.11 ac system [3]. An effective approach to amend the MAC efficiency of a high-data-rate WLAN is to combine random access with Orthogonal Frequency Division Multiplex Access (OFDMA) technology. In several recent works, it has been shown that systems exploiting this approach can improve the throughput in certain scenarios. In the systems proposed by [4,5], the channel contention among terminals is resolved in the frequency domain by randomly dispersing their channel access requests into different busy/idle state patterns of subcarriers. This frequency-domain backoff mechanism is

http://dx.doi.org/10.1016/j.comnet.2015.08.008 1389-1286/© 2015 Published by Elsevier B.V.

Please cite this article as: J. Yin et al., Access granularity control of multichannel random access in next-generation wireless LANs, Computer Networks (2015), http://dx.doi.org/10.1016/j.comnet.2015.08.008

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Fig. 1. An illustration for the influence of access granularity. Two contenders are assumed, say A and B.

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proved to efficiently decrease the probability of frame collision. However, because the terminals transmit a frame with the whole frequency band, a low channel utilization may still arise when applications frequently generate small packets (e.g. in VoIP and M2M communications). In the works [6–8], the presented systems can substantially mitigate the impact of protocol overhead by the sub-channelization and the multichannel concurrent data transmissions. Nonetheless, the systems cannot efficiently control the network congestion, because the terminals always intend to access all the idle subchannels. This drawback has been amended by an approach proposed in [9], whereby the number of accessible subchannels (i.e. the number of subchannels that a terminal is allowed to access) can be dynamically tuned. However, the appropriateness of the dynamic tuning in [9] cannot be ensured, because the work lacks the support of an accurate throughput model. Furthermore, we notice that all the existing Multichannel Random Access (MRA) systems1 adopt static sub-channelization approach. As a consequence, when packet size decreases or data rate rises, the reduced data transmission time will magnify the impact of protocol overhead and degrade the throughput performance. The loss of channel utilization in the preceding MRA systems can be ascribed to their static and inappropriate access granularity control schemes. The access granularity here is defined as the amount of spectrum resource that each terminal attempts to use. It influences MAC efficiency in 1 Multichannel random access in this paper is used to exclusively signify the systems which combine random access with OFDMA technology. In addition, we regard the single-channel system as a special case of multichannel system.

the aspects of channel contention intensity and the ratio of protocol overhead to data transmission time. As shown in Fig. 1a, if the access granularity is large, the frame collision may frequently occur due to the intense resource contention. At the same time, the protocol overhead may become notable because of the short data transmission time. In contrast, if the access granularity is small as shown in Fig. 1b, the extended data transmission time can effectively weaken the impact of protocol overhead. Nonetheless, the small access granularity also can cause a large proportion of idle spectrum, which degrades the system throughput. From above all, to maintain an efficient channel access, the system should adaptively control the access granularity according to the current traffic pattern and the network congestion status. In this paper, we focus on the analysis of access granularity control in an OFDMA Carrier Sensing Multiple Access (CSMA) system, which extends the traditional time-domain backoff in single-channel CSMA to the frequency domain (the detail can be found in Section 3). To reflect the influence of access granularity, our analytic model takes the channel contention and the varying impact of protocol overhead into account jointly. The given definition and Fig. 1 indicate that the access granularity is subject to the subchannel bandwidth and the number of accessible subchannels in an MRA system. Therefore, how to tune them for access granularity control is the major issue that this paper will address. It needs to be pointed out that a system with static sub-channelization can mitigate the inefficiency of constant subchannel bandwidth, by making the subchannel bandwidth relatively small and adaptively assigning few or many subchannels to a node. However, this solution cannot displace the dynamic tuning of subchannel bandwidth

Please cite this article as: J. Yin et al., Access granularity control of multichannel random access in next-generation wireless LANs, Computer Networks (2015), http://dx.doi.org/10.1016/j.comnet.2015.08.008

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because of its two defects. On the one hand, a node probably needs to send a large frame by using multiple narrow-band subchannels for delay requirement. Such multichannel transmission increases the risk of collision, since the frame will fail in reception as long as a collision occurs on any one of the accessed subchannels. On the other hand, the dense sub-channelization exacerbates the imbalance of contention intensity on different subchannels if the subchannels are randomly accessed. This imbalance causes much loss of channel statistic multiplexing gain, which negatively influences the spectral utilization. As an important conclusion obtained in this paper, the inefficiency of dense sub-channelization will be mathematically proved in Section 6. In summary, this paper has the following primary contributions. (1) We introduce the concept of dynamic access granularity control, which is useful to amend the channel utilization of various OFDMA-based MAC protocols. (2) We provide a throughput analysis of the OFDMA-CSMA system and reveal the relation between access granularity and MAC efficiency. In particular, this paper points out that sub-channelization is conducive to mitigating the impact of protocol overhead in the time domain but is adverse to resolving contention in the frequency domain. (3) We establish an access granularity optimization model towards the maximum channel utilization. By using it, we derive the condition for the optimal number of accessible subchannels and the one for the optimal subchannel bandwidth. (4) We propose two dynamic access granularity control algorithms, which adaptively tune the number of accessible subchannels and the subchannel bandwidth (dynamic subchannelization), respectively. Simulation results indicate the near-optimal performance and good compatibility of the proposed algorithms. Moreover, the algorithms totally rely on the local statistics, and therefore there is no extra communication overhead incurred. To the best of our knowledge, this is the first work that proposes dynamic sub-channelization and investigates the issue of access granularity control in MRA systems. The rest of the paper is organized as follows. The related works are reviewed in Section 2. Section 3 introduces the system model. Section 4 formulates the saturated throughput of the system. The tuning issues of the number of accessible subchannels and the subchannel bandwidth are investigated in Sections 5 and 6, respectively. Section 7 evaluates the proposed algorithms. Finally, Section 8 concludes the whole paper.

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2. Related works

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The latest ratified standards IEEE 802.11ac [3] and IEEE 802.11ad [10] have boosted the data rate of WLAN beyond 1Gb/s. Both the standards adopt the traditional frame aggregation technology to mitigate the MAC efficiency degradation problem in high-data-rate communications. Nonetheless, such time-domain access granularity control is subject to traffic load, because the number of backlogged packets may not always meet the requirement of the expected aggregated frame size. For this reason, a terminal may have to wait a long time for enough packets to arrive at its sending buffer, which can severely deteriorate the overall quality of delay-critical applications [9,11].

