Throughput Optimization of Multi-BSS IEEE 802.11 ... - IEEE Xplore

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 8, AUGUST 2017

3399

Throughput Optimization of Multi-BSS IEEE 802.11 Networks With Universal Frequency Reuse Yayu Gao, Member, IEEE, Lin Dai, Senior Member, IEEE, and Xiaojun Hei, Member, IEEE

Abstract— For IEEE 802.11 networks with multiple basic service sets (BSSs), most studies have focused on how to allocate different frequency sub-channels to BSSs for minimizing the co-channel interference. With the significant increase of the subchannel bandwidth, however, it becomes increasingly important to study the network performance with universal frequency reuse. In this paper, we focus on an uplink M-BSS IEEE 802.11 network, where all the BSSs share the frequency band rather than operate at different sub-channels. By dividing the nodes in each BSS into multiple groups according to the set of access points (APs) they can be heard by, the steady-state points of M BSSs in saturated conditions are obtained as the functions of the number of nodes in each group and the initial backoff window size of nodes of each BSS. The maximum network throughput is further characterized by optimally choosing the initial backoff window sizes of all the nodes and shown to be closely dependent on the percentage of nodes that can be heard by multiple APs. The comparison with orthogonal frequency division reveals that although the maximum network throughput is degraded due to interference among BSSs, a higher network data rate can still be achieved by universal frequency reuse, which makes it a preferable option for multi-BSS IEEE 802.11 networks. Index Terms— IEEE 802.11 networks, multiple basic service sets (BSSs), distributed coordination function (DCF), carrier sensing multiple access (CSMA), universal frequency reuse, throughput optimization.

I. I NTRODUCTION

I

EEE 802.11 wireless local area networks (also known as Wi-Fi networks) have gained worldwide popularity in the past decade due to the low cost and easy deployment [1]. Similar to cellular networks, an IEEE 802.11 network consists of multiple basic service sets (BSSs), within each of which a number of wireless stations (referred to as “nodes” in this paper) transmit to or receive from a Manuscript received January 18, 2017; revised May 5, 2017; accepted May 10, 2017. Date of publication May 19, 2017; date of current version August 14, 2017. The work of Y. Gao was supported by the National Natural Science Foundation of China (No. 61402186). The work of L. Dai was supported by the CityU Strategic Research Grant 7004428 and the contract research project (Project No. 9231159) from City University of Hong Kong with Huawei Technologies Co. Ltd. The work of X. Hei was supported by the National Natural Science Foundation of China (No. 61370231). The associate editor coordinating the review of this paper and approving it for publication was V. Aggarwal. (Corresponding author: Yayu Gao.) Y. Gao and X. Hei are with the School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: [email protected]; [email protected]). L. Dai is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCOMM.2017.2706280

single access point (AP). Different from cellular networks, a random access protocol, i.e., distributed coordination function (DCF), is adopted in IEEE 802.11 networks at the medium access control layer [2]. With minimum coordination from the AP, each node independently decides when to transmit and backs off if its transmission fails or it senses other on-going transmissions. Due to the broadcast nature of wireless medium, BSSs may interfere with each other if they share the frequency spectrum. To mitigate the so-called co-channel interference, a frequency division approach is adopted in the current IEEE 802.11 standard, where the frequency band is divided into multiple subchannels for BSSs to operate on. By assuming that a central controller is available for optimizing the channel assignment, various algorithms have been proposed to minimize the highest effective channel utilization [3] or the interference among APs [4], [5] and nodes [6]. For scenarios where BSSs do not coordinate with each other, distributed channel allocation schemes were developed where each AP determines its best sub-channel based on the estimation of network parameters such as the number of active nodes in its BSS [7], [8] or measurements on its local energy [9], interference levels [10], [11] and quality-of-experience [12]. The above frequency-division approach, nevertheless, may not be in line with the recent evolution of IEEE 802.11 standards, which tends to enlarge the bandwidth of each sub-channel to support the ever-increasing data rate, and the resulting number of sub-channels is greatly reduced. In the latest IEEE 802.11ac standard [13], for instance, each BSS could occupy a channel up to 160 MHz wide, leading to a data rate as high as 867 Mbps. However, as only one 160 MHz wide channel is provided in the 5 GHz band, BSSs longing for high data-rate transmissions would have to share the frequency band rather than operate at different subchannels. Such a universal frequency reuse approach has in fact been widely adopted in cellular networks, and extensive studies have shown that compared to orthogonal frequency division among cells, it can achieve much higher spectral efficiency and provide more flexibility in cell planning [14]. Different from cellular networks where centralized access is adopted in each cell, however, IEEE 802.11 DCF networks are based on random access. It is therefore of great theoretical and practical importance to characterize the performance of multiBSS IEEE 802.11 DCF networks with universal frequency reuse.

