Achieving Maximum Throughput in Random Access ... - IEEE Xplore

IEEE TRANSACTIONS ON MOBILE COMPUTING,

VOL. 13,

NO. 3,

MARCH 2014

497

Achieving Maximum Throughput in Random Access Protocols with Multipacket Reception Yun Han Bae, Bong Dae Choi, Member, IEEE, and Attahiru S. Alfa, Member, IEEE Abstract—This paper considers random access protocols with multipacket reception (MPR), which include both slotted-Aloha and slotted -persistent CSMA protocols. For both protocols, each node makes a transmission attempt in a slot with a given probability. The goals of this paper are to derive the optimal transmission probability maximizing a system throughput for both protocols and to develop a simple random access protocol with MPR, which achieves a system throughput close to the maximum value. To this end, we first obtain the optimal transmission probability of a node in the slotted-Aloha protocol. The result provides a useful guideline to help us develop a simple distributed algorithm for estimating the number of active nodes. We then obtain the optimal transmission probability in the -persistent CSMA protocol. An in-depth study on the relation between the optimal transmission probabilities in both protocols shows that under certain conditions the optimal transmission probability in the slotted-Aloha protocol is a good approximation for the -persistent CSMA protocol. Based on this result, we propose a simple -persistent CSMA protocol with MPR which dynamically adjusts the transmission probability  depending on the estimated number of active nodes, and thus can achieve a system throughput close to the maximum value. Index Terms—Multipacket reception, slotted-Aloha, CSMA, optimal transmission probability, WLAN, 802.11 DCF

Ç 1

INTRODUCTION

1.1

Motivation access protocols based on carrier sensing or noncarrier sensing have been used for coordinating the channel access of users in various wireless networks. One of the representative noncarrier sensing protocols is the slottedAloha. Various forms of slotted-Aloha protocols are widely used in most of the current digital cellular networks, such as the Global System for Mobile Communications (GSM),1 and in RFID systems as one of the popular anticollision algorithms. CSMA-based algorithms constitute a heart of contemporary wireless media access control technology. In the last decades, various enhancements have been introduced to basic CSMA schemes, first of all, to support collision avoidance. One of the widely used CSMA algorithms is the -persistent CSMA protocol. The performance analysis of the -persistent CSMA has gained a renewed interest recently, since the behavior of many CSMA protocols such as IEEE 802.11 DCF [2] and IEEE 802.15.4 [3] might be studied by a corresponding -persistent CSMA protocol. With advanced PHY-layer multipacket reception techniques, it is possible for a receiver to receive multiple packets transmitted concurrently. This new concept, referred to as multipacket reception (MPR), opens up new possibilities for

R

ANDOM

1. In the GSM network, the control channels of the TDM channels use slotted-Aloha.

. Y.H. Bae and B.D. Choi are with the Department of Mathematics Education, Sangmyung University, 7, Hongji-dong, Jongno-gu, Seoul 110743, Korea. E-mail: [email protected], [email protected]. . A.S. Alfa is with Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada. E-mail: [email protected]. Manuscript received 13 Mar. 2012; revised 9 Aug. 2012; accepted 6 Dec. 2012; published online 12 Dec. 2012. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference IEEECS Log Number TMC-2012-03-0125. Digital Object Identifier no. 10.1109/TMC.2012.254. 1536-1233/14/$31.00 ß 2014 IEEE

enhancing the capacity of wireless networks. Some recent efforts for standardization [1] and system design [5] to implement MPR have convinced researchers that it will not take much time to deploy MPR-based wireless networks in the near future. When the MPR-capability is employed in the physical layer, it is expected that MAC layer behaves differently from what is commonly believed in the MAC protocols with conventional single-packet reception. Therefore, to fully exploit the MPR-capability, it is important to find the optimal transmission probability of a node to maximize the system throughput in random access protocols with MPR. In addition, it is strongly required to develop a random access protocol with MPR which achieves a throughput close to the maximum value.

1.2 Related Work and Contribution of This Paper The performance of random access protocols such as the slotted-ALOHA and the -persistent CSMA is characterized by the transmission probability of a node. System throughput depends on the transmission probability of a node. We can expect that there might exist a transmission probability (called optimal transmission probability) that maximizes the system throughput. Under the assumption of single-packet reception capability, much effort has been devoted to figuring out the optimal transmission probability of a node. The performance of random access protocols with singlepacket reception has been extensively investigated and well understood during the last decade. In particular, since the seminal work of Bianchi [9], there have been extensive efforts to characterize the performance of 802.11-based WLANs under saturation and nonsaturation conditions, respectively, for example, see [11], [10], [12], [13]. In the context of study on random access protocols with MPR, Ghez et al. [15], [16] made the first attempt to model a general MPR channel in random-access-based wireless networks, in which stability properties of conventional slotted-Aloha with MPR were studied under an infinite-node Published by the IEEE CS, CASS, ComSoc, IES, & SPS

498

IEEE TRANSACTIONS ON MOBILE COMPUTING,

assumption. Their study was extended to CSMA protocols by Chan et al. in [17] and to Aloha systems with MPR under a finite-node assumption by Naware et al. [18]. Chan et al. [17] focused on the value of the maximum stable throughput and extended the framework in [15], [16] to grasp the effects of CSMA with MPR and then found that carrier sensing facilitates an improvement in capacity. Recently, [19] investigated a system throughput in contention-based MAC protocols with MPR. Chan et al. [14] investigated the impact of MPR on CSMA protocol and proposed a performanceimproving cross-layer designed CSMA protocol for wireless networks with MPR. Zhang et al. [19] showed that when the channel has an M-MPR capability, which means that the AP can decode successfully up to M simultaneous packet transmissions, the throughput scales linearly with M in the slotted-Aloha protocol with an infinite population and a finite population. Specifically, their result implies that the achievable throughput per unit cost (MPR-capability) increases with the MPR capability of the channel. Their result provides a strong incentive to deploy MPR in the next generation wireless networks. Zhang et al. extended their investigation in [19] to the nonsaturation case [20] and showed that superlinear scaling also holds for safe-bounded-mean-delay and safe-bounded-delay-jitter throughput, which are defined as the maximum throughput that can be safely sustained with finite-mean delay and delay jitter, respectively. Among all the works on MPR mentioned above, some [14], [19], [20], [24] focused on the theoretical analysis of the achievable maximum throughput. The others [15], [17], [18] dealt with the stability properties of MAC protocols with MPR. Little attention has been devoted to the study of the property of the optimal transmission probability which maximizes system throughput. Besides, the relationship between the optimal transmission probabilities for both the slotted-Aloha and the -persistent CSMA protocols with MPR has not been addressed so far. To the best of our knowledge, this paper is the first attempt to deal with the optimal transmission probability to maximize the system throughput in random access protocols, including both the slotted-Aloha and the -persistent CSMA protocols with MPR. The optimal transmission probability clearly depends on the number of active nodes which compete with each other for the channel access. Apart from the historical works presented above, some researchers have focused on designing dynamic random access protocols which dynamically adjust the transmission probability of a node (e.g., via tuning the backoff window size) depending on the estimated number of active nodes in the network. To perform such a tuning optimally in a way that the MAC protocol capacity is close to its theoretical bound, it is essential to design an algorithm which is able to estimate the number of active nodes at runtime. To realize such algorithms under single-packet reception assumption, the main idea adopted in the literature [6], [7], [8] is to use a feedback from the channel status to tune the transmission probability. The algorithm proposed in [8] computes an estimate of the collision cost and of the number of active nodes, which are obtained by observing the three events that may occur on the channel: idle slots, collisions, and successful transmissions. It seems that the extension of the algorithm to the MPR case is not straightforward. Different