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In contrast to the aforementioned single-channel systems, the access granularity in an MRA system has more flexible controllability. Unfortunately, this advantage is underused in the existing systems due to their static subchannelization and unsuitable tuning to the number of accessible subchannels. In the OFDMA Aloha system proposed by [12], a terminal can fast retransmit on a randomly selected subchannel once a frame fails in reception. The subchannelization of this system is static. Moreover, each terminal deployed in it can only access one subchannel. Due to these two characteristics, the system spectral utilization will drastically reduce if the number of contention nodes is less than the total number of subchannels [13,14]. This kind of static access granularity control is also extensively applied in the existing OFDMA-CSMA protocols. For example, a terminal attempts to transmit immediately once it senses an idle subchannel in the systems proposed by [6,7,15]. Owing to such a greedy and static access granularity control, the systems have to resolve the intense channel contention by using the improved time-domain backoff algorithm. However, because the time-domain backoff and the frequent frame collisions waste a large proportion of time, this scheme is harmful for the enhancement of channel utilization when equipped transceivers work at very high data rate. In the works [9,16], the static access granularity control was extended to the semi-dynamic case (i.e. the number of accessible subchannels is adjustable, but the sub-channelization is static). Specifically, [16] proposed an OFDMA-Aloha system, whereby a terminal can simultaneously access multiple subchannels of which the number is tuned by a frequency-domain Binary Exponential Backoff (BEB) algorithm. Using this algorithm, a terminal is allowed to access all the subchannels if its last transmissions do not encounter any collision, or else it reduces the number of accessible subchannels by half. Compared with BEB, the algorithm proposed by [9] tunes the number of accessible subchannels more smoothly and can obtain the larger throughput. This algorithm is termed as “Additive Increment and Multiplicative Decrement with Probability p” (pAIMD). It increases the number of accessible subchannels by one if no collision occurs, or else it multiplicatively decreases the number of accessible subchannels according to the current frequency of frame collision. Both BEB and pAIMD mitigate frame collision by rapidly reducing the number of accessible subchannels. For this drastic adjustment, they may be oversensitive to collisions when only a few terminals contend for the channel. The inefficiency of existing MRA systems is mainly caused by their inappropriate access granularity control. To overcome this problem, an MRA system should adaptively tune the subchannel bandwidth and the number of accessible subchannels according to the current channel contention intensity and protocol overhead. The rest of this paper will investigate the issue of access granularity control by analyzing the OFDMA-CSMA system proposed in [9], which resolves collision with a frequency-domain backoff mechanism. In the original work, the quantitative relation between the access granularity and the channel utilization is unknown. We will fill this vacancy by formulating and optimizing the system throughput with considering the dynamic nature of channel contention and traffic pattern.

Please cite this article as: J. Yin et al., Access granularity control of multichannel random access in next-generation wireless LANs, Computer Networks (2015), http://dx.doi.org/10.1016/j.comnet.2015.08.008

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Fig. 2. An illustration of the system model.

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3. System model

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As illustrated in Fig. 2, let us suppose that u homogeneous terminals (indexed by 1 to u) are deployed in an OFDMA system, where the communications are coordinated by an Access Point (AP). The system equally divides the whole band (K subcarriers) into NCH subchannels, each of which consists of NSC subcarriers. In our analytic model, NCH and NSC can be dynamically adjusted within [1, K] by modifying the mapping and inverse mapping between data bits and subcarriers. Therefore, the system has single channel if NSC = K (NCH = 1) and has multiple channels if NSC < K (NCH > 1). For simplicity, we assume that the channel is ideal without capture effect, channel fading and interference. Fig. 2b illustrates a complete channel contention process in the OFDMA system. The access opportunities appear when the entire channel is idle for a Distributed Interframe Space (DIFS) time. At the moment, if a terminal intends to send M frames and is allowed to access W (1 ≤ W ≤ NCH ) subchannels, it first randomly selects min (W, M) subchannels. Then, on each selected subchannel, the terminal uses a random subcarrier to send a busy tone to the AP. The busy tones transmitted by different contenders compose a special signaling termed as Multi-tone Request to Send (M-RTS). When it arrives at the AP, by comparing the energy of the busy subcarriers against a threshold, the AP selects the busy subcarriers with the smallest indexes on each subchannel as the winners of contention. The selection results are announced to the terminals by broadcasting a Multi-tone Clear to Send (M-CTS) signaling. After the M-CTS is decoded, the terminals

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Table 1 Notations of Major Parameters. Notation Parameter

Notation Parameter

R tmcts tmack tdifs K

tmrts tdata tprm tsifs NSC

M-RTS symbol duration Data symbol duration Preamble duration SIFS time Subchannel bandwidth

lmtu

Maximum transmission unit size

NCH

Maximum data rate M-CTS symbol duration M-ACK symbol duration DIFS time Total number of subcarriers Number of subchannels

access the assigned subchannels for data transmission. We suppose that a terminal can only send a data frame on each subchannel in every access period. The data frames may collide with each other, because two or more terminals may obtain the access opportunity on a same subchannel once they send busy tones on a same subcarrier. To confirm the correct receptions, the AP broadcasts a Multi-tone Acknowledgment (M-ACK) signaling after all the terminals end data transmissions. In the access process, M-CTS, data frames and M-ACK can be transmitted only if the whole channel is idle for a Short Interframe Space (SIFS) time. It is clear that the access granularity in the system is equal to NSC × W. In the next sections, we will compute the uplink throughput of the system and investigate how to appropriately tune W and NSC to achieve efficient channel access. Throughout this paper, saturated load is assumed. For the sake of conciseness, Table 1 gives the notations of the major parameters.

Please cite this article as: J. Yin et al., Access granularity control of multichannel random access in next-generation wireless LANs, Computer Networks (2015), http://dx.doi.org/10.1016/j.comnet.2015.08.008

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4. Throughput formulation

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The system throughput (S) is defined as the mean data bits successfully transmitted per unit time in the network. For its computation, we characterize the channel contention as a renewal process. The renewal period of the channel contention process is termed as a Transmission Period (TP). As illustrated in Fig. 2b, it is the time interval (TTP ) between the end moments of two adjacent M-ACK signalings. Let us suppose that an arbitrarily given terminal successfully accesses Ns subchannels in a TP and that the mean size of frame is l.¯

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According to the homogeneity of nodes and the renewal theorem, S can be expressed as

¯ (Ns )/E (TT P ) S = ulE 270 271 272

where E( · ) denotes the expectation of a stochastic variable. Without losing generality, the Cumulative Distribution Function (CDF) of the frame size is F(l). Hence, we have

l¯ = 273 274

(1)



lmtu

ldF (l )

0

(2)

In (1), TTP is determined by the longest data transmission time on NCH subchannels. Its expectation can be obtained by

E (TT P ) = tdi f s + tmrts + 3tsi f s + tmcts + t prm + NCH l¯max /R + tmack 275 276 277

where l¯max is the mean largest size of the transmitted data frames. Let k denote the mean number of transmitted frames during a TP. By using it, l¯max can be expressed as

l¯max = 278 279 280 281 282 283 284 285 286



lmtu

0

ldF (l )

k

288 289

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To compute the throughput, we now need to determine E(Ns ) and k. Let us observe an arbitrary subchannel. According to the description of the system model, it is known that the contention results in different TPs are independent of each other. Hence, if w = E (W ) and the xth node wins the contention with the probability pa, x , the mean number of times that the xth node accesses the observed subchannel is equal to pa,x Nw . Exploiting the homogeneity of nodes, we CH can express k as

i=0

(5)



j=0

and u−1 N SC −1  fi (τ) ps (τ) = NSC i=1



j=1

NSC − j NSC

NSC − j NSC

i

+ f0 (τ )

pa = E ( pa (τ)) =

i (1 − τ ) (u−1 i )τ

u−1−i



NSC

j=0

NSC −1 

1  = NSC

wj 1− NCH NSC

j=0

NSC − j NSC

NSC −1

 j=0

and

ps (i) =

1 NSC

⎧ ⎪ ⎨1

N SC −1

⎪ ⎩

j=1

1 NSC

NSC − j NSC



i

NSC − j NSC

(7)

i

for i = 0 for i > 0

(8)

299

301 302 303 304 305 306

u−1 (11) 307

ps = E ( ps (τ)) =

NSC u−1   i=0 j=1

i (1 − τ ) (u−1 i )τ

u−1−i

NSC



NSC 1  wj = 1− NSC NCH NSC



NSC − j NSC

i

u−1 (12)

j=1

Let t be the constant time overhead excluding data transmission during a TP. By exploiting the preceding derivations, S can be formulated as

lmtu

S=

t+

uw 0

1 lmtu R 0

ldF (l ) ps

ldF (l )

uwpa

NCH

W,NSC

subject to

308 309 310

(13)

maximize



297 298

i

E (Ns ) = E (W ) ps,x = wps

Let us suppose that the other i terminals contend for the observed subchannel besides the xth node. Then, pa and ps can be respectively expressed as a function of i, i.e.

