Cross-layer design of multiple antenna multicast combining AMC with ...

Ann. Telecommun. (2010) 65:803–815 DOI 10.1007/s12243-010-0190-2

Cross-layer design of multiple antenna multicast combining AMC with truncated HARQ Tan Tai Do · Jae Cheol Park · Iickho Song · Yun Hee Kim

Received: 1 January 2010 / Accepted: 30 June 2010 / Published online: 18 July 2010 © Institut Télécom and Springer-Verlag 2010

Abstract Combining adaptive modulation and coding with truncated hybrid automatic request, this paper presents a cross-layer design for multiple antenna multicast over a common radio channel. In the design, the modulation and coding scheme of a multicast packet is selected based on the minimum signal-to-noise ratio (SNR) in the multicast group in such a way that the constraint on the packet loss rate is satisfied for all users in the group. A general expression for the throughput of the proposed design is derived in frequencyflat fading channel environment and specific results in Rayleigh, Nakagami, and Rician fading channels are provided. It is shown that the proposed multicast design provides a significant throughput gain compared to the unicast counterpart, in particular, in the mid- to high

This work was supported by the IT R&D program of MKE/KEIT [KI001814, Game Theoretic Approach for Cross-layer Design in Wireless Communications] and by the National Research Foundation of Korea, with funding from the Ministry of Education, Science, and Technology, under Grant KRF-2008-314-D00311. T. T. Do · J. C. Park · Y. H. Kim (B) Department of Electronics and Radio Engineering, Kyung Hee University, Yongin, Gyeonggi 446-701, Korea e-mail: [email protected] T. T. Do e-mail: [email protected] J. C. Park e-mail: [email protected] I. Song Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea e-mail: [email protected]

SNR region. It is also shown that a larger value of the diversity order, Nakagami parameter, and Rician factor is more beneficial to multicast than to unicast. Keywords Wireless multicast · Cross-layer design · AMC · Truncated HARQ · SNR threshold

1 Introduction Recently, multicast has been introduced into wireless networks to support such group-oriented applications as audio-video conference, distance education, and intelligent transportation [1, 2]. In these applications, identical contents can be requested by a group of users located in the same cell. Using a common radio channel overheard by the group of users, multicast can deliver the contents requested more efficiently than unicast assigning a separate radio channel to each user in the group [3]. Initial studies on multicast have focused mainly on the design of efficient routing protocols to resolve the bottleneck at the network layer. As the bottleneck has moved toward the physical layer, multicast methods in the physical layer and their performance from the aspect of information theory have been addressed in the literature [4–9]. These studies have attempted to improve the statistics of the minimum signal-to-noise ratio (SNR) of the user group which determines the physical layer performance of multicast. In doing so, the simplest yet most viable approach is the employment of multiple antennas with and without channel state information at the transmitter. In essence, information theoretic study has shown that the capacity limits of multiple antenna multicast

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 1

are in the order of O K− d in Rayleigh fading channel, where K is the number of users in the group and d is the order of antenna diversity [5, 8]. Further enhancement of throughput can be achieved with the antenna subset selection [9], but the method requires a large amount of the channel quality indication (CQI) feedback, which is likely to make its practical implementation difficult. In the meantime, cross-layer design approaches have drawn a lot of interest to utilize radio resources more efficiently, meeting the quality of service (QoS) requirements at the same time [10]. In the approaches, more than two layers are taken into account for joint optimization of the performance. For unicast, adaptive modulation and coding (AMC) in the physical layer [11–13] has been combined with truncated automatic repeat request (ARQ) in the medium access control (MAC) layer [14] to improve the spectral efficiency under the constraints of packet loss rate and delay [15]. In the method, a modulation and coding scheme (MCS) in AMC can be chosen more aggressively (i.e., at a lower SNR level) by allowing certain time delay until the correct reception of a data packet. In [16] and [17], the performance is shown to be enhanced further by incorporating hybrid ARQ (HARQ) [18, 19] instead of ARQ in the cross-layer design. For multicast also, some cross-layer design approaches have been proposed [20– 23], where the combining occurs at layers higher than those in the unicast methods: specifically, the designs in [20, 22] optimize the rate of multicast taking erasure coding in the transport layer into account and the design in [21] combines AMC for multicast [24] with video streaming in the application layer. In this paper, by combining AMC in the physical layer and truncated HARQ in the MAC layer, we propose a cross-layer design for multicast guaranteeing the QoS of all users in the group at reasonable feedback overhead. The proposed design selects an MCS of AMC by comparing the minimum SNR with appropriate SNR thresholds, which are in turn determined to satisfy the constraint on the packet loss rate of all users when the number of retransmissions in HARQ is limited. For the design proposed, we derive a general expression of the throughput which can be applied in various fading channel models and for typical retransmission methods. Specifically, we provide analytical results when type-I (Chase combining) and type-II (incremental redundancy) HARQ methods are employed in Rayleigh, Rician, and Nakagami fading channels, and investigate the effect of various channel parameters on the performance of the proposed design. In addition, the performance of the proposed design is compared with that of the unicast counterpart, with which some