0090-6778 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

3400

Fig. 1.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 8, AUGUST 2017

Graphic illustration of (a) multiple access and (b) ad-hoc scenarios.

A. Modeling of Multi-BSS IEEE 802.11 DCF Networks DCF is a variant of Carrier Sensing Multiple Access (CSMA) [15], one of the most representative random access protocols. There have been extensive studies on the modeling of CSMA-based networks, which mainly focus on two scenarios: 1) Multiple Access: CSMA was originally proposed as a multiple access protocol with which multiple nodes transmit to a single receiver, as Fig. 1(a) shows. Representative applications include uplink IEEE 802.11 networks, where in each BSS, multiple users transmit to the AP. 2) Ad-hoc: CSMA can be also applied to an ad-hoc network where multiple transmitter-receiver pairs exist and compete with each other, as Fig. 1(b) illustrates. Representative applications include wireless sensor networks and mobile ad-hoc networks. Both categories have a long line of research. Though relatively new, studies on CSMA in the ad-hoc scenario gained significant attention in the past decade [16]–[22]. As performance of node pairs closely depends on the interference they receive, various models have been adopted to describe the spatial interference for given network topology [16]–[19], where interference graph [17], [18] is a representative one. By assuming random spatial locations of nodes, stochastic geometry [24]–[26] was also widely adopted to characterize the distribution of interference for evaluating the average network performance [20]–[23]. The multiple access scenario, in contrast, leads to a much simpler network topology. In this case, the network can be regarded as a multi-queue-single-server system, and the key to performance analysis is the characterization of the service time distribution [27]. Most of the studies have followed the channel-centric modeling approach adopted by Kleinrock and Tobagi in their original paper [15] by focusing on the state transition process of the aggregate channel [28]–[35]. Due to ignoring nodes’ queueing behavior, however, it is difficult to analyze the second-order statistics of service time based on the channel-centric models, especially when sophisticated backoff schemes are involved. As we pointed out in [27], the service time distribution is crucially determined by the aggregate activities of head-of-line (HOL) packets of nodes, which requires proper modeling of each HOL packet’s behavior. In our recent work, a unified analytical framework was proposed for CSMA networks in the multiple access scenario [27], and applied to single-BSS IEEE 802.11 DCF networks in [36]. Different from the classical Bianchi model [37] where only

Fig. 2.

Graphic illustration of a multi-BSS IEEE 802.11 DCF network.