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(10)

From (13), the issue of access granularity control can be described as

(6)

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300

i

which are two different linear combinations of fi (τ). If the applied algorithm independently adjusts W, the elements of τ have the independent and identical distribution. In this case, we can compute pa and ps respectively by using (11) and (12). After that, E(Ns ) and k can be further determined by using (5) and (6).

u−1 N SC −1 

294

(9)

Likewise, if the xth node successfully accesses the given subchannel with the probability ps, x . E(Ns ) can be obtained by

pa (i) = 293

u−1 N SC −1  fi (τ) NSC

pa (τ) =

(4)

k = uwpa,x = uwpa 287

We denote the index of a terminal by subscript x and use τ to signify the probability that a given terminal intends to access a subchannel. Therefore, the Probability Density Function (PDF) of i can be expressed as fi (τ), where the vector τ consists of τ 1 , τ2 . . . τu , and τx = Wx /NCH . Substituting fi (τ) into (7) and (8), we obtain

i=0

(3)

5

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S K NSC ≤K

1≤W ≤ 1 ≤ NSC

(14)

which is a constraint optimization problem. In the following, we will address it in terms of the tuning of W and the tuning of NSC .

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5. Tuning of accessible subchannels

316

The number of accessible subchannels (W) is static or is tuned empirically in the existing MRA systems. In this section, we will investigate two empirical algorithms, i.e. pAIMD

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Fig. 3. The evolution of W when 0.5AIMD algorithm is applied.

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and “Additive Increment and Multiplicative Decrement with probability 0.5” (0.5AIMD). These two algorithms both are the variants of frequency-domain BEB [9], but neither of them has been mathematically analyzed yet. The optimal W towards the maximum throughput will be solved in this section as well. We will propose a heuristic algorithm to help the system obtain the optimal W with little communication and computation overhead.

of ADP indicates that TCF is a geometrically distributed variable, of which the distribution depends on pnc . For tractability, we approximate that the contention results on different subchannels are independent of each other. Hence, it follows that

E (TCF ) = (1 − pnc )−1 − 1 = [1 − (1 − pc )w ]−1 − 1

 w u−1



1 1− 1− NSC NCH

5.1. 0.5AIMD

pc = pa − ps =

329

In 0.5AIMD algorithm, W is increased by one if no collision occurs, or else it is reduced by half. Adopting 0.5AIMD, a terminal only cares about its own transmissions [9,16]. Based on this characteristic, we approximate that W tuned by different nodes are independent of each other so that the system throughput can be computed by solving w and (13). The W adjustment process can be characterized as a renewal process. The renewal period is termed as an Adjustment Period (ADP), which is the interval between two adjacent W reductions. As shown in Fig. 3, W continues to increase for TCF collision-free TPs and finally stops at WC due to frame collision. Let subscription i denote the index of an ADP. Then, the initial W of the ith ADP is equal to 0.5WC(i−1) , which yields the recursive relation:

We establish the equation of w by using (15)–(18). Let nc be the mean number of the subchannels where frame collisions occur. Using the independence approximation again, we can compute nc by

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WCi = 0.5WC(i−1) −  + TCF i 343 344 345 346 347

348 349

(16)

Let pnc be the probability that no collision occurs on the subchannels accessed by the observed terminal. The definition

(18)

nc =

E (WC ) Pr (collision|W = E (WC )) E (TCF ) + 1

=

357 358 359

360 361 362 363

(20)

from which, we further derive

wpc =

356

(19)

Let Pr (collision|W = E (WC )) denote the conditional probability that a collision occurs on a given subchannel when some collisions have happened on E(WC ) subchannels. Then, nc can be alternatively expressed as

(15)

In (15),  is the fractional part omitted by the floor function (i.e. ·). We set it to 0.25 since only an odd W can cause error. According to [17], the variation of WC can be considered as a stationary stochastic process. Therefore, WCi and WC(i−1) have the following identical mean value (E(WC )):

E (WC ) = max (1, min (2E (TCF ) − 2 , NCH ))

nc = wpc

353 354

355

328

330

351 352

(17)

where the collision probability pc is expressed as



350

364

E (WC ) Pr (collision|W = E (WC )) E (TCF ) + 1



E (WC ) pc



1 − (1 − pc )W =E (WC ) (E (TCF ) + 1)

(21)

In (21), Pr (collision|W = E (WC )) is rewritten by exploiting Bayes formula, and the unknown w can be solved by numerical techniques. Appendix A proves that (21) has an unique solution. After obtaining w, we can directly compute the system throughput by using (13).

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Fig. 4. The Markov chain of the states of W.

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5.2. pAIMD

371

pAIMD increases W by one if no frame collision occurs, or else it reduces W according to the current frequency of frame collision [9]. The terminals applying this algorithm tune W by means of the identical reduction ratio (p), regardless of its access result (success, collision or deferring). Based on this characteristic, we can deduce that all the terminals have the identical W. Hence, the mean throughput of the network can be obtained by

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T

S = SW 379 380 381 382 383 384 385 386 387 388 389 390

(22)

where S = (S(W = 1), S(W = 2), . . . , S(W = NCH )) and W = ( Pr (W = 1), Pr (W = 2) . . . Pr (W = NCH )). To determine W, we characterize the variation of W as a Markov chain. As the diagram shown in Fig. 4, the states of W can transit to min (W + 1, NCH ) or max ((1 − p)W , 1) with the one-step transition probability pW,min (W +1,NCH ) and pW,max ((1−p)W ,1) , respectively. In the transitions, the mean value of the reduction ratio (p) is equal to the probability (qc ) that a collision occurs on a given subchannel. By using the relation among the collision probability, successful access probability and idleness probability of a subchannel, we can derive



qc = 1 − 1 − 391 392

W u

NCH

−

uW ps NCH

(23)

Let qc be equal to qc, i if W = i. The one-step transition probabilities (pi, j ) of the Markov chain are expressed as

⎧ 1 − (1 − qc,1 )NCH ⎪ ⎪ ⎪ ⎪(1 − qc,NCH )NCH ⎨ pi, j = (1 − qc,i )NCH ⎪ ⎪ 1 − (1 − qc,i )NCH ⎪ ⎪ ⎩ 0

for i = 1, j = 1 for i = NCH , j = NCH for j = i + 1 for j = max ((1 − qc,i )i, 1) else (24)