system conditions favorable to the proposed design are described tangibly. The remaining of this paper is organized as follows. Section 2 presents the system model of the proposed cross-layer design. The selection criteria of MCSs meeting the QoS are proposed in Section 3. We then derive a general expression of the throughput for arbitrary fading channel models and provide specific expressions of the throughput for some well-known fading channels in Section 4. The performance of the proposed design is investigated under various conditions in Section 5, followed by conclusions in Section 6.

2 System model Consider a wireless system with a base station (BS) and K users joined in a multicast group. In the system, multicast data common to the group are delivered over a common radio channel shared by the group. The detailed system model is illustrated in Fig. 1, where the system exploits multiple antennas at the transmitter and receiver for spatial diversity: specifically, the BS is equipped with nt transmit antennas and each of the K users is equipped with nr receive antennas. The system supports AMC and HARQ functions combined in a cross-layer approach, with AMC performed with L + 1 MCSs constructed by reasonable combinations of code rates and modulation sizes. The normalized data rates L per unit frequency of the MCSs are denoted as {Rl }l=0 bps/Hz in the ascending order, where R0 = 0 represents no transmission. It is also assumed that the HARQ allows a maximum number Nmax of retransmissions. In the system, orthogonal space-time block coding (OSTBC) is employed at the BS for multiple antenna transmission and maximal ratio combining is employed at the multicast group for multiple antenna reception. Here, we limit nt to be 1 or 2 with which OSTBC can be constructed without any loss in rate. Then, the instantaneous received SNR of user k is given by γk =

nt  nr ρ  |hk, j,i |2 , nt i=1 j=1

(1)

where ρ is the average received SNR per receive antenna and hk, j,i is the complex fading coefficient from the ith transmit antenna to the jth receive antenna of user k. We assume a frequency-flat block-fading channel model with {hk, j,i } invariant over the initial transmission and retransmissions of a data packet. We also assume that {hk, j,i } are independent and identically (i.i.d.) with mean zero and variance  distributed  2 E |hk, j,i | = 1.

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Fig. 1 System model of the proposed cross-layer design with truncated HARQ

User K Common Channel

User 2 User 1

Base Station Data Buffer HARQ Controller

Transmitter

Receiver

AMC Controller

CQI Generator

Data Buffer HARQ Generator

CQI Feedback Channel HARQ Feedback Channel

For the BS to choose the MCS of the data packet, the minimum SNR γmin = min γk k

(2)

of the multicast group is delivered to the BS over the CQI feedback channel common to the multicast group. The feedback mechanism of the minimum SNR can be designed more efficiently than that of the whole SNR values {γk } using an opportunistic feedback method proposed in [21].1 The BS then selects the MCS of the data packet by comparing the minimum SNR in the group with the SNR thresholds: the determination of the SNR thresholds is described in Section 3. Once the MCS is selected, the BS accordingly channelencodes and modulates a data packet containing cyclic redundancy check (CRC) bits for transmission over the wireless channel. After receiving the packet, each user performs coherent demodulation and channel decoding and then detects errors in the recovered data packet via the CRC checksum. If a packet error is detected, a retransmission request, typically in the form of no acknowledgement (NACK), is generated by the HARQ generator and is sent to the BS over the HARQ feedback channel. If the HARQ controller at the BS observes any retrans-

1 In

the method, each user of the group generates a CQI packet containing its SNR and sets the back-off timer of the CQI packet proportional to the SNR for a contention-based access. Then, the worst conditioned user with the minimum SNR can transmit the CQI packet the fastest and the other users sensing the CQI packet transmitted by the worst conditioned user do not transmit their CQI packet any more. Therefore, much of wireless resources can be saved by allowing only one user in the worst condition, instead of all users in the group, to transmit the CQI packet.

mission request from the users in the group, it selects a segment of the codeword stored in the buffer for retransmission: the segment of the codeword identical to that in the initial transmission is retransmitted in the cases of ARQ and type-I HARQ [15] while an incremental segment of parity bits is transmitted in the case of type-II HARQ [17]. The retransmission process continues until either all the users receive the data packet successfully or the number of retransmissions reaches the maximum value Nmax .

