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On hierarchical modulation to increase flow-level capacity in OFDMA-based networks Anis Jdidi1 , Tijani Chahed1 , Salah-Eddine Elayoubi2 , Hichem Besbes3 1 Institut TELECOM; TELECOM & Management SudParis; UMR CNRS 5157 9 rue C. Fourier - 91011 Evry CEDEX - France {anis.jdidi, tijani.chahed}@it-sudparis.eu 2 Orange Labs 38, rue du G´en´eral Leclerc - 92794 Issy les Moulineaux - France
[email protected] 3 Ecole Sup´ erieure des Communications de Tunis (Sup’com) Cit´e Technologique des Communications - Route de Raoued Km 3,5 - 2083 El Ghazala Ariana, Tunisia
[email protected]
Abstract— Hierarchical modulation is a means to better use the overall resources of a given system by superposing, in terms of modulation, a user with better radio conditions on one with inferior radio conditions so as to advantageously transfer some resources from the latter to the former. This in turn impacts the overall performance of the system, which we consider in this work at the flow level, for a realistic dynamic setting where users come to the system and leave it after a finite duration corresponding, for instance, to the completion of a file transfer. We quantify, in this work, analytically and via simulations, the gain thus achieved and propose a new scheme in which users with bad radio conditions are also superposed on ones with better radio conditions so as to enhance the system performance even further.
Fig. 1.
4QAM and 16QAM
Fig. 2.
Hierarchical constellation
Index Terms— Embedded constellations, Hierarchical modulation, OFDMA, capacity, flow-level modeling.
I. I NTRODUCTION Hierarchical Modulation (HM) [1], also known in the literature as Embedded Constellations (ECs), is an old technique that has been defined notably in [2]. It did not however attract much attention until recently when DVB-T [3] adopted it as a standard owing to its efficiency in the context of digital broadcasting. In an OFDMA-based system [4], as is IEEE802.16e-based Mobile WiMAX [5], the way HM works is as follows. Consider two users, one with good radio conditions, whom we call user of type 1, and one with lower radio conditions, user of type 2. In OFDMA-based networks, Adaptive Modulation and Coding (AMC) implies that user of type 1 would be assigned a good modulation scheme, say 16QAM (Quadrature Amplitude Modulation) where 4 bits define the quadrant occupied by the subcarrier (please refer to Figure 1 - left), and user of type 2 would be assigned a lesser one, say 4QAM, with 2 bits for each quadrant (same Figure - right). Now, adding up the two constellations1 results in the embedded one shown in Figure 2. 1 the
term addition refers to real and imaginary components of the subcarrier
To user of the first type, the received signal is clear as it enjoys good radio conditions: the constellation of user of type 2 appears as a single point inside its own constellation and it can thus easily filter it out before decoding its own signal. To user of the second type, the constellation of user of type 1 will appear as (Gaussian) noise which it can also filter before decoding its own signal [4][6]. And hence the possibility to transmit simultaneously to two users on the same subcarrier without degrading the capacity of the link. A surprising result found by [6] and known as writing on dirty paper which states that the addition of a sequence in the channel known only to the transmitter does not change the capacity of the link. Our aim in this work is twofold. We first use such a result to quantify the potential gain in terms of system performance for a realistic setting where (data) users come to the system and leave it after a finite duration, i.e., finite service duration
2
and hence file size. Using HM would intuitively enable users with good radio conditions to finish their service earlier than in the case without HM and in turn, users of type 2 will thus enjoy more resources and be able to finish their service earlier as well. We second propose to extend the HM algorithm to the case where users with good radio conditions would also assist those with worse radio conditions. In doing so, not only users of good radio conditions would finish their service earlier and leave more resources to users with worse radio conditions, as is the case of classical HM, but we would also accelerate even further the service rate of the latter. As mentioned earlier, our work is carried out at the flow level, with users wishing to transfer data files of, again, finite size. This is in contrast with other works, such as the one contained in Reference [8] and which implements HM at the packet level. This work is also in contrast to other works on cooperative communications in which users with better radio conditions would help those with lesser ones by relaying partial or total information to them. The remainder of this work is organized as follows. In Section 2, we describe the OFDMA-based system and the way resources are allocated in such a setting: subcarriers, power, modulation and coding. In Section 3, we model our system at the flow level and derive some performance metrics. In Section 4, we consider HM, its modeling, its implementation as well as its impact on the system level performance. Section 5 contains some numerical and simulation results and Section 6 eventually concludes the paper. II. S YSTEM OFDMA is a multiple access technique which divides the total Fast Fourier Transform (FFT) space into a number of sub-channels (set of sub-carriers that are assigned for data exchange) whereas the time resource is divided into time slots and a frame is constructed by a number of slots. Subchannel allocation is done in the time-frequency domain: a call may share a subchannel with other users. Depending on the Signal-to-Noise Ratio (SNR), each user will be assigned a coding and modulation scheme, as dictated by the Adaptive Modulation and Coding (AMC) feature, in use in IEEE802.16 WiMAX; with a lesser modulation and coding for users with higher SNR. If we consider the path loss only, this results in the division of the cell into r co-centric regions, as shown in Figure 3, each of radius rj , j = 1...r. In real, the shape of each region is more chaotic if we take into account other phenomena, such as fast-fading. In all cases, users in each region experience the same radio conditions and have thus the same coding and modulation scheme. III. F LOW- LEVEL MODELING A. Flow dynamics Let us first consider the system dynamics without HM. Let users arrive to the system following a Poisson process with mean intensity λ. Users in region j are granted Nj (k) subchannels for a (finite) mean time duration Tj . k is a vector with entry j representing the number kj of users present in
Fig. 3.