the backoff process of each saturated node was characterized, the complete behavior of each HOL packet, including backoff, collision and successful transmission, was modeled as a discrete-time Markov renewal process, based on which the throughput, delay and stability performance in both saturated and unsaturated conditions can be characterized, and optimized by tuning the backoff parameters such as the initial backoff window size of each node. The analysis was further extended to study the effect of backoff [38], the maximum network throughput with heterogeneous traffic input rates of nodes [39], and the optimal throughput differentiation for IEEE 802.11e EDCA networks [40], all in the single-BSS scenario. In this paper, the proposed analytical framework will be extended to the multi-BSS case. As Fig. 2 illustrates, with universal frequency reuse among BSSs, there exist certain nodes (such as Node A and Node B in Fig. 2) whose transmissions can be heard, and thus cause collisions, in neighboring BSSs. They can also sense the transmissions in neighboring BSSs and therefore have a higher chance of sensing a busy channel. For multi-BSS IEEE 802.11 DCF networks, a celllevel model was proposed in [41] by assuming that nodes in each cell have identical sensing results, i.e., they either all sense an idle channel or a busy channel. Yet in fact, the probability of sensing an idle/busy channel of each node is crucially determined by transmissions in which BSSs they can sense, which varies from node to node. As we will demonstrate in this paper, in this case, a multi-group model needs to be established, where nodes in each BSS are divided into different groups according to the set of APs they can be heard by, and each group has a distinct probability of sensing an idle channel. B. Throughput Optimization of Multi-BSS IEEE 802.11 DCF Networks Network throughput, which is defined as the average number of successfully transmitted packets of the network in each time slot, is an important performance metric for randomaccess networks. It has been demonstrated in [37] that for a single-BSS IEEE 802.11 DCF network, the network throughput closely depends on the initial backoff window size of nodes if the network is saturated. Explicit expressions of the maximum network throughput and the corresponding optimal

GAO et al.: THROUGHPUT OPTIMIZATION OF MULTI-BSS IEEE 802.11 NETWORKS WITH UNIVERSAL FREQUENCY REUSE

3401

TABLE I M AIN N OTATIONS

initial backoff window size were obtained in [36], which showed that the initial backoff window size should be linearly increased with the number of nodes to optimize the network throughput performance. Throughput analysis of a multi-BSS IEEE 802.11 DCF network, in contrast, has received little attention. Intuitively, if orthogonal sub-channels are allocated to neighboring BSSs, then the throughput of each BSS is approximately equal to that in the single-BSS case by ignoring the out-of-BSS interference. That is why most of efforts have been devoted to channel assignment design for minimizing the interference among neighboring BSSs [3]–[10]. With universal frequency reuse, however, the throughput performance of each BSS is crucially determined by the transmissions/interference from neighboring BSSs, which needs to be carefully characterized. In this paper, throughput analysis will be presented for a saturated uplink M-BSS IEEE 802.11 DCF network with universal frequency reuse. Specifically, the steady-state points (i) of M BSSs in saturated conditions { p A }i=1,··· ,M are obtained by jointly solving M fixed-point equations of the steady-state probability of successful transmission of HOL packets given that the channel of each BSS is idle. The network throughput is (i) further derived as an explicit function of { p A }i=1,··· ,M , which is crucially determined by the initial backoff window sizes of nodes {W (i) }i=1,··· ,M . By optimally choosing {W (i) }i=1,··· ,M , the maximum network throughput is shown to be closely dependent on the percentage of nodes that can be heard by multiple APs. Practical implications, including the effect of individual BSS optimization and performance comparison of universal frequency reuse and orthogonal frequency division, are also discussed. The remainder of this paper is organized as follows. Section II establishes the new analytical model of M-BSS IEEE 802.11 DCF networks. Throughput analysis is presented in Section III, and verified by simulation results provided in Section IV. Implications to practical network design are discussed in Section V. Finally, concluding remarks are summarized in Section VI. Table I lists the main notations used in this paper. II. M ODELING AND P RELIMINARY A NALYSIS Consider the uplink of an IEEE 802.11 DCF network with M BSSs, where in each BSS, multiple nodes transmit to a

Fig. 3. Graphic illustration of a two-BSS network. Nodes in the non-shadow area are denoted as Group Gi,{i} , who belong to BSS i and can only be heard by AP i, i = 1, 2. Nodes in the shadow area are denoted as Group Gi,{1,2} , who belong to BSS i, but can be heard by both AP 1 and AP 2, i = 1, 2.

