Fair and efficient resource allocation in IEEE 802.11ah WLAN with heterogeneous data rates

Computer Communications 151 (2020) 154–164

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Fair and efficient resource allocation in IEEE 802.11ah WLAN with heterogeneous data rates U. Sangeetha ∗, A.V. Babu Department of Electronics and Communication Engineering, National Institute of Technology, Calicut, Kerala, India

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Keywords: IEEE 802.11ah Throughput Restricted access window Data rate based grouping Wireless LAN

ABSTRACT For dense wireless LANs (WLANs), IEEE 802.11ah has specified restricted access window (RAW) as the channel access scheme. Here the competing stations (STAs) are divided into groups and STAs belonging to each group attempt to access the channel during their designated RAW slot by invoking the distributed coordination function (DCF) protocol. This paper describes an analytical model to evaluate the throughput performance of IEEE 802.11ah WLAN, when STAs use distinct data rates and the RAW mechanism is implemented for medium sharing. When STAs are grouped randomly (i.e., without considering their data rates), a group will contain STAs operating at distinct data rates. Since all these STAs simultaneously contend for channel access based on DCF protocol, the throughput of high data rate STAs are down-equalized to that of lower data rate STAs; thus the aggregate network throughput is degraded significantly. To resolve the resulting performance anomaly problem, we consider data rate based grouping where STAs operating at the same data rate are grouped together. We describe an algorithm for implementing data rate based grouping at the access point (AP). Further, we describe an analytical procedure to find the network throughput under data rate based grouping. Through numerical and experimental investigations, we establish that data rate based grouping can significantly improve the aggregate network throughput performance, as compared to the conventional random grouping strategy. Further, we use Jain’s fairness index (JFI) to establish that data rate based grouping can also provide fair resource allocation among the STAs operating at distinct data rates, by ensuring that all the competing STAs in the network achieve throughput proportional to their data rates.

1. Introduction IEEE 802.11ah specifies the physical (PHY) and medium access control (MAC) layer protocols for the implementation of sub-1 GHz wireless LAN (WLAN) for machine-to-machine (M2M) and Internet of Things (IoT) applications involving a large number of connected devices or stations (STAs) [1]. In the legacy 802.11 based WLANs, STAs use either distributed coordination function (DCF) or enhanced distributed channel access (EDCA) protocol for getting the channel access. Notice that the DCF/EDCA protocols are based on the carrier sense multiple access-collision avoidance (CSMA-CA) mechanism. When the legacy DCF/EDCA protocol is used in dense WLAN scenarios, all the STAs having frames in their MAC queue ready for transmission, compete simultaneously for channel access, which leads to very high collision probability. This ultimately leads to significant reduction of network throughput [2]. To reduce the level of contention among the competing STAs in a dense WLAN scenario, IEEE 802.11ah has specified a novel channel access scheme that utilizes the notion of restricted access window (RAW). The RAW duration consists of many number of RAW slots and

during each RAW slot, a group of STAs will attempt to get channel access by using the DCF/EDCA protocol. During this time slot, the remaining STAs belonging to other groups remain idle. Within each beacon interval, multiple RAWs are present as can be seen in Fig. 1, where each RAW consists of many RAW slots. By limiting the channel contention to relatively lower number of STAs during a particular RAW slot, the RAW-based channel access scheme supported by 802.11ah MAC is anticipated to significantly enhance the channel access efficiency and scalability of IEEE 802.11ah dense WLANs. Further, at the PHY layer, 802.11ah has specified the use of various modulation coding schemes (MCS) such that the STAs can operate at distinct data rates depending on the applications handled; furthermore, 802.11ah standard specifies that each STA has to support rate adaptation as well [1]. In the legacy WLANs, where STAs use DCF/EDCA protocol, all the STAs within the cell (i.e., basic service set) are allowed to access the medium simultaneously. Each STA will choose a backoff (BO) counter, whose initial value is uniformly distributed in the interval (0, 𝑊0 − 1). The BO counter will be decremented whenever the channel is sensed as idle and would be frozen when the channel is sensed as busy.

∗ Corresponding author. E-mail addresses: [email protected] (U. Sangeetha), [email protected] (A.V. Babu).

https://doi.org/10.1016/j.comcom.2019.12.043 Received 7 September 2019; Received in revised form 15 November 2019; Accepted 23 December 2019 Available online 7 January 2020 0140-3664/© 2020 Elsevier B.V. All rights reserved.

U. Sangeetha and A.V. Babu

Computer Communications 151 (2020) 154–164

with the help of simulation studies. An analytical model for finding the non-saturation/saturation throughput of 802.11ah WLAN under the RAW-based channel access mechanism has been presented in [8], assuming that STAs use homogeneous data rates. Here the authors assume that the STAs, which defer their transmission at the end of a given RAW slot (i.e., due to non-availability of sufficient time for completing the frame transmission in the current RAW slot) will renew their BO parameters (i.e; BO counter value and BO stage when STA is deferred) at the beginning of the next designated RAW slot. In [9], the authors describe an analytical model to evaluate the MAC layer performance of differentiated services in IEEE 802.11ah WLAN under the RAW-based scheme. In [10], the authors describe an analytical model to evaluate the throughput of IEEE 802.11ah WLAN under the RAW mechanism and present detailed results for idle probability, backoff time and frame transmission probability. In these papers, all the STAs in the WLAN are assumed to operate at the same data rate. Recently, several authors have proposed efficient schemes for forming groups, (i.e; STA groups) in IEEE 802.11ah WLAN to improve the network performance [11–15]. In [11], the authors propose an algorithm for finding the optimal RAW parameters for forming STA groups, considering heterogeneous traffic characteristics at each node. In [12], the authors propose a traffic-adaptive RAW optimization algorithm for real time STA grouping. Here the RAW parameters such as the number of groups, the number of STAs per group etc., are adapted according to the current traffic conditions. The authors of [13] propose an efficient algorithm, which can be used to adapt the RAW parameters for improving the energy efficiency of uplink communications in 802.11ah WLANs. In [14], the authors propose a load-balanced STA grouping algorithm for improving the channel utilization efficiency of each group of STAs in 802.11ah WLANs. A sector-based STA grouping scheme has been proposed in [15] to improve the efficiency of channel access in IEEE 802.11ah based IoT networks. Apart from the above papers, some researchers have focused on proposing modifications for the legacy 802.11ah MAC protocol. In [16], the authors have proposed a Quality of Service (QoS)-aware priority grouping and RAW scheduling algorithm for IEEE 802.11ah based networks. Hybrid, contention reservation MAC protocol has been proposed in [17] to realize energy efficient uplink communication in IEEE 802.11ah WLANs. In [18], the authors study an enhancement for the RAW scheme by including a reservation slot to ensure that the QoS requirements of the applications are satisfied. Detailed literature survey has shown that the existing research papers on the performance evaluation of IEEE 802.11ah WLAN under the RAW mechanism assume that all the competing STAs use the same data rate for transmitting the frames. The STAs may use distinct data rates either due to the implementation of suitable rate adaptation protocols at the data link layer or depending on the application handled by the STA. In such scenarios, a group will contain STAs operating at distinct data rates. The objective of this work is to develop an analytical model for finding the throughput performance of IEEE 802.11ah WLAN when STAs employ distinct data rates for frame transmission under the RAWbased medium sharing approach. First of all, we assume that the STAs are grouped randomly, i.e., grouping is done by randomly picking the STAs. In this case, a group may contain STAs operating at distinct data rates. During the RAW slot assigned to this group, the STAs within the group will contend together for getting the transmission opportunity. In this scenario, the STAs operating at lower data rates will occupy the medium for a longer time as compared to STAs operating at higher data rates. This degrades the overall throughput of the group as well as that of the network. Further, the throughput of the high data rate STA is significantly reduced and becomes almost equal to that of the low data rate STAs (assuming that all of them use same MAC parameters). To resolve this performance anomaly problem, we consider data rate based grouping, i.e., STAs using the same data rate are grouped together. Here, all the STAs within a given group will be operating at the same data rate. They all will compete together for getting the

Fig. 1. RAW based channel access scheme [1].