VOL. 13,

NO. 3,

MARCH 2014

from the algorithm in [8], we introduce a new metric, which turns out to be a quantity related to the length bias concept, to develop an algorithm for estimating the number of active nodes at runtime. Our proposed algorithm simply requires each node to obtain the information on the number of nodes involved in each successful transmission slot. Moreover, under certain conditions the proposed algorithm works well in both the slotted-Aloha and the -persistent CSMA protocols with MPR. In this paper, we consider random access protocols with MPR, where there are an access point (AP) and multiple nodes in a wireless network. The AP has an M MPRcapability. The study includes both the slotted-Aloha and the -persistent CSMA protocols. For both protocols, each node makes a transmission attempt with a given probability in a generic slot. The goals of this paper are to figure out the relation between the optimal transmission probabilities for both protocols and, based on the result, to develop a random access protocol which achieves a throughput close to the maximum value in the MPR scenario. The main contributions of this work are summarized as follows: First, we derive the optimal transmission probability in the slotted-Aloha with MPR. The result reveals that the optimal transmission probability in the slotted-Aloha protocol with MPR is directly related to a discrete version of the length biasing concept. . Second, based on the derived optimal transmission probability in the slotted-Aloha protocol with MPR, we provide a simple distributed algorithm for estimating the number of active nodes (i.e., nodes that have packets ready for transmission) at runtime. By observing the number of successful transmissions in a slot, each node can get an estimate of the number of active nodes and use this estimate to tune its transmission probability. The results show that the proposed algorithm can estimate the real value that we want to find. . Third, we obtain the optimal transmission probability in the -persistent CSMA protocol with MPR under the general settings Ti < Ts and Ti < Tc , where Ti , Ts , and Tc is the length of an idle slot, a successful transmission slot, and a collision slot, respectively. We show that under certain conditions the optimal transmission probability in the slottedAloha protocol with MPR is a good approximation of the one in the -persistent CSMA protocol with MPR. This finding is very useful because it allows us to directly apply the proposed algorithm for estimating the number of active nodes in the slotted-Aloha protocol to the case of the -persistent CSMA protocol with MPR. . Finally, to improve the system throughput in the MPR case, we propose a simple -persistent CSMA protocol with MPR, where the transmission probability  of a node is dynamically and optimally tuned based on the estimated number of active nodes in the network. The proposed protocol can achieve a system throughput close to the theoretical maximum value. The remainder of this paper is organized as follows: The system model is presented in Section 2. In section 3, we .

BAE ET AL.: ACHIEVING MAXIMUM THROUGHPUT IN RANDOM ACCESS PROTOCOLS WITH MULTIPACKET RECEPTION

derive the optimal transmission probability in the slottedAloha and then propose a new algorithm for estimating the number of active nodes at runtime. Section 4 provides the optimal transmission probability in the -persistent CSMA protocol. In addition, we show that the optimal transmission probability in the -persistent CSMA protocol is well approximated by the one in the slotted-Aloha protocol. In Section 5, we investigate the performance of IEEE 802.11based WLAN with MPR and then show that IEEE 802.11 DCF with MPR gives a poor throughput. In Section 6, we propose a simple -persistent CSMA protocol which achieves a throughput close to the maximum value in WLANs with MPR. Some discussion is given in Section 7, and finally Section 8 concludes this paper.

2

SYSTEM MODEL

We consider a fully connected one-hop wireless network consisting of N active nodes (nodes that always have packets ready for transmission) and an AP. For the time being, for both the slotted-Aloha and the -persistent CSMA protocols, we begin with the assumption that the number N of active nodes is known to all the nodes, and then this assumption will be relaxed. The time is divided into slots and packet transmissions start only at the beginning of a slot. After each transmission, we assume that the transmitting nodes know the result of the transmission whether it is successful or not. In addition, all the nodes in the network get the information on how many nodes are involved in each successful transmission by employing an appropriate feedback mechanism. The AP has the capability to decode up to M simultaneous transmissions. The MPR model used in this paper is described by X ¼ Y 1fY Mg ;

3.1

Optimal Transmission Probability in Slotted-Aloha Protocol with MPR We assume that the number N of active nodes is known a priori. In the slotted-Aloha protocol, the channel time is divided into time slots of an equal length. We assume that the length of a packet is constant. A time slot is long enough to accommodate a packet transmission and the corresponding acknowledgement packet. Each node attempts to transmit a packet with a probability  at the beginning of a time slot, 0 <  < 1. Since each node makes a transmission attempt with probability  in a time slot, Y follows a binomial distribution with parameters N and . Thus, its distribution is given by IPðY ¼ kÞ ¼ N Ck  k ð1  ÞNk . Let X denote the number of packets received successfully in a time slot. Then, the probability mass function of X is given by IPðX ¼ 0Þ ¼ IPðY ¼ 0 or Y > MÞ ¼ ð1  ÞN þ

N X

N Ck 

k

ð1  ÞNk

k¼Mþ1

IPðX ¼ kÞ ¼ IPðY ¼ kÞ ¼ N Ck  k ð1  ÞNk ; for 1  k  M: ð2Þ Then, for a given , the system throughput Saloha ðÞ, which is measured in the number of successful transmissions in a slot, is obtained as Saloha ðÞ ¼

M X

kIPðX ¼ kÞ ¼

k¼1

M X

k  N Ck  k ð1  ÞNk :

ð3Þ

k¼1

We want to find the optimal transmission probability  , which maximizes a system throughput Saloha ðÞ. aloha Differentiating Saloha ðÞ with respect to , we have

ð1Þ

where M is the MPR capability of the AP, Y is the number of nodes which make a transmission attempt in a time slot, X is the number of packets successfully received by the AP in a time slot, and 1fg is the indicator function. The above model means that the AP can receive up to M packets simultaneously, and concurrent transmissions more than M packets result in collisions and so all the packets are lost. Note that the conventional single-packet reception model is the special case of M ¼ 1. When N  M, it is obvious that the optimal transmission probability which enables the maximal throughput to be achieved is equal to 1. To obtain a nontrivial result, we assume N > M.

