Erik G. Larsson is with the Signals, Sensors & Systems, Commu- nication Theory, School of Electrical Engineering, Royal Institute of. Technology (KTH).
Exploiting Multiuser Diversity in Reservation Random Access Nan Zhang, Branimir Vojcic, Senior Member, IEEE, Michael Souryal, Member, IEEE, Erik G. Larsson, Member, IEEE
Abstract–In this paper, we study the uplink of a cellular network where reservation random access schemes such as reservation slotted ALOHA and reservation slotted nonpersistent ISMA are used for multiple access. When channel state information is not available to users when they contend for channel access, we find that capture effect inherent in wireless communications can exploit the channel variations and reserve users with favorable channels for data transmission. We investigate two channel models where in the first model, channel variations come from time-varying Nakagami-m fading and in the second model, mobile users have both time-varying fading and path loss. Index Terms–Random access, multiuser diversity, Nakagami-m fading I. I NTRODUCTION Multiuser diversity is a concept introduced in [1] that can exploit fading in a multiuser system. In a wireless network with multiple users and where each user is subject to independent fading, there is a high probability that one or a few users have very good links to the base station (BS). By allowing only these users to transmit, the system throughput can be increased. Multiuser diversity is often studied assuming centralized scheduling, in which case the BS estimates the users’ channels and chooses users with favorable channels to transmit; e.g., see [2]. An application of multiuser diversity can be found in cdma2000 high rate packet data (HRPD) system [3], also known as 1xEV-DO. There are several approaches to exploit multiuser diversity in the uplink of a cellular network. One approach is the centralized scheme as in [4], [5] which selects the user with the most favorable channel to transmit. But to obtain the channel state information (CSI) of all users, the centralized schemes requires pilot signal for each channel, which causes scaling problem when the number of users increases. In order to constrain the channel estimation overhead of exploiting multiuser diversity, distributed approaches are introduced. One approach called opportunistic random access was studied in [6], [7], [8]. Opportunistic random access scheme assumes distributed channel state information (CSI) of Nan Zhang and Branimir Vojcic are with the Department of Electrical and Computer Engineering, the George Washington University. Michael Souryal is with the Advanced Network Technologies Division, National Institute of Standards and technology. Erik G. Larsson is with the Signals, Sensors & Systems, Communication Theory, School of Electrical Engineering, Royal Institute of Technology (KTH).
the uplink in each user, which is achieved by the basestation broadcasting pilot signal and each user estimating its uplink channel by this downlink pilot. An inherent assumption of this scheme is the symmetric CSI at the uplink and the downlink channels. With CSI information, each user can adjust its transmission strategy to jointly achieve multiuser diversity in the system. However, this assumption of symmetric uplink and downlink is only satisfied in a time division duplex (TDD) system, but not in a frequency division duplex (FDD) system where the uplink and downlink channels occupy different bands. Similar symmetric CSI in TDD system has also been exploited in [9], [10] for a transmitter to predict the channel to a user according to the received signal from the same user. When distributed CSI is not always available, in this paper, we find that by exploiting the benefit of capture effect, a simple distributed approach, reserve random access, can also achieve the multiuser diversity gain. Capture effect is an event that if multiple users compete for the channel access and generate a collision, the signal of some user is strong enough that it can be recovered despite the collision. With multiple users contending for the channel access with equal transmitter power, capture effect selects the user with the best channel by competition, and by allowing that user to reserve the channel and transmit adaptively according to the channel condition, multiuser diversity gain is achieved. By plotting cumulative density function (cdf) of the channel gain with or without capture, we show the relation between capture and more favorable channel during data transmissions. Assuming time-varying Nakagami-m fading [11] with or without time-varying path loss, for reservation slotted ALOHA [12] and reservation slotted nonpersistent inhibit sense multiple access (ISMA) [14], [15], [16], we obtain the analytical results for the spectral efficiency. Reservation slotted ALOHA was proposed in [12] based on the well-known slotted ALOHA scheme [13]. ISMA [14], [15], [16] is a variation of carrier sense multiple access (CSMA) [17]. In CSMA, each user will sense the channel before transmission. If the channel is sensed busy, the user will backoff its transmission. Instead of channel sensing, in ISMA, the basestation transmits a “busy” signal to inhibit all other users from transmitting as soon as an inbound packet is being received [14]. By transmitting channel status by the basestation, the hidden terminal problem, where a user may not be able to sense the transmission of another user due to fading and path loss of wireless channels, is avoided. Another benefit of ISMA is that capture is hard to sense in CSMA but can be notified to users by