The duration of time between two consecutive decrements of the BO counter is a random variable as it may contain an idle slot, a successful transmission slot or a collision slot. Once the BO counter value becomes zero, the STA is allowed to transmit the frame over the medium. If the frame is received correctly, the destination STA transmits an ACK frame. Upon receiving the ACK, the sender initiates a new transmission procedure for the next frame in the MAC layer queue. Non-receipt of ACK frame leads to STA repeating the frame transmission attempts till the retry limit is reached, after which the frame is discarded [2]. Compared to the legacy DCF/EDCA protocol employed in 802.11 WLANs, the RAW mechanism specified by 802.11ah has certain distinct features. Under the RAW-based channel access mechanism, the STAs in 802.11ah WLAN that use DCF/EDCA protocol, are restricted to access the medium during the RAW slot allotted to them. Accordingly, the STA within a group that undergoes the BO counter decrement process during the designated RAW slot, has to verify that sufficient time is available in the current RAW slot for completing the frame transmission [1]. If the residual time in the current RAW slot is not enough, the STAs have to defer their attempt and wait till the next designated RAW slot arrives. In this case, the STAs will reset the current value of the BO counter and restarts it at the beginning of the next designated RAW slot. In particular, when the next designated RAW slot begins, the STA creates a new BO function and initializes the BO counter with a random integer value drawn from the uniform distribution over the interval (0, 𝑊0 − 1). Further, 802.11ah specifies two methods to handle the handover of RAW slots among two distinct groups of STAs. With non-cross slot boundary (i.e., NCSB), frame transmission of the STAs are not allowed to cross the boundary of the designated RAW slot allotted to them during a given slot, while for the cross slot boundary case (i.e., CSB), an on-going transmission is allowed to cross the RAW slot boundary; but STAs in the group cannot start a new transmission in the RAW slot allocated to the other group [1]. Owing to these important differences, it is very essential to develop an analytical model to study the performance of 802.11ah WLANs, where STAs having heterogeneous data rates use RAW-based channel access scheme. 1.1. Related work Recently, a few authors have made attempts to evaluate the effectiveness of RAW-based channel access scheme through analytical/simulation studies, in the context of IEEE 802.11ah WLANs [3– 10]. In [3] the authors evaluate the saturation throughput of WLAN under RAW-based channel access scheme with NCSB, while in [4], the authors propose that the RAW size need to be selected according to the group size for improving the network throughput. In [5], the authors propose an algorithm to find the optimal size for the RAW slot as a function of network size. In [6], the authors describe a mean value based analysis for assessing the performance of group synchronized DCF in the context of 802.11ah WLAN. The authors of [7] have evaluated the impact of RAW size on network performance 155

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Computer Communications 151 (2020) 154–164

transmission opportunity during the RAW slot assigned to them. The proposed strategy can ensure equal channel access opportunity for all the STAs in the network, irrespective of their data rates. With the help of analytical and simulation studies, we establish that the aggregate throughput of the network can be significantly improved under the data rate based grouping as compared to the random grouping method. 1.2. Contributions The main contributions of the paper are outlined as follows: • We develop an analytical model to evaluate the saturation throughput of IEEE 802.11ah WLAN under the RAW-based channel access scheme assuming non-cross slot boundary case. The analytical model is a modification of the Markov chain based models of [2,19] which takes into account: (i) STAs operating at distinct data rates within a group and (ii) the distinct features of 802.11ah RAW-based MAC protocol. Here, we consider both random grouping as well as data rate based grouping. With the help of analytical and simulation studies, we establish that the aggregate throughput of the network can be improved under data rate grouping as compared to the random grouping. Further, it is established that data rate grouping improves the throughput of high data rate STAs, without penalizing the low data rate STAs. • We present an algorithm that describes the procedure for realizing data rate based grouping in 802.11ah WLAN. • We use the Jain’s fairness index (JFI) to evaluate the fairness in resource allocation among the STAs operating at distinct data rates. Under the conventional random grouping method, STAs within a given group operating at distinct data rates compete against each other for getting the channel access. Due to this, they all achieve the same throughput. Thus random grouping provides highly unfair resource allocation among STAs of distinct data rates. With the help of JFI, we establish that the data rate based grouping ensures fair resource allocation among STAs operating at distinct data rates. The rest of the paper is organized as follows: Sections 2 and 3 describe analytical models for network throughput under random grouping and data rate based grouping respectively. The algorithm for data rate based grouping is described in Section 3. The numerical and simulation results are described in Section 4. The paper is concluded in Section 5.

Xiao [20] with appropriate modifications to suit the distinct features of 802.11ah MAC protocol. At time t, the BO stage and the BO counter of the tagged STA are represented as {𝑆𝑖(𝑚) (𝑡)} and {𝐵𝑖(𝑚) (𝑡)} respectively. Assuming the frame collision to be Bernoulli, {𝑆𝑖(𝑚) (𝑡)𝐵𝑖(𝑚) (𝑡)} can be modelled as a discrete time Markov Chain (DTMC) [2]; Fig. 2. shows the state transitions associated with the DTMC. Let 𝑊𝑖,0 be the min(𝑚) imum contention window of group 𝑖 STAs; further let 𝑝(𝑚) 𝑐,𝑖 and 𝜏𝑖 respectively are the collision and transmission attempt probabilities of the tagged group 𝑖 STA operating at data rate 𝑅𝑖(𝑚) . We assume that NCSB is employed; thus group i STAs are allowed to access the medium for duration 𝑇𝑖 ; however they are not allowed to cross the RAW slot boundary. When DCF protocol is used, a competing STA will transmit the frame when its BO counter becomes zero. However, in 802.11ah supported RAW-based scheme, a STA intending to transmit a frame, after counting down to zero, must ensure that the remaining time available in the current RAW slot is sufficient for a frame transaction with the AP. Now the RAW slot duration for group i, 𝑇𝑖 is given by [1]. 𝑇𝑖 = 𝑇𝑎,𝑖 + 𝑇ℎ + 𝑇𝑔

(1)

where 𝑇𝑎,𝑖 is the free access period; 𝑇ℎ and 𝑇𝑔 represent the holding time and the guard time respectively. The guard time ensures that a frame transmission initiated during the current RAW slot 𝑇𝑖 does not cross the RAW slot boundary. During the holding period, the STAs within the group remain idle by freezing their BO counters, knowing that the current RAW slot will elapse sooner [1]. STA sense the The probability that the tagged group i, rate 𝑅(𝑚) 𝑖 ) is given medium as busy during the BO decrementing procedure (𝑝(𝑚) 𝑏,𝑖 by ( )𝑛(𝑢) ( )𝑛(𝑚) −1 (𝑚) 𝑀 𝑝𝑏,𝑖 = 1 − 𝛱𝑢=1 1 − 𝜏𝑖(𝑢) 𝑖 1 − 𝜏𝑖(𝑚) 𝑖

(2)

𝑢≠𝑚

The BO counter is frozen with probability (𝑝(𝑚) ) either when the 𝑓 ,𝑖 ) or medium is observed to be busy (that occurs with probability 𝑝(𝑚) 𝑏,𝑖 when the access time of the designated RAW slot (𝑇𝑎,𝑖 ) has expired (with probability equal to unity), and is given by ( ( ) ) 𝑇𝑎,𝑖 (𝑚) 𝑇𝑅 − 𝑇𝑎,𝑖 = 𝑝𝑓(𝑚) + 𝑝 (3) ,𝑖 𝑏,𝑖 𝑇𝑅 𝑇𝑅 As seen in Fig. 2, the decrementing of the BO counter occurs with ), when the BO counter value y lies in the range the probability (1 − 𝑝(𝑚) 𝑓 ,𝑖 (2, 𝑊𝑖𝑥−1 ); 𝑥 = 1, ..𝐿. However, after counting down to unity, the BO counter gets decremented to zero if and only if the tagged STA finds the residual time in the current RAW slot to be sufficient enough for a frame transaction with the AP. Otherwise, the STA enters a defer state (𝑚) as shown in Fig. 2. The STA will 𝐷𝑥 𝑥 ∈ (0, 𝐿) with probability 𝑝0,𝑖

2. Analytical model for network throughput under random grouping of stations

continue to be in the defer state 𝐷𝑥 𝑥 ∈ (0, 𝐿) with probability 𝑝(𝑚) . 1,𝑖 However, as soon as the next designated RAW slot for group 𝑖 STAs arrive, the tagged STA resets the current BO counter to zero and starts a new BO counter decrementing cycle. For this, it selects the BO counter value uniformly within the range [0, 𝑊𝑖,0 − 1] as shown in Fig. 2. Let us assume that the remaining time available in the current RAW slot is uniformly distributed over (0, 𝑇𝑎,𝑖 ). The defer probability 𝑝(𝑚) is 0,𝑖 the probability that the residual time in the current RAW slot is not sufficient for a successful frame transmission from rate 𝑅(𝑚) STA. Thus 𝑖