3

499

M dSaloha ðÞ X ¼ kN Ck  k1 ð1  ÞNk1 ðk  NÞ: d k¼1

ðÞ ðÞ It is easy to see dSaloha > 0 for   1=N and dSaloha qðuÞ for any u þ v; v 2 ð0; 1Þ, and u > 0 and v > 0. Letting bk ¼ kak , then we have qðu þ vÞ  qðuÞ ( ) M M M M X X X X k k k k ¼ bk u  kbk ðu þ vÞ  kbk u  bk ðu þ vÞ k¼1

k¼1

N

M X

k¼1

k¼1

) M X k k bk ðu þ vÞ bk u :

k¼1

k¼1

ð7Þ Since the denominator of (7) is always positive, we only have to show the numerator of (7) is positive. Denoting the numerator of (7) by cðu; vÞ, simple algebra then yields ! ! M M X M X X j i cðu; vÞ ¼ bi u  bj ðu þ vÞ i¼1



k¼1 j¼k

M X M X

!

bj u

k¼1 j¼k

¼

M X k¼1



"

:

j¼k

It is easy to see that

,(

j

M X

bi u

j¼k

i

bj u

! bi ðu þ vÞ

i¼1



M X



M X i¼1

i

!

bj ðu þ vÞ

j¼k

! j

M X



!

i¼1 M X

j

j

ðÞ ¼

M  X

bi ui bj ðu þ vÞj  bi ðu þ vÞi bj uj



j¼k

¼

M X

i

i

bi bj u ðu þ vÞ ðu þ vÞ

ji

u

ji



ð9Þ :

j¼k

Since j > i and ðu þ vÞji > uji . Thus, we have ðÞ > 0. The proof of the third statement can be done trivially from the second result. u t Remark 1. It seems that dd hðÞ < 1 holds, which is assumed in Theorem 1-(3). We can prove that dd hðÞ < 1 for M ¼ 1; 2; 3, but we cannot prove it for an arbitrary M because it involves extremely complex algebraic manipulations. To show the plausibility that dd hðÞ < 1 holds, a numerical example is presented in Fig. 1, which plots hðÞ for the varying  and M when the number N of contending nodes is 10. To emphasize that hðÞ is a function of  and M for a given N, let hð; MÞ ¼ hðÞ. We see from the following facts that dd hðÞ < 1 holds in the numerical example: .

.

First, we note that for a fixed , hð; MÞ increases with M. This is because IE½Z increases with M as explained in Remark 3. Second, when M ¼ 10, i.e., N ¼ M, then 1 hð; 10Þ ¼ N1 N  þ N . This is due to the following reasoning: when M ¼ N, Y and X are identically distributed where Y has binomial distribution with parameters N and ; therefore, IE½X;  ¼ N, Var½X;  ¼ Nð1  Þ, a n d s o IE½X2 ;  ¼ Nð1  Þ þ N 2  2 ; hence,

!#

hð; 10Þ ¼

i

bi ðu þ vÞ

Thus,

d d

IE½X2 ;  N  1 1 ¼ þ : NIE½X N N

hð; 10Þ  N1 N < 1 for any M.

BAE ET AL.: ACHIEVING MAXIMUM THROUGHPUT IN RANDOM ACCESS PROTOCOLS WITH MULTIPACKET RECEPTION

501

which explains the fact that the successful transmission slot including the tagged packet involves more packet transmissions than an ordinary successful transmission slot.

3.2

IE½X; for the varying M and  when the Fig. 1. y ¼ hðÞ ¼ IE½X ;= N number of nodes is 10. 2

. Third, for any M, dd hð; MÞ  dd hð; 10Þ  N1 N 0Þ ¼ M k¼1 N Ck  ð1  Þ where X is the number of packets successfully received by AP in a generic time slot. Let Pc be the probability that a given slot is a collision slot. Then, Pc  IPðY > MÞ ¼ PN Nk k , and clearly Pi þ Ps þ Pc ¼ 1. k¼Mþ1 N Ck  ð1  Þ Define system throughput SðÞ as the average amount of bits transmitted successfully per second. Then, SðÞ can be obtained as the ratio between the average payload bits transmitted per generic slot to the average length of a generic slot as follows: P Pdata M k¼1 kIPðX ¼ kÞ SðÞ ¼ P T þ Ps Ts þ Pc Tc ( i i ) M X Nk k ¼ Pdata k  N Ck  ð1  Þ (

k¼1

Ti ð1  ÞN þ Ts

M X

ð14Þ N Ck 

k¼1

þ Tc

N X

N Ck 

k

k

ð1  ÞNk

) Nk

ð1  Þ

:

k¼Mþ1

We want to find the optimal transmission probability   that maximizes the system throughput SðÞ by solving d d SðÞ ¼ 0. The following result gives a functional equation for such .

504

IEEE TRANSACTIONS ON MOBILE COMPUTING,

Theorem 2. A solution of dd SðÞ ¼ 0 satisfies the following equation: 0 1, 2 2 gðÞ  ðTs  Tc Þ ðIE½X;Þ 2 IE½X ;  IE½X ; A  N; ð15Þ  ¼@ IE½X;  Tc where gðÞ ¼ Pi Ti þ Ps Ts þ Pc Tc . Proof. For notational convenience, let ak ¼ N Ck , fðÞ  PM Nk k , and k¼1 kak  ð1  Þ gðÞ  Ti ð1  ÞN þ Ts

M X

þ Tc

MARCH 2014

Let

ak  k ð1  ÞNk

kðÞ ¼

ak  k ð1  ÞNk :

k¼Mþ1 df dg Noting that, dSðÞ d ¼ 0 () d gðÞ  fðÞ d ¼ 0, simple algebraic manipulations yield

Theorem 3.

2.

k¼1

   IE½X 2 ;  Ti 1  Pi 1  N: IE½X;  Tc

The following result shows the uniqueness of the solution of (18).

1.

df dg gf ¼0 d d M X () gðÞ k2 ak  k1 ð1  ÞNk1 M X

NO. 3,

We discuss about the uniqueness of the solution of (15) to obtain the optimal transmission probability which maximizes SðÞ. Note that, if Ti ¼ Tc ¼ Ts , then the -persistent CSMA becomes the slotted-Aloha, and thus (15) reduces to (5). We consider two cases separately; Case 1: Ti < Ts ¼ Tc and Case 2: Ti < Tc < Ts . First, we consider Case 1. Then, (15) reduces to    IE½X 2 ;  Ti 1  Pi 1  N: ð18Þ ¼ IE½X;  Tc

k¼1 N X

VOL. 13,

Equation(18) has at least one solution and kðÞ is a monotone increasing function of . 2 If dd 2 kðÞ < 0, which implies that kðÞ is concave in , then (18) has a unique solution and so SðÞ has a maximum value at such .

