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Cooperative Pseudo-Bayesian Backoff Algorithms for Unsaturated CSMA Systems with Multi-Packet Reception Hu Jin, Member, IEEE, Jun-Bae Seo, Member, IEEE, and Victor C.M. Leung, Fellow, IEEE Abstract—This paper proposes efficient backoff algorithms for uplink multi-packet reception (MPR) capable IEEE 802.11 systems in order to maximize the system throughput. According to the proposed algorithms, each station (STN) estimates, in a Bayesian manner under an unsaturated channel traffic condition, the number of backlogged STNs sharing the multiple access channel to obtain an optimal (re)transmission probability. Additionally, an access point and associated STNs cooperate by exchanging information piggybacked in transmitted data packets and the corresponding acknowledgment packets. The mean and variance of the queuing delays of the proposed algorithms are extensively evaluated via simulations under various environments such as time-varying populations and various asymmetric traffic conditions and compared to those of the conventional binary exponential backoff (BEB) algorithm. Furthermore, the queuing performance of the proposed algorithms is compared to the queuing delay lower bound obtained from a system that has perfect knowledge of the backlog size. Numerical results demonstrate the robustness of the proposed algorithms in various environments, and that they outperform the BEB algorithm. Index Terms—CSMA, IEEE 802.11, multi-packet reception, backoff algorithm
Ç 1
INTRODUCTION
1.1
Motivations EEE 802.11 wireless local area networks (WLANs) have been widely deployed to provide the last hop to the Internet. Along with such widespread deployments, IEEE 802.11 standards have gradually evolved to increase the physical transmission capacity in order to meet the explosive demands on high-speed Internet access. In addition, due to low installation cost and easy implementation, IEEE 802.11 standards have been extended to cover wireless access in the vehicular networking environment and mesh networking [1]. For infrastructure WLANs, the recent IEEE 802.11n standard supporting single-user multiple-input multipleoutput (MIMO) transmissions and IEEE 802.11ac standard supporting multi-user MIMO (MU-MIMO) downlink transmissions between access points (APs) and user stations (STNs) are increasingly deployed as they support higher data rates than the older IEEE 802.11a/b/g standards. Furthermore, in order to meet increasing demands on uplink capacity, MU-MIMO for STN to AP transmissions in IEEE 802.11 systems has also been discussed [2], [3], in which packets simultaneously transmitted by multiple STNs can be successfully received by an AP; this capability is referred to as multi-packet reception (MPR). While the physical layer
I
H. Jin is with the Department of Electronics and Communication Engineering, Hanyang University, Ansan 426-791, Korea. E-mail:
[email protected]. J.-B. Seo and V.C.M. Leung are with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada. E-mail: {jbseo, vleung}@ece.ubc.ca.
Manuscript received 1 Oct. 2013; revised 15 Apr. 2014; accepted 21 Apr. 2014. Date of publication 30 Apr. 2014; date of current version 23 Dec. 2014. For information on obtaining reprints of this article, please send e-mail to:
[email protected], and reference the Digital Object Identifier below. Digital Object Identifier no. 10.1109/TMC.2014.2321374
capacity of MPR capable 802.11 systems would be improved by MIMO, the backoff algorithm in the medium access control (MAC) layer of such systems is still based on the existing binary exponential backoff (BEB) algorithm specified in the IEEE 802.11 standard, which has been developed for single-packet reception (SPR) systems. Since we have learned in [2], [4] that in IEEE 802.11 systems with SPR a proper choice of the initial contention window size of the BEB algorithm can significantly improve MAC layer performance, e.g., access delay, it is expected that in MPR capable IEEE 802.11 systems the initial contention window size would greatly affect MAC layer performance as well. Moreover, considering that multiple packets can be simultaneously received over the uplink in an MPR capable IEEE 802.11 system, one might reasonably expect that a more aggressive retransmission control scheme could be used to encourage retransmissions so that the MPR capability could be fully exploited. These expectations motivate us to investigate how well the conventional BEB algorithm works in MPR capable IEEE 802.11 systems and its performance limitations. The insights gained from this investigation motivate us to develop three enhanced backoff algorithms, which can overcome the possible performance limits of the BEB algorithm in MPR capable IEEE 802.11 systems. Our design assumes that the number of packets successfully received by an AP is proportional to the number of receive antennas owing to the use of MU-MIMO technology in the AP. In our proposed algorithms, STNs perform a Bayesian-like estimation on the number of backlogged STNs, and use an optimal transmission probability that is a function of the estimated backlog size. In particular, each data packet transmitted by an STN carries also a one-bit indication to the AP, based on which the AP cooperatively assists the STNs’ backlog estimation by providing feedback information in the corresponding acknowledgment (ACK) packet. Thus, the
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JIN ET AL.: COOPERATIVE PSEUDO-BAYESIAN BACKOFF ALGORITHMS FOR UNSATURATED CSMA SYSTEMS WITH MULTI-PACKET...
proposed algorithm is called either cooperative PseudoBayesian backoff (C-PBB), or cooperative recursive Bayesian backoff (C-RBB) algorithm. The additional overhead introduced by our proposal is minimal and requires minor protocol modifications in the existing standard.
1.2 Related Works and Contributions Since Bianchi’s seminal paper [4], the analytical framework of slotted carrier-sensed multiple access (CSMA) systems (including IEEE 802.11) under saturated traffic, where a finite number of STNs always have packets to send, has been widely used in evaluating the performance of such systems due to its simplicity and good accuracy. It is important to note that under saturated traffic condition, all the parameters such as packet transmission success probability and retransmission probability can be expressed as functions of the population size and initial contention window size, because the uncertainty caused by new packet arrivals (or by the number of STNs with an empty queue) can be ignored. This implies that the population size can be readily estimated without considering the new packet arrival rate, if such probabilities are properly estimated. Thus, once the population size is estimated, the initial contention window size of the BEB algorithm can be readily optimized in order to maximize the system throughput. Accordingly, based on such an analytical framework for saturated traffic condition, Cali et al. [5], [6], [7] proposed an asymptotically optimal backoff algorithm that controls the retransmission probability based on the population size estimation in order to maximize the system throughput. On the other hand, some signal processing filtering algorithms used for state estimation have also been applied in order to estimate the number of backlogged STNs, e.g., Kalman filter [8], sequential Monte Carlo method and Gibbs sampler [9], [10], and particle filter [11]. However, since [5], [6], [7], [8], [9], [10] have been developed under the assumption of saturated traffic condition, these algorithms may not be applicable to unsaturated systems, in which STNs’ queues are often empty. Our cooperative backlog estimation method fills this gap. Furthermore, this paper shows how the estimated backlog information can be used by exploiting the characteristics of MPR systems to maximize the system throughput. It is anticipated that this latter aspect of our work may also help to extend the previous work [8], [9], [10], [11] to MPR systems. Beside these backoff algorithms based on backlog estimation, optimization of the initial contention window size of the BEB algorithm is considered in [4] under saturated traffic condition, and its dynamic control is also considered with radio channel errors in [12], [13] and with a heuristic backlog size estimation [14]. In addition, [15] provides a MAC protocol to enhance IEEE 802.11 systems with uplink MPR capability using multiple antennas, whereas [16] proposes a suboptimal contention window setting for IEEE 802.11 systems supporting multiple communications under saturated conditions. The retransmission controlling algorithm in [17] is intended for unsaturated traffic and studied using a system model that is quite similar to ours; however, only SPR is considered. This algorithm is based on keeping the channel busy ratio below some threshold, which is difficult to apply to MPR systems. Furthermore, the performance of this algorithm has not been compared to the throughput upper-bound or the access delay lower-bound.