2
the basestation in ISMA. We also compare the spectral efficiencies of reservation random access schemes with that of round-robin, where users transmit their signals in a fixed order where multiuser diversity can not be exploited. From these results, we observe that with capture effect, variations from fading or path loss can be exploited to increase the network throughput. However, exploiting the variations of path loss will have a more dramatic effect on capacity increase. Capture effect often occurs in the wireless channel where the received power of different users varies. Capture effect has received extensive studies, e.g., [15], [16], [18], [19], [20], [21], [22], [23], [24]. [16], [18], [19], [20], [23] have investigated the potential of capture as a means of increasing the probability of successful packet transmission in random access without reservation. Nearfar effect, the impact of different distance of users, for random access with capture was studied in [15], [21], [22], [24]. It is found that when both distance and fading are time-varying, the variable distance is usually the main reason contributing to capture [21], [22]. Since these previous research studied the throughput of random access schemes without reservation, they focused on the probability of capture, which determines the throughput. Another important aspect of capture effect, however, is largely ignored. That are the channel conditions of the captured users. Although the channel conditions have no impact on the network throughput of random access such as slotted ALOHA, for reservation random access with adaptive transmission, since packets are transmitted after channel captured, the channel conditions will determine the spectral efficiency of packet transmission. In our paper, assuming time-varying Nakagami-m fading with or without time-varying path loss, we derive the distribution of channel signal-to-noise ratio (SNR) conditioned on capture, and then obtain the network throughput for different reservation random access schemes based on the distribution. To our knowledge, few, if any, papers have analyzed the statistics of the channel conditioned on capture and its impact on throughput of reservation random access schemes; consequently, our analysis is believed to be novel. The paper is organized as follows. The next section describes the system model. Reservation random access schemes with Nakagami-m fading and users located at equal distance are analyzed in Section 3. Reservation random access schemes with mobile users experiencing time-varying fading and path loss are analyzed in Section 4. Finally, conclusions are summarized in Section 5. II. S YSTEM M ODEL As shown in Figure 1, the total transmission duration is divided into frames, and each frame consists of a variable duration reservation interval and a fixed duration data interval. We will study two approaches to provide channel access in the reservation interval, which are slotted ALOHA [13] and slotted nonpersistent ISMA [14],
[15], [16]. For synchronization of the slotted scheme, the basestation could broadcast pilot signal to all users. Note that the pilot can not be used for channel estimation of the uplink because the downlink and uplink channels are not necessarily symmetric. The reservation interval is used to determine which user has the right to transmit its packets. As soon as one user gains channel access, the data interval starts where the reserved user is able to transmit packets without interference from other users. We specify T as the (normalized) time unit and call it time slot. The duration of reservation interval is uncertain due to the nature of random access. But similar to 1xEVDO, the data interval has fixed length, M time slots as we assume, but variable data rates. The reserved user achieves adaptive transmission by estimating the channel at the beginning of the data interval and then use adaptive coding and modulation to transmit the packets. To measure the throughput, we use spectral efficiency of the data interval calculated with Shannon’s formula multiplied by the average occupancy of the data interval in a frame. The Shannon capacity is a theoretical upper bound but can be closely approximated by well-design adaptive coding and modulation schemes.
Fig. 1.