Proof. We prove the first statement. We have    k¼1 IE½X 2 ;  Ti Ti ! M N 1  Pi 1  N¼ >0 lim X X Nk1 Nk1 !0 IE½X;  T NT k k c c þ Ts ak  ð1  Þ þ Tc ak  ð1  Þ ð19Þ     IE½X 2 ;  Ti M k¼1 k¼Mþ1 1  P < 1: N ¼ lim 1  i M !1 IE½X;  X N Tc ¼ NgðÞ kak  k ð1  ÞNk1 Therefore, (18) has at least one solution by intermediate k¼1 value theorem. Note that, ð1  Pi ð1  TTci ÞÞ is a monotone M M X X Nk Nk1 k k1 increasing function of  and þ kak  ð1  Þ kak  ð1  Þ Ts þN

kak  k ð1  ÞNk Ti ð1  ÞN1

k¼1

þ Tc

k¼1 N X

kak 

k1

ð1  Þ

!

IE½X 2 ;  IE½X; 

Nk1

k¼Mþ1 2

is also a monotone increasing function of . Therefore, kðÞ is a monotone increasing function of . The proof of the second statement can be done trivially from the fact (19) together with the assumption. u t

IE½X ;  NIE½X;  NIE½X;  gðÞ þ gðÞ ¼ gðÞ ð1  Þ 1 1 N X þ IE½X;  Tc kak  k1 ð1  ÞNk1

()

k¼1

þ ðTs  Tc Þ

M X

2

!

kak  k1 ð1  ÞNk1 :

k¼1

ð16Þ Keeping in mind that IE½Y  ¼ N, (16) can be written as df dg gf ¼0 d d  IE½X2 ;  Tc () gðÞ ¼ IE½X;  IE½Y ;  ð1  Þ ð1  Þ  Ts  Tc þ IE½X;  ð1  Þ

IE½X 2 ;  IE½X; 

() IE½X2 ; gðÞ ¼ IE½X; ðTc N þ ðTs  Tc ÞIE½X; Þ; ð17Þ which completes the proof.

Remark 5. It seems that dd 2 kðÞ < 0 holds, which is assumed 2 in Theorem 3. We are able to prove that dd 2 kðÞ < 0 holds for M ¼ 1; 2. But, we cannot prove it for general M. To show the plausibility of the assumption, we provide some numerical examples. In Fig. 4, kðÞ is plotted for the varying M and  when the number N of nodes is 10. First, we see that the solution of (18) is unique, since Ti . Second, we observe that for a fixed kðÞ kð0þÞ ¼ NT c increases with M. This is because

u t

increases with M and Pi is constant over M. Third, note that, for a given M, kðÞ looks concave in . 2 Hereafter, we assume that dd 2 kðÞ < 0. Remark 6. (Fixed-Point Iteration) Let   be the unique solution of (18). The solution   can be found using the following fixed-point iteration method. Define

BAE ET AL.: ACHIEVING MAXIMUM THROUGHPUT IN RANDOM ACCESS PROTOCOLS WITH MULTIPACKET RECEPTION



kðM; NÞ ¼



505

IE½X2 ;  Ti  1  Pi 1  IE½X;  Tc



 Ts  Tc þ N: Tc ð21Þ

Using (20), in the subsequent sections we investigate approximations of the optimal transmission probabilities for two cases; Case 1 Ti < Ts ¼ Tc and Case 2 Ti < Tc < Ts .

4.2

Approximations of Optimal Transmission Probabilities for Case 1 and Case 2 In this section, we show that the optimal transmission probability in the -persistent CSMA protocol with MPR is well approximated by the one in the slotted-Aloha protocol with MPR. 2

; Ti Fig. 4. y ¼ kðÞ ¼ IE½X IE½X; ð1  Pi ð1  Tc Þ=N for the varying  and M when the number of nodes is 10.

n ¼ kðn1 Þ for n ¼ 1; 2; . . . and 0 2 ð  ; 1Þ. If kðÞ is concave, then this implies that kðÞ < 1 for  2 ð  ; 1Þ. Thus, the fixed-point iteration theorem ensures that n converges to   . We recommend to use the value 1 or a value close to 1 as an initial value 0 . Next, we consider Case 2; Ti < Tc < Ts . By checking many numerical examples, we observe that (15) has a unique solution when the difference Ts  Tc is small. We next obtain more useful version (20) which is equivalent to (15). Applying Chebyshev’s version of the second moment method, i.e., IPðU ¼ 0Þ 

IE½ðU  IE½UÞ2  IE½U2

:

where U is a nonnegative random variable, and keeping in mind that IPðX > 0Þ ¼ Ps , we have ðIE½XÞ2  Ps : IE½X 2  Let ðÞ ¼ Ps 

ðIE½XÞ2 ; IE½X2 

then 0    Ps , where  depends on a given . Then, (15) can be rewritten as 0 1 2  2 gðÞ  ðTs  Tc Þ ðIE½X;Þ 2 ; IE½X ;  IE½X @ A  N ¼ IE½X;  Tc 0 1 2  2 Pi Ti þ Ps Ts þ Pc Tc  ðTs  Tc Þ ðIE½X;Þ 2 IE½X ;  IE½X ; A  N ¼@ IE½X;  Tc   IE½X2 ;  Ti Ts  Tc  Ps þ Pc þ Pi þ  N ¼ IE½X;  Tc Tc    IE½X2 ;  Ti  Ts  Tc ¼ N: þ  1  Pi 1  IE½X;  Tc Tc ð20Þ To emphasize that the right-hand side of (20) is a function of M and N, we let

4.2.1 Case 1 (Ti < Ts ¼ Tc ) Since Pi þ Pc þ Ps ¼ 1, kðM; NÞ becomes    IE½X2 ;  Ti 1  Pi 1  N: kðM; NÞ ¼ IE½X;  Tc

ð22Þ

 Let ð1Þ be the optimal transmission probability for Case 1  is the optimal which is a solution of (22). Note that, aloha transmission probability in the slotted-Aloha protocol with MPR satisfying the equation  IE½X 2 ;  N: ¼ IE½X; 

Then, since IE½X2 ;  kðM; NÞ  IE½X; 

 N;

  it is obvious that ð1Þ  aloha . Note that, if M ¼ N, then    increases with M. This ð1Þ ¼ aloha ¼ 1. Also, note that ð1Þ  N implies Pi ¼ ð1  ð1Þ Þ decreases and is close to 0 for a   is close to aloha for a relatively large large M. Thus, ð1Þ M. On the other hand, let us consider the probability Pi for a fixed M. We investigate how Pi varies depending on the number N of nodes. Fig. 6 depicts Pi versus N for given M ¼ 4; 5; 7, where Pi is evaluated at  satisfying  N Þ . It is obvious to see that Pi (22), i.e., Pi ¼ ð1  ð1Þ decreases with M for a given N. It is worth noting that Pi 0 for M  4 regardless of N. This implies that the right-hand side of (22) is well approximated by

IE½X 2 ;  IE½X;     near ð1Þ . Thus, ð1Þ is close to aloha for M  4 regardless of N. Numerical results show below that as long as M  4 the  Þ achieved at the optimal  ¼ maximum throughput Sðð1Þ    ð1Þ is quite close to the one Sðaloha Þ achieved at  ¼ aloha . Figs. 5a and 5b display the maximum throughput versus MPR-capability M for the various number of active nodes in the case of Ts ¼ 500 and Ts ¼ 50, respectively. In figures, “approximation” means the maximum throughput  satisfying (5) SðÞ given by (14) achieved at the  ¼ aloha and “exact” means the actual maximum value of SðÞ  satisfying (20). As shown in Figs. 5a achieved at  ¼ ð1Þ  and 5b, note that as long as M  4, aloha serves as a good approximation for ð1Þ .