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In comparison with [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [16], this paper considers CSMA systems with the following two striking features. First, since most of the previous studies assumed saturated systems, their algorithms do not need to estimate new packet arrival rates at the STNs, which is related to the number of STNs newly joining the system backlog. Because IEEE 802.11 systems are expected to mostly work under unsaturated conditions in practice, our algorithms considering unsaturated traffic condition are expected to be more suitable for practical systems. Under unsaturated conditions, our algorithms only need to know the total population size including the number of backlogged and non-backlogged STNs, which can be readily obtained from the service association and disassociation procedures [18]. Our simulation results illustrate the performance of our proposed algorithms under more practical scenarios where the population size varies over time, and STNs have different new packet arrival rates. Second, we consider MPR capable IEEE 802.11 systems with MUMIMO, in which all the packets simultaneously transmitted are assumed to be successfully received as long as no more than M packets are concurrently transmitted, where M is a system specific parameter. Otherwise, none of them can be successfully received. Therefore, our system model is a generalization of the previous models [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], which considered the special case M ¼ 1. Along with these two striking features, the main contributions of this paper can be summarized as follows:
We derive the optimal retransmission probability as a function of the number of backlogged STNs and propose the C-PBB algorithm and C-RBB algorithm for MPR capable IEEE 802.11 systems, whose update rules are based on backlog size estimation. The mean and variance of queuing delay are evaluated by simulations under symmetric/asymmetric traffic conditions and time-varying population size. To compare our proposed algorithms with the existing BEB algorithm, we also evaluate by simulations the mean and variance of the queuing delay of the BEB algorithm, when the initial contention window size is optimized under saturated condition for throughput maximization. We further present the queuing delay lower bound obtained from a system that has perfect knowledge of the backlog size.
1.3 Notations and Organization A binomial distribution with parameters n and p is denoted by Bin ðpÞ ¼ ni pi ð1 pÞni , while a Poisson distribution with mean x is denoted by fn ðxÞ ¼ ðxn =n!Þex . The rest of this paper is organized as follows. Section 2 introduces the system model. The proposed C-PBB and C-RBB algorithms and the existing BEB algorithm are described in Section 3. Performance evaluations are presented in Section 4 and concluding remarks are given in Section 5.
2
SYSTEM MODEL
We consider a slotted CSMA system in which N STNs communicate with an AP, and time is equally divided into backoff slots of s msec long as shown in Fig. 1. We
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Fig. 1. Channel state of slotted CSMA and IEEE 802.11 systems.
assume that the AP knows the value of N through the service association and disassociation procedures, each STN has a queue of unlimited length to store incoming packets, and packets arrive at each STN according to a Poisson process with mean rate (packets/msec). We further assume that the AP has MPR capability over the uplink, which makes it possible to receive at most M packets successfully at a time without an error, as long as the number of packets simultaneously transmitted from the STNs is less than or equal to M [19]. If more than M packets are simultaneously transmitted, none of the packets can be received in the resulting collision. As marked by the down-arrows in Fig. 1, we define the boundary of a backoff slot at the end of an idle slot, a successful transmission, or a collision as an embedded point. By the proposed backoff algorithm introduced in Section 3, regardless of backlogged or non-backlogged, each STN observes the channel and updates its own backlog estimation based on the channel outcome, i.e., idle, success or collisions at every embedded epoch. At each backoff slot, a backlogged STN senses the channel and defers its packet transmission until an ongoing transmission ends. For an idle backoff slot sensed, a backlogged STN transmits its packet based on a Bernoulli trial with a retransmission probability, denoted by p, which is a function of the estimated backlog size. Note that a newly arriving packet to an STN with an empty queue is also transmitted with this retransmission probability p. When packet transmissions occur at the beginning of a backoff slot, we assume that it takes Ts msec for STNs to complete successful packet transmissions, while Tc msec is taken for collisions. As shown in Fig. 1, we can see that this slotted CSMA system represents an IEEE 802.11 system with the basic access mechanism if Ts ¼ Tc , or the RTS/CTS mechanism if Ts > Tc . In Section 4 we show how Ts and Tc are related to the basic and the RTS/CTS access mechanisms in details.
3
PROPOSED BACKOFF ALGORITHMS
In this section we first show how the proposed algorithms work and derive the optimal retransmission probability given the MPR capability, if the mean number of backlogged STNs is known. We then present the update rules of two C-PBB algorithms for the mean backlog size estimation, which are based on two different a priori distributions on the number of backlogged STNs, and those of C-RBB algorithm that does not assume a specific form of a priori distribution. Finally, the BEB algorithm and its initial window size optimization are presented for comparisons.