single AP. Assume that universal frequency reuse is adopted among BSSs, that is, all the BSSs share the same spectrum. 1) Coverage: Define the coverage of a BSS as an area centered on its AP, within which the AP can “hear”1 the transmissions of all the nodes. With multiple BSSs, their coverage areas may overlap. Nodes who fall into the overlapping areas can be heard by multiple APs, among which they choose one to associate with. For instance, in Fig. 3, Node A in the shadow area belongs to BSS 1 but can be heard by both AP 1 and AP 2. 2) Sensing: In this paper, we assume that each node senses the channel based on the feedback of the APs, rather than by itself. As each AP can hear all the transmissions in its coverage area, it can inform the nodes about the channel availability via broadcast messages. As a result, if a node is in the coverage area of BSS i , it can correctly “sense” the transmissions of all the nodes in the coverage area of BSS i , even though they may not hear from each other directly. In the current IEEE 802.11 standard, the Requestto-Send/Clear-to-Send (RTS/CTS) access mechanism has been designed to implement the above feedback process [42]. Specifically, each AP would broadcast a CTS frame upon receiving the RTS frames, and the CTS frame includes the information of how long the channel would be occupied. As all the nodes in the same BSS can hear the AP, they can infer the channel availability based on the CTS frame, and avoid being the hidden nodes. Note that nodes in the overlapping areas can hear multiple APs, and are therefore able to sense the transmissions of nodes from multiple BSSs. As a result, they have a higher chance of sensing a busy channel. Similar [36] and [38]–[40], in this paper we use “the channel of BSS i ” to refer to the aggregate transmissions of nodes 1 Specifically, “hear” means that the average received power of the received packets is higher than the threshold for successful decoding. In the IEEE 802.11n standard, for instance, the minimum requirement of received power for an AP to successfully decode a packet is to detect the start of a valid transmission signal at a power level of at least -82/-79 dBm for a 20/40 MHz channel [2].

3402

that can be heard by AP i . For Node A in Fig. 3, for instance, as it can sense the channels of both BSS 1 and BSS 2, it has a lower chance to transmit than those in the non-overlapping area. 3) Collision Model: The classical collision model is assumed at the receivers, i.e., the APs. That is, a packet transmission is successful if and only if its AP hears no other concurrent transmissions. Otherwise, a collision occurs and none of the packets can be successfully decoded. Note that for nodes in the overlapping area, their transmissions may cause collisions not only to their own BSS, but also to the neighboring BSSs. In this paper, we aim to optimize the network throughput performance. Therefore, we consider the saturated conditions where each node always has a packet to transmit. Moreover, assume that the maximum number of retransmission attempts of each head-of-line (HOL) packet is infinite, i.e., a HOL packet stays in the queue until it is successfully transmitted. Let M = {1, 2, . . . , M} denote the set of APs/BSSs. As each node may be heard by multiple APs, we divide the nodes into groups according to the BSS i it belongs to, and the BSS set S that consists of all the APs/BSSs that it can be heard by, denoted as Group Gi,S . Denote the size of Group Gi,S as | Gi,S |= n (i,S) and the initial backoff window size of nodes of BSS i as W (i) . In this paper, we assume that all the nodes have the same cutoff phase (which was also called the maximum backoff stage in the literature [37]), i.e., K (i) = K , i ∈ M , and focus on how to optimally choose the initial backoff window sizes {W (i) }i∈M to maximize the network throughput.2 As we have mentioned in Section I-A, each BSS can be regarded as a multi-queue-single-server system, where the service time distribution is determined by the aggregate activities of HOL packets. In the following subsections, we will first characterize the state transition process of HOL packets of each group, and then derive the steady-state point of each BSS. A. State Characterization of HOL Packets Let us establish  a discrete-time Markov renewal  process  (i,S) (i,S)  (i,S) (i,S) , j = 0, 1, . . . to model = X j , Vj ,V X the behavior of each HOL packet of nodes in Group Gi,S of (i,S) BSS i , i ∈ M , S ⊆ M . X j denotes the state of a tagged HOL packet in Group Gi,S at the j -th transition and V j(i,S) denotes the epoch at which the j -th transition occurs.   . Fig. 4 shows the embedded Markov chain X(i,S) = X (i,S) j   (i,S) The states of X j can be divided into three categories: 1)   waiting to request a transmission  state Rk , k = 0, . . . , K , 2) collision  state Fk , k = 0, . . . , K , and 3) successful transmission state T . As Fig. 4 illustrates, a HOL packet moves from state Rk to State T if the transmission is successful; otherwise, it stays at state Fk until the end of the collision 2 Note that both the cutoff phase and the initial backoff window size are important backoff parameters that can be tuned. In this paper, we only focus on the optimal tuning of initial backoff window size because when the cutoff phase is large enough, its effect on the throughput performance becomes negligible [36].