506

IEEE TRANSACTIONS ON MOBILE COMPUTING,

VOL. 13,

NO. 3,

MARCH 2014

Fig. 5. The maximum throughput (measured in the number of packets successfully transmitted per slot) versus MPR-capability M for different N in case of carrier sensing protocol of Case 1.

4.2.2 Case 2 ( Ti < Tc < Ts ) We first note that the relation between Ts and Tc in IEEE 802.11 DCF basic access mechanism falls on this case, where Ts , Tc , and Ti are given by (13). Specifically, Ti Ts , Ti Tc , Ts  Tc ¼ , Ti =Tc 0, and ðTs  Tc Þ=Tc ¼ Tc 0.  the optimal Let us consider (20). We denote by ð2Þ transmission probability for Case 2 which is a solution of (20). From (20), we have 



 IE½X2 ;  Ti N  kðM; NÞ 1  Pi 1  IE½X;  Tc   IE½X 2 ;  Ts  Tc   1þ N: IE½X;  Tc

ð23Þ

As in Case 1, we claim that the lower bound (the first term in (23)) of kðM; NÞ is close to IE½X2 ;  IE½X; 

 N

 in the neighborhood of ð2Þ regardless of N as long as M  4. Fig. 7a depicts the idle probability Pi versus N for  given M ¼ 4; 5; 7, where Pi is evaluated at ð2Þ satisfying (20) 5 with Ts ¼ Tc þ 1 and Tc ¼ 30. Obviously, Pi decreases with M for a given N. From Fig. 7a, we notice that Pi 0 at   ¼ ð2Þ regardless of N as long as M  4. This implies that the lower bound of kðM; NÞ in (23) is close to

IE½X2 ;  IE½X; 

IE½X2 ;  IE½X; 

 N

 in the neighborhood of  ¼ ð2Þ . From the above two observations,  IE½X2 ;  N kðM; NÞ IE½X;    and as a result ð2Þ aloha regardless of N as long as M  4. Figs. 8a and 8b show the maximum throughput versus ðTs  Tc Þ=Tc for the varying N, and Tc ¼ 50 in case of M ¼ 5 and M ¼ 9, respectively. In figures, “approximation” means the maximum throughput SðÞ given by (14) achieved at   ¼ aloha satisfying (5) and “exact” means the actual  maximum value of SðÞ achieved at  ¼ ð2Þ satisfying (20). As ðTs  Tc Þ=Tc increases, which means the difference Ts  Tc becomes big, the discrepancy between the approximated and exact values becomes big as expected. We can observe that as long as ðTs  Tc Þ=Tc < 0:4, the difference is very small, and thus the optimal transmission probability  aloha in the slotted-Aloha protocol can be used as a good  approximation of the optimal transmission probability ð2Þ in the -persistent CSMA protocol.

 N

 . Next, consider the upper bound in the neighborhood of ð2Þ of KðM; NÞ in (23). Note that, ðÞ  Ps . Fig. 7b displays  with N for a given M. We first how ðÞ varies at  ¼ ð2Þ notice that ðÞ decreases with M for a given N. For a given M, ðÞ increases with N, but it seems bounded below by a c is certain value less than Ps as N increases. Moreover, if TsTT c Ts Tc negligibly small, then  Tc 0. Thus, this implies that the upper bound of KðM; NÞ in (23) is well approximated by

5. This setting is the case of IEEE 802.11 DCF basic access mechanism.

Fig. 6. The probability Pi ¼ ð1  ÞN at  satisfying the equation 2 ; Ti kðM; NÞ ¼ IE½X IE½X; ð1  Pi ð1  Tc ÞÞ=N for a given M.

BAE ET AL.: ACHIEVING MAXIMUM THROUGHPUT IN RANDOM ACCESS PROTOCOLS WITH MULTIPACKET RECEPTION

507

 Fig. 7. Pi and ðÞ at  ¼ ð2Þ satisfying the (20) with Ts ¼ Tc þ 1 and Tc ¼ 30.

Fig. 8. The maximum throughput (measured in the number of packets successfully transmitted per slot) versus ðTs  Tc Þ=Tc for varying N when Tc ¼ 50 backoff slots.

With the practical settings of system parameters Ti , Ts , and Tc used in IEEE 802.11 DCF basic access mechanism, we next investigate how well the optimal transmission  satisfying (5) approximates the optimal probability aloha  in the -persistent CSMA transmission probability ð2Þ protocol with MPR. The system parameters are presented in Table 1 and are adopted from IEEE 802.11g. Fig. 9 compares the maximum approximated system throughput and the exact system throughput for the -persistent CSMA protocol with the settings of system parameters Ti , Ts , and Tc given by (13) for the varying MPR-capability M, when N ¼ 10; 40 and packet payload size Pdata ¼ 8184 bits, 4,092 bits. Specifically, in Fig. 9, “exact” denotes the actual maximum throughput of SðÞ given by (14), i.e.,   Þ achieved at  ¼ ð2Þ , and the throughput value Sðð2Þ  Þ “approximated” denotes the throughput value Sðaloha  achieved at  ¼ aloha satisfying (5). As shown in Fig. 9, as long as M  4, with the practical settings of system parameters used in IEEE 802.11g, we again see that there is a good match between the approximated and actual  serves as a good values. Keeping in mind that aloha

 approximation for ð2Þ as long as ðTs  Tc Þ=Tc 0, the  transmission probability aloha satisfying (5) can be used as a good approximation for the optimal transmission  probability ð2Þ for the -persistent CSMA protocol with

TABLE 1 System Parameters Used in Carrier Sensing Protocol (IEEE 802.11g)

508

IEEE TRANSACTIONS ON MOBILE COMPUTING,

VOL. 13,

NO. 3,

MARCH 2014

  Fig. 9. The comparison of the approximated throughput (the value Sðaloha Þ achieved at  ¼ aloha satisfying (5)) and the exact maximum throughput   (the throughput value Sðð2Þ Þ achieved at  ¼ ð2Þ satisfying (15)) for the -persistent CSMA protocol with the settings of system parameters given by (13) when packet payload size is set as 8,184 bits and 4,092 bits, respectively.

system parameters such as IEEE 802.11 DCF basic access mechanism. Summary. We would like to conclude this section with the brief summary of the obtained results and the emphasis on its practical meaning. .