System model
We assume two possible scenarios for uplink channels. For the first case, all users are equidistant from the common receiver. With block fading channel, the fading amplitude of each user is fixed during a frame but varies independently from frame to frame and from user to user, and has a Nakagami-m distribution with m restricted to be a positive integer. For a Nakagami-m fading channel, without CSI, each user will transmit with the same power, and the received signal-to-noise ratio (SNR) of a randomly selected user has a Gamma distribution [11] as 1 m m m−1 −mp/Ω p e , p≥0 (1) fP (p) = Γ(m) Ω where P is the SNR, m is a positive integer, Γ(m) = (m − 1)! and the average received SNR, Ω, is fixed. Note that when m = 1, the fading is Rayleigh. For the second case, we assume mobile users and the distance from one user to the basestation changes with time. Thus, both the fading and path loss (Ω) are time-varying. To characterize the variations of distance, we use the model proposed in [20], which has been widely used, for example, in [15], [18], [24], and the probability density function (pdf) of Ω is written as 1 π (2) fΩ (Ω) = Ω−3/2 exp(− ). Ω ≥ 0 2 4Ω
3
III. R ESERVATION R ANDOM ACCESS WITH T IME -VARYING FADING In order to find the throughput, we at first obtain the distribution of channel SNR conditioned on capture. Let us assume that the received SNR for a random chosen user is ps and that the sum of the received SNRs of n interferers is pn . The distribution of p s is given by Eq. (1). The distribution of p n , the sum of n independent and identically Gamma-distributed random variables, can be found using the characteristic function to be
where P (zn > β) was given in Eq. (3). The cumulative distribution function of P s can be obtained by
ps FPs (ps |zn > β) = fPs (t|zn > β)dt (5) 0
Figure 2 shows the cdf of the received SNR of a captured user when Ω = 10dB, β = 6dB, m = 1 (Rayleigh fading) and different n. 6dB is a common value used for β as in [20], [15], [24]. 1 F (p )
mp 1 m nm nm−1 n , pn ≥ 0 pn exp − fPn (pn ) = Γ(nm) Ω Ω
P
0.9
s
s
FP (ps|z1>β) s
FP (ps|z5>β)
0.8
s
FP (ps|z10>β) s
0.7
fZn ,Ps (z, ps )
=
m (n+1)m 1 × Γ(nm)Γ(m) Ω (n+1)m−1
ps z nm+1
exp(−
m (ps + ps /z)) Ω
0.6 CDF of Ps
The ratio of the received power of the user to the total received power of the interferers plus thermal Gaussian noise is called the signal-to-interference-plusnoise ratio (SINR). When Gaussian noise can be ignored (i.e., in interference-limited systems), the SINR can be approximated by z n = ps /pn . Let β denotes the capture ratio. When zn > β, we assume that the signal with the received SNR ps is captured. In order to find P (z n > β), one can calculate the pdf of z n from fPs (ps ) together with fPn (pn ), and then integrate the pdf of z n from β to ∞. For Nakagami-m fading, the probability of capture in a given slot with n interferers was found in [23] to be n=0 1 Γ((n+1)m) 1 − Γ(nm)Γ(m) × P (zn > β) = β m−1 z (z + 1)−(n+1)m dz otherwise 0 (3) Since ps and pn are independent, their joint probability density function (pdf) is f Ps ,Pn (ps , pn ) = fPs (ps )fPn (pn ). With zn = ps /pn , the joint distribution of ps and zn can be obtained from f Ps ,Pn (ps , pn ) by the variable change p n = ps /zn ,
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
Ps (dB)
Fig. 2. SNR for the captured user, for Ω = 10dB, β = 6dB, m = 1 and different n
To understand Figure 2, note that when n increases, the chance for capture diminishes because more interference is generated and p n is larger in average. However, once a user captures the channel, statistically, its p s will be larger than the p s when n is smaller because ps > βpn and pn becomes larger. This improvement of channel quality for the captured user can be seen from Figure 2 in that the cdf of p s shifts to the right when n increases, which indicates a larger probability to have a better channel. Also, all F Ps (ps |zn > β), (n ≥ 1) are to the right of F Ps (ps ). The higher SNR conditioned on capture is crucial for using our approach to exploit the multiuser diversity. It shows even without any forms of CSI, the capture effect inherent in wireless communications still can select users with channels better than average, and the selected channels will be more favorable when more users participate in the selection process. During packet transmission in data interval, the higher SNR can be exploited together with adaptive coding and modulation to achieve a higher throughput.