3 Proposed cross-layer design 3.1 MCS selection criteria The QoS of multimedia services is specified typically by the maximum packet loss rate and delay time allowed in the services [15]. Therefore, the parameter Nmax of L the truncated HARQ and the decision regions {Al }l=0 of the MCSs in AMC are so designed as to satisfy the delay time and maximum packet loss rate allowed, where Al is a function of the vector γ = [γ1 γ2 · · · γ K ] of instantaneous received SNR. More specifically, we first choose a value of Nmax not exceeding the maximum delay time at the complexity tolerable [15]. For the value Nmax chosen, we then determine the SNR regions {Al } for which the packet error rate (PER) after Nmax retransmissions does not exceed the target packet loss tar with the selected MCS. In deriving {Al }, we rate Ploss assume in this paper that (1) the BS has the information only of the minimum SNR γmin in the multicast group, (2) channel estimation and packet error detection are perfect at the receivers, and (3) the frequencyflat fading channel coefficient is invariant during the transmission of a data packet and is independent for each packet.

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Let us assume that a data packet is transmitted by the lth MCS to the multicast group with the instantaneous SNR vector γ : we denote the PER of user k after all possible retransmissions by Pl(k) (γ ). Considering the constraint on the packet loss rate, we should have Pl(k) (γ )

≤

tar Ploss

(3)

for any γ in Al . Since a retransmission occurs if there exists any user detecting a packet error in the group, Pl(k) (γ ) can be expressed as Pl(k) (γ ) = Pl,Nmax +1 (γk )

N max 

gr

Pl,n (γ ),

(4)

n=1 gr

where the user PER Pl,n (γk ) and group PER Pl,n (γ ) are the probabilities of detecting a packet error at the nth transmission of user k and of any user in the group, respectively. From the assumption of i.i.d. fading, the group PER is expressed as gr

Pl,n (γ ) = 1 −

K 



 1 − Pl,n (γk ) .

(5)

al,Nmax +1 e−gl,Nmax +1 γmin

N max 



We now obtain the SNR regions {Al } based on the minimum SNR information available at the BS by utilizing the inequality

n=1

(9) We can then determine l by searching for the smallest value of γmin satisfying (9). For the exact value of l , we may resort to a computer-aided numerical search method. Since such exact value of l is rather difficult to obtain, we will consider three approximations to Pl,n (γ min ) = 1 − (1 − al,n e−gl,n γmin ) K , gr

(6)

1) Single-user thresholds {lsu }: For small K, we might use

Pl,Nmax +1 (γmin )



tar 1 − {1 − Pl,n (γmin )} K ≤ Ploss

⎛N



⎞

max +1

lsu =

1 Nmax +1

gl,n

⎜ ⎜ ln ⎜ ⎝

al,n ⎟ ⎟ ⎟, ⎠

n=1 tar Ploss

(12)

n=1

(7)

n=1

from (4) and (5). Therefore, if we choose the SNR  region of the lth MCS as Al = γ |l ≤ γmin < l+1 , where l is the smallest value of γmin satisfying (7) with 0 = −∞ and  L+1 = ∞, the constraint (3) on the packet loss rate can be always satisfied. 3.2 SNR thresholds In this section, we obtain an explicit expression of l by employing the approximation [15, 17] Pl,n (γ ) = al,n e−gl,n γ

(11)

in (9) assuming that there exists only one user. This approximation represents for example the case where only the user with the minimum SNR requests retransmission as in [25]. Replacing (11) in (9), we have the SNR thresholds

tar tically γmin . Because of (6), if Pl(k) (γ min ) ≤ Ploss , (3) is

satisfied. Now, replacing Pl(k) (γ ) with Pl(k) (γ min ) in (3), we have

(10)

which will eventually result in three sets of the thresholds {l }.

where γ min is the 1 × K vector whose entries are iden-

N max 

 tar 1−(1−al,n e−gl,n γmin ) K ≤ Ploss .