.

5

We have the result that in the MPR case, the optimal transmission probability   satisfying (15) in the -persistent CSMA protocol is very close to  satisfying (5) under the conditions; the one aloha c 0, and 2) M  4. This 1) Ti < Tc  Ts and TsTT c  satisfying (5) as a implies that we can use the aloha good approximation for the optimal transmission probability   , instead of resorting to solving (15), which is very complex,6 to obtain   in the -persistent CSMA protocol with MPR. The above result (A) gives us a useful implication in terms of practical algorithmic aspects. Recalling the proposed algorithm presented in Section 3.2, for the purpose of estimating the number of nodes the proposed algorithm does not require a node to obtain any additional information, such as idle slots and collision slots, except the estimates for the quantity IE½X2  IE½X2  IE½X . It is worth noting that IE½X is a invariant quantity that can be obtained regardless of whether a protocol is based on carrier sensing or not. Thus, the algorithm proposed in Section 3.2 can be directly applied to the -persistent CSMA protocol with MPR to estimate the number of active nodes. This issue will be discussed in Section 6.

 -PERSISTENT CSMA WITH MPR 802.11 DCF WITH MPR

VERSUS

For the conventional IEEE 802.11 DCF with single-packet reception, Bianchi [9] developed an analytical model to analyze the performance of the IEEE 802.11 DCF, and thus obtained the expression for saturation throughput. His 6. We use the term “complex” in the sense that we cannot prove that (15) for general case has a unique fixed solution.

derivation is based on modeling the stochastic behavior of a tagged node by a bidimensional embedded Markov chain represented by its backoff stage sðtÞ and backoff counter bðtÞ at time t. The key approximation made in [9] is that at each transmission attempt, each packet collides with constant probability p regardless of the number of retransmissions suffered, where p is referred to as conditional collision probability, meaning that this is the probability of a collision seen by a packet being transmitted on the channel. Based on this assumption, complex CSMA/CA protocol with BEB can be approximated as simple -persistent CSMA, where  is given by [9], [10] 2ð1  2pÞ ð1  2pÞðW þ 1Þ þ pW ð1  ð2pÞm Þ 2 ; ¼ P k W þ 1 þ pW m1 k¼0 ð2pÞ

¼

ð24Þ

where W is the minimum contention window size, m is the maximum backoff stage, and p ¼ ð1  ÞN1 . The steadystate behavior of IEEE 802.11 DCF with M-MPR capability can also be approximated by -persistent CSMA [19], where PN1 N1k k  is given by (24) with p : ¼ . k¼M N1 Ck  ð1  Þ Keeping in mind the result given in (24), for any given minimum contention window W and maximum backoff stage m in IEEE 802.11 DCF, if we choose  satisfying (24), then the system throughput for IEEE 802.11 DCF is well approximated by the one for the corresponding -persistent CSMA protocol. Thus, the system throughput of the optimal   -persistent CSMA protocol, where   is the solution satisfying (15) with the settings of system parameters Ti , Ts , and Tc given by (13), is a theoretical upper bound for the one of 802.11 DCF.

6

PROPOSED OPTIMAL  -PERSISTENT CSMA PROTOCOL WITH MPR

In this section, we develop a simple -persistent CSMA protocol with MPR which achieves the system throughput close to the maximum value with the configuration of system parameters such as Ti , Ts , and Tc used in IEEE 802.11. The operation principle of the proposed protocol is

BAE ET AL.: ACHIEVING MAXIMUM THROUGHPUT IN RANDOM ACCESS PROTOCOLS WITH MULTIPACKET RECEPTION

509

Fig. 10. The estimated number of active nodes versus update interval.

to adjust the transmission probability  optimally depending on the number of active nodes in the network. The proposed protocol requires each node to regularly update the number of active nodes at the end of every update interval by employing a proper feedback mechanism for each node to get the information7 on the number of nodes involved in each successful transmission slot. The update interval can be set as a multiple of the interval between two consecutive beacons (for example, a beacon interval is 100 ms as a default value in 802.11 WLAN). Consider the scenario that the number of active nodes in the network is unknown and varies over time. The proposed  algorithm is the aloha -persistent CSMA protocol which  tunes the transmission probability aloha depending on the estimate of the number of active nodes which can be obtained by using the algorithm proposed in Section 3.2. It is easy to see that the proposed protocol operates almost optimally if 1) the communication time until an active node completes the transmissions of its whole data packets is relatively long8 (e.g., longer than several update intervals) and 2) the protocol promptly reacts to the change in the number of active nodes in the network. Through simulation, we show below that, after the proposed protocol detects the change in the number of active nodes, it can estimate the number of active nodes promptly. We next investigate how well the proposed algorithm to estimate the number of active nodes works under the realistic WLANs environment. The system parameters used in carrier sensing protocols are presented in Table 1 and are adopted from IEEE 802.11g standards. In Fig. 10, two cases are considered; 1) L ¼ 300 ms and  ¼ 0:3, and 2) L ¼ 100 ms and  ¼ 0:7. The results are obtained under the following assumptions: M ¼ 5; all the nodes initially believe there are 50 active nodes (Note that, to emphasize the promptness of the proposed algorithm, a relatively large initial value 50 is set.); there are actually 7, 15, and 30 active nodes, respectively, which we want to estimate. Each figure contains 30 real trajectories to display the results of running 30 independent simulations. To reach a good compromise between accuracy and promptness of 7. One of the ways to implement it is to insert the information into the ACK packet. 8. This assumption is usually valid in the scenario of file upload of nodes.

the algorithm, as identified in Section 3.2, a small  ¼ 0:3 is used with a long update interval L ¼ 300 ms, and a relatively large  ¼ 0:7 with a short update interval L ¼ 100 ms. In Fig. 10, we notice that, starting from the initial value 50, the estimate for N sharply decreases to the real value. After the transient period, the algorithm is in quasi-stationary period and the estimate for N oscillates near the real value. It is worth noting that the error between the estimate and ideal value in the quasistationary period is at most three for all cases. Moreover, the length of the transient period is at most 1,000 ms (¼ 10  100 ms) in the case of L ¼ 100 ms and 1,200 ms in the case of L ¼ 300 ms, respectively. To investigate how the proposed algorithm promptly reacts to the change in the number of actual active nodes, we simulate the proposed algorithm under the following scenario: initially, there are 15 active nodes in the network; all the nodes assume that there might be 50 active nodes initially; in the middle of simulations, five new active nodes are injected into the network at the beginning of the 20th update interval and then five active nodes are terminated at the beginning of the 50th update interval. Fig. 11 plots the 30 real trajectories of results of running 30 independent simulations. The result shows that the proposed algorithm promptly copes with the change in the number of active nodes. Specifically, recalling that L ¼ 200 ms, after the change in the number of actual active nodes, the transient period (the time which it takes for the proposed algorithm to get accurate estimates close to the real value) is at most 1,000 ms. Besides, the error between the estimates and the real value in each quasi-stationary period is bounded above by two. We conclude this section with the following two remarks. Remark 7. The proposed optimal -persistent CSMA protocol can be also implemented on contention-window basis. Let W be the contention window size of a node. Then, from the optimal transmission   obtained, the contention window W can be determined by W ¼ d2  1e, where dae is the least integer not less than a. Remark 8. When M < 4, the approximation of the -persistent CSMA to the slotted-Aloha may not be accurate as identified in Section 4.2. We therefore need an iterative algorithm to obtain the optimal transmission