where z > 0 and p s > 0. Then, we derive the pdf of the received SNR for the captured user as ∞ β fZn ,Ps (z, ps )dz fPs (ps |zn > β) = P (zn > β) m (n+1)m 1 A. Reservation Slotted ALOHA p(n+1)m−1 × = s P (zn > β)Γ(nm)Γ(m) Ω For the reservation ALOHA [12], both the reservation
1 mps mps β nm−1 interval and the data interval consist of multiple time t dt t exp − exp − slots. During the reservation interval, users with packets Ω Ω 0 transmit and retransmit their preambles in the manner m 1 1 m = pm−1 × of the well-known slotted ALOHA scheme [13]. At the s P (zn > β) Γ(m) Ω beginning of a time slot in the reservation interval, all
∞ mp 1 1 mps k s exp − 1+ (4) users with packets will transmit preambles, and they will Ω β k! βΩ receive the feedback of channel status before the end k=nm
4
of the same time slot. If a collision occurs, all users involved in the collision will back off and retransmit in successive time slots with certain probability. The feedback of channel status is achieved by the basestation broadcasting the channel status after receiving preambles. The broadcast includes whether a preamble is received and if received, it is which user’s preamble. Since the propagation delay is usually brief in local networks, users are able to receive the channel status information within the same time slot. Once one user’s preamble is detected by the basestation, data interval begins and only the reserved user can transmit in the data interval. In the reservation interval, if the retransmission probability of ALOHA scheme is a constant, it was shown in [25] that the traffic in the reservation interval can be modelled as a Poisson process with rate G preambles/slot, which includes the newly generated and the retransmitted preambles. For ALOHA without capture, the probability of success is p succ = Ge−G [25]. With capture, according to [20], psucc
= =
∞
lP (l arrivals)P (zl−1 > β)
l=1 ∞
G
n=0
e−G
Gn P (zn > β) n!
the overhead of reservation interval to coordinate the multiple ∞ access. In data interval, the spectral efficiency is 0 log2 (1 + ps )fps (Ps |succ)dps bits/s/Hz with fPs (ps |succ) defined in Eq. (9). Since the reservation interval has average length 1/p succ time slots and the data interval has fixed length M time slots, the portion of M . Therefore, time used in data transmission is M+1/p succ the overall spectral efficiency is
∞ M S= 1 log2 (1 + ps )fPs (ps |succ)dps psucc + M 0 bits/s/Hz (10) In later part, this measure will be repeatedly used for other schemes. From Eq. (10), we observe that S becomes larger when the spectral efficiency of data interval increases, and becomes smaller when the duration of reservation interval lasts longer. Therefore, one would expect that S achieves the maximum when reservation interval selects users with favorable channels with limited time. Note that at the beginning of data interval, the selected user’s channel will be estimated. However, since the time required to estimate one user’s channel is short, we do not count this overhead when calculating S.