1 − (1 − al,n e−gl,n γmin ) K ≈ al,n e−gl,n γmin

k=1

Pl(k) (γ ) ≤ Pl(k) (γ min ),

retransmission methods employed: these numbers can be obtained for instance by fitting the PER curves obtained at the nth transmission of a packet using the lth MCS. By replacing Pl,n (γmin ) in (7) with (8), we have

which coincide exactly with those in the unicast counterpart [15]: nonetheless, it should be noted that the SNR thresholds are compared with the minimum SNR of the group in multicast while they are compared with the SNR of the single user in unicast. It should be mentioned that the MCS selection with {lsu } does not always satisfy the constraint (9) on the packet loss rate, in particular as K increases. 2) Upper-bound thresholds {lub }: An upper-bound lub on l is obtained by applying   1 − (1 − al,n e−gl,n γmin ) K ≤ min Kal,n e−gl,n γmin , 1

(8)

(13)

to the user PER. Here, al,n and gl,n are real numbers depending on the modulation, channel coding, and

in (9). The proof of (13) is straightforward from the facts that a) the probability p = al,n e−gl,n γmin

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cannot exceed 1 and b) 1 − (1 − p) K ≤ pK for 0 ≤ p ≤ 1 since g( p) = 1 − (1 − p) K − pK is a nonincreasing function of p with g(0) = 0, g(1) = −K, and g ( p) = K(1 − p) K−1 − K ≤ 0. From the inequality (13), the constraint (9) is always satisfied if N max 

al,Nmax +1 e−gl,Nmax +1 γmin

tar min(Kal,n e−gl,n γmin ,1) ≤ Ploss .

n=1

(14) Since retransmission provides an identical or enhanced performance, the user PER Pl,n (γ ), and thus al,n e−gl,n γ , is a non-increasing function of n with l and γ fixed. Therefore, there exists a number t ∈ {1, 2, · · · , Nmax + 1} such that, if n ≥ t, Kal,n e−gl,n γmin < 1 for γmin given. Hence, (14) can be rewritten as −gl,Nmax +1 γmin

al,Nmax +1 e

N max 

tar Kal,n e−gl,n γmin ≤ Ploss ,

n=t

subject to Kal,t e−gl,t γmin < 1. (15)

=

⎧ ⎪ ⎪ ⎪ ⎨ min

1≤t≤Nmax +1

max

1 n=t

Nmax +1−t

Nmax +1 n=t

tar Ploss

⎫ ⎪ ⎪ ⎪ al,n ⎟ ⎬ ln(Ka ) l,t ⎟ (16) ⎟, ⎠ gl,t ⎪ ⎪ ⎪ ⎭ ⎞

is obtained as the smallest γmin satisfying (15). We would like to note that the MCS selection using {lub } always satisfies the constraint (9) on the packet loss rate since the upper-bound threshold lub is always larger than the exact threshold l . asy

3) Asymptotic thresholds {l }: When K tends to infinity, we have 1 − (1 − al,n e−gl,n γmin ) K → 1. Thus, we obtain asy l

=

1 gl,Nmax +1

In this section, we derive the throughput of the proposed cross-layer design for an arbitrary probability density function (pdf) fγ (x) of the received SNR γk and provide specific results in such well-known fading channels as Nakagami, Rician, and Rayleigh fading channels. 4.1 General expression of the throughput We define the user throughput Su by the data rate per unit frequency delivered to a user and system throughput Ssys by the total data rate per unit frequency delivered to the multicast group over the common multicast channel. The system throughput is then expressed as Ssys = K · Su ,

Su =

Nmax ⎪ +1 ⎪ ⎪ gl,n ⎩

⎛ ⎜K ⎜ × ln ⎜ ⎝

4 Throughput analysis

(18)

where the user throughput

Thus, the value

lub

always satisfies the constraint (9) on the packet loss asy rate since l ≥ lub . In addition, note that the use asy of {l } would incur some throughput loss when K asy is small since the asymptotic threshold l is more or less larger than the exact SNR threshold l .



al,Nmax +1 ln tar Ploss

 (17)

L 1 

N

πl Rl

(19)

l=1

can be obtained by adopting the results derived in the unicast with truncated ARQ [15], πl is the probability that the lth MCS is chosen, and N is the average number of transmissions per data packet. In the proposed design, since the lth MCS is chosen when γmin ∈ Al , πl is obtained as  l+1 πl = dFγmin (x) = Fγmin (l+1 ) − Fγmin (l ), (20) l

where Fγmin (x) = 1 − {1 − Fγ (x)} K

(21)

is the cumulative distribution function (cdf) of γmin [26, 27] derived from the cdf Fγ (x) of γk . The average number N of transmissions per data packet is also obtained similarly to that in unicast [15] except that the retransmission probability is given by the group PER in multicast instead of the user PER. In other words, we have gr

gr

gr

gr

gr

gr

N = 1 + P1 + P1 P2 + · · · + P1 P2 · · · P Nmax asy

from (9). The asymptotic threshold l can also be obtained from (16) by letting K → ∞. It should asy be mentioned that the MCS selection with {l }

= 1+

j Nmax   j=1 n=1

gr

Pn ,

(22)

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where the average group PER at the nth transmission gr Pn

=

!"