510

IEEE TRANSACTIONS ON MOBILE COMPUTING,

Fig. 11. The performance of the estimation algorithm with L ¼ 200 ms and  ¼ 0:6; initially N ¼ 15 and an initial guess for N is 50, in the middle of simulations five new active nodes are injected at the beginning of the 20th update interval, and then five active nodes are terminated at the beginning of the 50th update interval.

probability when M < 4. A heuristic algorithm which may be considered is as follows: As we can estimate IE½X2  and IE½X in each update interval, we can also estimate the probabilities Pi and Ps . Thus, a similar iteration method proposed in Section 3.2 may be applied using (15).

7

DISCUSSIONS

It is well-known and commonly believed that in the case of single-packet reception scenario the -persistent CSMA protocol outperforms the slotted-Aloha in terms of the achievable maximum throughput. In this section, we discuss whether this result is still valid in the MPR scenario or not. To this end, we consider three kinds of protocols; ideal-CSMA, CSMA-DCF, and slotted-Aloha as presented in Fig. 12. First, “ideal-CSMA” is the optimal   -persistent CSMA protocol, where Ti ¼ , Ts ¼ H þ Pdata =R þ SIF S þ ACK, and Tc ¼ H þ Pdata =R þ SIF S, and   is the solution satisfying (15). Therefore, the ideal-CSMA protocol is able to know immediately whether the channel is busy or not. Second, “CSMA-DCF” is the optimal   -persistent protocol, where Ti ¼ , Ts ¼ H þ Pdata =R þ SIF S þ ACK þ DIF S, and Tc ¼ H þ Pdata =R þ DIF S, and   is the solution satisfying (15). The only difference between the idealCSMA protocol and the CSMA-DCF protocol is that the ideal-CSMA protocol does not need DIFS for carrier sensing. Finally, “slotted-Aloha” is the slotted-Aloha pro tocol with the optimal transmission probability aloha , where  Ts ð¼ Ti ¼ Tc Þ ¼ H þ Pdata =R þ SIF S þ ACK and aloha is the solution satisfying (5). Fig. 12 plots the maximum throughput for the three protocols versus the MPRcapability M when data payload sizes are 8,184 bits and 4,092 bits, respectively, for N ¼ 20. We see that the idealCSMA protocol gives the best throughput among the three protocols for all cases. On the other hand, it is interesting to note that when M  3 the CSMA-DCF protocol outperforms the slotted-Aloha protocol, but with the increase of M the slotted-Aloha protocol gives higher throughput than the CSMA-DCF protocol regardless of the length of data

VOL. 13,

NO. 3,

MARCH 2014

Fig. 12. The comparison of the maximum system throughput between slotted-Aloha and -persistent CSMA protocols when Pdata ¼ 8;184 bits and Pdata ¼ 4;092 bits, respectively, for N ¼ 20.

payload. This result is expected because with the increase of M the DIFS overhead for carrier sensing more negatively affects the system throughput. In addition, it is worth noting that with the increase of MPR-capability M the throughput gap between the ideal-CSMA protocol and the slotted-Aloha becomes less. This finding provides us with a new guideline to be considered when designing MAC protocols in MPR scenario.

8

CONCLUSIONS

In this paper, we investigate the impact of MPR on the MAC layer behavior where the study is carried out by considering both slotted-Aloha and -persistent CSMA protocols. Based on the result, we eventually develop a simple -persistent protocol with MPR which can achieve a system throughput close to the maximum value. The main results obtained in this paper are summarized as follows: We first obtain the optimal transmission probability in slotted-Aloha protocol with MPR. We show that the optimal transmission probability in slotted-Aloha with MPR is related to the length bias concept. Second, the result 2  provides us with a useful metric (IE½X , where X is the IE½X number of nodes involved in each successful transmission slot) which enables each node to estimate the number of active nodes in a distributed manner. Third, we obtain the optimal transmission probability in the -persistent CSMA protocol with MPR. We show that under certain conditions the optimal transmission probability in the -persistent CSMA protocol is well approximated by the one in slottedAloha protocol. This finding allows us to directly apply the estimation algorithm developed in the slotted-Aloha protocol to the -persistent CSMA protocol. Thus, fourth we develop a -persistent CSMA protocol where the transmission probability of each node is dynamically and optimally tuned depending on the estimated number of active nodes. The proposed protocol can achieve a system throughput close to the maximum value in the case that the completion time of an active node (e.g., file transfer) is relatively long. As a by-product of our study, we identify that in the MPR scenario the -persistent CSMA protocol is not always superior to slotted-Aloha, contrary to what is known in the

BAE ET AL.: ACHIEVING MAXIMUM THROUGHPUT IN RANDOM ACCESS PROTOCOLS WITH MULTIPACKET RECEPTION

single-packet reception model. This finding is useful in that it provides a new guideline when designing a MAC protocol with MPR.

ACKNOWLEDGMENTS This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada and was supported by a National Research Foundation of Korea grant funded by the Korean Government (Ministry of Education, Science and Technology), Korea (NRF-3552011-1-C00015), and by National Research Foundation of Korea grants funded by the Korean goverment (MEST) (no. 2012-008099).