3.1 Procedure Before introducing the optimal retransmission probability and the backlog update rules of the proposed backoff algorithms, we first describe the protocol that allows STNs and the AP to cooperate with each other. Note that backlogged and non-backlogged STNs update their backlog estimation at each embedded point based on the channel outcome. Suppose in Fig. 1 that at an embedded point the system has n backlogged STNs, each of which (re)transmits a packet with probability p. Then, k out of n STNs for k M will transmit their packets successfully. In updating the backlog size at the next embedded point after the successful packet transmissions, since k packets are successfully transmitted, it might be expected that these k packets should be subtracted from the current estimation and the number of newly joining backlogged STNs is added. However, due to possibly remaining packets in each backlogged STN’s queue, the number of packets successfully transmitted is not necessarily equal to the number of backlogged STNs to be removed. Only those backlogged STNs that have transmitted the last packets in their queues become nonbacklogged at the end of the embedded point. Accordingly, it is necessary for STNs to know how many STNs having only one packet in their respective queues transmit their packets successfully. To this end, in the proposed algorithms, each STN adds a one-bit feedback to its data packets, which is denoted by bL . If the STN is sending the last packet in its queue, it sets bL ¼ 1, otherwise bL ¼ 0. The AP sums up bL s from the packets successfully transmitted. Let m and mb denote the number of packets successfully received by the AP and the sum of bL s of those packets. Then, the AP inserts m and mb into the ACK-frame to notify the STNs the values of m and mb . Although the ACK frame will be addressed specifically to the STNs that have transmitted their packets successfully, other STNs can also receive the ACK frame due to the broadcast nature of the wireless channel. This is also feasible in the case of IEEE 802.11, when STNs update the network allocation vector (NAV) by indicating the beginning of the ACK frame, instead of indicating the end of the ACK frame in the current standards. Now, given mb , STNs can determine the number of backlogged STNs that have become non-backlogged. In addition to mb , the AP also broadcasts any change of N in the ACK frame. The cooperation is enabled by the bL bit sent by backlogged STNs, and m, mb and N values broadcast by the AP, only when successful packet transmissions occur. It is important to note that based on our proposed algorithms, the AP cannot give a retransmission probability to STNs using the ACK-frame, since STNs also need to determine the retransmission probability even when the channel is idle.
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TABLE 1 Optimal xy over Different Values of s=Ts
3.2 Optimal (Re)Transmission Probability This section derives the optimal (re)transmission probability p for backlogged STNs to use, when the system has n backlogged STNs. In Fig. 1, the channel state alternates between idle backoff slot, successful packet transmissions, or collisions. Let Xnk and Ynk respectively denote random variables for the lengths of an idle period and a packet transmission period (irrespective of success and collision) at the kth renewal epoch, when the system has n backlogged STNs. We then have a renewal process N ðtÞ for t 0 with interarrival times Lkn ¼ Xnk þ Ynk . The mean values of Xnk and Ynk can be written Pminðn;MÞ i Bn ðpÞ þ as E½Xnk ¼ sB0n ðpÞ and E½Ynk ¼ Ts i¼1 Pminðn;MÞ i k Tc ½1 i¼0 Bn ðpÞ, respectively, for all k. Let Rn denote the reward earned at the kth renewal given n backlogged STNs, which is an independent identically distributed random variable with respect to k, but depends on Lkn . The total reward PN ðtÞ earned up to t is denoted by Rn ðtÞ ¼ k¼0 Rkn . Then, by renewal reward theory [20], the system throughput is the long-run average reward expressed as E Rkn E½Rn ðtÞ E Rkn ¼ k ¼ k Sn ¼ lim t!1 t E Ln E Xn þ E Ynk Pminðn;MÞ i Bin ðpÞ i¼1 h i: ¼ Pminðn;MÞ i P minðn;MÞ i Bn ðpÞ þ Tc 1 i¼0 Bn ðpÞ sB0n ðpÞ þ Ts i¼1
(1) Note that this system throughput can also be understood as the total service rate for the queues in n backlogged STNs. Accordingly, maximizing the system throughput is equivalent to maximizing this overall service rate for the queues in n backlogged STNs. Let pn be the optimal (re)transmission probability that maximizes Sn given n backlogged STNs. In order to obtain a closed form of pn , by assuming n ! 1 in (1), we rewrite Sn as S^
PM
sf0 ðxÞ þ Ts
i¼1 i fi ðxÞ ; PM f i¼1 i ðxÞ þ Tc 1 i¼0 fi ðxÞ
PM
(2)
where the binomial distributions in (1) are approximated by Poisson distributions with mean x ¼ npn . We can interpret x as the average number of (re)transmitting STNs at each embedded point. Then, the optimal x that maximizes S^ can ^ be obtained by setting dS=dx ¼ 0. For M ¼ 1, by using x 2 e 1 x þ 0:5x , we have x
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2s=Ts ðTc =Ts Þ2 ¼ 2s=Tc :
(3)
Thus, the approximated optimal transmission probability, denoted by p^n , is expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffi 2s=Tc p^n ¼ : (4) n
Fig. 2. Optimal x =x M versus Tc =Ts .
It is notable that for M ¼ 1, p^n depends only on Tc , even for Ts 6¼ Tc . We now consider an MPR capable system, i.e., M 2. We first account for the basic access mechanism, i.e., ^ ¼ 0, we have Ts ¼ Tc . From dS=dx PM
i¼1 e ð1 s=Ts Þ þ PM x
i¼1
fi1 ðxÞ ifi1 ðxÞ
x 1 ¼ 0:
(5)
Let xy denote the solution of (5). Since it is difficult to find a closed form expression of xy , we find xy numerically for different values of s=Ts and M. The results are tabulated in Table 1. As M increases, xy becomes insensitive to s=Ts . Particularly for M ¼ 5, xy is almost a constant. Since we are only interested in s=Ts 0:1, which is application to most practical systems, by letting s=Ts ! 0 we rewrite (5) as PM i¼1 fi1 ðxÞ fðxÞ ¼ ex þ PM x 1: i¼1 ifi1 ðxÞ
(6)
We denote the solution of fðxÞ ¼ 0 by x M , whose values are given by the entries in the last row of Table 1. Then, we can set p^n ¼ x M =n for any M and s=Ts 0:1. Later in Section 4, we use s=Ts ¼ 0:0197 (basic access) and 0:0155 (RTS/CTS). We next consider the system with the RTS/CTS mechanism, i.e., M 2 and Ts > Tc . Since it is hard to analyti^ cally find the solution for dS=dt ¼ 0, we first numerically search for the optimal x that maximizes (2) when s=Ts ¼ 0:01. The optimal x normalized to x M is plotted with symbols in Fig. 2 for various values of Tc =Ts in [0.1, 1]. Second, in order to mathematically describe the obtained value of x =x M for any Tc =Ts in [0.1, 1], we adopt the nonlinear curve-fitting algorithm with least-squares [21]. In doing this, we would like to simplify fðTc =Ts Þ by preserving the same form of (3). Since x in (3) decreases in proportion to a negative exponent, we assume that the fitting function has the form of fv ðTc =Ts Þ ¼ ðTc =Ts Þv , where v is a parameter to be determined. Then, the nonlinear curve-fitting algorithm with least-squares requires v to satisfy the following condition:
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3.3 Update Rules Based on Binomial Distribution Assumption In the above we have obtained the optimal retransmission probability given the number of backlogged STNs. This section provides the procedure to estimate the backlog size after each channel outcome based on a posteriori estimation, which is used in Algorithm 1. In other words, we are interested in finding the a posteriori distribution of the number of backlogged STNs after observing the channel outcome. For instance, when I denotes an idle slot and n denotes the number of backlogged STNs, the a posteriori distribution on n after I is observed is expressed as Pr½njI ¼ Pr½I; n=Pr½I ¼ Pr½IjnPr½n=Pr½I;
Fig. 3. Optimal throughput versus Tc =Ts .