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where (1 − qc,i )NCH represents the probability that no collision happens. By solving the stationary distribution of the Markov chain, W and the system throughput can be finally determined. 5.3. Optimal number of accessible subchannels The derivative of (13) is useful to find the optimal W for throughput maximization. Eq. (13) indicates that W affects throughput both in terms of l¯max and wps . Nonetheless, we can approximate l¯max as a constant, since many frames are concurrently transmitted on multiple subchannels, and a large proportion (more than 20%) of Internet packets have

the size about 1500 bytes [18]. Based on this approximation, we substitute the maximization of throughput by the maximization of wps (i.e. E(Ns )) so that the optimization problem can be substantially simplified. As shown in Fig. 5, wps rises with the increment of w at first. Then, it keeps falling down when w exceeds the optimal value (w∗ ). This variation is a consequence of the fact that the subchannel idleness dominatingly influences the throughput if w < w∗ , whereas the frame collision is the dominating factor if w > w∗ . Clearly, the probabilities of subchannel idleness and frame collision both are monotonous functions of w. Therefore, the maxima point of wps is unique. Let E indicate a special event that satisfy the two conditions: (1) a given subchannel is successfully accessed; (2) the observed terminal sends busy tone on the given subchannel by using the subcarrier which has the second smallest index among the busy subcarriers. It follows that Proposition 5.1. The OFDMA-CSMA system can obtain the maximum channel utilization, if the number of accessible subchannels makes ps = Pr (E) hold. Proof. Taking the derivative of wps with respect to w and imposing it equal to 0, we obtain NSC −1

 j=1



jw 1− K

u−1

After multiplying by

ps =

NSC −1

=

 j=1

1 NSC



jw jw 1− ( u − 1) K K

u−2

NSC −1 1  jw jw 1− (u − 1) NSC K K

405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425

(25)

in both sides, (25) becomes



404

426

u−2 (26)

j=1

We denote the right side of (26) as the probability Pr (E), which has the meaning given by Appendix B.  Exploiting Proposition 5.1, we design a W adjustment algorithm to maximize the system throughput. Eq. (12) can yield that ps is a monotone-decreasing function of w. Hence, in our algorithm, W increases if ps > Pr (E), whereas it decreases if ps < Pr (E). We suggest that the AP centrally operates the W adjustment algorithm and broadcasts the updated W periodically. Otherwise, due to the hidden terminal effect, some of the terminals probably have a very large W, but some others may have a small one. Such inconsistency is apparently bad for fairness. Directly using the preceding algorithm, the AP needs to collect the busy tone pattern sent by each terminal for the computation of Pr (E). This collection process will incur a lot of communication overhead in a large-scale network. For this reason, it is necessary to solve the optimal W by exploiting the statistic parameters which can be locally obtained at the

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Fig. 5. The change of wps with respect to w when u = 30, NCH = 29 and NSC = 16.

445 446 447 448 449 450 451 452 453

AP side. Let N¯ c , N¯ c1 and N¯ 1 denote the mean number of the collision subchannels, the mean number of the collision subchannels where only one subcarrier is busy in the M-RTS signaling, and the mean number of the subchannels where only a node sends busy tone in the M-RTS signaling, respectively. We have the proposition as follows. Proposition 5.2. The OFDMA-CSMA system can obtain the maximum channel utilization, if the following condition is satisfied.

N¯1 = N¯c − N¯ c1

(27)

454

where all the statistics can be locally obtained at the AP side.

455

Proof. See Appendix C.

456 457 458 459



Extending the proof in Appendix C to ps < Pr (E ) and ps > Pr (E), we can accordingly obtain that N¯ 1 < N¯ c − N¯ c1 and that N¯ 1 > N¯ c − N¯ c1 . From this property and the monotonicity of ps , we know that the AP should decrease W if N¯ 1 < N¯ c − N¯ c1 , whereas it should increase W if N¯ 1 > N¯ c − N¯ c1 . Algorithm 1

6. Tuning of subchannel bandwidth

469

In the above analysis, we focus on the tuning issue of the number of accessible subchannels (W). The investigation of 0.5AIMD and pAIMD shows that the throughput is influenced by the mean value and distribution of W. The derivation of the optimal W indicates that such influence is primarily related to frame collision and subchannel idleness. By the tradeoff between the factors, we proposed an AOWA algorithm, which is able to control the intensity of channel contention to an appropriate extent. In this section, we will address another important issue of access granularity control, i.e., the tuning of subchannel bandwidth (NSC ). NSC is closely related to the data rate on each subchannel. As a result, its tuning needs to consider not only the impact of contention results, but also the impact of protocol overhead. This paper uses time utilization (St ) and spectral utilization (Sf ) (defined in Sec V-A) to reflect how NSC influences the channel utilization separately in the time domain and the frequency domain. Through the tradeoff between St and Sf , we will propose a dynamic subchannelization algorithm to maximize the throughput.

470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489

Algorithm 1 Asymptotic Optimal W Adjustment (AOWA). 1: 2: 3: 4: 5: 6: 7:

460 461 462 463 464 465 466 467 468

procedure Update W (W , Nc , Nc1 ) if N1 < Nc − Nc1 then W = max (W − 1, 1); else W = min (W + 1, NCH ) end if end procedure

6.1. Time utilization and spectral utilization Time utilization (St ) reflects the impact of protocol overhead. As shown by (28), it is defined as the mean ratio of data transmission time to a TP duration without considering the idle subchannels.

St := gives the pseudo-code of this Asymptotic Optimal W Adjustment (AOWA) algorithm, in which the tuning step is set to 1, and we use N1 , Nc and Nc1 in each TP as the substitutions of their mean values. Using AOWA algorithm, the AP can directly obtain the required statistics from the busy tone pattern of a M-RTS signaling. Additionally, AOWA algorithm only costs two comparisons and one addition/subtraction. Therefore, it has very low complexity and is easy to implement.

490

l¯ R t NCH

+

lmax R

491 492 493 494

(28)

Proposition 6.1. Time utilization is negatively related to the subchannel bandwidth but is positively related to the number of subchannels.

495

Proof. We prove this proposition in two aspects. The prolonged data transmission time can reduce the proportion of the constant protocol overhead (i.e. t) in a TP. Therefore, St

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9

Fig. 6. The time waste caused by the stochastic packet size.

Fig. 7. The improvement of time utilization caused by the dense sub-channelization.

501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517

rises with the increasing NCH as expressed by (28). Furthermore, the dense sub-channelization can amend St , because it is able to mitigate the misalignment among the end moments of data transmissions. As illustrated in Fig. 6, time will be wasted on the subchannels where the data transmissions end earlier. However, if we suitably increase NCH and reduce MTU size simultaneously, this waste can be avoided. For instance, by doubling NCH and reducing lmtu to 750 bytes, the system shown in Fig. 7 can fragment the original packets (A1 and A2 in Fig. 6) into four segments (A1−1 , A1−2 , A2−1 and A2−2 ). Depending on the concurrent transmissions of the segments, the time waste is avoided at the cost of two idle subchannels.  By contrast, spectral utilization (Sf ) reflects the impact of channel contention intensity. As shown in (29), it is defined by the proportion of the subcarriers that successfully carry data bits in the whole channel band.