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 8, AUGUST 2017

  (i,S) of the state transition process of Fig. 4. Embedded Markov chain X j an individual HOL packet in Group Gi,S .

Fig. 5. Graphic illustration of successful transmission and collision in IEEE 802.11 DCF networks with the RTS/CTS access mechanism.

and then moves to state Rk+1 . Here k denotes the number of collisions experienced by the HOL packet of Group Gi,S and is incremented until it reaches the cutoff phase K . Let p(i) represent the limiting probability of successful transmission of HOL packets of BSS i given that the channel of BSS i is idle, i ∈ M .3 The steady-state probability distribution of the embedded Markov chain shown in Fig. 4 can then be obtained as ⎧ (i,S) k = 0, . . . , K − 1 ⎨(1 − p(i) )k πT (i,S) (i) K (1) π Rk (1 − p ) (i,S) ⎩ π k = K , T p(i) and

π F(i,S) = π R(i,S) · (1 − p(i) ), k = 0, . . . , K . k k (i,S)

(2) (i,S)

The interval between successive transitions, i.e., V j +1 −V j

,

(i,S) Xj ,

is called the holding time in state which solely depends (i,S) on state X j , j = 0, 1, . . .. In IEEE 802.11 DCF networks, the holding time τT in state T and the holding time τ F in state Fk , k = 0, . . . , K , are determined by the access mechanism. A graphic illustration of τT and τ F with the RTS/CTS access mechanism is shown in Fig. 5. (i,S) The mean holding time τ Rk in state Rk , k = 0, . . . , K , of a HOL packet of Group Gi,S , on the other hand, is determined by the backoff window size in that state. When a HOL packet enters state Rk , it randomly selects a value from {0, . . . , Wk(i) − 1}, where Wk(i) is the backoff window size, and then counts down at each idle time slot. It leaves state Rk and makes a transmission request when the channels of all the BSSs it can sense are idle and the counter is zero. By following a similar derivation to [36, Appendix A], the mean holding time τ R(i,S) k 3 Note that an implicit assumption here is that all the nodes of BSS i have an identical limiting probability of successful transmission given that the channel of BSS i is idle, p(i) . Intuitively, nodes in different groups may have distinct probabilities of successful transmission. Yet the modeling complexity would be sharply increasing with the number of groups if a pergroup limiting probability of successful transmissions of HOL packets is adopted. The accuracy of this assumption will be examined by simulation results presented in Section IV.

GAO et al.: THROUGHPUT OPTIMIZATION OF MULTI-BSS IEEE 802.11 NETWORKS WITH UNIVERSAL FREQUENCY REUSE

in state Rk of a HOL packet of Group Gi,S can be obtained as = τ R(i,S) k

1 · (1 + Wk(i) ), 2α (i,S)

(3)

k = 0, . . . , K , where α (i,S) denotes the steady-state probability of sensing the channels of BSS set S idle for each node in Group Gi,S . Note that for nodes in Group Gi,{i} , they can only sense the transmissions of BSS i . The steady-state probability of sensing the channel of BSS i idle for nodes in Group Gi,{i} , α (i,{i}) , is thus equal to the probability that the channel of BSS i has neither a successful transmission nor a collision, which can be obtained as 1 , (4) α (i,{i}) = (i) 1 + τ F − τ F p − (τT − τ F ) p (i) ln p(i) i ∈ M , by following the derivation presented in [36, Appendix B]. Finally, the limiting state probabilities of the Markov  renewal process X(i,S) , V(i,S) are given by π˜ g(i,S) =

(i,S)

(i,S)

πT

·τT +

πg

K k=0

(i,S)

· τg

(i,S)

π Fk ·τ F +

K k=0

(i,S)

(i,S)