min v
K X ðfv ðzðkÞÞ x ðkÞ=x M Þ2 ;
(7)
k¼1
where zðkÞ is a sample of Tc =Ts , i.e., a value of the x-axis in Fig. 2, while x ðkÞ is the corresponding optimal value of x obtained numerically. We used zðkÞ ¼ 0:1 þ 0:05ðk 1Þ for k 2 f1; 2; . . . ; 19g, with which we finally obtained the values of v as 0.4677, 0.3182 and 0.2549 for M ¼ 2; 3 and 4, respectively. The related fitting curves with dashed lines shown in Fig. 2 closely approximate the corresponding symbols which are obtained through numerical search. An additional simplification could be performed by comparing the values of v (0.4677, 0.3182 and 0.2549) to the MPR capability M. Since 0:4677 1=2, 0:3182 1=3 and 0:2549 1=4, we can further simplify v as 1=M; then, x can be approximated as 1
x ¼ x M ðTc =Ts ÞM :
(10)
where Pr½I can be obtained by summing Pr½I; n over n. While we need to have the a priori distribution of the backlogged STNs, i.e., Pr½n in (10), it is impossible to know Pr½n in advance. It is reasonable to choose Pr½n among some known distributions that can reflect the actual physical behaviour of n as closely as possible, whereas Pr½njI could have the same form as Pr½n. Then, it would be possible to use Pr½njI recursively for the a priori distribution at the next epoch. Even though the a priori distribution on the number of backlogged STNs Pr½n can be derived from N-STN’s queuing process, here we assume that Pr½n follows a binomial distribution with parameters N and r. The rationale behind this assumption is that the utilization of each STN’s queue would be r and each STN’s queuing process is identical and independent, which might not be true in practice. It can be, however, expected that such binomial distribution works well for Pr½n, if N-STN’s queuing process are very weakly correlated. Although a theoretical justification for this assumption is hard to find, we shall evaluate in Section 4 the validity of this assumption by comparing its a posteriori distribution resulting from success, idle and collisions against simulations.
(8)
Note that the corresponding solid fitting curves in Fig. 2 also closely approximate the results from numerical searches. The closeness of the corresponding throughput to the optimal one will be examined in this section. Using (8), we then get the approximated p^ for M 2 as p^n ¼ x =n:
(9)
To see the accuracy of this approximated p^n , we depict S^ against various values of Tc =Ts , for s=Ts ¼ 0:01 and s=Ts ¼ 0:1 in Fig. 3. Note that in this figure Tc =Ts ¼ 1 corresponds to the basic access mechanism, while Tc =Ts ! 0:1 corresponds to the more efficient RTS/CTS mechanism. In Fig. 3, the symbols show S^ with the optimal pn found numerically, while the solid curves indicate S^ with the approximated p^n . It is notable that S^ is insensitive to s=Ts as M increases, which agrees with the results shown in Table 1. We have checked that the throughput from applying (8) and (9) is very close to the optimal throughput, for M values up to 10, although the results are not provided here.
Before proceeding further, recall that Sn in (1) is derived for a system that has n backlogged STNs. Since we assume that the number of backlogged STNs is distributed as a binomial distribution with mean Nr, we can write the
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average system throughput as
Second, we consider the case of m simultaneous successful transmissions. The joint probability that m simultaneous Pminðn;MÞ i PN n iBn ðpÞ transmissions are made from n backlogged STNs is i¼1 n¼0 BN ðrÞ n h S ¼ PN Pminðn;MÞ i io Pminðn;MÞ i n 0 expressed as Bn ðpÞ þ Tc 1 i¼0 Bn ðpÞ n¼0 BN ðrÞ sBn ðpÞþTs i¼1 PM iBiN ðprÞ ¼ 0 : PM i¼1 P m n m nm ð1 pÞr i i B ðprÞ sBN ðprÞ þ Ts i¼1 BN ðprÞ þ Tc 1 M : (19) ðpÞB ðrÞ ¼ B ðprÞB Pr½S; m; n ¼ B i¼0 N n N N Nm 1 pr (11) Summing this over n yields When N ! 1, we can write (11) as N X PM Pr½S; m ¼ Pr½S; m; n ¼ Bm (20) N ðprÞ: i¼1 ifi ðxÞ S (12) ; PM PM n¼m sf0 ðxÞ þ Ts i¼1 fi ðxÞ þ Tc 1 i¼0 fi ðxÞ By using Pr½njS; m ¼ Pr½S; m; n=Pr½S; m, one can get where the binomial distributions are approximated by Poisð1 pÞr Pr½njS; m ¼ Bnm ; (21) son distributions with mean x ¼ Npr ¼ pn. It should be Nm 1 pr noted that S in (12) is identical to (2), from which we obtain the optimal value x as (8). From x ¼ pn, the optimal transwhich turns out to be a binomial distribution as well. The mission probability p is written as average backlog size is obtained by x (13) p¼ : ð1 pÞr n E½njS; m ¼ ðN mÞ þ m: (22) 1 pr This means that if the average number of backlogged STNs n is obtained, we can simply set the (re)transmission probability to x =n, in order to maximize the system throughput. We are now in a position to introduce the update rules for backlog estimation based on each channel outcome. Let I, S and C denote an idle slot, m successful packet transmissions and a collision, respectively. As mentioned previously, when the a priori distribution of the backlog size is assumed to be a binomial distribution with mean Nr, for an idle backoff slot we have ð1 pÞr : (14) Pr½I; n ¼ ð1 pÞn BnN ðrÞ ¼ B0N ðprÞBnN 1 rp Summing this over n, we obtain the probability that the channel is idle as Pr½I ¼
N X
Pr½I; n ¼ B0N ðprÞ:
(15)
As in (18), by putting p ¼ x=n and r ¼ n=N, we have E½njS; m ¼ ðN mÞ
which turns out to be a binomial distribution similar to the a priori distribution. Consequently, the average backlog size after an idle slot is expressed as E½njI ¼ N
ð1 pÞr : 1 rp
(17)
Pr½C; n ¼ Pr½n Pr½I; n
(18)
This suggests that we can approximate the mean number of backlogged STNs by (18), if its previous estimate is n. Note that x is given in (3) for M ¼ 1 and in (8) for M 2, respectively.