(6)
where the probability of capture with n interferers, P (zn > β), is given in Eq. (3). Since the probability of success in each slot is equal and the arrivals are independent, the duration of the reservation interval is a geometric random variable with average
B. Reservation Slotted Nonpersistent ISMA
ISMA [14], [15], [16] is a variation of CSMA [17]. CSMA is designed in the manner that before transmission, a user will sense the channel to detect whether there are other users transmitting [17]. If the channel is sensed busy, the user will delay its transmission to some ∞ later time based on retransmission delay distribution. 1 ipsucc (1 − psucc )i−1 = slots (7) If the channel is sensed idle, the user will transmit Nr = psucc i=1 its preamble. The only difference between CSMA and In addition, the pdf of the received SNR for users ISMA is that in ISMA, instead of sensing the channel, the user will obtain the information of channel status successfully reserving the data interval is based on messages sent by the basestation as soon as 1 P (no interf erer)fPs (ps ) + the basestation receives preambles. The performance of fPs (ps |succ) = psucc ISMA is largely determined by the inhibit time, which ∞ is the time interval between the beginning of a preamble P (n interf erers)(n + 1)P (zn > β) × transmission and the detection of channel status by n=1 users. We assume the inhibit time consumes a time (8) slots whereas a β) 1 −G nonpersistent ISMA, where the time axis is divided Ge fPs (ps ) + = to intervals with the length of a time slots and users psucc ∞ can only transmit in the beginning of each interval. In n G Ge−G (9) nonpersistent ISMA, users will immediately run backoff P (zn > β)fPs (ps |zn > β) n! n=1 algorithm when detecting the channel is busy. Since the where psucc is given by Eq. (6). In Eq. (8),(n+1) is mul- slot of ISMA is much shorter than that of ALOHA, the tiplied with P (zn > β) because P (zn > β) is the prob- traffic intensity offered in each slot of ISMA is also ability of capture for a selected user and in total there smaller. Nevertheless, collision still occurs if multiple are n + 1 such probabilities. Also, P (no interf erer) = preambles are scheduled to transmit in the same slot. Gn+1 −G Following [17]’s approach, the probability of success of e . Ge−G and P (n interf erers) = (n+1)! To evaluate the performance of the reservation slot- ISMA can be calculated as ¯ ted ALOHA, we measure its spectral efficiency, which U psucc = ¯ ¯ counts both the spectral efficiency of data interval and B+I
5
3.5
where a cycle of preamble transmissions alternate be¯ and tween busy and idle states with average duration B ¯ ¯ I. U is the probability of success in a cycle. With =
I¯ = ¯ U
=
1+a ae−aG 1 − e−aG ∞ aGe−aG (aG)n P (zn > β) 1 − e−aG n=0 n!
3
Spectral efficiency (bits/sec/Hz)
¯ B
Reservation slotted ALOHA Reservation slotted nonpersistent ISMA Round robin transmission
aGe−aG 1 + a − e−aG
(aG)n P (zn > β) n! n=0
P (no
interf erer)
= n+1
aGe−aG 1+a−e−aG −aG
(aG) e (n+1)!(1+a−e−aG )
0 −3 10
(11)
One can also find that the average duration of the reser¯ I¯ 1 vation interval is B+ = psucc . By (8), we obtain the ¯ U pdf of the received SNR for users successfully reserving the data interval as 1 fps (ps |succ) = × (1 + a − e−aG )psucc ∞ (aG)n −aG aGe fPs (ps ) + aGe−aG × n! n=1 P (zn > β)fPs (ps |zn > β) (12) Here,
1.5
0.5
and
P (n interf erers) = . The spectral efficiency of ISMA can be obtained by substituting (11), (12) to (10). In Figure 3, the spectral efficiencies of reservation slotted ALOHA and reservation slotted nonpersistent ISMA are compared with the spectral efficiency of round robin transmission. In round robin transmission, the users transmit to the basestation one by one with a fixedorder regardless of their SNR. ∞ Its spectral efficiency is 0 fPs (x) log2 (1 + x)dx. With reference to 1xEV-DO [3], we choose M = 32. a for reservation slotted nonpersistent ISMA normally is valued between 0.1 and 0.01, and here it is set as 0.02. From Figure 3, we observe that the spectral efficiencies gradually attain their maximum, and then fall sharply when the traffic intensities increase further. We measure that the maximal spectral efficiencies of ALOHA and ISMA are approximately equal, and both are approximately 7% higher than the spectral efficiency of round robin. The high spectral efficiency of reservation random access schemes can be achieved by adjusting the parameters such as retransmission probability in random access. The figure also shows that ISMA reaches its maximum with larger traffic intensity than ALOHA. The reason is ISMA has shorter slot than ALOHA. With equal traffic intensity, the traffic offered in each slot of ISMA is lower than that of ALOHA. To further illustrate the improved channel quality of reserved users by capture effect, in Figure 4, we plot the cdfs of the received SNR for users successfully reserving the data interval in ALOHA and ISMA. The
−2
10
−1
10
0
10 Traffic intensity, G
1
10
2
10
3
10
Fig. 3. Spectral efficiencies of reservation slotted ALOHA, reservation slotted nonpersistent ISMA and round robin transmission under timevarying fading, for m = 1, β = 6dB, Ω = 10dB.