In this section, we present an analytical model to find the throughput of IEEE 802.11ah WLAN with STAs operating with heterogeneous data rates under the RAW-based scheme. Consider a fully connected WLAN that operates according to 802.11ah specifications with N number of STAs uniformly distributed around the access point (AP). Here we assume that the STAs are grouped randomly without considering parameters such as the data rates. Further, assume that no hidden STAs are present in the network and a centralized grouping strategy has been followed with 𝑁 STAs divided into 𝐾 groups; there are 𝑛𝑖 STAs per 𝐾 𝑛 = 𝑁. Let 𝑇 be the duration of one RAW, which group such that 𝛴𝑖=1 𝑖 𝑅 is divided into 𝐾 RAW slots. Assume that duration of the RAW slot assigned to group i STAs is equal to 𝑇𝑖 . Each STA in group 𝑖 can operate (𝑚) at distinct data rates 𝑅(𝑚) STAs operating at 𝑖 , 𝑚 = 1, … , 𝑀; there are 𝑛𝑖 (𝑚) (𝑚) 𝑀 rate 𝑅𝑖 bits per sec (bps) such that 𝛴𝑚=1 𝑛𝑖 = 𝑛𝑖 . The STAs belonging to group 𝑖 contend for channel access during 𝑇𝑖 by following the DCF protocol. To evaluate the throughput of the groups and that of the network, we consider a tagged group 𝑖 STA operating with data rate 𝑅(𝑚) 𝑖 (𝑖 = 1, 2, … , 𝐾; 𝑚 = 1, 2, ..𝑀). To develop the analytical model for throughput under RAW mechanism, we rely on the model proposed by Y.

𝑇

(𝑚)

(𝑚) we set 𝑝(𝑚) = 𝑇𝑠 ; 𝑇𝑠(𝑚) ≤ 𝑇𝑎,𝑖 and 𝑝0,𝑖 = 1; 𝑇𝑠(𝑚) > 𝑇𝑎,𝑖 ; where 𝑇𝑠(𝑚) 0,𝑖 𝑎,𝑖 is the time required for the successful transmission of a frame from rate 𝑅(𝑚) STA. Thus the probability of BO counter decrementing from 𝑖 unity to zero is (1 − 𝑝(𝑚) )(1 − 𝑝(𝑚) ) ≊ 1 − 𝑝(𝑚) − 𝑝(𝑚) , as shown in Fig. 2. 𝑓 ,𝑖 𝑓 ,𝑖 0,𝑖 0,𝑖 The corresponding one-step transition probabilities are represented as follows: { (1 − 𝑝𝑓(𝑚) ); 𝑦 ∈ [2, 𝑊𝑖,𝑥 − 1], 𝑥 ∈ [0, 𝐿] ,𝑖 𝑃 (𝑥, 𝑦 − 1|𝑥, 𝑦) = (4) (𝑚) (1 − 𝑝𝑓 ,𝑖 − 𝑝(𝑚) ); 𝑦 = 1 0,𝑖

where 𝑊𝑖,𝑥 = 2𝑥 𝑊𝑖,0 , 𝑥 = 1, 2, … , 𝐿 − 1 with 𝐿 selected such that 𝑊𝑖,𝑚𝑎𝑥 = 2𝐿 𝑊𝑖,0 . Further, we set 𝐿 as the retry limit as well. Now 156

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Computer Communications 151 (2020) 154–164

Fig. 2. Frame transmission procedure under DCF: Markov Chain model for group 𝑖, rate 𝑅(𝑚) STA. 𝑖

= 1− referring to Fig. 2, 𝑝(𝑚) 1,𝑖 duration of one RAW.

𝑇𝑖 𝑇𝑅

= and 𝑝(𝑚) 2,𝑖

𝑇𝑖 , 𝑇𝑅

where 𝑇𝑅 is the total

While (5a) represents the BO counter freezing probability, (5b) is the probability of tagged STA entering the next BO stage after it encounters a collision, and (5c) is the probability that the STA enters the BO stage 0 after a successful transmission. Further, (5d)–(5f) represent the transitions associated with the deferring of STA, while in BO stage 𝑥. Here (5d) represents the transition associated with deferring of STA, i.e; the STA after counting down to unity, checks for the availability of sufficient residual time in the current RAW slot for completing the transmission. If enough time is available and the channel is also sensed as idle, it will count down to zero so that STA can transmit immediately, before another STA occupies the channel. If sufficient time is not available, the STA enters the defer state with probability 𝑝(𝑚) . Furthermore, (5e) represents the probability that the tagged STA 0,𝑖 remains in the defer state and (5f) represents the probability that the STA, on finding that the next designated RAW slot for the group to which it belongs has arrived, resets the BO counter and begins a new BO counter decrementing cycle. After resetting, it selects the BO counter value to be uniformly in the range (0, 𝑊𝑖,0 − 1).

According to the DCF protocol, the tagged STA will enter the transmission state once the BO counter becomes zero. Successful transmission is inferred from the reception of ACK frame at the STA. In case collision happens, the STA enters the next BO stage and restarts the BO counter decrementing process. If the transmission is unsuccessful even after 𝐿 attempts, the frame is dropped from the MAC queue. More details of the DCF protocol can be seen in [2]. Various other one-step transition probabilities are as given below. 𝑃 (𝑥, 𝑦|𝑥, 𝑦) = 𝑝(𝑚) ; 𝑓 ,𝑖 𝑃 (𝑥, 𝑦|𝑥 − 1, 0) = {

𝑦 ∈ [0, 𝑊𝑖,𝑥 − 1], 𝑝(𝑚) 𝑐,𝑖

𝑊𝑖,𝑥

;

𝑥 ∈ [0, 𝐿]

𝑦 ∈ [0, 𝑊𝑖,𝑥 − 1],

𝑥 ∈ [1, 𝐿]

(5a) (5b)

(𝑚)

(1−𝑝𝑐,𝑖 ) 𝑊𝑖,0

𝑃 (0, 𝑦|𝑥, 0) =

;

1 ; 𝑊𝑖,0

𝑦 ∈ [0, 𝑊𝑖,0 − 1],

𝑥 ∈ [0, 𝐿 − 1]

𝑦 ∈ [0, 𝑊𝑖,0 − 1],

𝑥=𝐿

(5c)

𝑃 (𝐷𝑥 |𝑥, 1) = 𝑝(𝑚) , 0,𝑖

𝑥 ∈ [0, 𝐿]

(5d)

𝑝(𝑚) , 1,𝑖 (𝑚) 𝑝2,𝑖

𝑥 ∈ [0, 𝐿]

(5e)

𝑃 (𝐷𝑥 |𝐷𝑥 ) = 𝑃 (0, 𝑦|𝐷𝑥 ) =

𝑊𝑖,0

,

𝑥 ∈ [0, 𝐿], 𝑦 = (0, 𝑊𝑖,0−1 )

2.1. Finding the transmission attempt probability Let 𝜋𝑖(𝑚) (𝑥, 𝑦) = lim𝑡→∞ 𝑃 {𝑆𝑖(𝑚) (𝑡) = 𝑥, 𝐵𝑖(𝑚) (𝑡) = 𝑦} be the steady state probability distribution of the DTMC corresponding to tagged group i rate 𝑅𝑖(𝑚) STA. Further, let 𝜋𝑖(𝑚) (𝐷𝑥 ) be the steady state probability of

(5f)

157

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The conditional probability of successful transmission of tagged STA is

the tagged STA entering the defer stage 𝐷𝑥 ; (𝑥 = 0, ..𝐿). The following relations can be obtained from Fig. 2. 𝑥 (𝑚) 𝜋𝑖(𝑚) (𝑥, 0) = (𝑝(𝑚) 𝑥 ∈ (1, 𝐿) 𝑐,𝑖 ) 𝜋𝑖 (0, 0); ) (𝑚) 𝑥 (𝑚) ( (𝑝 ) 𝜋𝑖 (0, 0) 𝑊𝑖,𝑥 − 1 𝑐,𝑖 𝜋𝑖(𝑚) (𝑥, 𝑦) = ; 𝑊𝑖,𝑥 1 − 𝑝(𝑚) 𝑓 ,𝑖

( )𝑛(𝑚) −1 ( )𝑛(𝑢) 𝑖 (𝑚) 𝑀 1 − 𝜏 (𝑢) 𝑖 𝑛(𝑚) 1 − 𝜏𝑖(𝑚) 𝛱𝑢=1 𝑖 𝜏𝑖 𝑖

(6a)

𝐿 ( 1 − 𝑝(𝑚) ) ∑ 𝑐,𝑖

𝑊𝑖,0

𝜋𝑖(𝑚) (𝑥, 0) +

𝑥=0

𝐿 𝑝(𝑚) 2,𝑖 ∑

𝑊𝑖,0

𝐿 (𝑊∑ 𝑖,𝑥 −1) ∑ 𝑥=0

𝜋𝑖(𝑚) (𝑥, 𝑦) +

𝑘=2

𝐿 ∑ 1 ∑

( )𝑛(𝑚) (𝑚) 𝑖 𝑝(𝑚) 𝑡𝑟,𝑖 = 1 − 1 − 𝜏𝑖

𝑥=0

𝑥 ∈ [0, 𝐿]