M X
Pr½S; k; n:
(24)
k¼1
Summing this over n, we have the collision probability as Pr½C ¼ 1 Pr½I
M X
Pr½S; k:
(25)
k¼1
After a collision, the resulting distribution is expressed as P Pr½C; n Pr½n Pr½I; n M Pr½S; k; n Pr½njC ¼ ; ¼ PM k¼1 Pr½C 1 Pr½I k¼1 Pr½S; k (26) which is no longer a binomial distribution. However, we make the simplifying assumption that the resulting distribution still follows a binomial distribution with the following mean
If we set p ¼ x =n and r ¼ n=N (from n ¼ Nr), we have n x : E½njI ¼ N N x
(23)
which implies that the mean number of backlogged STNs after m successful transmissions can be estimated by (23) with a previous estimate n. As previously mentioned and also shown in the 7th line in Algorithm 1, we subtract mb from (23), i.e., the number of backlogged STNs that transmitted the last packets in their queues successfully. Finally, when a collision is observed, we have
n¼0
After observing an idle slot, by Bayes’ rule we write the a posteriori distribution as ð1 pÞr ; (16) Pr½n j I ¼ Pr½I; n=Pr½I ¼ BnN 1 rp
n x þ m; N x
E½njC ¼
Nr E½njI Pr½I
PM Pk¼1 M
E½njS; k Pr½S; k
1 Pr½I k¼1 Pr½S; k P k Bk ðprÞ ð1 pÞr 1 r Nrp M ¼N : þ PMk¼1 k N 1 pr 1 pr 1 k¼0 BN ðprÞ (27)
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Again, if p ¼ x =n and r ¼ n=N are applied, we have PM n x N n x k¼1 kBkN xN E½njC ¼ N þ : P k x N x N x 1 M k¼0 BN N
E½njS; m ¼ (28)
We shall examine in Section 4 how the binomial distribution with the mean derived from (28) differs from the empirical distribution of the number of backlogged STNs obtained by simulations, after a collision is observed. The whole update rules are summarized in Algorithm 1, where b, ds , di and dc are the parameters used for estimating new packet arrivals and will be introduced in the next section.
3.4 Estimation of New Packet Arrivals At each embedded point, the non-backlogged STNs may become backlogged if new packets arrive at their empty queues. To estimate this, we denote by t the estimated mean packet arrival rate to the system at time slot t. At each slot it can be updated as t ¼
bTs t1 þ mds ; bTs þ sdi þ Ts ds þ Tc dc
(29)
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n Pr½njS; m ¼ ð1 pÞn þ m:
(33)
n¼1
Using p ¼ x =n, we have E½njS; m ¼ n x þ m:
(34)
Here, we also subtract mb from (34) as we have done in (23), which is given in the 6th line in Algorithm 2. When a collision occurs, we approximate the resulting distribution of the number of backlogged STNs by a Poisson distribution with mean value E½njC ¼ n þ
1
np fM ðnpÞ : P M i¼1 fi ðnpÞ
enp
(35)
When p ¼ x =n is used, we have E½njC ¼ n þ
x fM ðx Þ : P 1 M i¼0 fi ðx Þ
(36)
The overall procedure is summarized in Algorithm 2.
in which b is the observation window for estimating the packet arrival rate and m denotes the number of packets successful transmitted. In (29), di , ds , and dc , denote indicator functions, which are equal to one for an idle, success, and collision slot, respectively. These indicator functions are equal to zero for irrelevant events, e.g., dc ¼ 0 for an idle or success slot.
3.5
Update Rules Based on Poisson Distribution Assumption Recall that in (12), the binomial distribution of the number of backlogged STNs is approximated by a Poisson distribution. Compared to the update rules based on the binomial distribution assumption presented in the last section, when a Poisson distribution is used for the a priori distribution of the number of backlogged STNs, the update rules become simpler than those in the previous section. Furthermore, we shall see how sensitive the system performance is to the a priori distribution of the backlog estimation in Section 4. The derivation of each update rule is similar to that in the last section except for the use of Poisson distributions instead of binomial distributions. When an idle slot is observed, starting from the following joint probability Pr½I; n ¼ B0n ðpÞfn ðnÞ ¼ fn ðnð1 pÞÞf0 ðnpÞ;
(30)
we have the a posteriori probability that the system has n backlogged STNs given an idle slot as Pr½njI ¼ fn ðnð1 pÞÞ:
(31)
By substituting p ¼ x =n into (31), we have E½njI ¼ n x ;
(32)
which implies that we only need to subtract x from the previously estimated mean number of backlogged STNs after an idle backoff slot is observed. Similar to (19)-(23), after m successful packet transmissions in a slot, we have
3.6 Cooperative Recursive Bayesian Backoff Algorithm In C-PBB algorithms, we have assumed that the a priori distribution is binomial (or Poisson) and further assumed that the a posteriori distribution has the same form as the a priori distribution. In this section, we relax these assumptions by proposing the C-RBB algorithm, with which STNs recursively use the a posteriori distribution on backlogged STNs as the a priori distribution at the next embedded point without assuming that the a priori distribution should have a specific form as before. We shall see in Section 4 that C-PBB algorithms will show disagreements between their estimated a posteriori distributions on the number of backlogged STNs and simulation results particularly after collisions since we have assumed the a priori distributions arbitrarily, whereas C-RBB will show good agreements between them at the cost of a higher complexity. Thus, we can see later how the queuing performances of these algorithms could be improved based on accuracy of those a posteriori distributions.
JIN ET AL.: COOPERATIVE PSEUDO-BAYESIAN BACKOFF ALGORITHMS FOR UNSATURATED CSMA SYSTEMS WITH MULTI-PACKET...
Let VðtÞ ½n denote the a priori distribution on the number of backlogged STNs at an embedded point t, which will be initialized. Upon an idle slot, the a posteriori distribution for the number of backlogged STNs is calculated as Pr½njI ¼
Pr½I; n Pr½IjnVðtÞ ½n ¼ PN ðtÞ Pr½I n¼0 Pr½IjnV ½n
B0 ðpÞVðtÞ ½n : ¼ PN n ðtÞ 0 n¼0 Bn ðpÞV ½n
(37)
Upon a success slot with m successful packet transmissions, the a posteriori distribution is calculated as Pr½S; m; n ¼ Pr½njS; m ¼ Pr½S; m
ðtÞ Bm n ðpÞV ½n : PN ðtÞ m n¼m Bn ðpÞV ½n
If there are mb backlogged STNs having only one packet in each of their queues, as their queues become empty after the success slot, the a posteriori distribution should be changed to Pr½n mb jS; m ¼ Pr½njS; m for mb n N:
As Vðtþ1Þ ½n is changed at each embedded point, it is hard to get a closed-form of the optimal transmission probability p that maximizes S. When we approximate VðtÞ ½n by either a binomial or Poisson distribution as in Sections 3.3 or 3.5, respectively, we could obtain the approximate closed-form of the optimal p which is expressed as x =n (n is the average number of backlogged STNs). Inspired by this observation, in C-RBB we still use x =n as the (re) transmission probability although it is not the optimal solution for (42). The average number of backlogged STNs n is calculated as n¼
(38)
309
N X
nVðtþ1Þ ½n:
(43)
n¼0
The details of C-RBB algorithm are shown in Algorithm 3. Note that we initialize Vð0Þ ½n ¼ 0 for 0 n N 1 and Vð0Þ ½N ¼ 1.