cdfs can be derived from corresponding pdfs in (9) with G = 2.95 and (12) with G = 141.25, for which traffic intensities the spectral efficiencies achieve their maximum, respectively. The cdfs of the received SNR for users successfully reserving the data interval are compared with F Ps (ps ). The comparisons show both reservation schemes give similar F Ps (ps |succ), and these cdfs are around 2dB better than F Ps (ps ). That explains why reservation schemes give higher throughput than round robin transmission. 1 F (p ) P
0.9
s
s
FP (ps|succ), Reservation slotted ALOHA s
FP (ps|succ), Reservation slotted nonpersistent ISMA s
0.8
0.7
0.6 cdf
psucc =
2
1
With (11), the probability of success, p succ , for slotted non-persistent ISMA is [16] ∞
2.5
0.5
0.4
0.3
0.2
0.1
0 −10
−5
0
5 Ps (dB)
10
15
20
Fig. 4. Cdfs of the received SNR for users successfully reserving the data interval in reservation slotted ALOHA, reservation slotted nonpersistent ISMA and round robin transmission, for m = 1, β = 6dB, Ω = 10dB, with G = 2.95 for ALOHA and G = 141.25 for ISMA.
IV. R ESERVATION R ANDOM ACCESS WITH T IME -VARYING FADING AND PATH L OSS In this section, we will focus on a cellular network with mobile users. Taking into account the mobility, the distance from a user to the basestation changes from time to time, which generates time-varying path loss in
6
fPn (pn ) =
F (p ) P
s
0.9
s s
0.8
and F (, , , ) is hypergeometric function [26, Chap. 9.1]. Therefore, for this case, [24] gives 1 n=0 (13) P (zn > β) = 1 − FZn (β) otherwise With the above equations, we derive f Ps (ps |succ). At first, we have ps ps fZn ,Ps (z, ps ) = 2 fPs (ps )fPn ( ) z z √ n Γ(m + 0.5)Γ(nm + 0.5) × = 4m Γ(m)Γ(nm) psm−1.5 1 π m+0.5 √ πnz nm+0.5 (14) (ps + 4m ) z(1 + 4mps ) With (14), we can find f Ps (ps |zn > β) as ∞ β fZn ,Ps (z, ps )dz fPs (ps |zn > β) = P (zn > β) √ n Γ(m + 0.5)Γ(nm + 0.5) 1 × = P (zn > β) 4m Γ(m)Γ(nm) β −nm psm−1.5 π m+0.5 nπ nm+0.5 × (ps + 4m ) nm( 4mp ) s −4mps 1 ) (15) F (nm + , nm; nm + 1; 2 nπβ where the last equation uses the result of definite integral in [26, 3.194-2]. The cdfs of P s in condition of capture are plotted in Fig. 5. Similar to Fig. 2, F Ps (ps |zn > β) shifts to the right when n increases. However, we observe that the improvement of channel quality due to capture effect becomes more significant with the variations of path loss. For example, in Fig. 2, F Ps (ps |z10 > β) is around 11dB better than F Ps (ps ). But in Fig. 5, the gap increases to around 30dB. With this difference, we
FP (ps|z10>β) s
0.7
0.6
0.5
0.4
0.3
pm−1 1 Γ(m + 0.5) s √ π m+0.5 ) 2 m Γ(m) (ps + 4m pnnm−1 n Γ(nm + 0.5) 0.5 πn nm+0.5 m Γ(nm) (pn + 4m )
The cdf of z n = ps /pn was also found in [24] as