π Rk ·τ Rk

,

(5) g ∈ {T, F0 , . . ., Fk , R0 , . . ., Rk }. By substituting (1)-(3) into (5), the probability of a HOL packet of Group Gi,S being in state T can be obtained as

K −1 (i)  1−p (i,S) = α (i,S) τT / α (i,S) τT+τ F · (i) + (1 − p(i) )k π˜ T p k=0

(i) 1 + Wk(i) (1 − p(i) ) K 1 + W K + . (6) · · 2 2 p(i) Note that is also the service rate of each node’s queue as each queue has a successful output if and only if the HOL packet stays at state T. In IEEE 802.11 DCF networks, Binary Exponential Backoff (BEB) is adopted. That is, the backoff window size Wk(i) is set as (7) Wk(i) = W (i) · 2k , k = 0, . . . , K , where W (i) is the initial backoff window size of BSS i , i ∈ M . By substituting (7) into (6), we have

(i) 1−p 1  (i,S) = α (i,S) τT / α (i,S) τT+τ F · (i) + (i) 1+W (i) π˜ T p 2p



p (i) p (i) (i) K · + 1 − (i) (2 − 2 p ) . 2 p (i) − 1 2p − 1 (8) For a saturated network, it has been shown in [36] that the probability of each node sensing an idle channel is close to zero due to the high contention level. With a large cutoff phase K , the first term of the denominator in the right-hand side of (8) is much smaller than the second term and can be ignored. Specifically, with K = ∞, we have π˜ T(i,S),K =∞ ≈

B. Steady-State Point of Each BSS Under Saturated Conditions The analysis has shown that the network performance is crucially determined by p(i) , the limiting probability of successful transmission of HOL packets of BSS i given that the channel of BSS i is idle, i ∈ M . In this subsection, the steady-state point of BSS i in saturated conditions will be further characterized based on the fixed-point equations of p(i) , i ∈ M . Specifically, when saturated, each node in Group Gi,S must be in one of the following four states: S1 : the HOL packet is in state Rk , k = 0, . . . , K , and not requesting any transmission; S2 : the HOL packet is in state Rk , k = 0, . . . , K , and requesting a transmission; S3 : the HOL packet is in state T; S4 : the HOL packet is in state Fk , k = 0, . . . , K . For a given HOL packet in Group Gi, of BSS i , its transmission request is successful if and only if all the other nodes that can be heard by AP i , including 1) nodes in Group Gi, , 2) nodes in Group Gi,S , S = , and 3) nodes in Group G j,S , j ∈ M , j = i and S ⊇ {i }, are in state S1 . The steady-state probability of successful transmission of HOL packets of BSS i given that the channel of BSS i is idle, p(i) , can then be written as  p(i) = Pr{node in Group Gi, is in S1 |channel of BSS i is (i,) −1

idle})n  · S=S˜



 Pr{node in Group Gi,S is in S1 |

{i},S˜ ⊆M \{i} S=

channel of BSS i is idle})n

(i,S) π˜ T

2τT

α (i,S) (2 p (i) W (i)

− 1)

.

(9)

3403

(i,S)

·





 j ∈M S=S˜ {i, j }, j =i S⊆M ˜ \{i, j }

 Pr{node in Group G j,S is in S1 |channel of BSS i is idle})n

( j,S)

.

(10)

With a large group size n (i,)  1, n (i,) −1 ≈ n (i,) . (10) can be approximately written as    p(i) ≈ Pr{node in Group G j,S is in S1 | j ∈M

 S=S˜ {i, j } ˜ S⊆M \{i, j }

channel of BSS i is idle})n

( j,S)

. (11)

If the channel of BSS i is idle, each node in Group G j,S , j ∈ M , S ⊇ {i }, must be in either state S1 or S2 with the probabilities Pr{node in Group G j,S is in S1 } =

K 

( j,S)

π˜ Rk

  ( j,S) 1 − rk ,

k=0

(12) Pr{node in Group G j,S is in S2 } =

K 

( j,S) ( j,S)

π˜ Rk rk

,

(13)

k=0 ( j,S)

respectively, where π˜ Rk is the probability of the HOL packet ( j,S) is the conditional of Group G j,S being in state Rk . rk