(39)
When a collision slot happens, the a posteriori distribution is given as PminðM;nÞ k ðtÞ 1 k¼0 Bn ðpÞ V ½n Pr½C; n ¼P Pr½njC ¼ PminðM;nÞ k ðtÞ : N Pr½C 1 B ðpÞ V ½n n¼0
k¼0
n
(40) In (40), we can easily find that Pr½njC is zero for n < M. Now to find the a priori distribution for the next slot, i.e., Vðtþ1Þ ½n, the number of STNs that become newly backlogged should be taken into account. Let qB denote the probability that a STN becomes backlogged. We then have qB ¼ 1 et s , 1 et Ts or 1 et Tc when an idle, success, or collision slot is observed, respectively. Consequently, the a posteriori distribution for the next embedded point, Vðtþ1Þ ½n, can be updated as n X N k nk ðtþ1Þ q ð1 qB ÞNn Pr½kjE; V ½n ¼ (41) nk B k¼0 where E denotes either I, S, or C depending on the channel outcome. In using C-RBB, the values of Pr½njE for n ¼ 0; 1; 2; . . . ; N should be calculated and stored by each STN. This may incur a very high implementation complexity. Once Vðtþ1Þ ½n is obtained, in order to maximize the average system throughput we can calculate the optimal transmission probability as P
PN minðn;MÞ ðtþ1Þ ½n iBin ðpÞ i¼1 n¼0 V ; (42) S¼ PN ðtþ1Þ ½nD½n n¼0 V in which D½n in the denominator is D½n ¼
sB0n ðpÞ
þ Ts
minðn;MÞ X i¼1
Bin ðpÞ
þ Tc 1
minðn;MÞ X i¼0
! Bin ðpÞ
:
3.7 BEB Algorithm In the BEB algorithm, upon a collision, the backlogged STN doubles its contention window size until the number of doubling reaches a certain limit. Let Wi denote the contention window size of the ith backoff stage. Then, after the ith collision, Wi is determined by Wi ¼ 2i W0 for 0 i R, while Wi ¼ Wi1 for i > R. At the ith backoff stage, the backlogged STN randomly chooses a backoff counter in the range ½0; Wi 1, and then decrement the backoff counter very slot. When it reaches zero, the backlogged STN transmits its packet. Upon a (re)transmission failure, the STN increments the backoff stage by one and chooses the next backoff counter again. While the BEB algorithm works in a distributed way, it still remains to determine W0 in the BEB algorithm. So far in the literature, the optimal W0 has been derived under the saturated traffic condition only. Under saturated traffic condition, the transmission probability for each STN is obtained by solving the discrete Markov chain model of the backoff state and it is given as [4] t¼
2ð1 2qÞ ; ð1 2qÞðW0 þ 1Þ þ qW0 1 ð2qÞR
(44)
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Fig. 4. A posteriori distributions, M ¼ 3.
where q is the probability that a transmitted packet collides with another STN’s packet transmission. We can write q as [2] q ¼1
M 1 X i¼0
BiN ðtÞ:
(45)
Hence if t is provided, we can reversely calculate W0 as 1 2q 2 W0 ¼ 1 ; (46) 1 q 2R qRþ1 t in which dye denotes the smallest integer not less than y. On the other hand, the saturation throughput is expressed as
JIN ET AL.: COOPERATIVE PSEUDO-BAYESIAN BACKOFF ALGORITHMS FOR UNSATURATED CSMA SYSTEMS WITH MULTI-PACKET...
311
PminðN;MÞ SN ¼
i BiN ðtÞ h P i: minðN;MÞ i BiN ðtÞ þ Tc 1 i¼0 BN ðtÞ
i¼1
sB0N ðtÞ þ Ts
PminðN;MÞ i¼1
From the similarity between SN and (1), by the same derivation in Section 3.2 we can obtain the optimal t that maximizes SN . The optimal t can be approximated as t¼
x ; N
(47)
where x is give by (8). Substituting this t ¼ x =N into (46), we obtain the optimal W0 as 1 2q 2N W0 1 (48) 1 q 2R qRþ1 x PM1 with q 1 ex i¼0 fi ðx Þ, and x is given in (8).
4
PERFORMANCE EVALUATIONS
In the simulation study, unless otherwise stated, 10 STNs, i.e., N ¼ 10, are considered with a payload size of 1,000 bytes and s ¼ 9 msec. Results are averaged over 10 simulation runs, each of which runs for 106 Ts . At the start of each simulation, all STNs have empty queues. We set the data rate to 24 Mbps at the physical layer and the following system parameters are based on IEEE 802.11n. For the basic access mechanism, i.e., Ts ¼ Tc , we have Ts ¼ Packet TX ð364 msecÞ þ SIFS ð16 msecÞ þ ACKTime ð44 msecÞ þ DIFS ð34 msecÞ ¼ 458 msec; Tc ¼ Packet TX þ ACK Time-out ð94 msecÞ ¼ 458 msec: For the RTS/CTS mechanism, i.e., Ts > Tc , we assume that the control packets of RTS, CTS and ACK are transmitted at the lowest data rate of 6 Mbps. Then the system has Ts ¼ RTS ð48 msecÞ þ SIFS þ CTS ð44 msecÞ þ SIFS þ Packet TX þ SIFS þ ACKTime þ DIFS ¼ 582 msec; Tc ¼ RTS þ CTS Time-outð94 msecÞ ¼ 142 msec: For the BEB algorithm with the retry limit R ¼ 5, in order to make the BEB algorithm work optimally, from (48) we use W0 ¼ 78, 20, 7 and 5 for M ¼ 1, 2, 3, 4 in the basic access mechanism, while for M ¼ 1, 2, 3, 4, in the RTS/CTS mechanism, we use W0 ¼ 33, 4, 2 and 2, respectively. A posteriori distributions. For the system with M ¼ 3 and ¼ 2 (packets/msec), Figs. 4a and 4b depict the a posteriori distributions for each event with the binomial distribution as the a priori distribution for the number of backlogs using (18), (23) and (28) in comparison with simulations, while Figs. 4c and 4d depict them with the Poisson distribution as the a priori distribution with (32), (34) and (36), in order to see the sensitivity of the proposed algorithms to the assumed a priori distributions. Moreover, Figs. 4e and 4f depict the a posteriori distributions with (37)-(41) from CRBB algorithm. In Figs. 4a, 4b, 4c, and 4d, after idle and success slots, the simulation results for the a posteriori distributions are very close to the analytical results based on the assumed a priori distributions, thus validating these assumptions. However, after collision slots, the empirical and analytical
Fig. 5. Queuing delays of RTS/CTS mechanism.