REFERENCES [1] [2] [3]

[4]

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15] [16] [17]

[18]

IEEE Std, Task Group 802.11ac Timetable, http://grouper.ieee.org/ groups/802/11/Reports/tgacupdate.htm, 2013. IEEE 802.11 WG, IEEE 802.11-2007, Wireless LAN MAC and PHY Specifications, Revised of IEEE 802.11-1999, IEEE LAN/MAN Standards Committee, June 2007. IEEE Std 802.15.4-2003, IEEE Standard for Information Technology Part 15.4: Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Low-Rate Wireless Personal Area Networks (LR-WPANs), IEEE, 2003. IEEE Std 802.15.1-2005, Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications for Wireless Personal Area Networks (WPANs), IEEE, http://standards.ieee.org/getieee802/ download/802.15.1-2005.pdf, June 2005. S. Yoon, I. Rhee, B.C. Jung, B. Daneshrad, and J.H. Kim, “Contrabass: Concurrent Transmissions without Coordination for Ad Hoc Networks,” Proc. IEEE INFOCOM, 2011. M. Gerla and L. Kleinrock, “Closed Loop Stability Control for S-Aloha Satellite Communications,” Proc. ACM Special Interest Group on Data Comm. (SIGCOMM), pp. 2.10-2.19, Sept. 1977. B. Hajek and T.V. Loon, “Decentralized Dynamic Control of a Multiaccess Broadcast Channel,” IEEE Trans. Automatic Control, vol. 27, no. 3, pp. 559-569, June 1982. F. Cali, M. Conti, and E. Gregori, “IEEE 802.11 Protocol: Design and Performance Evaluation of an Adaptive Backoff Mechanism,” IEEE J. Selected Areas Comm., vol. 18, no. 9, pp. 1774-1786, Sept. 2000. G. Bianchi, “Performance Analysis of the IEEE 802.11 Distributed Coordination Function,” IEEE J. Selected Areas Comm., vol. 18, no. 3, pp. 535-547, Mar. 2000. A. Kumar, E. Altman, D. Miorandi, and M. Goyal, “New Insights from a Fixed Point Analysis of Single Cell IEEE 802.11 WLANs,” Proc. IEEE INFOCOM, vol. 3, pp. 1550-1561, Mar. 2005. G. Bianchi and I. Tinnirello, “Remarks on IEEE 802.11 DCF Performance Analysis,” IEEE Comm. Letters, vol. 9, no. 8, pp. 765767, Aug. 2005. Y. Xiao, “Performance Analysis of Priority Schemes for IEEE 802.11 and IEEE 802.11e Wireless LANs,” IEEE Trans. Wireless Comm., vol. 4, no. 4, pp. 1506-1515, July 2005. T. Sakurai and H.L. Vu, “MAC Access Delay of IEEE 802.11 DCF,” IEEE Trans. Wireless Comm., vol. 6, no. 5, pp. 1702-1710, May 2007. D.S. Chan and T. Berger, “Performance and Cross-Layer Design of CSMA for Wireless Networks with Multipacket Reception,” Proc. Record of the Thirty-Eighth Asilomar Conf Signals, Systems, and Computers, 2004. S. Ghez, S. Verdu, and S. Schwartz, “Stability Properties of Slotted ALOHA with Multipacket Reception Capability,” IEEE Trans. Automatic Control, vol. 33, no. 7, pp. 640-649, July 1988. S. Ghez, S. Verdu, and S. Schwartz, “Optimal Decentralized Control in the Random Access Multipacket Channel,” IEEE Trans. Automatic Control, vol. 34, no. 11, pp. 1153-1163, Nov. 1989. D.S. Chan, T. Berger, and L. Tong, “On the Stability and Optimal Decentralized Throughput of CSMA with Multipacket Reception Capability,” Proc. Allerton Conf. Comm., Control, and Computing, Sept./Oct. 2004. V. Naware, G. Mergen, and L. Tong, “Stability and Delay of Finite - User Slotted ALOHA with Multipacket Reception,” IEEE Trans. Information Theory, vol. 51, no. 7, pp. 2636-2656, July 2005.

511

[19] Y.J. Zhang, P.X. Zheng, and S.C. Liew, “How Does MultiplePacket Reception Capability Scale the Performance of Wireless Local Area Networks?” IEEE Trans. Mobile Computing, vol.8, no. 7, pp. 923-935, July 2009. [20] Y.J. Zhang, S.C. Liew, and D.R. Chen, “Sustainable Throughput of Wireless LANs with Multipacket Reception Capability under Bounded Delay-Moment Requirements,” IEEE Trans. Mobile Computing, vol. 9, no. 9, pp. 1226-1241, Sept. 2010. [21] M.H. Mahmood, C. Chang, D. Jung, Z. Mao, H. Lim, and H. Lee, “Throughput Behavior of Link Adaptive 802.11 DCF with MUD Capable Access Node,” Int’l J. Electronics and Comm., vol. 64, pp. 1031-1041, 2010. [22] B.J. Kwak, N.O. Song, and L.E. Miller, “Performance Analysis of Exponential Backoff,” IEEE/ACM Trans. Networking, vol. 13, no. 2, pp. 343-355, Apr. 2005. [23] M. Lotfinezhad, B. Liang, and E.S. Sousa, “Adaptive ClusterBased Data Collection in Sensor Networks with Direct Sink Access,” IEEE Trans. Mobile Computing, vol. 7, no. 7, pp. 884-897, July 2008. [24] R.H. Gau and K.-M. Chen, “Probability Models for the Splitting Algorithm in Wireless Access Networks with Multi-Packet Reception and Finite Nodes,” IEEE Trans. Mobile Computing, vol. 7, no. 12, pp. 1519-1535, Dec. 2008. Yun Han Bae received the BS, MS, and PhD degrees in mathematics from Korea University, Seoul, in 2003, 2005, and 2009, respectively. He is an assistant professor in the Department of Mathematics Education, Sangmyung University, Seoul, Korea. His research interests include queueing theory and its applications to communication systems, and performance analysis of protocols and wireless networks.

Bong Dae Choi received the BS and MS degrees in mathematics from Kyungpook University, Daegu, Korea, and the PhD degree in mathematics from Ohio State University, Columbus, 1980. He is a professor in the Department of Mathematics, Sungkyunkwan University, Seoul, Korea. He worked as a professor at the Korean Advaced Institute of Science and Technology from 1983 to 1999, and Korea University from 1999 to 2012. He received the Best Paper Award from the IEE in 2000 and the Seoul Culture Prize in science in 2001. He was an editor of the Journal of Communications and Networks and an associate editor of Queueing Systems, and is an associate editor of Telecommunication Systems. His research interests include queueing theory and its applications to communication systems. His current research interests include performance evaluation of IEEE 802.11, 15.4, 16e, power saving scheme, cognitive radio networks, and IEEE 802.11p/1609.4 WAVE. He has published about 115 papers in refereed journals. His papers have appeared in Queueing Systems, Journal of Applied Probability, Performance Evaluation, Telecommunication Systems, Computer Networks, and other IEEE, IEE, and IEICE publications. He is a fellow of Korea Academy of Science and Technology and a member of the IEEE. Attahiru S. Alfa is a professor in the Department of Electrical and Computer Engineering at the University of Manitoba, Winnipeg, Canada. He was the NSERC industrial research chair of Telecommunications from 2004 to 2012. His research interests include queueing and network theories with applications mostly to telecommunication systems, wireless communication networks with recent focus on cognitive radio networks, mobility, Internet traffic, channel modeling, stochastic models, performance analysis, and teletraffic forecasting models. He has contributed significantly in the area of matrix-analytic methods for stochastic models used in telecommunications. He belongs to the following organizations: APEGM and INFORMS. He is a member of the IEEE.