3404

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 8, AUGUST 2017

probability that a state-Rk HOL packet of Group G j,S makes a transmission request given that the channel of BSS i is idle, which can be obtained as

2) Fully-Overlapped: In a fully-overlapped network, the coverage area of each BSS fully overlaps with others’. In this case, each node can be heard by all the APs in the network, and we have n (i,S) = 0 if and only if S = M , i ∈ M . The fixed-pointed equation of p(i) in (16) can be then written as ⎧

⎨  n ( j,M ) α ( j,M ) 1− p( j ) (i) ( j,M ) τT +τ F · ( j ) − (i,{i}) (i) / α p ≈ exp ⎩ α p p j ∈M p( j ) 1 1 + W ( j) + (i) ( 2p 2 p j) − 1



p( j ) ( j) K , (18) + 1 − ( j) (2 − 2 p ) 2p − 1

( j,S)

rk

=

2 1+

( j) Wk

·

α ( j,S) , α (i,{i})

(14)

k = 0, . . . , K . By substituting (12)-(13) into (11), the steadystate probability of successful transmission of HOL packets of BSS i given that the channel of BSS i is idle, p(i) , can be obtained as   ⎫n( j,S) ⎧ ( j,S) ( j,S) K ⎬ 1 − rk   ⎨ k=0 π˜ Rk p(i) = . (15)  ( j,S) K ⎩ ⎭  π ˜ ˜ k=0 Rk j ∈M S=S {i, j } ˜ S⊆M \{i, j }

By combining (1), (3), (5) and (14), we have ⎫ ⎧ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨   −1 (i) ( j,S) ( j,S) p ≈ exp , (16) · n π ˜ T ⎪ ⎪ τT α (i,{i}) p(i)  ⎪ ⎪ ˜ j ∈M S=S {i, j } ⎭ ⎩ S˜ ⊆M \{i, j }

i ∈ M , by applying (1 − x)n ≈ exp{−nx} for 0 < x < 1.4 It can be clearly seen from (16) that the M fixed-point equations are closely dependent on the group sizes {n (i,S) }. According to different values of group sizes, the network can be categorized as non-overlapped, fully-overlapped and partially-overlapped, respectively. 1) Non-Overlapped: Let us first consider a non-overlapped network, where the coverage area of each BSS does not overlap with others’. In this case, each node can only be heard by its associated AP, and we have n (i,S) = 0 if and only if S = {i }, i ∈ M . The fixed-point equation of p(i) in (16) can be then written as     (i) p ≈ exp n (i,{i}) / α (i,{i}) τT p(i) + τ F (1 − p(i) ) 1 p(i) (i) + 1+W 2 2 p (i) − 1



p (i) (i) K , (2 − 2 p ) + 1 − (i) 2p − 1 (17) by substituting (8) into (16). For each i ∈ M , (17) has one (i),N single non-zero root p A . We can clearly see from (17) that (i),N the steady-state points { p A } of M BSSs are independent of each other. In this case, the network can be regarded as a combination of M independent single-BSS networks, and (17) is consistent with the steady-state point of a singleBSS IEEE 802.11 DCF network, which was derived in [36, eq. (25)]. 4 Note that (16) is derived based on the assumption that the group size

n ( j,S) , j ∈ M , S ⊆ M , is large. With a small group size, i.e., n ( j,S) < 5, for instance, the approximation error may become noticeable. It, nevertheless, rapidly diminishes as the group size grows.

by substituting (8) into (16), where α ( j,M ) denotes the probability of sensing the channels of all the BSSs idle. As the transmissions of all the nodes can be heard by any AP in a fully-overlapped network, all the BSSs have an identical channel. α ( j,M ) is then equal to the probability that this channel has neither a successful transmission nor a collision, which can be written as α ( j,M ) =

1 , 1 + τ F − τ F p( j ) − (τT − τ F ) p( j ) ln p( j )