L 

gr πl Pl,n

L 

l=1

! πl

(23)

l=1

4.2 Throughput in specific fading channels

is obtained the conditional group PER  grby averaging  gr Pl,n = E Pl,n (γ )|l over all possible MCSs except for no transmission. As the conditional pdf fγ |l (γ ) of γ when the lth MCS is chosen can be expressed as fγ |l (γ ) =

⎧ ⎨

1 πl

⎩

K 

fγ (γk ), if γ ∈ Al ,

(24)

k=1

otherwise,

0,

we have  gr gr Pl,n = Pl,n (γ ) fγ |l (γ )dγ 1 = πl

#



1−

γ ∈Al

K 

$% {1 − Pl,n (γk )}

k=1

K 

& fγ (γk ) dγ

k=1

k=1 k = j

K = 1− πl

l

1) Nakagami fading: For i.i.d. Nakagami fading channels with Nakagami parameter m, the SNR γk defined in (1) is a gamma random variable with α  nt nr m and β  nt m/ρ [28]. The pdf and cdf of γk are then given by

{l,n (γ j)} K−1 ψl,n (γ j)dγ j,

β α α−1 −βx x e

(α)

(30)

(α, βx) ,

(α)

(31)

and Fγ (x) = 1 −

respectively, where  ∞ tα−1 e−t dt

(α) =

(32)

0

·{1 − Pl,n (γ j)} fγ (γ j)dγ j l+1

As the two functions Fγmin (x) and l,n (x) are the key elements in the evaluation of the throughput, we obtain these two functions in three well-known fading channels.

fγ (x) =

⎡ ⎤ K  ∞ K  l+1   1 (a) ⎢ ⎥ = 1− {1− Pl,n (γk )} fγ (γk )dγk⎦ ⎣ πl j=1 l γj 

from (8) and (26), πl from (20) using (12), (16), or (17) gr appropriately, and Pl,n from (28): the evaluations will gr lead to Pn in (23), N in (22), and Su in (19) sequentially.

(25)

is the complete gamma function and  ∞

(α, x) = tα−1 e−t dt

(33)

x

where 

∞

l,n (x) =

{1 − Pl,n (u)} fγ (u)du

(26)

x

and

and

ψl,n (x) = −

d l,n (x). dx

(27)

In (25), (a) is obtained by partitioning Al into the mutually exclusive regions {Al, j} Kj=1 , where Al, j = {γ j ≤ γk( = j) , l ≤ γ j < l+1 } denotes the SNR region in which user j has the minimum SNR and the lth MCS has been chosen. By changing a variable u = l,n (γ j) in the integral of (25), we finally have gr

Pl,n = 1 −

is the incomplete gamma function. We then have / 0

(α, βx) K (34) Fγmin (x) = 1 −

(α)

K  K . 1 - l,n (l ) − l,n (l+1 ) . πl

(28)

In summary, once the pdf fγ (x) is given, we can evaluate Fγmin (x) from (21),  ∞ e−gl,n u fγ (u)du (29) l,n (x) = 1 − Fγ (x) − al,n x

 al,n β α ∞ α−1 −(gl,n +β)u l,n (x) = 1 − Fγ (x) − u e du

(α) x  

(α, βx) al,n α, β(1 + gl,n /β)x  α = − . (35)

(α) 1 + gl,n /β (α) 2) Rician fading: For i.i.d. Rician fading channels with Rician factor K R , the SNR γk is a non-central chisquare random variable with 2nt nr degrees of freedom. By defining d  nt nr , s2  K R nt nr /(1 + K R ), and σ 2  1/(2(1 + K R )), the pdf of γk is written as [29] !   d−1 2 s2 + nρt x nt nt fγ (x) = x exp − 2σ 2 ρ ρs2 2σ 2  1  s nt ×Id−1 x , (36) σ2 ρ

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we also have

where Iν (x) =

∞ 



(x/2)ν+2 j j! (ν + j + 1)

j=0

(37)

is the νth-order modified Bessel function of the first kind. Using the series representation of Iν (x), we can express the cdf of γk as   1 s2 Fγ (x) = exp − (2σ 2 )d 2σ 2 2 j ∞   s/2σ 2 (nt /ρ)d+ j × j! (d + j) j=0  nt × u exp − 2 u du 2σ ρ 0 2 j   ∞ s2  s /2σ 2 = exp − 2 2σ j! (d + j) j=0 



x

d+ j−1

/



nt × (d + j) − d + j, 2 x 2σ ρ   s2 = 1 − exp − 2 2σ  j ∞  s2 /2σ 2 (d + j, 2σn2t ρ x) × j! (d + j) j=0 