Adaptive Modulation and Coding (AMC)
region j, j = 1, ..., r. The service duration T depends on the quantity of resources the users get which in turn depends on the number of users k that are simultaneously in progress in the system as well as how well they can take advantage of the resources they are granted, i.e., their modulation and coding. Let us assume that users are served on a Round Robin (RR) basis. This ensures fairness in time among all users in the cell. B. Processor Sharing analysis Based on the previously formulated flow dynamics, we can now model the number of users in the system as a Continuous Time Markov Chain (CMTC) with state vector k again denoting the number of users kj in each region of the system, j = 1, ..., r. This model is P completely described by a Processor Sharing queue with k = j kj being the total number of users in the system [7]. We also apply an admission control scheme on the maximal number of users kmax that can be admitted to the system. This would in turn guarantee every admitted user some minimal QoS in terms of mean throughput for instance. The steady-state probabilities are given by [7]: ρ¯x π(x) = 1 + ρ¯ + ... + ρ¯kmax for 0 ≤ x ≤ kmax and where: r X 2 ) ; ρ = λE[F ] ρ¯ = ρ¯j ; ρ¯j = ρj /cj ; ρj = ρπ(rj2 − rj−1 j=1
where rj is the radius of region j, j = 1, ..., r, E[F ] is the mean file size and cj is the rate a user in region j would achieve if he was alone in the system and was thus given all capacity, subcarriers and power. It is given by: cj =
N X W log2 (1 + SN Rj,n ) N n=1
(1)
where W is the total bandwidth, N is again the total number of subcarriers and SN Rj,n is the SNR experienced by user in region j on subcarrier n given by: Pj,n |hj,n |2 Γσ 2 where Pj,n is the transmit power of user j on subcarrier n, hj,n is its channel gain witch includes the effects of flat fading,
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path loss and shadowing and σ is the variance of the Additive White Gaussian Noise (AWGN) with zero mean. Γ represents the SNR-gap which is used to ensure some QoS for an uncoded QAM-AMC system and is given by: · µ ¶¸2 Pe 1 −1 Q Γ= ,Γ ≥ 1 3 4 where Q−1 (.) is the inverse standard Gaussian Q-function and Pe is the target Bit Error Rate (BER). C. Performance metrics The mean transfer time Tj in ring j is given by: Tj = E[F ]
1 − (kmax + 1)¯ ρkmax + kmax ρ¯(kmax +1) cj (1 − ρ¯)(1 − ρ¯kmax )
(2)
and the mean transfer time T in all the cell is given by: Pr j=1 Tj λj (3) T = Pr j=1 λj where λj is the mean arrival intensity to region j. The blocking probability B of a new flow is given by: ρ¯kmax B= 1 + ρ¯ + ... + ρ¯kmax
(8)
where hj,n represents the channel gain for the user of type j on subcarrier n. B. Algorithm We denote by HK×N , the channel gain matrix, and tot Pn = PN , the power to be assigned for each subcarrier n, n = 1, ..., N ; Ptot being the total transmit power on subcarrier n. HM can be implemented as follows. For each subcarrier n: • For user k (to be served by the scheduler on the Round Robin basis): •
Calculate bk,n using Equation (5); (this step indicates the number of bits achieved by user k on subcarrier n)
If user k is of type 2, i.e., belongs to the edge of the cell, determine user k ∗ , k ∗ = arg maxk∈K,k∗ 6=k hk,n and hk∗ ,n > hk,n and Pk,n +Pk∗ ,n < Pn ; (user k ∗ is the user of type 1 to be superposed on the same subcarrier as k. The constraint on powers ensures that HM can in effect take place when the power required by the two users does not exceed the total power on the shared subcarrier) where dk,n , Pk,n , dk∗ ,n , bk∗ ,n and Pk∗ ,n using Equations (7), (8), (5) and (6), respectively. The extension we propose to the case of superposing users of type 2 on ones of type 1 follows the same rationale and algorithm. •
(4)
IV. A DDING HIERARCHICAL MODULATION A. Modeling We now add HM. As described in the Introduction, the idea behind HM is to have two users, one with good radio conditions and one with lower ones share the same subcarrier, meant initially to the second one. The final transmitted constellation will thus be composed of two hierarchical levels. With HM, and for 4/M hierarchical QAM constellation M denotes the size of the final transmitted constellation and is equal to 64 in our case (see Figure 2) - log2 (M ) − 2 bits are assigned to the user with good radio conditions which represents the second level of hierarchy while 2 bits are assigned to the user with inferior radio conditions and which represents the first level of hierarchy [9]. For the user of type j, (j = 1, 2 in our example for two regions), the number of bits that can be transmitted on the shared subcarrier n is given by: Pj,n |hj,n |2 ) (5) Γσ 2 The amount of power allocated to the users of type j is given by [10], [11], [12]: bj,n = log2 (1 +