∞ πn −(nm+0.5) ) xnm+m−1 (x + × FZn (z) = K 4m 0 −4mzx )dx F (m + 0.5, m, m + 1, π where √ m m−1.5 nz 2m−1 Γ(m + 0.5)Γ(nm + 0.5)m K =2 Γ(m)Γ(nm)π m+0.5
s
FP (ps|z1>β) FP (ps|z5>β)
0.2
0.1
0 −10
−5
0
5
10
15 Ps (dB)
20
25
30
35
40
Fig. 5. SNR for the captured user, for β = 6dB, m = 1 and different n
expect a larger capacity gain by exploiting multiuser diversity for mobile users. In Figure 6, the spectral efficiency of reservation slotted ALOHA and reservation slotted nonpersistent ISMA are plotted, and compared with the spectral efficiency of round robin transmission. This figure shows by exploiting the multiuser diversity, the maximal spectral efficiency of these two reservation random access schemes is approximately 250% higher than that of round robin. The maximal spectral efficiencies in Figure 6 are also around three times of the maximum in Figure 3, which shows that mobility does increase throughput in reservation random access schemes. 10
9
Reservation slotted ALOHA Reservation slotted nonpersistent ISMA Round robin transmission
8
Spectral efficiency (bits/sec/Hz)
fPs (ps ) =
1
CDF of Ps
addition to fading. In [27], it was shown that mobility increases the capacity of the wireless ad hoc network. To test whether mobility will also increase the throughput of reservation random access schemes, we will evaluate the spectral efficiencies and compare them with the results obtained in the previous section. Following the approach in [20], we use the quasiuniform spatial distribution as in (2). Combining (1) and (2) gives fPs (ps ) and fPn (pn ). According to [24],
7
6
5
4
3
2
1
0 −3 10
−2
10
−1
10
0
10 Traffic intensity, G
1
10
2
10
3
10
Fig. 6. Spectral efficiency of reservation slotted ALOHA, reservation slotted nonpersistent ISMA and round robin transmission under timevarying fading and path loss, for m = 1, β = 6dB.
To investigate the improved channel quality by multiuser diversity, we plot the cdfs of the received SNR for users successfully reserving the data interval under traffic intensities which maximize the spectral efficiencies in Figure 7. This figure shows the cdfs of reserved users are around 27dB better than F Ps (ps ), which explains why the spectral efficiencies of the reservation schemes
7
are much higher than that of round robin. 1 FP (ps) s
0.9
FP (ps|succ), Reservation slotted ALOHA s
FP (ps|succ), Reservation slotted nonpersistent ISMA s
0.8
0.7
cdf
0.6
0.5
0.4
0.3
0.2
0.1
0 −10
−5
0
5
10
15 Ps (dB)
20
25
30
35
40
Fig. 7. Cdfs of the received SNR for users successfully reserving the data interval in reservation slotted ALOHA, reservation slotted nonpersistent ISMA and round robin transmission, for m = 1, β = 6dB, with G = 10.23 for ALOHA and G = 501.19 for ISMA.