a posteriori distributions no longer follow each other. The empirical distributions show that the number of backlogged STNs causing a collision is never less than 3 upon a collision, whereas the analytical distributions show that the number of backlogged STNs causing a collision can be less than 3, which contradicts with our MPR channel model with M ¼ 3. In particular, the means of Pr½njC in (28) and (36) are slightly overestimated, e.g., as 5:116 and 5:172 against the measured values, 4:864 and 4:629 in Figs. 4a and 4b, in the basic and RTS/CTS access mechanisms, respectively, since those probability distributions are overestimated in analysis when n > 6. However, when it comes to those distributions with the C-RBB algorithm, the analytical results agree well with the simulation results. In what follows, we shall see that this discrepancy between analytical and simulation results of a posteriori distributions coming from the assumed binomial and Poisson distribution shows only negligible degradation in terms of queuing performance. Queuing delay performance. In Figs. 5a and 5b, we compare the mean and variance of queuing delays of Algorithms 1, 2, and 3, which are denoted by C-PBB(B), CPBB(P), and C-RBB, respectively, against the performance of the BEB algorithm and the optimal performance, for a system with the RTS/CTS access mechanism. The optimal performance, which is not realizable in practice, is
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Fig. 6. Queuing delays of RTS/CTS mechanism when N ¼ 5.
Fig. 7. Effect of cooperation in RTS/CTS mechanism.
obtained from the simulations in which STNs know the number of backlogged STNs perfectly at every embedded points and use the optimal (re)transmission probability based on this perfect knowledge of the backlog size. We can see that the performances of Algorithm 1, 2 and 3 are indistinguishable, which means that the binomial or Poisson assumption on the a posteriori distribution is a reasonable approach to replace the complexity of C-RBB algorithm. It is notable that compared to the optimal algorithm that perfectly knows the backlog size at each embedded point, the proposed algorithms use the mean of the backlogged STNs with estimated distributions. It is observed for M ¼ 1 that the performance of the proposed algorithms is very close to the optimal performance, while the proposed algorithms mostly outperform the BEB algorithm for M > 1. Since W0 is optimized for the saturated traffic condition, the performance of the BEB algorithm gets close to that of the proposed algorithms and the optimal performance as the system gets saturated. For M ¼ 4, the BEB algorithm, however, cannot reach the optimal performance even for the saturated traffic condition. This means that while an aggressive (re)transmission probability is needed for higher MPR capability, e.g., M ¼ 4, the BEB algorithm cannot yield such a (re)transmission probability. This is explained by the fact that we cannot set of W0 2 ð1; 2Þ as a real number in the BEB algorithm, whereas C-PBB(B), C-PBB(P), and C-RBB allow STNs to access the
channel with a (re)transmission probability greater than 0:5. From the observation that the variance of queuing delays by the BEB algorithm is also higher than that of CPBB(B), C-PBB(P), and C-RBB, it can be concluded that more fair access opportunities are achieved by the proposed algorithms. When we examine the mean and variance of queuing delays for the system with the basic access mechanism, the observations drawn from Figs. 5a and 5b are also similarly applied. Note that C-PBB(P) is algorithmically simpler than C-PBB(B) and C-RBB, while its performance is as good as C-PBB(B) and C-RBB. Thus, in Figs. 7, 8a, 10a, and 10b we use C-PBB(P) only for simplicity of the presentation. Fig. 6 shows the delay and variance performance for the system with N ¼ 5. This shows how effectively C-PBB(P) algorithm works when the system has a smaller population size compared to Figs. 4c and 4d with N ¼ 10. For the BEB algorithm, the values of W0 are set to 17, 2, 1, 1, respectively, for M ¼ 1; 2; 3; 4 based on (48). We can see that Fig. 6 shows similar trends as shown in Fig. 5, from which we can conclude that the Poisson assumption on the a priori distribution works well for the system with a small population size. Effects of cooperation. Since the proposed C-PBB(P) algorithm enables cooperation between the AP and STNs via information exchange, in Figs. 7a and 7b, the effects of this cooperation are examined. In the case without cooperation, denoted as PBB(P), we set mb ¼ m. This cooperation
JIN ET AL.: COOPERATIVE PSEUDO-BAYESIAN BACKOFF ALGORITHMS FOR UNSATURATED CSMA SYSTEMS WITH MULTI-PACKET...
Fig. 8. Estimated backlog size in time-varying population size in RTS/ CTS mechanism.
has significant effects on the mean and variance of queuing delays versus the arrival rate, in that cooperation becomes more effective for the saturated arrival rates at which the queuing delay means and variances increase steeply, and the gap between PBB and C-PBB(P) becomes larger with increasing M. Therefore, it seems that such cooperation is needed particularly for MPR capable IEEE 802.11 systems. Time-varying population size. Fig. 8a shows the backlog size estimated by the proposed algorithm for the RTS/CTS access mechanism with M ¼ 3, when the population size N varies over time. In the simulation, during the first 1,000 sec, the system has only 10 STNs with ¼ 1:5 (packets/ msec). Then, 10 STNs are added to this system, which results in ¼ 3 (packets/msec). At 2,000 sec, those 10 STNs are removed from the system for the rest of the simulation run time. For N ¼ 10 the proposed algorithm slightly underestimates the number of backlogged STNs in comparison with the actual number of backlogged STNs. In this case, backlogged STNs are encouraged to (re)transmit their packet aggressively. When the population size increases, apparently the estimation process, which is based on observations of the channel states such as idle, success or collision, keeps track of the actual backlog size quite well. In addition, for each interval of 1,000 sec long, the average queuing delay that STNs have experienced is
313
Fig. 9. Delay performance of RTS/CTS mechanism with asymmetric traffic scenario 1.