d2j,n (2bj,n − 1) (6) 6 where dj,n is the minimum distance between the flectious symbol in the constellation of user j on subcarrier n (please refer to Figure 1). For the square constellation 4/M -QAM, and for the case of two types of users: 1 and 2, the former belonging to the center of the cell and the latter to the edge, d1,n is related to d2,n by: √ 0 M − 1)d1,n (7) d2,n = dn + ( 2 Pj,n =
0
where d1,n (respectively dn ) is given by: s s 6Γσ 2 6Γσ 2 (resp. ) 2 h1,n h22,n
C. Markovian analysis With HM, the previously mentioned Processor Sharing analysis does not hold because the resource sharing depends, not only on the total number of users in the cell, but also on the exact number of users or type 1 and type 2. Indeed, users of type 1 will take advantage of HM only if there are users of type 2 in the cell. The state of the system is thus described by (k1 , k2 ) where ki is the number of users of type i, and the exact analysis needs the solving of the CMTC mentioned in Section III-B. This requires construction of the infinitesimal generator matrix Q of the CMTC and resolution of the system of equations πQ = 0 and π.e = 1, where π is the steady-state probability vector and e a vector of ones of appropriate dimension. Elements of matrix Q describe the transitions between states of the system that correspond to the following: 2 • Arrivals of calls of type 1 or 2, with rates λπr1 and 2 2 λπ(r2 − r1 ), respectively. • Departures of users of type 1. These users profit from their throughput c1 , in a Round Robin way, and thus 1 . They will each user of type 1 obtain a throughput k1c+k 2 however be also helped by HM each time a user of type 2 2 of the time and produces is scheduled; this happens k1k+k 2
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PN W P1,n |h1,n |2 ), a throughput equal to n=1 N log2 (1 + Γσ 2 shared by all users of type 1. Here we note by P1,n the power for user of region 1 obtained from Equation (6). The resulting additional user throughput due to HM is PN W P1,n |h1,n |2 2 thus: k11 k1k+k ), giving the n=1 N log2 (1 + Γσ 2 2 total type 1 user throughput:
V. N UMERICAL RESULTS
We now show the system performance, with and without HM, through numerical as well as simulation results. We have developed for this end a new system simulator that implements detailed features of the channel, slow and fast fading, as well as complete HM operation, in terms of constellations and N powers. 2 1 k2 X W P1,n |h1,n | t1 (k1 , k2 ) = [c1 + log2 (1+ )] The system under consideration corresponds to a typical k1 + k2 k1 n=1 N Γσ 2 office environment for Non Line-of-Sight (NLOS) conditions (9) with a decaying exponential intensity profile [13]. The paramthe departure rate for users of type 1 is thus equal to eters are as described in Table I. 1 ,k2 ) k1 t1 (k E[F ] . k2 c2 Carrier frequency 3.5 GHz • Departures of mobiles of type 2, with rate k +k . 1 2 E[F ] System channel bandwidth 1.25 MHz Numerical resolution of this system of equation is possible FFT 128 OFDMA symbol duration 102,9 µs and gives the performance measures. Γ
D. Approximate PS analysis Equation (9) suggests that the behaviour of the system is like if users of type 1 have a peak throughput equal to PN P1,n |h1,n |2 )], and they can profit c1 + kk21 n=1 W N log2 (1 + Γσ 2 from it in a round robin way. We can thus approximate the performance using the Processor Sharing model by changing the rate c1 of users of type 1 to a new total rate which is the obtained sum of the previous rate c1 plus the new rate cHM 1 from HM. Formally, N X W P1,n |h1,n |2 k¯2 log2 (1 + )P r(k1 > 0, k2 > 0) ¯ 2 N Γσ k1 n=1 (10) where k¯j is the mean number of users of type j present in the cell given by: ρ¯j 1 − ρ¯
cHM = 1