S f :=

uwps NCH

(29)

520

Proposition 6.2. Spectral utilization is positively related to the subchannel bandwidth but is negatively related to the number of subchannels.

521

Proof. Expanding (29), we have

518 519



NSC uw  wj Sf = 1− K K

u−1 (30)

j=1

522 523 524 525 526 527

which is a monotone-increasing function of NSC (i.e. a monotone-decreasing function of NCH ).  The positive relation between NSC and Sf can be interpreted by the concept of statistic multiplexing gain. In a system resolving contention in the frequency domain, the channel bandwidth is positively related to the collision resolution

performance. For this reason, we can regard a multichannel system as a group of relatively low-performance subservers (subchannels), which provide collision resolution service. Since the traffic load is randomly distributed, some of the sub-servers may be idle, but some of them may be very busy. Such potential imbalance of load distribution degrades the sub-server utilization (spectral utilization) and makes it lower than that of a single server (channel) system. The increment of NCH is equivalent to the repartition of each subchannel. When it happens, the statistic multiplexing gain of a subchannel loses again, which causes the further drop of spectral utilization.

528

6.2. Optimal subchannel bandwidth

540

From the above analysis, we can conclude that subchannelization is conducive to mitigating the impact of protocol overhead in the time domain (i.e. St ), but it is adverse to the frequency-domain contention resolution (i.e. Sf ). According to (13), we have

S = RSt S f

529 530 531 532 533 534 535 536 537 538 539

541 542 543 544 545

(31)

which means that the system should make the tradeoff between St and Sf for the enhancement of system efficiency. The tradeoff problem has two fundamental constraints: both NSC and NCH must be integers; NCH × NSC must be less than K. To satisfy the constraints, the system can divide the whole band into NCH + 1 subchannels, of which the NCH subchannels have the equal bandwidth NSC , but the remaining one has the bandwidth consisting of K − NCH NSC subcarriers (NSC ≥ K − NCH NSC ). To achieve such sub-channelization result, the system optimizes NSC first and then computes NCH by K/NSC . Proposition 6.3. The OFDMA-CSMA system can obtain the maximum channel utilization if the subchannel bandwidth

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satisfies the following condition:



wNSC uw 1 − K −t 560



w 1− K

563 564

l¯max + K R

Algorithm 2 Dynamic Sub-channelization (DS).



u



wNSC − 1− K

u 



NSC



wj 1− K

1

l¯ uw R tNSC + lmax K R



NSC 1

=0

(32)

u−1

dj

(33)



wj K

1−

(34)

u−2  tN

l¯max + K R



SC

(35) dg(NSC ) dNSC

< 0. Thus, g(NSC ) is a monotone-

565

Eq. (35) shows that

566

570

decreasing function. By means of this property and that K ∗ ). g( w ) ≤ 0, we can infer that (32) has an unique real root (NSC According to the monotonicity of g(NSC ), it can be proved that ∗ ) is the maxima point. Moreover, N ∗ satisfies the necS(NSC SC ∗ ≥ w, since g(N essary condition that NK∗ = NCH CH = w) ≤ 0,

571

and g(NCH ) monotonously increases with respect to NCH . 

569

SC

572 573 574 575 576 577

∗ by directly using Proposition It is unwise to compute NSC 6.3 due to the time-consuming solution and the dynamic nature of a real network. For this, it is necessary to develop a heuristic Dynamic Sub-channelization (DS) algorithm. According to the definition given in Section 5.3, N¯ 1 can be formulated as



N¯ 1 = uw 1 − 578 579 580

582 583 584 585 586 587 588 589 590 591 592 593 594 595

u−1

(36)

After substituting (36) into (32) and doing some algebraic manipulations, we rewrite the condition for the optimal NSC as

g(NSC ) = N¯ 1 581

wNSC K

6:

if g > 0 then NSC = NSC + 1; end if if g < 0 then NSC = NSC − 1 end if   NCH = NK

CH

7:

10:

dj

E (TT P )NSC − t (δ − qi ) = 0 K

(37)

where δ is the probability that a given subcarrier is idle. Note that all the parameters in (37) can be locally obtained at the AP side. Hence, g(NSC ) is easy to calculate. g(NSC ) is a monotonic function. Its comparison with zero guides the tuning of NSC . In brief, the AP should increase NSC if g(NSC ) > 0, whereas it should decrease NSC if g(NSC ) ≤ 0 (see Algorithm 2). The tuning result of NSC or NCH will be broadcasted periodically. According to the received updating information, the terminals redivide channel by modifying the mapping and inverse mapping between data bits and subcarriers. This adjustment to subchannel bandwidth can be fulfilled by a normal OFDMA transceiver. Moreover, the proposed DS algorithm has a very low complexity. It only costs a few simple arithmetic operations. In addition to DS, (37) can yield the following proposition.

i −N i NSC CH K

g = N1 NT P − t

9:

u−1

Taking the derivative of (34) with respect to NSC and imposing it equal to 0, we obtain (32). Let g(NSC ) denote the left side of (32). We have



567

T

5:

8:

u(u − 1)w2 wNSC dg(NSC ) =− 1− dNSC K K

568

i ). Count the number of idle subchannels (NCH i 2: Count the number of idle subcarriers (NSC ). 3: procedure Update the channel division(W , NSC , NCH , N1 , i , Ni , T ) NCH SC T P 4: K =NSC NCH



1:

SC

Accordingly, the throughput can be approximated as

S≈ 562

u−1  tN

Proof. Let us consider NSC as a continuous variable. Then,

uw Sf ≈ K 561

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11: 12: 13:

SC

end procedure

Proposition 6.4. The OFDMA-CSMA system should reduce the subchannel bandwidth to mitigate congestion in a massive user network. Proof. We consider g(NSC ) as a function of u and define it as g(u). According to (32) and (37), the condition for the optimal channel division can be expressed as

g(u) = A − B = 0 where





w B =t 1− K

597 598 599 600 601

(38)

TP wNSC A= uw 1 − NCH K and

596

u−1



wNSC +t 1− K

602

u (39)

603

u (40)

In a congestion network, E(TTP ) is approximately equal to a constant, because a large number of nodes will attempt to access the channel simultaneously. In this circumstance, we have



u+1 wNSC g(u + 1) ≈ A 1− u K



w −B 1− K

604 605 606 607

(41) K−wN

u+1 SC Note that A = B and that u+1 u ≈ 1 yields u K−w < 1. As a result, g(u + 1) < 0 = g(u). By incorporating this property with the monotonicity of g(NSC ), it can be proved that the increment of u will cause the reduction of the optimal NSC (the increment of the optimal NCH ). This conclusion points out that a system should enhance time utilization and sacrifice spectral utilization to mitigate congestion. 