2



s 2σ 2 ⎫K ⎧ 2 j ∞ ⎨ s /2σ 2 (d + j, 2σnt2xρ ) ⎬

Fγmin (x) = 1 − exp −K

Since  ∞

(38)

⎩

j! (d + j)

j=0

⎭

2 j   ∞  1 s2  s2 /σ 2 (nt /ρ)d+ j = exp − 2 (2σ 2 )d 2σ j! (d + j) j=0 ×

u

d+ j −

e

j=0

s2 /2σ 2 j!

j

 ⎧  ⎨ d + j, 2σn2t ρ x ⎩

(d + j)

  ⎫ 2σ 2 ρ ⎬ al,n d + j, 1 + gl,n nt x ⎪ − .  d+ j 2 ⎪ ⎭ 1 + gl,n 2σnt ρ

(d + j) 

nt 2ρσ 2

(41) 3) Rayleigh fading: We can obtain Fγmin (x) and l,n (x) for i.i.d. Rayleigh fading channels either from (34) and (35) obtained for Nakagami fading channels by letting m = 1 (leading to α = nt nr and β = nt /ρ) or from (39) and (41) obtained for Rician fading channels by letting K R = 0 (leading to s = 0 and σ 2 = 1/2). Either of the two approaches will produce %n n −1    & K t r  nt 1 nt i (42) Fγmin (x) = 1 − x exp − x i! ρ ρ i=0 and

 

nt nr , nρt γ j (nt nr − 1)!   al,n nt nr , (gl,n + nρt )γ j − nt nr .  ρ (nt nr − 1)! 1 + gl,n nt

(43)

5 Numerical results

x

∞

 ∞ 

. (39)

  exp −gl,n u fγ (u)du



×

l,n (γ j) =

which leads to

×

0

s2 l,n (x) = exp − 2 2σ



nt +gl,n 2σ 2 ρ

 u

du

x

  s2 = exp − 2 2σ    2 j  2 ∞ s /2σ 2 d + j, nt 1+gl,n 2σnt ρ x  2σ 2 ρ , ×  d+ j 2 j=0 j! 1+gl,n 2σnt ρ

(d + j) (40)

In this section, we evaluate the performance of the proposed cross-layer design by applying the analytical retar sults derived in Section 4 when Ploss = 10−3 and Nmax = 2. We employ the MCSs used in [17] with L = 4: the MCSs are constructed with turbo codes for channel coding and QAM for modulation. The MCSs and their PER parameters {al,n } and {gl,n } required for (8) are provided in Table 1 when type-I HARQ and type-II HARQ are employed in the retransmission method. Since the performance of the initial transmission is determined only by the MCS employed, both retransmission methods have the same sets of {al,1 } and {gl,1 }. On the other hand, {al,n = al,1 } and {gl,n = ngl,1 } for n ≥ 2 when type-I HARQ (providing only the energy gain through retransmission) is employed while both {al,n } and {gl,n } vary with n when type-II HARQ

810 Table 1 Specification of the proposed cross-layer design for performance evaluation

Ann. Telecommun. (2010) 65:803–815 MCS (l) Modulation Code rate Data rate (Rl )

1 QPSK 1/2 1

2 QPSK 2/3 4/3

3 16QAM 1/2 2

4 16QAM 2/3 8/3

All HARQ Schemes

al,1 gl,1

2.5 1.0

1.8 0.4

3.1 0.3

1.5 0.1

Type-I HARQ

al,2 gl,2 al,3 gl,3

2.5 2.0 2.5 3.0

1.8 0.8 1.8 1.2

3.1 0.6 3.1 0.9

1.5 0.2 1.5 0.3

Type-II HARQ

al,2 gl,2 al,3 gl,3

7.5 4.3 6.8 7.1

6.9 3.1 7.6 4.9

4.0 1.1 5.7 2.0

4.0 0.8 4.5 1.3

(providing both the energy and code gain through retransmission) is employed.