𝑥=0 𝑦=0

𝐿 ∑

𝜋𝑖(𝑚) (𝐷𝑥 ) = 1

2.2. Finding the network throughput

(6d)

Under random grouping (RG), the throughput of rate 𝑅(𝑚) STAs be𝑖 longing to group 𝑖 is the average number of bits successfully transferred per second and is calculated as follows: ] [ (𝑚) 𝐸[𝑋] 𝑝𝑠,𝑖 𝑝(𝑚) 𝑡𝑟,𝑖 𝑈𝑖(𝑚),𝑅𝐺 = (12) 𝐸[𝜎]

(6e)

(6f)

𝑀 𝑈 (𝑚) The total throughput of group 𝑖 STAs is given by 𝑈𝑖𝑅𝐺 = 𝛴𝑚=1 𝑖 𝑅𝐺 ) and the aggregate network throughput under random grouping (𝑈𝑡𝑜𝑡 is given as

𝑥=0

Notice that (6f) represents the normalization condition for steady state probabilities of the DTMC. Now 𝜋𝑖(𝑚) (0, 0) can be obtained as follows, after combining and simplifying (6a)–(6f): 𝜋𝑖(𝑚) (0, 0)

1 = 𝐴1 + 𝐴2 + 𝐴3 + 𝐴4 + 𝐴5

𝑅𝐺 𝑈𝑡𝑜𝑡 =

(7a)

(7b) 1 − 𝑝(𝑚) 𝑐,𝑖 [ (1 − 𝑝(𝑚) )𝐿+1 𝑊 2𝐿 (2 − 𝑝(𝑚) ) − [(1 − 𝑝(𝑚) )(2𝐿+1 − (𝑝(𝑚) )𝐿+1 )] ] 𝑖,0 𝑐,𝑖 𝑐,𝑖 𝑐,𝑖 𝑐,𝑖 (𝑚) 𝐿 (1 − 𝑝(𝑚) 𝑐,𝑖 )𝑊𝑖,0 2 (2 − 𝑝𝑐,𝑖 )

(2) (1) (1) (2) (2) (2) (1) (1) (1) 𝐸[𝜎] = (1 − 𝑝𝑡𝑟,𝑖 )𝜎0 + 𝑝(1) 𝑡𝑟,𝑖 𝑝𝑠,𝑖 𝑇𝑠 + 𝑝𝑡𝑟,𝑖 𝑝𝑠,𝑖 𝑇𝑠 + 𝑝𝑡𝑟,𝑖 (1 − 𝑝𝑠,𝑖 )(1 − 𝑝𝑡𝑟,𝑖 )𝑇𝑐

(7c)

𝐴3 =

𝐴4 =

𝐴5 =

[

(1) (2) (1) (2) (1) (2) (2) +𝑝(2) 𝑡𝑟,𝑖 (1 − 𝑝𝑠,𝑖 )(1 − 𝑝𝑡𝑟,𝑖 )𝑇𝑐 + 𝑝𝑡𝑟,𝑖 𝑝𝑡𝑟,𝑖 𝑚𝑎𝑥(𝑇𝑐 , 𝑇𝑐 )

(14)

[ (1 − 2𝑝(𝑚) )𝐿+1

3𝑝(𝑚) 𝑐,𝑖 𝑐,𝑖 𝑊𝑖,0 2𝑝(𝑚) 𝐵1 − 𝑐,𝑖 (𝑚) (𝑚) 2 1 − 𝑝𝑓 ,𝑖 1 − 2𝑝𝑐,𝑖 ( 2𝐿+1 − (𝑝(𝑚) )𝐿+1 )]] 𝑝(𝑚) 𝑐,𝑖 𝑐,𝑖 (7d) + 𝐿 2 𝑊𝑖,0 2 − 𝑝(𝑚) 𝑐,𝑖 𝐿+1 𝑝(𝑚) 𝛽 [ (1 − 𝑝(𝑚) 𝑐,𝑖 ) 0,𝑖 0 + 𝑝(𝑚) 𝑐,𝑖 𝐴1 𝑊𝑖,0 𝑊𝑖,0 (𝑚) 𝐿 (𝑚) 𝐿 ] 𝛽0 (𝑊𝑖,0 − 1) 1 𝑝𝑐,𝑖 (2 − (𝑝𝑐,𝑖 ) ) − ( )+ (7e) 2 𝑊𝑖,0 𝐿 𝑊𝑖,0 (2 − 𝑝(𝑚) 𝑐,𝑖 )2 (𝑚) 𝐿 [ [ ( 2𝐿 − (𝑝(𝑚) )𝐿 )]] 𝑝(𝑚) 𝑝(𝑚) 𝑊𝑖,0 − 1 𝑐,𝑖 𝑐,𝑖 (1 − (𝑝𝑐,𝑖 ) ) 𝑐,𝑖 1 𝑝(𝑚) 𝛽 ( )+ − 0,𝑖 0 (𝑚) (𝑚) 𝑊𝑖,0 𝑊𝑖,0 2𝐿 (2 − 𝑝(𝑚) ) 𝑝2,𝑖 1 − 𝑝𝑐,𝑖 𝑐,𝑖 1

where (1 − 𝑝𝑡𝑟,𝑖 ) is the probability that the medium is empty, i.e.; there are no transmissions from STAs belonging to group 𝑖, which occurs with (𝑢) 𝑀 (1 − 𝜏 (𝑢) )𝑛𝑖 . Further, 𝑇 (𝑚) and 𝑇 (𝑚) , (𝑚 = 1, 2) are given probability 𝛱𝑢=1 𝑠 𝑐 𝑖 by (𝑚) 𝑇𝑠(𝑚) = 𝑇𝑅𝑇 𝑆 + 3𝑇𝑆𝐼𝐹 𝑆 + 𝑇𝐶𝑇 𝑆 + 𝑇𝐻(𝑚) + 𝑇𝐸[𝑋] + 𝑇𝐴𝐶𝐾 + 3𝛿

𝑇𝑐(𝑚) = 𝑇𝑅𝑇 𝑆 + 𝑇𝐷𝐼𝐹 𝑆 + 𝛿

Here 𝛽0 = 1∕(1 − 𝑝(𝑚) − 𝑝(𝑚) ) and the transmission attempt probability 𝑓 ,𝑖 0,𝑖 for group 𝑖, rate 𝑅(𝑚) STA is given by 𝑖 𝐿 ∑ 𝑥=0

𝜋𝑖(𝑚) (𝑥, 0) =

𝐿+1 1 − (𝑝(𝑚) 𝑐,𝑖 )

1 − 𝑝(𝑚) 𝑐,𝑖

𝜋𝑖(𝑚) (0, 0)

(15)

(𝑚) Here 𝛿 is the propagation delay; 𝑇𝑅𝑇 𝑆 , 𝑇𝐶𝑇 𝑆 , 𝑇𝐻(𝑚) , 𝑇𝐸[𝑋] and 𝑇𝐴𝐶𝐾 are the time durations corresponding to the transmission of RTS, CTS, frame header, frame payload and ACK respectively; 𝑇𝐷𝐼𝐹 𝑆 , 𝑇𝑆𝐼𝐹 𝑆 are the durations of SIFS and DIFS time periods. To find the throughput using (12) and (13), we need to determine various probabilities. First (𝑚) of all, 𝜏𝑖(𝑚) , 𝑝(𝑚) 𝑐,𝑖 and 𝑝𝑠,𝑖 , which are given by (7)–(10), along with relevant equations for 𝑝𝑓(𝑚) , 𝑝(𝑚) , 𝑝(𝑚) , 𝑝(𝑚) and 𝑝(𝑚) form a system ,𝑖 𝑏,𝑖 0,𝑖 1,𝑖 2,𝑖 of non-linear equations that can be numerically solved to find the (𝑚) (𝑚) probabilities 𝜏𝑖(𝑚) , 𝑝(𝑚) 𝑐,𝑖 , 𝑝𝑡𝑟,𝑖 , 𝑝𝑠,𝑖 etc., which are required for finding the throughput. We use the Levenberg–Marquardt algorithm [21,22], which is a combination of steepest descent and Gauss–Newton method to solve the system of linear equations. After finding these probabilities, the throughput can be determined using (12)–(15).