(19)

by following the derivation presented in [36, Appendix B]. By substituting (19) into (18), we can see that the M fixedpoint equations are identical. Denote the single non-zero root of (18) as p FA . It can be clearly seen from (18) that p FA is determined by the number of nodes n ( j,M ) of each BSS in the network and the backoff parameters including the initial backoff window sizes W ( j ) and the cutoff phase K . 3) Partially-Overlapped: If the network does not fall into the above two categories, we have a partially-overlapped network, where some nodes can only be heard by their associated AP while those in the overlapping areas of multiple BSSs can be heard by multiple APs. In this more general case, the fixedpoint equation of (16) can be written as ⎧ ⎪ ⎪ ⎨  n (i,S) α ( j,S) (i) − (i,{i}) (i) / p ≈ exp ⎪ α p ⎪ ⎩ j ∈M S=S˜ {i, j } ˜S⊆M \{i, j }

1  1 − p( j ) ( j,S) τT + τ F · + 1 + W ( j )· α p( j ) 2 p( j ) ⎫



⎪ ⎪ ⎬ p( j ) p( j ) ( j) K ) + 1− (2 − 2 p , ⎪ 2 p( j ) − 1 2 p( j ) − 1 ⎪ ⎭ (20) by substituting (8) into (16), where α (i,S) denotes the probability of sensing the channels of BSS set S idle by nodes in Group Gi,S , which can be written as  α ( j,{ j }) . (21) α (i,S) = j ∈S

GAO et al.: THROUGHPUT OPTIMIZATION OF MULTI-BSS IEEE 802.11 NETWORKS WITH UNIVERSAL FREQUENCY REUSE

By substituting (21) into (20), we can see that in contrast to the non-overlapped and fully-overlapped cases, here M fixed-point equations need to be jointly solved to obtain the steady-state points { p (i),P }. It should be pointed out that (21) is based on A an implicit assumption that the channel of BSS i being sensed idle is independent of the channel of BSS j being sensed idle, i = j . When the coverage area of BSS i overlaps with that of BSS j , however, both channels would be sensed busy when nodes in their overlapping area transmit. It can be expected that due to ignoring the dependency of channel sensing, deviation would be observed especially when the number of nodes in the overlapping area is large. A detailed discussion on the validity of this assumption will be presented in Section IV. III. T HROUGHPUT O PTIMIZATION In this section, we will characterize the throughput performance of an M-BSS IEEE 802.11 DCF network at the steadystate points, and demonstrate how to maximize the network throughput by properly selecting the backoff parameters. A. Node Throughput, BSS Throughput and Network Throughput In this paper, the node/BSS/network throughput is defined as the average number of successfully transmitted packets per time slot of a node/BSS/network. With the collision model, as there is at most one successfully transmitted packet in each time slot, it is also the percentage of time that is used for successful transmissions. Denote λ(i,S) out as the throughput of a node in Group Gi,S . The throughput of BSS i can be then written as  λˆ (i) n (i,S) λ(i,S) (22) out = out ,  S=S˜ {i} S˜ ⊆M \{i}

which is the sum of throughput of all the nodes associated with AP i . The total network throughput, which is the sum of throughput of all the BSSs, can be further written as  (i)   (i,S) λˆ out = n (i,S) λout . (23) λˆ out = i∈M

i∈M

 S=S˜ {i} ˜ S⊆M \{i}

In saturated conditions, the throughput of each node is deter(i,S)

(i,S)

(i,S)

mined by its service rate, i.e., λout = π˜ T , where π˜ T is given in (8). The BSS throughput and the network throughput can be then obtained by substituting (8) into (22) and (23), respectively.

Note that although the network throughput λˆ out is obtained as an explicit function of the initial backoff window sizes {W (i) } (i) and the steady-state points { p A } by combining (8) and (23), (i) for given {W (i) }, the steady-state points { p A } need to be numerically calculated by jointly solving M fixed-point equations given in (20) in a general partially-overlapped network. It is therefore difficult to search for the optimal initial backoff window sizes {Wm(i) }. To efficiently solve (24), we propose to first convert it into an equivalent M-variable optimization problem: λˆ max = λˆ out , max (25) (i)

0< p A