The SNR thresholds of the proposed cross-layer design using type-I HARQ are shown in Fig. 2 for l = 1 and 3 as the number K of users varies. In this figure, ‘mar’ denotes the marginal (minimum) SNR threshold l satisfying (9) which is obtained by a numerical search method, and ‘su’, ‘ub’, and ‘asy’ denote the SNR threshasy olds lsu , lub , and l , respectively, obtained in Subsection 3.1. It is quite interesting to note that the SNR threshold lub obtained rather simply is very close to the marginal SNR threshold. In the meantime, the thresholds l and lub vary with K and are always upper- and asy lower-bounded by l and lsu , respectively, which do not depend on the number K of users.

Figures 3 and 4 compare the average packet loss rate Ploss = E{Pl(k) (γ )} and user throughput Su , respectively, where the four types of SNR thresholds shown in Fig. 2 are employed in the proposed design employing type-I HARQ. The performance is evaluated in the Rayleigh fading channel with nt = 1 and nr = 1 as the average SNR ρ varies when K = 2 (representing a small number of users) and 20 (representing a large number of users). It is observed in Fig. 3 that all the SNR thresholds except for {lsu } when K = 20 satisfy tar the constraint Ploss ≤ Ploss on the packet loss rate. In Fig. 4, it is observed that the throughput obtained with simple upper-bound thresholds {lub } and that with {l } obtained from the numerical search method are indiscernible. Since the SNR thresholds {lub } can be obtained quite easily and provide almost the same throughput as the marginal SNR thresholds without QoS degradation, they should be appropriate for

Fig. 2 The SNR thresholds {l } for the proposed cross-layer design employing type-I HARQ as a function of the number K of users

Fig. 3 Average packet loss rate Ploss of the cross-layer design employing type-I HARQ with four sets of SNR thresholds in Rayleigh fading channel when nt = 1 and nr = 1

5.1 SNR thresholds

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Fig. 4 User throughput Su of the cross-layer design employing type-I HARQ with four sets of SNR thresholds in Rayleigh fading channel when nt = 1 and nr = 1

multicast as anticipated from an observation in Fig. 2. In the performance evaluation from now on, we thus assume only {lub }. In passing, we would like to mention that, if K asy is expected to be varying, {l } can be used as an ub alternative to {l } to meet the target packet loss rate at a slight throughput loss. In addition, it is noteworthy that the gain in the throughput obtained with thresholds {lsu } is a consequence of sacrificing the QoS of the users: thus, employing {lsu } in multicast is less attractive. Although we have not shown here explicitly, we have also confirmed that the same observation can be made when the type-II HARQ is employed instead of the type-I HARQ in the proposed design.

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Fig. 5 User throughput for various retransmission methods when K = 1 and 10 in Rayleigh fading channel with nt = 1 and nr = 1

the order of ‘AMC’, ‘AMC-ARQ’, and ‘AMC-HARQ’. Among the two HARQ types, type-II exhibits a slight loss of throughput in the high SNR region but provides a better performance in the low to mid SNR region by adopting more aggressive MCS selection with lower SNR thresholds. Another observation is that a crossover occurs between the system throughput for K = 1 and that for K = 10: the cross-over point defines the SNR over which multicast outperforms unicast. Since the cross-over SNR of ‘AMC-HARQII’ is the smallest among the four schemes, ‘AMC-HARQII’ (resulting in a multicast gain in the widest SNR region) is the best choice in the proposed design.

5.2 Effects of cross-layer design on multicast The effect of the retransmission method on the performance is shown in Figs. 5 and 6 from the aspects of the user throughput Su and system throughput Ssys = KSu , respectively, in the Rayleigh fading channel with nt = 1 and nr = 1. In these figures, ‘AMC-ARQ’, ‘AMCHARQI’, and ‘AMC-HARQII’ denote the proposed design which combines AMC with truncated ARQ, type-I HARQ, and type-II HARQ, respectively. In addition, ‘AMC’ denotes the conventional AMC without retransmission. As the number K of users increases, the user throughput in Fig. 5 decreases since the minimum SNR determining the rate tends to be smaller: on the other hand, the system throughput in Fig. 6 increases in the mid to high SNR region because of the multicast gain obtained from sharing the channel. It should be noted that the throughput increases in

Fig. 6 System throughput for various retransmission methods when K = 1 and 10 in Rayleigh fading channel with nt = 1 and nr = 1

812 Fig. 7 User and system throughputs as functions of K when ρ = 15 dB in Rayleigh fading channels with antenna diversity: a user throughput Su , b system throughput Ssys

Fig. 8 User and system throughputs as functions of K when ρ = 15 dB in Nakagami fading channels with nt = 2 and nr = 1: a user throughput Su , b system throughput Ssys