(7f)

𝜏𝑖(𝑚) =

(13)

𝑇𝑅

In (12), 𝐸[𝑋] is the average frame size and 𝐸[𝜎] is the average duration of the slot time, which is computed as follows. Let 𝜎0 be the duration of the empty slot time; 𝑇𝑠(𝑚) represents the average time the channel is sensed busy due to successful transmission of rate 𝑅𝑖(𝑚) STA; 𝑇𝑐(𝑚) be the average time the channel is sensed as busy due to transmission failure of rate 𝑅(𝑚) STA. Assuming that STAs within group 𝑖 can operate at two 𝑖 (2) distinct data rates 𝑅(1) 𝑖 and 𝑅𝑖 (bps), 𝐸[𝜎] is given by

𝐿+1 (1 − 𝑝(𝑚) 𝑐,𝑖 )

𝐴2 = 𝛽0

𝐾 ∑ 𝑈𝑖𝑅𝐺 𝑇𝑖 𝑖=1

where parameters 𝐴1 –𝐴5 are defined as follows: 𝐴1 =

(11)

(6c)

0,𝑖

𝜋𝑖(𝑚) (𝑥, 𝑦) +

(10)

(𝑚) where 𝑝(𝑚) 𝑡𝑟,𝑖 is the probability that at least one STA with rate 𝑅𝑖 belonging to group 𝑖 transmits in a given slot time, which is given by

𝜋𝑖(𝑚) (𝐷𝑥 );

𝑦 ∈ (0, 𝑊𝑖,0 − 1) ( ) (𝑚) (𝑚) 𝑊𝑖,𝑥 − 1 (𝑝𝑐,𝑖 )𝑥 𝜋𝑖 (0, 0) 𝜋𝑖(𝑚) (𝑥, 1) = ; 𝑥 ∈ (0, 𝐿) 𝑊𝑖,𝑥 1 − 𝑝(𝑚) − 𝑝(𝑚) 𝑓 ,𝑖 0,𝑖 ) (𝑚) (𝑚) ( 𝑊𝑖,𝑥 − 1 (𝑝𝑐,𝑖 )𝑥 𝜋𝑖 (0, 0) ( (𝑚) (𝑚) ) 𝑝0,𝑖 ∕(𝑝2,𝑖 ) ; 𝜋𝑖(𝑚) (𝐷𝑥 ) = 𝑊𝑖,𝑥 1 − 𝑝(𝑚) − 𝑝(𝑚) 𝑓 ,𝑖

𝑝(𝑚) 𝑡𝑟,𝑖

𝑥 ∈ (0, 𝐿), 𝑦 = (2, 𝑊𝑖,𝑥 − 1) (6b)

𝜋𝑖(𝑚) (0, 𝑦) =

𝑢≠𝑚

𝑝(𝑚) 𝑠,𝑖 =

(8)

The probability that the tagged group 𝑖, rate 𝑅(𝑚) STA suffers collision 𝑖 is [ ] (𝑢) (𝑚) (𝑢) ( ( (𝑢) )𝑛𝑖 ( (𝑚) (𝑚) )𝑛𝑖 −1 𝑀 (𝑢) )𝑛𝑖 𝑚−1 𝑝𝑐,𝑖 = 1 − 𝛱𝑢=1 1 − 𝜏𝑖 1 − 𝜏𝑖 𝛱𝑢=𝑚+1 1 − 𝜏𝑖 (9)

3. Analytical model for network throughput under data rate based grouping In this section, we present a simple algorithm for realizing the data rate based grouping in IEEE 802.11ah WLAN. After this, we present an 158

U. Sangeetha and A.V. Babu

Computer Communications 151 (2020) 154–164

analytical model to find the throughput of the group as well as that of the network under data rate based grouping, based on the analytical model described in Section 2. Here the STAs operating at the same data rate are grouped together.

𝑝(𝑚) 𝑠 =

Here, we present an algorithm for data rate based grouping in 802.11ah WLAN. Algorithm 1 describes the procedure for implementing data rate based grouping. During the network initialization phase, the AP broadcasts beacon frames with information elements. Association of the STAs with the AP is carried out by listening to the beacon information. The PLCP (i.e; PHY layer convergence protocol) header of the association request frame includes information about the data rates of the STAs. Thus the AP collects the information regarding the data rates of the STAs, and thus it can form groups according to the data rate. Define 𝐴(𝑚) as an array associated with rate 𝑅(𝑚) STAs. A tagged STA is appended to this array if it transmits at rate 𝑅(𝑚) . Assume that M such arrays are formed corresponding to rates 𝑅(𝑚) ; 𝑚 = 1, ..𝑀. Once the association procedure is over, the AP assigns 13-bit association identifier (AID) to every STA within group 𝑚(𝑚 = 1, … , 𝑀), where the last ⌈(𝑙𝑜𝑔2 𝑀)⌉ LSB bits uniquely represent the ID of group 𝑚. After the procedure is finished, the AP transmits RAW parameter set (RPS) down streams, which include information such as the number of groups and the corresponding data rates. Notice that the proposed procedure for data rate based grouping can be realized without any major modifications to IEEE 802.11ah specifications.

𝜋 (𝑚) (0, 0) =

𝐵1 =

(20a)

𝐿+1 (1 − 𝑝(𝑚) 𝑐 )

𝐵2 = 𝛼0

(20b) 1 − 𝑝𝑐(𝑚) [ 𝐿+1 𝑊 2(𝐿) (2 − 𝑝(𝑚) ) − [(1 − 𝑝(𝑚) )(2𝐿+1 ) − (𝑝(𝑚) )𝐿+1 ] ] (1 − 𝑝(𝑚) 𝑐 ) 𝑐 𝑐 𝑐 0 (𝑚) 𝐿 (1 − 𝑝(𝑚) 𝑐 )𝑊0 2 (2 − 𝑝𝑐 )

(20c)

𝐵3 =

[

[

𝐿+1 (1 − 2𝑝(𝑚) 𝑐 ) − (𝑚) (𝑚) 1 − 𝑝𝑓 1 − 2𝑝𝑐 ( 𝐿+1 𝐿+1 )]] 𝑝(𝑚) (2 ) − (𝑝(𝑚) 𝑐 ) + 𝐿𝑐 2 𝑊0 2 − 𝑝(𝑚) 𝑐

1

𝑊0 2𝑝𝑐(𝑚)

3𝑝𝑐(𝑚) 𝐵1 2

[ (𝑚) (1 − 𝑝𝑐(𝑚) )𝐿+1 𝑝0 𝛼0 (𝑚) + 𝑝𝑐 𝐵1 𝑊0 𝑊0 ( (𝑚) 𝐿 ] (𝑚) 𝐿 ) 𝛼 (𝑊 − 1) 1 𝑝𝑐 (2 − (𝑝𝑐 ) ) + 0 0 − 𝑊0 𝑊02 (2 − 𝑝𝑐(𝑚) )2𝐿 [( ) [ (𝑚) 𝐿 𝑊0 − 1 𝑝 (1 − (𝑝(𝑚) 1 𝑐 ) ) 𝐵5 = 𝑝(𝑚) 𝛼0 + 𝑐 0 𝑊0 𝑝(𝑚) 1 − 𝑝(𝑚) 𝑐 2 (𝑚) ( 𝐿 (𝑚) 𝐿 )]] 𝑝𝑐 2 − (𝑝𝑐 ) − 𝑊0 2𝐿 (2 − 𝑝(𝑚) ) 𝑐

(20d)

𝐵4 =

(20e)

(20f)

Here 𝛼0 = 1∕(1 − 𝑝(𝑚) − 𝑝(𝑚) ) and 𝑝(𝑚) , 𝑝(𝑚) , 𝑝2(𝑚) are the probabilities 𝑓 0 0 1 associated with the deferring of STAs belonging to group 𝑚, when the BO counter values reach unity, as described in Section 2. The transmission attempt probability of group 𝑚 STA can be determined as follows:

(16)

𝜏 (𝑚) =

𝑝(𝑚) 𝑡𝑟

𝐿 ∑ 𝑥=0

where is the probability that at least one STA within group 𝑚 transmit the frame and 𝑝(𝑚) is the probability that this transmission is 𝑠 (𝑚) successful. Now 𝑝(𝑚) and 𝑝 are given by 𝑠 𝑡𝑟 (𝑚)

1 𝐵1 + 𝐵2 + 𝐵3 + 𝐵4 + 𝐵5

where the parameters 𝐵1 to 𝐵5 are defined as:

Here, we describe an analytical model to find the throughput of the network under data rate based grouping strategy. Consider 802.11ah WLAN, where STAs are configured to operate at distinct data rates. Assume that the STAs in the network can operate at M distinct data rates, 𝑅(𝑚) (𝑚 = 1, 2, … , 𝑀). Suppose we consider a grouping method for the STAs in the network, in which each STA operating with the same data rate are grouped together. Thus there can be at most M groups in the network; all the STAs in a given group 𝑚 operate at the same data rate 𝑅(𝑚) , 𝑚 = 1, 2, … , 𝑀. Let there be 𝑛(𝑚) STAs per group so that 𝑀 𝑛(𝑚) = 𝑁, where N is the total number of STAs in the network. Let 𝛴𝑚=1 𝑇𝑅 be the total RAW duration available within a beacon interval and it is assumed that 𝑇 (𝑚) is the RAW slot duration allotted to group 𝑚 such 𝑀 𝑇 (𝑚) = 𝑇 . The saturation throughput of group 𝑚 under data that 𝛴𝑚=1 𝑅 rate grouping (DRG) is computed as follows:

(𝑚) 𝑛 𝑝(𝑚) ) 𝑡𝑟 = 1 − (1 − 𝜏

(19)

To determine the frame transmission probability corresponding to STAs belonging to group 𝑚, the DTMC model shown in Fig. 2 can be used with appropriate modifications. Let {𝑆 (𝑚) (𝑡), 𝐵 (𝑚) (𝑡)} be the twotuple that represents the BO stage and the BO counter of the tagged group 𝑚 STA undergoing the DCF protocol. Now the RAW slot duration allotted to group 𝑚 is 𝑇 (𝑚) = 𝑇𝑎(𝑚) + 𝑇ℎ + 𝑇𝑔 , as given in (1). The DTMC model applicable for the tagged STA can be solved as in Section 2. Let 𝜋 (𝑚) (𝑥, 𝑦) == lim𝑡→∞ 𝑃 {𝑆 (𝑚) (𝑡) = 𝑥, 𝐵 (𝑚) (𝑡) = 𝑦} be the steady state probability distribution of the DTMC corresponding to a STA within group 𝑚. Analytical equation for 𝜋 (𝑚) (𝑥, 𝑦) can be derived by following the steps described in Section 2. The following expression can be obtained for 𝜋 (𝑚) (0, 0);which is the steady state probability of reaching the state (0,0):

3.2. Throughput calculation under data rate based grouping

𝐸[𝜎 (𝑚) ]

(𝑚) −1

𝜏 (𝑚)

Initially, AP creates M arrays with index 1 to M Initialize arrays 𝐴(𝑚) 𝑚 = 1..𝑀 as empty ; for k = 1 to N do if 𝑠𝑘 transmits with rate 𝑅(𝑚) then 𝑖𝑛𝑑𝑒𝑥𝑝𝑡𝑟 = index(𝑅(𝑚) ) Choose array 𝐴(𝑚) pointed by 𝑖𝑛𝑑𝑒𝑥𝑝𝑡𝑟 𝐴(𝑚) <- {𝐴(𝑚) ,𝑠𝑘 } end if end for STAs in 𝐴(𝑚) forms group 𝐺(𝑚) AP assigns 13-bit AID for STA within group 𝑚 (m=1,.M) Set Group ID as last ⌈(𝑙𝑜𝑔2 𝑀)⌉ LSB-bit of AID of every STA is unique to the corresponding group.

(𝑚) 𝑝(𝑚) 𝑡𝑟 𝑝𝑠 𝐸[𝑋]

(18)

𝑝(𝑚) 𝑡𝑟

(𝑚) 𝑛 𝑝(𝑚) ) 𝑐 = 1 − (1 − 𝜏

Algorithm 1 Procedure for data rate based grouping

𝑈 (𝑚),𝐷𝑅𝐺 =

𝑛(𝑚) 𝜏 (𝑚) (1 − 𝜏 (𝑚) )

where 𝜏 (𝑚) (𝑚 = 1, 2, … , 𝑀) is the transmission attempt probability of a tagged STA belonging to group m, which is computed as follows. Consider STAs belonging to group m, which operate with data rate 𝑅(𝑚) (𝑚 = 1, 2, … , 𝑀). Let 𝑊0 be the minimum CW of STAs belonging to group m. During the allocated RAW slot duration 𝑇 (𝑚) , STAs belonging to group m compete for channel access based on the DCF protocol. Simultaneous transmissions by two STAs within a group 𝑚 will lead to collisions. Let 𝑝(𝑚) be the conditional collision probability experienced 𝑐 by the tagged STA belonging to group m, which is given by

3.1. Algorithm for data rate based grouping

1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

𝑛(𝑚) −1

𝜋 (𝑚) (𝑥, 0) =

𝐿+1 1 − (𝑝(𝑚) 𝑐 )

1 − 𝑝(𝑚) 𝑐

𝜋 (𝑚) (0, 0)

(21)

In (22), 𝑝(𝑚) is the BO counter freezing probability which is given by 𝑓 𝑝(𝑚) = 𝑓

(17) 159

(

) ( ) 𝑇𝑅 − 𝑇𝑎(𝑚) 𝑇𝑎(𝑚) (𝑚) 𝑝𝑏 + 𝑇𝑅 𝑇𝑅

(22)

U. Sangeetha and A.V. Babu

Computer Communications 151 (2020) 154–164

(𝑖 = 1, ..𝐾; 𝑚 = 1, ..𝑀). Define the normalized throughput of rate 𝑅(𝑚) 𝑖

Table 1 System parameters [1].

STAs belonging to group 𝑖 as 𝑦𝑖(𝑚),𝑅𝐺 =

Parameters

Value

Packet payload (bytes) MAC header 𝑇𝑃 𝐿𝐶𝑃 RTS (bytes) CTS (bytes) 𝑇𝑆𝐼𝐹 𝑆 𝑇𝐷𝐼𝐹 𝑆 Slot time (𝜎0 ) 𝐶𝑊𝑚𝑖𝑛 𝐶𝑊𝑚𝑎𝑥 ACK (bytes) Propagation Loss Model Modulation coding schemes

768, 512 224 bits 20 μs 20 14 160 μs 264 μs 52 μs 16 1024 14 Outdoor, macro [3] MCS0, MCS1, MCS2, MCS4, MCS7, MCS9

𝑝(𝑚) , 𝑝(𝑚) , 𝑝(𝑚) and 𝑝(𝑚) . We use [21,22] to solve the system of linear 𝑏 0 2 1 equations. Once 𝜏 (𝑚) and 𝑝(𝑚) are known, various probabilities such as 𝑐 (𝑚) 𝑝(𝑚) etc. can be determined and 𝑈 (𝑚),𝐷𝑅𝐺 can be calculated by 𝑡𝑟 , 𝑝𝑠 using (16), provided 𝐸[𝜎 (𝑚) ] is known. Now 𝐸[𝜎 (𝑚) ] is the average slot duration relating to the DCF protocol used by the STAs within group m for channel access. Since all the STAs within group m use the same data rate, 𝐸[𝜎 (𝑚) ] can be determined as follows: (23)

𝑇𝑐(𝑚)

where and are given by (15). Finally, the aggregate network 𝐷𝑅𝐺 ) is given by throughput under data rate based grouping (i.e; 𝑈𝑡𝑜𝑡 𝐷𝑅𝐺 𝑈𝑡𝑜𝑡 =

𝑀 ∑ 𝑚=1

𝑈 (𝑚),𝐷𝑅𝐺 𝑇 (𝑚) 𝑇𝑅

Notice that 𝐹𝐽 = 1 iff 𝑦(𝑚),𝐷𝑅𝐺 = 𝑦 ∀ 𝑚 = 1, ..𝑀, which implies that each STA belonging to rate 𝑅(𝑚) will get a throughput proportional to its data rate. Thus we compare the fairness in resource allocation among the STAs of distinct data rates under random grouping as well as data rate based grouping scheme. Next, we present the performance evaluation results. First of all, we assume that the STAs can operate at any one of the two data rates, i.e; 600 kbps and 2.4 Mbps. Let us assume that K = 2 and each group contains equal number of STAs. We consider distinct values of N as well. Table 2 shows the throughput of the group as well as the aggregate throughput of the network under RG and DRG. The percentage improvement of aggregate throughput can also be seen in Table 2. Table 3 shows the results for the three data rates case with STAs operating at any one of the three data rates, 600 kbps, 2.7 Mbps and 4 Mbps. All the MAC layer parameters are assumed to be the same for all the STAs in the network regardless of their data rates. The results show that the aggregate throughput of the network improves significantly when the DRG scheme is employed. The reason for this behaviour can be explained as follows: In random grouping, a group will contain STAs operating at distinct data rates. A lower rate STA will compete for channel access with a higher rate STA within the same group. Once a lower rate STA wins the channel, it occupies the channel for a longer time as compared with a higher data rate STA. Thus the throughput performance of the group is dominated by the lower data rate STAs. The aggregate throughput of the network also gets degraded significantly under random grouping. When data rate based grouping is employed, all the STAs within a group operate at the same data rate. Thus the issue of lower data rate STAs occupying the channel for a longer time, when competing with higher rate STA can be avoided. Thus the throughput performance of each group depends on the data rate at which each STA within the group operate. Accordingly, the aggregate throughput improves under data rate based grouping. Results in Tables 2 and 3 further show that under RG, all the groups in the network achieve the same throughput. This happens owing to the