0.8581 msec for the first 1,000 sec, 2.8099 msec for the second 1,000 sec, and 0.8521 msec for the last 1,000 sec. To see the behavior of our algorithm in tracking the actual backlog size particularly when the population size is changed, e.g., after 1,000 and 2,000 sec, Fig. 8b shows the comparisons between the actual and estimated backlog size for the time period of [1,000, 1,200] and [2,000, 2,200], respectively. We can see that the estimated backlogs track the trends of the actual backlog size in [1,000, 1,200] and [2,000, 2,200] quite well as the actual backlog size increases or decreases, while being less accurate regarding small scale variations of the backlog size. Asymmetric traffic condition. In Figs. 9a and 9b, we number each STNs from 1 to 10 in the system with M ¼ 4 and set the arrival rate of the STN i to i 0:0727 (packets/msec), which implies that the total arrival rate of the 10 STNs is 4 (packets/msec). We call this asymmetric traffic scenario 1. These figures show that while the BEB algorithm is very sensitive to the packet arrival rate variation at each STN, the proposed algorithms are robust to those variation with respect to both the mean and variance of queuing delays. In Figs. 10a and 10b we divide the 10 STNs into two groups in what we refer as asymmetric traffic scenario 2. For the first group of five STNs, say C1, each STN has a packet arrival rate of 1=15 (packets/msec), while for the
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condition. Furthermore, when the MPR capability is increased, the performance of the proposed algorithms gets close to the optimal performance due to the significant effect of the cooperation, while the BEB algorithm shows limited performance even for saturated systems.
ACKNOWLEDGMENTS This work was supported by the Canadian Natural Sciences and Engineering Research Council under grants RGPIN 2014-06119 and RGPAS 462031-2014.
REFERENCES [1] [2]
[3] [4] [5] [6] [7] Fig. 10. Delay performance of RTS/CTS mechanism with asymmetric traffic scenario 2.
rest of the STNs, say C2, each has double the packet arrival rate. As one might expect, the larger the packet arrival rate at a STN, the larger queuing delay is observed. The figure shows that with the proposed algorithm, the worst queuing delays, i.e., those of C2, are much lower than the best queuing delays of the BEB algorithm experienced by C1.
5
[8] [9]
[10]
[11]
CONCLUSIONS
This paper has proposed novel backoff algorithms called CPBB and C-RBB for IEEE 802.11 systems with uplink MPR capability, and evaluated their performance in comparison with the BEB algorithm and the optimal algorithm by simulations. While the C-RBB algorithm provides exact a posteriori distributions at the cost of a high computational complexity, C-PBB(P) algorithm showed a similar queuing performance but with a lower implementation complexity compared to C-RBB. The proposed algorithms particularly target unsaturated systems with the basic access and RTS/ CTS mechanisms and utilize cooperation between STNs and AP via information exchange. With the help of such cooperation, the proposed algorithms show lower mean queuing delays than the BEB algorithm and provide more fair access opportunities. The robustness of the proposed algorithms is examined by various simulation scenarios such as time-varying population size and asymmetric traffic
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[19] H. Jin, B. C. Jung, H. Y. Hwang, and D. K. Sung, “A MIMO-based collision mitigation scheme in uplink WLANs,” IEEE Commun. Lett., vol. 12, no. 6, pp. 417–419, Jun. 2008. [20] E. P. C. Kao, An Introduction to Stochastic Processes. Pacific Grove, CA, USA: Duxbury Press, 1997. [21] T. F. Coleman and Y. Li, “An interior, trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim., vol. 6, pp. 418–445, 1996. Hu Jin (S’07-M’12) received the BE degree in electronic engineering and information science from the University of Science and Technology of China (USTC), China, in 2004, and the MS and PhD degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), in 2006 and 2011, respectively. He was a postdoctoral fellow at the University of British Columbia, Canada from 2011 to 2013 and was a research professor at Gyeongsang National University, Korea, from 2013 to 2014. Since 2014, he is an assistant professor with the Department of Electronics and Communication Engineering, Hanyang University, Korea. His research interests include medium access control and radio resource management for random access networks and scheduling systems considering advanced signal processing and queueing performance. He is a member of the IEEE. Jun-Bae Seo (S’08-M’12) received the BS and MSc degrees in electrical engineering from Korea University, in 2000 and 2003, respectively, and the PhD degree from the University of British Columbia (UBC), Vancouver, Canada, in June 2012. He received the Natural Sciences and Engineering Research Council Postgraduate Scholarship from September 2009 to August 2011. From 2003 to 2006, he was with Electronics and Telecommunications Research Institute (ETRI), Korea, as a member of the engineering staff, carrying out research on IEEE 802.16 systems. He is now a posdoctoral fellow at UBC since July 2012. His research interests include queueing theory and its computational aspects, and optimization on queueing systems with application to wireless mobile networks. He is a member of the IEEE.
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Victor C.M. Leung (S’75-M’89-SM’97-F’03) received the BASc (Honors) degree in electrical engineering from the University of British Columbia (UBC) in 1977, and was awarded the APEBC Gold Medal as the head of the graduating class in the Faculty of Applied Science. He attended graduate school at UBC on a Natural Sciences and Engineering Research Council Postgraduate Scholarship and completed the PhD degree in electrical engineering in 1981. From 1981 to 1987, he was a senior member of technical staff and satellite system specialist at MPR Teltech Ltd., Canada. In 1988, he was a lecturer in the Department of Electronics at the Chinese University of Hong Kong. He returned to UBC as a faculty member in 1989, and currently holds the positions of professor and TELUS Mobility Research chair in Advanced Telecommunications Engineering in the Department of Electrical and Computer Engineering. His research interests are in the broad areas of wireless networks and mobile systems, in which he has co-authored more than 700 technical papers in international journals and conference proceedings, 27 book chapters, and coedited six book titles. Several of his papers had been selected for best paper awards. He is a registered professional engineer in the Province of British Columbia, Canada. He is a fellow of IEEE, the Royal Society of Canada, the Engineering Institute of Canada, and the Canadian Academy of Engineering. He was a distinguished lecturer of the IEEE Communications Society. He is a member of the editorial boards of the IEEE Wireless Communications Letters, Computer Communications, and several other journals, and has served on the editorial boards of the IEEE Journal on Selected Areas in Communications Wireless Communications Series, and IEEE Transactions on Wireless Communications, Vehicular Technology, and Computers. He has guest-edited many journal special issues, and provided leadership to the organizing committees and technical program committees of numerous conferences and workshops. He received the IEEE Vancouver Section Centennial Award and 2012 UBC Killam Research Prize. " For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib.