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Fig. 9 User and system throughputs as functions of K when ρ = 15 dB in Rician fading channels with nt = 2 and nr = 1: a user throughput Su , b system throughput Ssys

The effect of the channel fading parameters on the performance is investigated in Figs. 7–9 when typeII HARQ is employed in the proposed design. For comparison, we also provide the throughput of the conventional AMC without retransmission [24] and the

ergodic capacity E{log2 (1 + γmin )} of multicast when the data rate is selected with the minimum SNR [5, 9]. Figure 7 shows the effect of antenna diversity on the user and system throughputs as K varies when ρ = 15 dB in Rayleigh fading channel. The performance improves significantly irrespective of the methods employed as the diversity order d = nt nr increases

Fig. 10 System throughput of the multicast and unicast approaches as a function of ρ when K = 10 in Rayleigh fading channels with antenna diversity

Fig. 11 System throughput of the multicast and unicast approaches as a function of K when ρ = 10 dB in Rayleigh fading channels with antenna diversity

5.3 Effects of channel parameters

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since the antenna diversity reduces the fluctuation of the SNR, effectively leading to a higher minimum SNR on the average. It is also observed that, as K increases, the rate of change in the throughput of the proposed design (solid lines)    follows  roughly the theoretic rates 1 1 O K− d and O K1− d [5, 9] of the user and system throughputs, respectively. The results in Fig. 7 also show that the proposed design can provide a throughput closer to the theoretically achievable rate than the conventional AMC method. Figures 8 and 9 compare the throughput for four values of Nakagami parameter m in Nakagami fading channels and of Rician factor K R in Rician fading channels, respectively, when ρ = 15 dB, nt = 2, and nr = 1. With these parameters, we have α = 2m and β = 2m/ρ in (30) and d = 2, s2 = 2K R /(1 + K R ), and σ 2 = 1/(2(1 + K R )) in (36). Clearly, a larger m or K R implies a smaller fluctuation in the fading channel, and consequently, a higher minimum SNR on the average, similar to the increase of the average minimum SNR in Fig. 7 due to the diversity order: therefore, the system throughput increases with Nakagami parameter m and Rician factor K R as shown in Figs. 8 and 9. 5.4 Multicast versus unicast In Figs. 10 and 11, we have compared the system throughput between the proposed multicast design with type-II HARQ (‘Mul-HARQII’) and the unicast counterpart with opportunistic scheduling (‘UniHARQII’)2 in the Rayleigh fading channel. An interesting observation in Fig. 10 is that the multicast design performs better than the unicast design in the higher SNR region and the opposite is true in the lower SNR region. Another observation is that both transmit and receive antenna diversity significantly improve the performance of the multicast design while only the array gain at the receive antennas improves the performance in the case of the unicast design as observed also in [29]. This figure in addition indicates that, as the diversity order increases, the multicast design outperforms the unicast design in a wider SNR region. The performance of the multicast and unicast designs is also compared as the number K of users varies when ρ = 10 dB in Fig. 11. It is observed that the performance of the multicast design improves and then deteriorates as the value of K increases while that of

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the unicast design improves slightly as K increases: yet, the multicast design provides a considerably higher throughput over practical ranges (3 − 100) of K when the diversity order is at least 2.3 Although we do not provide the results explicitly in this paper, we can also expect that the multicast design is more suitable than the unicast design in Nakagami fading channels with m > 1 and Rician fading channels with K R > 0 since the smaller fading fluctuation in such cases than that in Rayleigh fading channel would result in higher system throughput.

6 Conclusion By combining AMC with truncated retransmission, we have proposed a cross-layer design to improve the throughput of wireless multicast under the constraints on the delay and packet loss rate. The proposed design selects MCSs by comparing the minimum SNR in the multicast group with SNR thresholds obtained to guarantee the QoS of all users. For the proposed design, we have derived three sets of the SNR thresholds, which can be employed in various scenarios. In addition, we have derived a general expression of the throughput for arbitrary fading channels and then obtained specific expressions of the throughput in Nakagami, Rician, and Rayleigh fading channels with antenna diversity. We have observed that the SNR thresholds appropriate for the proposed design can be obtained quite simply from the upper-bound approach. It is also observed that the proposed design significantly outperforms the conventional AMC without retransmission: among the retransmission methods employed in the proposed design, the type-II HARQ outperforms the unicast counterpart in a wider SNR region. In addition, we have found that the multicast gain can be significantly improved with a larger value of antenna diversity order, Nakagami parameter, and Rician factor. Acknowledgements The authors would like to thank the Associate Editor and anonymous reviewers for their constructive suggestions and helpful comments.

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