(24)

4. Numerical and simulation results In this section, we present the analytical and simulation results for the throughput performance of 802.11ah WLAN with heterogeneous data rates. The analytical results, which are obtained from the mathematical models of sections 2 and 3 are validated by conducting extensive simulations using network simulator version 3 (NS3) [23]. The RAW based channel access protocol has been simulated according to the PHY and MAC layer parameters of 802.11ah standard. The system parameters used for the analysis as well as simulations are listed in Table 1. An infrastructure based basic service set (BSS) has been simulated with one AP and N number of STAs, which are uniformly distributed around the AP. Uplink data transfer has been assumed with RTS/CTS scheme. We ignore the impact of hidden nodes on the analysis of network throughput. The total RAW duration has been selected as 500 ms and it is divided equally among the groups in the network. Notice that the duration of RAW has been selected such that at least one frame transaction can be ensured within a group, when the number of groups (K) is equal to 256. We consider data rate based grouping and random (i.e; data rate independent) grouping for the performance evaluation of the network throughput. In addition to the evaluation of network throughput, we evaluate the fairness in resource allocation among the STAs under random grouping and data rate based grouping strategies. Here we are trying to compare the fairness in resource allocation among the STAs of distinct data rates. In random grouping, each group contains STAs operating at distinct data rates. Recall that 𝑈𝑖(𝑚),𝑅𝐺 is the throughput of rate 𝑅(𝑚) 𝑖 STAs belonging to group 𝑖 under random grouping. Let 𝑢(𝑚),𝑅𝐺 = 𝑖

. We use the following

The above calculation assumes that all the STAs in group 𝑖 use the same frame size and keep all the MAC layer parameters to be the same regardless of their data rate. A resource allocation is fair if and only if 𝐹𝐽 ,𝑖 = 1, which implies that 𝑦𝑖(𝑚),𝑅𝐺 = 𝑦𝑖 ∀ 𝑚 = 1, ..𝑀, i.e.; all the STAs in the group achieve equal normalized throughput, normalized with reference to their data rate. The motivation for choosing normalized throughput as the metric for evaluating the fairness of resource allocation can be explained as follows. We are considering a heterogeneous environment in which each group contains STAs with distinct data rates 𝑅𝑖(𝑚) , 𝑚 = 1...𝑀. In this scenario, the resource allocation is fair if each STA gets a throughput proportional to its data rate. As mentioned before, 𝐹𝐽 ,𝑖 = 1 iff 𝑦(𝑚),𝑅𝐺 = 𝑦𝑖 ∀ 𝑚 = 1, ..𝑀. Thus fair resource allocation 𝑖 implies that throughput of each STA will become proportional to its data rate. We evaluate the fairness of resource allocation under data rate based grouping scheme as well. In this case, each group contains STAs operating at the same data rate. Let 𝑈 (𝑚) and 𝑢(𝑚) be the total and perSTA throughput corresponding to group 𝑚 (𝑚 = 1, ..𝑀). Further, let (𝑚),𝐷𝑅𝐺 𝑦(𝑚),𝐷𝑅𝐺 = 𝑢 𝑅(𝑚) be the normalized throughput of STAs belonging to (𝑚) rate 𝑅 , in group 𝑚. The Jain’s fairness index is defined as ∑ (𝑚),𝐷𝑅𝐺 )2 ( 𝑀 𝑚=1 𝑦 𝐹𝐽 = (26) 2 ∑ (𝑚),𝐷𝑅𝐺 ) 𝑀 𝑀 𝑚=1 (𝑦

(𝑚)

𝑇𝑠(𝑚)

(𝑚)

𝑅𝑖

Jain’s fairness index to measure the fairness in resource allocation under random grouping [24]: (∑𝑀 (𝑚),𝑅𝐺 )2 𝑚=1 𝑦𝑖 (25) 𝐹𝐽 ,𝑖 = 2 ∑𝑀 𝑀 𝑚=1 (𝑦𝑖(𝑚),𝑅𝐺 )

where 𝑝(𝑚) = 1 − (1 − 𝜏 (𝑚) )𝑛 . is the probability with which STAs 𝑏 belonging to group 𝑚 will sense the channel as busy. Now (19)–(21) can be solved numerically to determine 𝜏 (𝑚) and 𝑝(𝑚) for a given set 𝑐 of parameters such as 𝐿, 𝑊0 , 𝑛(𝑚) etc. However, solving (19)–(21) simultaneously requires knowledge of other probabilities such as 𝑝𝑓(𝑚) ,

(𝑚) (𝑚) (𝑚) 𝑚) (𝑚) 𝐸[𝜎 (𝑚) ] = (1 − 𝑝(𝑚) + 𝑝(𝑚) 𝑡𝑟 )𝜎0 + 𝑝𝑡𝑟 𝑝𝑠 𝑇𝑠 𝑡𝑟 (1 − 𝑝𝑠 )𝑇𝑐

(𝑚),𝑅𝐺

𝑢𝑖

(𝑚),𝑅𝐺

𝑈𝑖

(𝑚)

𝑛𝑖

be the per-STA throughput of rate 𝑅(𝑚) STAs belonging to group 𝑖 𝑖 160

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Computer Communications 151 (2020) 154–164

Table 2 Saturation throughput (kbps): Random grouping scheme (RG) vs Data rate based grouping scheme (DRG) (𝑅(1) = 600 kbps, 𝑅(2) = 2.4 Mbps, K = 2, E[X] = 768 bytes). Random Grouping (RG) No: of rate 𝑅(𝑚) (𝑚 = 1, 2) (2) STAs in group 2 (1) 𝑛2 , 𝑛(2) 2

Total no: of STAs in the network (N)

No: of rate 𝑅(𝑚) (𝑚 = 1, 2) (1) STAs in group 1 (1) 𝑛1 , 𝑛(2) 1

Total throughput of group 1 (𝑈1𝑅𝐺 ) Analysis

Simulation

80 160 180 240

20,20 40,40 45,45 60,60

738.55 736.12 730.14 728.92

737.40 735.67 729.14 727.45

Total throughput of group 2 (𝑈2𝑅𝐺 )

Aggregate throughput 𝑅𝐺 (𝑈𝑡𝑜𝑡 )

Analysis

Simulation

Analysis

Simulation

20,20 40,40 45,45 60,60

738.55 736.12 730.14 728.92

737.40 735.67 729.14 727.45

738.55 736.12 730.14 728.92

737.40 735.67 729.14 727.45

No: of rate 𝑅(2) STAs in group 2 𝑛(2)

Total throughput of group 2 (𝑈 (2),𝐷𝑅𝐺 ) Analysis

Simulation

Analysis

Simulation

40 60 90 120

1610.80 1603.20 1594.40 1587.2

1609.10 1602.24 1593.60 1586.30

1054.70 1050.30 1045.40 1041.11

1053.23 1049.80 1044.40 1040.69

Data rate based grouping (DRG) Total no: of STAs in the network (N)

No: of rate 𝑅(1) STAs in group 1 𝑛(1)

80 120 180 240

40 60 90 120

Total throughput of group 1 (𝑈 (1),𝐷𝑅𝐺 ) Analysis

Simulation

498.54 497.41 496.10 495.01

496.99 496.87 495.23 494.89

Aggregate throughput 𝐷𝑅𝐺 𝑈𝑡𝑜𝑡

Percentage improvement aggregate throughput: Random grouping vs Data rate based grouping scheme for three data rates No: of STAs (N)