608

7. Performance evaluation

615

By the simulation experiments on OPNET, this section evaluates the introduced access granularity control algorithms. Our evaluation begins with the simulations of 0.5AIMD, pAIMD and AOWA. Then, the performance of DS algorithm will be compared with that of Static Subchannelization (SS). Lastly, this section ends with the study of cooperation between AOWA and DS (termed as Joint Adjustment). The simulated network has a centralized architecture. Specifically, a number of terminals are deployed randomly

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Table 2 Simulation parameters. Parameter

Value

Parameter

Value

R tmcts tmack tdifs K NCH

71.8/145/290/580 Mb/s 28.4 μs 15.6 μs 50 μs 16 × 29 or 16 × 14 29 or 14

tmrts tdata tprm tsifs NSC lmtu

37.4 μs 15.6 μs 46.8 μs 10 μs 16 subcarriers 1500/750/100 bytes

NOTE: R = 71.8 Mb/s when the system works on 20 MHz band and equips 1 antenna. R = 145, 290 and 580 Mb/s when the system works on 40 MHz band and equips 1 antenna, 2 × 2 Multiple-Input-Multiple-Output (MIMO) and 4 × 4 MIMO [9], respectively. NCH = 29 if the bandwidth is 40 MHz. NCH = 14 if the bandwidth is 20 MHz.

635

in a single-cell network, where an AP is the common destination of data transmissions. The simulated terminals are configured according to the parameters listed in Table 2. The traffic load is saturated as the assumption. Unless otherwise specified, we suppose that 100, 750 and 1500 byte packets are generated with the probabilities 0.4, 0.2 and 0.4, respectively. Throughout the simulations, we measure both MAC efficiency and channel utilization by using the normalized uplink throughput, i.e. the ratio of network uplink throughput to the maximum data rate.

636

7.1. The study of W

637

We simulate the 71.8 and 580 Mb/s systems applying the different W tuning algorithms. Fig. 8 depicts the obtained mean W (denoted by w) with the different number of terminals. The analytic results (lines) coincide with the ones obtained from the simulation runs (marks), which verifies our analysis. As illustrated in the figure, w tuned by AOWA is in accordance with the optimal value curves. By contrast, w tuned by pAIMD and the one tuned by 0.5AIMD both deviate from the optimal value curves obviously. Fig. 9 reports the normalized throughput of the systems. Observing the curves indicating 0.5AIMD, we can find that the systems obtain the relatively poor throughput performance when a small number of nodes contend for the channel. This inefficiency is a result of the large and constant W reduction ratio of 0.5AIMD. With the increasing number of contenders, Fig. 8 shows that W rapidly converges. Since pc = N1 SC if u → ∞, the converged w is only related to NSC and is equal to the solution of that

626 627 628 629 630 631 632 633 634

638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654



w= 655 656 657 658 659 660 661 662 663 664 665 666



2−3 1−

1− 1−

1 NSC

1 NSC

w

2[1−(1− N1

SC

)w ]−1 −1

(42)

For the rapid convergence of w, the throughput of the 0.5AIMD systems first gradually rises because of the reducing number of idle subchannels. Then, it quickly decreases after reaching the peak due to the intense channel contention. Like the 0.5AIMD systems, there are notable fluctuations on the normalized throughput of the pAIMD systems. When a few terminals contend for the channel, the over-sensitivity to collision is a dominating factor influencing the performance. The over-sensitivity to collision can cause a small w. Consequently, we observe that the throughput of the pAIMD systems keeps falling down before the number of contenders exceeds 12. The intensified channel contention can mitigate

Fig. 8. The average W when the different W adjustment algorithms are respectively used. lmtu = 750 bytes in the simulation.

the descent rate and variance of w. This results in the reduction of the idle spectrum and the gradual increment of the throughput. After the throughput reaches the peak (W has already converged to 1), the probability of frame collision becomes the new dominating factor. Therefore, the increasing number of contenders deteriorates the performance again. In comparison with 0.5AIMD and pAIMD, AOWA shows the notable advantage in throughput performance. Especially, the channel utilization achieved by AOWA can be 30% larger than those achieved by 0.5AIMD and pAIMD in the 580 Mb/s case. We notice that the systems applying AOWA have the optimal mean W (see Fig. 8), but their throughput is slightly lower than the optimal theoretical performance (see Fig. 9). This performance gap is owing to the variance of W, which quickly reduces with the increasing number of terminals.

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7.2. DS versus SS

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We conduct the comparison between DS and SS (NSC = 16) in the 71.8 Mb/s and 580 Mb/s systems. In this simulation experiment, W is constantly set to 2 for the fairness of comparison. In addition, we use the different lmtu settings

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Fig. 9. The normalized throughput when the different W adjustment algorithms are respectively used. lmtu = 750 bytes in the simulation.

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(100 bytes and 750 bytes) to reflect the varying impact of protocol overhead. Fig. 10 reports the normalized throughput of the systems. We observe that DS obtains the quasi-optimal throughput and notably outperforms SS, depending on the tradeoff between the time utilization and the spectral utilization. The performance advantage of DS gradually increases with either the reduction of MTU size or the increment of terminals. This change implies that an appropriate sub-channelization can efficiently amend the throughput of a large-scale network where a massive number of short packets need to be sent (such as M2M communications). On the other hand, Fig. 10 reflects that the system with lmtu = 750 bytes has the higher channel utilization than that with lmtu = 100 bytes. Equation (32) shows that the optimal NSC is positively related to lmtu . Hence, the comparison result demonstrates that the MTU size should be sufficiently large to avoid the drastic reduction of spectral utilization, even though this is against the improvement of time utilization in random packet size case.

707

7.3. Joint Adjustment

708

AOWA and DS can be simultaneously applied. To prove that, we simulate the systems which adopt AOWA and DS jointly. Fig. 11 illustrates the comparison between the

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Fig. 10. The normalized throughput when the different channel division algorithms are respectively used.

Joint Adjustment (marks) and the optimal access granularity (lines) in terms of the mean W, the average subchannel bandwidth and the normalized throughput. We observe the close match between the simulation results and the optimal analytic values except the slight deviations on subchannel bandwidth and normalized throughput. These deviations are owing to the variance of W. It can be amended by the smoothing of adjustment results. In order to further prove the advantage of the Joint Adjustment, we simulate the systems with the diverse data rates and the different channel contention intensities (u = 20 and u = 50). The performances of the different access granularity control algorithms (i.e. 0.5AIMD, pAIMD and the Joint Adjustment) are reported in Fig. 12. Additionally, the channel utilization of traditional CSMA/CA system is also plotted as a performance benchmark. As shown in the figure, the MRA system notably outperforms the traditional CSMA/CA system depending on the fine-grained channel access. Furthermore, the system applying the Joint Adjustment has the best throughput performance due to its appropriate control to the access granularity. Observing the curves indicating 0.5AIMD and the Joint Adjustment, we notice that the MAC efficiency does not drop when the data rate augments from

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Fig. 12. The MAC efficiency when the system applies the different access granularity control algorithms and transmits data at the different rates.

Fig. 11. The mean W, average subchannel bandwidth, and normalized throughput when the system applies the Joint Adjustment and the optimal access granularity, respectively. lmtu = 1500 bytes in the simulation.