V. C ONCLUSION We have computed the spectral efficiency of reservation slotted ALOHA and reservation slotted nonpersistent ISMA for time-varying Nakagami-m fading with and without time-varying path loss. It was shown that the capture effect inherently reserves users with better channels, which increases the spectral efficiency and throughput. Numerical results illustrate that both reservation schemes successfully exploit multiuser diversity and result in a significantly higher throughput than roundrobin transmission. It also shows the time-varying path loss could provide a higher potential multiuser diversity gain than the time-varying fading. VI. ACKNOWLEDGEMENT The authors are grateful to the anonymous reviewers for their comments and suggestions. R EFERENCES [1] R. Knopp and P. Humblet, “Information capacity and power control in single cell multiuser communications,” Proc. 1995 IEEE Int. Conf. on Communications, pp. 331-335, June 1995. [2] R. Heath Jr., M. Airy and A. J. Paulraj, “Multiuser diversity for MIMO wireless systems with linear receivers,” Proc. 2001 IEEE Asilomar Conf. on Signals, Systems, and Computers, pp. 11941199, Nov. 2001. [3] Q. Wu and E. Esteves, “The cdma2000 high rate packet data system”, Advances in 3G Enhenced Technologies for Wireless Communications, Editors J. Wang and T.-S. Ng, Chapter 4, March 2002. [4] G. Holland, N. Vaidya, P. Bahl, “A rate-adaptive MAC protocol for multi-Hop wireless networks,” Proc. of ACM MOBICOM, July 2001. [5] B. Sadeghi, V. Kanodia, A. Sabharwal and E. Knightly, “Opportunistic media sccess for multirate ad hoc networks,” Proc. of ACM MOBICOM, Sept. 2002. [6] X. Qin and R. Berry, “Exploiting multiuser diversity for medium access control in wireless networks,” Proc. of IEEE INFOCOM, pp. 1084-1094, March-April 2003.
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Nan Zhang is a Ph.D. student of the George Washington University. He received his M.S. from Chinese Academy of Sciences (2000), and B.S. from Huazhong University of Science and Technology (1997). His research interests include MIMO, wireless network scheduling, and multiuser detection.
Branimir R. Vojcic is a professor in, and a past Chairman of, the Department of Electrical and Computer Engineering at the George Washington University. He has received his Dipl. Ing., M.Sc. and D.Sc. degrees from the University of Belgrade in Serbia and Montenegro in 1980, 1986 and 1989, respectively. His current research interests are in the areas of communication theory, performance evaluation and modeling mobile and wireless networks, mobile internet, code division multiple access, multiuser detection, adaptive antenna arrays, spacetime coding and ad-hoc networks. He has also been an industry consultant and has published and lectured extensively in these areas. He co-authored the book: The cdma2000 System for Mobile Communications. Dr Vojcic received NSF CAREER Award in 1995. He is a Senior Member of IEEE, was an Associate Editor for IEEE Communications Letters and is presently an Associate Editor for Journal on Communications and Networks.
Michael R. Souryal is an NRC postdoctoral research associate in the Wireless Communication Technologies group at the National Institute of Standards and Technology, Gaithersburg, MD. He received his D.Sc. in electrical engineering from The George Washington University (2003), M.S. in information networking from Carnegie Mellon University (1991), and B.S. in electrical engineering from Cornell University (1990). From 1991 to 1999, he was with Telcordia Technologies (formerly Bellcore), Red Bank, NJ, where he was involved in new service development for public network providers. His research interests include wireless ad hoc networks, spread spectrum systems, multiuser detection and adaptive transmission techniques.
Erik Larsson received his Ph.D. in Electrical Engineering from Uppsala University (2002). He has been an Assistant Professor at the University of Florida (Gainesville, FL, USA), and at the George Washington University (Washington DC). He also has worked for Ericsson Radio Systems, where he was involved in standardization and algorithm design for E-911 wireless location. Since 2005 he is Associate Professor and Docent at the Royal Institute of Technology in Stockholm, Sweden. His research interests include communication theory, wireless systems and signal processing. He has coauthored some 40 papers on these topics, he holds several U.S. patents on wireless technology, and he is a coauthor of the textbook Space-Time Block Coding for Wireless Communications (Cambridge University Press, 2003, with P. Stoica). He is an associate editor for the IEEE Transactions on Vehicular Technology and the IEEE Signal Processing Letters.