80

120

180

240

𝐷𝑅𝐺 𝑅𝐺 − 𝑈𝑡𝑜𝑡 ( 𝑈𝑡𝑜𝑡 ) *100 𝑅𝐺 𝑈𝑡𝑜𝑡

42.80%

42.60%

43.18%

42.83%

fact that the number of STAs operating with distinct data rates and the total number of STAs in each group, are assumed to be equal. Further, the MAC layer parameters such as the minimum contention window and frame size of all the STAs are assumed to be equal. Furthermore, we assume the RAW slot durations assigned to each group to be equal. Under these conditions, the DCF protocol provides throughput based fairness [2]. Another important observation is that, under DRG, throughput of group 1 is observed to be lower as compared to the throughput obtained under RG, as can be seen in Tables 2 and 3. Recall that, under DRG, the group 1 will contain STAs operating at the same data rate. Accordingly, when DRG is employed the throughput performance of each group depends on the data rate to which it belongs. In Tables 2 and 3, group 1 under DRG corresponds to a lower data rate group; thus the throughput of group 1 is observed to be lower under DRG, as compared to that obtained under RG. However, notice that group 1 does not suffer throughput degradation under DRG. In fact, the per-STA throughput of a tagged STA belonging to group 1 under DRG is the same as the throughput that it will achieve in a single rate WLAN with 𝑛(1) STAs operating at data rate 𝑅(1) . Thus the throughput of lower data rate STAs (i.e; lower data rate group) is not penalized under DRG. Figs. 3 and 4 show the results for the aggregate throughput (under RG and DRG) for the two data rate and the three data rate cases respectively. Here we consider distinct values for the number of groups (K) in the network. Results show that DRG improves the aggregate throughput significantly as compared to RG. Further, increase of K improves the aggregate throughput. As K increases, the number of STAs within each group reduces. Consequently the level of contention is reduced within each group, leading to enhancement of throughput. Figs. 5 and 6 respectively show the aggregate throughput variation against frame size for the two-data rate and the three data rate cases. The results show that aggregate throughput increases as the frame size is increased. Table 4 shows the comparison of JFI under RG and DRG for distinct values of N and K. We repeat the calculations of the throughput achieved under RG and DRG, for distinct values of N and K. The normalized throughputs achieved by each STA corresponding to distinct data rates are calculated. The JFI is determined as described in the initial part of this section. The results confirm that DRG leads to fair resource allocation among the STAs operating at distinct data rates as the JFI has been observed to be higher as compared to the

Fig. 3. Aggregate throughput vs no: of STAs [𝑅(1) = 600 kbps, 𝑅(2) = 1.2 Mbps, E[X] = 512 bytes].

Fig. 4. Aggregate throughput vs no: of STAs [𝑅(1) = 600 kbps, 𝑅(2) = 2.4 Mbps, 𝑅(3) = 3.6 Mbps, E[X] = 768 bytes].

161

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Table 3 Saturation throughput (kbps): Random grouping scheme (RG) vs Data rate grouping (DRG) (𝑅(1) = 600 kbps, 𝑅(2) = 2.7 Mbps, 𝑅(3) = 4 Mbps, K = 3, E[X] = 768 bytes). Random Grouping (RG) Total no: of STAs in the network (N)

No: of rate 𝑅(𝑚) (𝑚 = (1) 1, 2, 3) STAs in group 1 𝑛(1) , 𝑛(2) , 𝑛(3) 1 1 1

Total throughput of group 1 𝑅𝐺 (𝑈(1) ) Analysis

Simulation

180 270 360 450

20, 30, 40, 50,

1045.05 1040.43 1034.70 1030.92

1043.90 1039.34 1033.27 1029.67

20, 30, 40, 50,

20 30 40 50

No: of rate 𝑅(𝑚) (𝑚 = (2) 1, 2, 3) STAs in group 2 𝑛(1) , 𝑛(2) , 𝑛(3) 2 2 2 20, 30, 40, 50,

20, 30, 40, 50,

20 30 40 50

Total throughput of group 2 𝑅𝐺 (𝑈(2) ) Analysis

Simulation

1045.05 1040.43 1034.70 1030.92

1043.90 1039.34 1033.27 1029.67

No: of rate 𝑅(𝑚) (𝑚 = (3) 1, 2, 3) STAs in group 3 𝑛(1) , 𝑛(2) , 𝑛(3) 3 3 3

Total throughput of group 3 𝑅𝐺 (𝑈(3) ) Analysis

Simulation

Analysis

Simulation

20, 30, 40, 50,

1045.05 1040.43 1034.70 1030.92

1043.90 1039.34 1033.27 1029.67

1045.05 1040.43 1034.70 1030.92

1043.90 1039.34 1033.27 1029.67

20, 30, 40, 50,

20 30 40 50

Aggregate throughput 𝑅𝐺 (𝑈𝑡𝑜𝑡 )

162

Data rate based grouping (DRG) Total no: of STAs in the network (N)

No: of rate 𝑅(1) STAs in group 1 𝑛(1)

180 270 360 450

60 90 120 150

No: of rate 𝑅(2) STAs in group 2 𝑛(2)

Total throughput of group 1 (𝑈 (1) , 𝐷𝑅𝐺) Analysis

Simulation

497.39 496.07 495.01 494.10

496.78 495.78 494.80 493.32

60 90 120 150

Total throughput of group 2 (𝑈 (2) , 𝐷𝑅𝐺) Analysis

Simulation

1747.10 1736.80 1728.40 1721.20

1746.34 1735.45 1727.72 1720.56

No: of rate 𝑅(3) STAs in group 3 𝑛(3)

Total throughput of group3 (𝑈 (3) , 𝐷𝑅𝐺)

Aggregate throughput 𝐷𝑅𝐺 𝑈𝑡𝑜𝑡

Analysis

Simulation

Analysis

Simulation

60 90 120 150

2278.80 2262.20 2248.70 2237.78

2276.56 2261.80 2247.43 2235.80

1507.56 1497.70 1490.80 1484.13

1506.36 1497.67 1489.90 1483.22

Percentage improvement aggregate throughput: Random grouping vs Data rate based grouping scheme for three data rates 180

270

360

450

𝐷𝑅𝐺 𝑅𝐺 − 𝑈𝑡𝑜𝑡 ( 𝑈𝑡𝑜𝑡 ) *100 𝑅𝐺 𝑈𝑡𝑜𝑡

44.26%

43.95%

42.65%

42.02%

Computer Communications 151 (2020) 154–164

No: of STAs (N)

U. Sangeetha and A.V. Babu

Computer Communications 151 (2020) 154–164

5. Conclusion In this paper, an analytical model was proposed to find the saturation throughput of IEEE 802.11ah wireless LAN under the restricted access window (RAW) based channel access scheme, assuming that stations (STAs) operate at distinct data rates. Two strategies were considered for STA grouping, i.e., conventional random grouping and data rate based grouping. In the former case, the STAs within a group can operate at distinct data rates while under data rate based grouping, all the STAs within a group will operate at the same data rate. A simple procedure to implement data rate grouping was described. We have used the network simulator (NS3) to investigate the effectiveness of the considered grouping schemes in a WLAN scenario that follows 802.11ah PHY and MAC layer specifications. We have also compared the results obtained from the developed mathematical model against the simulation results. It was observed that data rate based grouping can provide significant improvement in aggregate network throughput as compared to the conventional random grouping scheme when STAs employ heterogeneous data rates for transmission. Further, it was shown that data rate based grouping scheme can ensure fair resource allocation among the competing STAs, i.e., all the competing STAs in the network achieve throughput proportional to their data rates.

Fig. 5. Aggregate throughput vs packet payload size [𝑅(1) = 600 kbps, 𝑅(2) = 2.4 Mbps, N = 320].

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement U. Sangeetha: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing original draft. A.V. Babu: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing - review & editing, Supervision. Acknowledgement Fig. 6. Aggregate throughput vs packet payload size [𝑅(1) = 600 kbps, 𝑅(2) = 2.7 Mbps, 𝑅(3) = 4 Mbps, N = 300].

Authors would like to thank Department of Science and Technology, Government of India for supporting this work under the FIST scheme No. SR/FST/ET-I/2017/68.

Table 4 Comparison of Jain’s fairness index (WLAN with two data rates: 𝑅(1) = 600 kbps, 𝑅(2) = 2.4 Mbps). Total no: of STAs (N)

K = 2

80 120 180 240

References

Jain’s fairness index K = 4

K = 8

RG

DRG

RG

DRG

RG

DRG

0.74 0.74 0.74 0.74

0.99 0.99 0.99 0.99

0.74 0.74 0.74 0.74

0.99 0.99 0.99 0.99

0.76 0.73 0.77 0.76

0.99 0.99 0.99 0.99

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values obtained for the RG case. Under RG, when lower rate STAs compete with higher rate STAs, channel is occupied for a larger time period by the lower rate STA. Thus the effective throughput of higher data rate STA becomes almost equal to that of lower rate STA. Under DRG, we avoid channel contention among lower and higher data rate STAs by grouping them separately in distinct groups. Thus each STA will achieve a throughput which is proportional to its data rate. Hence we propose that, when STAs use distinct data rates in IEEE 802.11ah WLAN, data rate based grouping method has to be employed to ensure fair and efficient resource allocation among the competing STAs. 163

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