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71.8 Mb/s (20 MHz bandwidth) to 145 Mb/s (40 MHz bandwidth). There are two justifications for this behavior: (1) if the adopted algorithm is 0.5AIMD, the simulation settings (NSC = 16 and NCH = 29) just result in a quasi-optimal access granularity; (2) if the algorithm is the Joint Adjustment, the used frequency-domain access granularity control can effectively prevent the reduction of channel utilization caused by the doubling of bandwidth. In addition to the channel utilization performance, the fairness of the proposed algorithms is investigated as well. AOWA and DS endow contenders with the identical amount of access opportunities by imposing the same constraint to their access granularities. Moreover, depending on an appropriate MTU size setting, the algorithms can effectively mitigate the throughput unfairness caused by the probable heterogeneity among frame sizes (c.f. Fig. 6). These features guarantee the good long-term fairness of the system. However, the short-term fairness problem may still exist. We evaluate the short-term fairness by using Jain’s fairness index [19], which quantizes the fairness during a given observation period (To ) with a value distributed in [0, 1]. If Jain’s index is close to 1, the channel sharing is fair; otherwise, an index close to 0 indicates a high level of unfairness. Fig. 13 illustrates the Jain’s index of the 71.8 and

Fig. 13. The Jain’s fairness index with respect to the observation duration To when the system applies the different access granularity control algorithms. N = 20 and lmtu = 750 bytes in the simulation.

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580 Mb/s systems with the different access granularity control algorithms. We observe that the short-term unfairness exists indeed, but it is very slight in an MRA system. Furthermore, compared with 0.5AIMD, pAIMD and CSMA/CA, the proposed Joint Adjustment obtains the best fairness, especially in the system with the higher data rate. The fairness advantage of the Joint Adjustment can be interpreted in two aspects. Firstly, its uniform access granularity control guarantees the fair access opportunity and the equal expected individual throughput. In the second place, the throughput enhancement by the Joint Adjustment increases the number of frames transmitted per unit time, which reduces the variance of the temporary individual throughput.

771

8. Conclusion

772

796

This paper investigated an OFDMA-CSMA system, where a terminal can access multiple subchannels simultaneously and the contention is resolved by the frequency-domain backoff. Exploiting this system model, we studied the access granularity control issue which widely exists in MRA systems. The tuning issue of the number of accessible subchannels was investigated via the instances of 0.5AIMD and pAIMD. We proposed AOWA algorithm to solve the related optimization problem. For the tuning issue of subchannel bandwidth, we proved that dense sub-channelization is positively related to the time utilization but is negatively related to the spectral utilization. To address this tradeoff problem, DS algorithm was proposed. The simulation results not only show the accuracy of the presented analytic model in predicting the system throughput, but also the advantage of the proposed dynamic access granularity control algorithms. When using 0.5AIMD or pAIMD, the system performance is sensitive to the collision probability and strongly depends on the number of contention nodes. However, when using the proposed AOWA and DS, the system obtains the stable and quasi-optimal throughput. The proposed dynamic access granularity control algorithms can help a system achieve the suitably fine-grained channel access. It would present more flexibility to the nextgeneration WLANs.

797

Acknowledgments

798

This work was supported by the National S&T Major Project of China under Grant no. 2014ZX03004-003, the National Natural Science Foundation of China under Grant (no. 61374189), the Fundamental Research Funds for the Central Universities (No. ZYGX2013J009), and EU FP7 Project CLIMBER (PIRSESGA-2012-318939).

773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795

799 800 801 802 803 804

805

Taking the derivative of φ , we have

(A.2)

which yields that φ is a monotone-increasing function with respect to E(TCF ). We denote the computation result of the right side of (A.1) as w . Clearly, if w increases, E(TCF ) decreases and pc increases. Hence, w is a monotone-decreasing function with respect to w, which means that (21) has the unique solution.  Appendix B. Proof of Proposition 5.1

CH SC

carrier, Pr (E) can be expressed as



Pr (E j ) =

0 u−1 jw NSC K



1−



jw u−2 K

for j = 0, j = NSC for 1 ≤ j ≤ NSC − 1

809 810 811 812

814 815 816 817 818 819 820 821

(B.1)

Summing all the Pr (E j ) ( j = 0, . . . , NSC ), we obtain the right side of (26). 

822 823

Appendix C. Proof of Proposition 5.2

824

Proof. Let N¯ sx be the mean number of the subchannels that the xth node successfully accesses. By means of the ergodicity and the node homogeneity, ps can be expressed as

825

ps =

N¯ s1 + N¯ s2 + · · · + N¯ su N¯ s = uNCH uNCH

Let N¯ Ex be the mean number of the subchannels where event E occurs. By using N¯ Ex , Pr (E) can be expressed as,

Pr (E) =

E[Uc ](N¯ s − N¯ 1 ) N¯ E1 + N¯ E2 + · · · + N¯ Eu = uNCH uNCH

uNCH

N¯ 1 = qc2 (N¯ s − N¯ 1 )

827

828 829

(C.2)

In (C.2), Uc is the number of the terminals which send busy tones on the subcarrier such that the flowing conditions hold. Firstly, the subcarrier locates on a successfully accessed subchannel; secondly, the subcarrier is in busy status and has the second smallest index among the busy subcarriers. If Uc has  the distribution {qc1 , qc2 . . . qcu }, we have E[Uc ] = ui=1 iqci . By neglecting the probability that three or more terminals send busy tones on the same subcarrier (this is available if NSC  Nuw ), (C.2) becomes

(qc1 + 2qc2 )(N¯ s − N¯ 1 )

826

(C.1)

830 831 832 833 834 835 836 837 838

(C.3)

According to (C.1) and (C.3), ps = Pr (E) is equivalent to

(A.1)

807 808

813

Proof. Through the derivation of S, we have known that a node attempts to access a given subchannel with the probability Nw . Hence, on any one of the subchannels the xth node CH contends for, the probability that a certain subcarrier is used by the xth node to send busy tone is N1 , whereas the probSC ability that the subcarrier is used by the other contenders is w N N . If the xth node sends the signal on the ( j + 1)th sub-

Pr (E ) =

Proof. By exploiting (16), (21) can be transformed as

2E (TCF ) − 2 [1 − (1 − pc )E (WC ) ][E (TCF ) + 1] 2φ = 1 − (1 − pc )E (WC )

1+ dφ = dE (TCF ) [E (TCF ) + 1]2

CH

Appendix A. Proof of the unique solution of (21)

w=

806

[m3Gdc;September 2, 2015;4:19]

839

(C.4)

Let σ be the joint probability that event E occurs and only the observed terminal sends a busy tone on the subcarrier which has the second smallest index among the busy subcarriers.

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843 844

By means of the ergodicity and the node homogeneity, qc2 can be obtained by

qc2 = 845

uw( Pr (E) − σ ) 2(N¯ s − N¯ 1 )

(C.5)

Using a proof as Appendix B, we have



σ=

Nsc−1 1  jw ( j + 1)w (u − 1) 1− NSC K K

u−2 (C.6)

j=1

846 847

After some transformation, Pr (E) − σ can be approximated by



Nsc 1  jw w (u − 1) 1− NSC K K

Pr (E) − σ =

j=1

−(u − 1) ≈ pc − pc1 848 849 850 851

15

[16] A. Mutairi, S. Roy, Exponential backoff in frequency-domain for random access in OFDMA femtocells, in: Proceedings of IEEE WCNC, 2013. [17] J. Padhye, V.F. V., D. Towsley, J.F. Kurose, Modeling TCP Reno performance: a simple model and its empirical validation user estimate, IEEE Trans. Wirel. Commun. 4 (4) (2005) 1506–1515. [18] R. Sinha, C. Papadopoulos, J. Heidemann, Internet packet size distributions: some observations, May 2007. http://www.isi. edu/˜johnh/PAPERS/Sinha07a.pdf. [19] R. Jain, D. Chiu, W. Hawe, A Quantitative Measure of Fairness and Discrimination for Resource Allocation in Shared Systems, Technical Report, 1984. DEC-TR-301.

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Jiechen Yin received the bachelor degree in communication science from University of Electronic Science and Technology of China (UESTC) in 2009. He is currently a Ph. D student with the network engineering department, UESTC. His primary research interests include wireless communication protocols, HetNet, queuing theory and game theory.

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Yuming Mao was born in Deyang, Sichuan Province, China in 1956. He received his B.Eng. degree in Communication Engineering from the University of Electronic Science and Technology of China (UESTC) in 1982, and received his M.Eng. degree in Information Engineering & Processing from UESTC in 1984. After graduation, he has been working with UESTC. He was employed as an associate Professor in 1992. He has been a Professor with the School of Communication & Information Engineering, UESTC since 1999. Currently, he serves as the head of the Department of Network Engineering, UESTC. He is also the executive vice-director of the National Communication & Information System Laboratory Center, UESTC. His research focuses on data communication and computer networks.

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Supeng Leng is a Professor in the School of Communication & Information Engineering, University of Electronic Science and Technology of China (UESTC). He received his Ph.D. degree from Nanyang Technological University (NTU), Singapore. He has been working as a Research Fellow in the Network Technology Research Center, NTU. His research focuses on resource, spectrum, energy, routing and networking in wireless ad hoc/sensor networks, broadband wireless access networks, smart grid, and the next generation mobile networks. He is a member of IEEE ComSoc and VTSoc. He published over 80 research papers in recent years. He serves as an organizing committee chair and TPC member for many international conferences, as well as a reviewer for over 10 international research journals.

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Yuming Jiang received his BSc from Peking University, China, in 1988, MEng from Beijing Institute of Technology, China, in 1991, and PhD from National University of Singapore, Singapore, in 2001. He worked with Motorola from 1996 to 1997. From 2001 to 2003, he was a Member of Technical Staff and Research Scientist with the Institute for Infocomm Research, Singapore. From 2003 to 2004, he was an Adjunct Assistant Professor with the Electrical and Computer Engineering Department, National University of Singapore. From 2004 to 2005, he was with the Centre for Quantifiable Quality of Service in Communication Systems (Q2S), Norwegian University of Science and Technology (NTNU), Norway, supported in part by the Fellowship Programme of European Research Consortium for Informatics and Mathematics (ERCIM). He visited Northwestern University, USA from 2009 to 2010. Since 2005, he has been with the Department of Telematics, NTNU, as a Professor. He was Co-Chair of IEEE Globecom2005 - General Conference Symposium, TPC Co-Chair of 67th IEEE Vehicular Technology Conference (VTC) 2008, General/TPC Co-Chair of International Symposium on Wireless Communication

946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966

(C.7)

(C.8) 

852

854 855 856 857 858 859 860 861 Q2 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894

u−2

where pc1 stands for the probability that only one terminal’s transmission collides with the observed terminal’s transmission on the given subchannel. Since N¯ c1 = 12 uwpc1 and N¯ c = 1 2 uwpc , (C.4), (C.5), (C.7) and the condition ps = Pr (E) yield

N¯ 1 = N¯ c − N¯ c1

853



w w 1− K NCH

u−2

[m3Gdc;September 2, 2015;4:19]

References [1] E. Magistretti, K.K. Chintalapudi, B. Radunovic, WiFi-nano: re- claiming WiFi efficiency through 800 ns slots, in: Proceeding of ACM SGIMOBILE MobiCom, 2011. [2] S. Sen, C.R. Roy, S. Nelakuditi, No time to countdown: migrating backoff to the frequency domain, in: Proceedings of ACM SIGMOBILE MobiCom, 2011. [3] Wireless LAN medium access control (MAC) and physical layer (PHY) specification; amendment 4: Enhancements for very high throughput for operation in bands below 6 ghz. [4] X. Feng, J. Zhang, Q. Zhang, B. Li, Use your frequency wisely: explore frequency domain for channel contention and ACK, in: Proceedings of the IEEE INFOCOM, 2012. [5] P. Huang, X. Yang, L. Xiao, WiFi-BA: Choosing arbitration over backoff in high speed multicarrier wireless networks, in: Proceedings of IEEE INFOCOM, 2013. [6] H. Kwon, H. Seo, S. Kim, B.G. Lee, Generalized CSMA/CA for OFDMA systems: protocol design, throughput analysis, and implement issue, IEEE Trans. Wirel. Commun. 8 (8) (2009) 4176–4187. [7] W. Xudong, W. Hao, A novel random access mechanism for OFDMA wireless network, in: Proceedings of IEEE Globecom, 2010. [8] R. Dong, M. Ouzzif, S. Saoudi, A two-dimension opportunistic CSMA/CA protocol for OFDMA-based in-home PLC networks, in: Proceedings of IEEE ICC, 2011. [9] J. Fang, et al., Fine-grained channel access in wireless LAN, IEEE/ACM Trans. Netw. 21 (3) (2013) 772–787. [10] Wireless LAN medium access control (MAC) and physical layer (PHY) specification; amendment 3: Enhancements for very high throughput in the 60 ghz band. [11] T. Peyman, D. Aresh, S. Khosrow, K. Kiseon, An optimal packet aggregation scheme in delay-constrained IEEE 802.11n WLANs, in: Proceedings of IEEE WiCOM, 2012. [12] Y.-J. Choi, S. Park, S. Bahk, Multichannel random access in OFDMA wireless network, IEEE J. Sel. Areas Commun. 24 (3) (2006) 603–613. [13] A. Mutairi, S. Roy, G. Hwang, Delay analysis of OFDMA-Aloha, IEEE Trans. Wirel. Commun. 12 (1) (2013) 89–99. [14] C.-H. Wei, R.-G. Cheng, S.-L. Tsao, Modeling and estimation of one-shot random access for finite-user multichannel slotted ALOHA systems, IEEE Commun. Lett. 16 (8) (2012) 1196–1199. [15] R. Dong, M. Ouzzif, S. Saoudi, Opportunistic random-access scheme design for OFDMA-based indoor PLC networks, IEEE Trans. Power Deliv. 27 (4) (2012) 2073–2081.

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Q3

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Systems (ISWCS) 2007-2010, and General Chair of IFIP Networking 2014 Conference. He is first author of the book Stochastic Network Calculus. His research interests are the provision, analysis and management of quality of service guarantees in communication networks. In the area of network calculus, his focus has been on developing models and investigating their basic properties for stochastic network calculus (snetcal), and recently also on applying snetcal to performance analysis of wireless networks.

Muhammad Asad Khan received the bachelor degree in computer science from University of Peshawar, Pakistan, in 2002 and master degree from Oxford Brookes University, UK in 2006. He is currently a Ph.D student with the network engineering department, UESTC. His primary research interests include wireless communication protocols, HetNet, backhaul link and resource management.

Please cite this article as: J. Yin et al., Access granularity control of multichannel random access in next-generation wireless LANs, Computer Networks (2015), http://dx.doi.org/10.1016/j.comnet.2015.08.008

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