The first term of the right hand side of Equation (10) gives the throughput that is achieved by type 1 users on a subcarrier allocated originally to a type 2 user. This is conditioned by the fact that HM effectively takes place which is ensured by the probability P r(k1 > 0, k2 > 0) that here are users of both types in the system. We compute it through the following expression:
TABLE I S YSTEM PARAMETERS
Note that we deliberately take small values for the channel bandwidth and FFT size so as to ease the simulation burden. These values yield a maximum of 5 Mbps throughput for users of region 1. Note also that Γ = 8.8dB ensures a probability of error equal to 10−6 . We assume that users arrive to the system following a Poisson process of mean intensity λ. The file size is exponentially distributed with mean 40Kb. Figures 4, 5 and 6 show the mean transfer time for all the users in the system, users of type 1 and users of type 2, respectively, as a function of increasing offered traffic (in bps) - corresponding to a system utilization ranging from 0.1 to 0.8 - using the theoretical models we have developed above and which we term PS for Processor Sharing and Markovian for the exact analysis, as well as Monte Carlo simulations, with and without HM. 0.06 Without HM, simulations With HM, simulations Without HM, PS With HM, PS With HM, Markov
0.055
which is the stationary distribution of the number of users of each type for a system with no admission control. Our numerical results shown in the next section show the validity of this approximation. ¯ The last term kk¯21 gives the average number of type 2 users that are helping type 1 users through HM and is also equal to ρ¯2 /ρ¯1 . Similarly, for the case where superposition is also extended to users of type 1 assisting those of type 2, the new rate c2 for the latter also becomes: N X P2,n |h2,n |2 k¯1 W log2 (1 + )P r(k1 > 0, k2 > 0) ¯ c2 + 2 N Γσ k2 n=1 (11)
Mean Transfer Time (s)
0.05
(k1 + k2 )! k1 k2 π(k1 , k2 ) = ρ¯ ρ¯ (1 − ρ¯) k1 !k2 ! 1 2
8.8 dB
0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01
Fig. 4.
0
0.5
1
1.5 2 Offered traffic (bps)
2.5
3 6
x 10
Mean transfer time for all users
We first observe in all these cases that the mean transfer times are, trivially, increasing with increasing load. We second
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the users are of type 2; Scenario 3: 75% of the users are of type 1 and 25% of the users are of type 2. Figures 7, 8 and 9 show the gain achieved by using HM for all scenarios for all users, for users of type 1 and for users of type 2, respectively.
0.04
•
Without HM, simulations With HM, simulations Without HM, PS With HM, PS With HM, Markov
Mean Transfer Time (s)
0.035
0.03
0.025
0.02
15 Scenario 1 Scenario 2 Scenario 3
0.015
0.01
0.005
Fig. 5.
0
0.5
1
1.5 2 Offered traffic (bps)
2.5
3 6
x 10
Gain (%)
10
Mean transfer time for users of type 1
5
0.08 Without HM, simulations With HM, simulations Without HM, PS With HM, PS With HM, Markov
0.06
0 0.8
Fig. 7.
0.04
0.03
0.02
0.01
Fig. 6.
0
0.5
1
1.5 2 Offered traffic (bps)
2.5
3 6
x 10
Mean transfer time for users of type 2
observe the quite good match between theory and simulations. The differences between the theoretical and simulation curves can be explained by the fact that our simulator implements fast fading as well as detailed operation of HM in terms of powers and channel gains, whereas the theoretical model assumes slow fading only and constant channel gains, averaged for each region of the cell. We note however the good match between the Markovian analysis and the PS approximation in the case of HM. We third observe that there is a gain in using HM and it is even bigger as the system becomes more loaded, as in this case more users of type 2 will be helping more users of type 1. The gain is, as expected, more to the advantage of users of type 1 (Figure 5) but it does exist for users of type 2 also (Figure 6). We now investigate to which extent this gain (in mean transfer time) is dependent on the proportion of users of each type. We consider there scenarios: • •
1
1.2
1.4 1.6 1.8 Offered traffic (bps)
2
2.2
2.4 6
x 10
Gain for all users
0.05
Scenario 1: 25% of the users are of type 1 and 75% of the users are of type 2; Scenario 2: 50% of the users are of type 1 and 50% of
We observe that with respect to all users in the system considered at once (Figure 7), scenario 2 achieves the highest gain because: i. when there are more users of type 1 (scenario 3), the amount of individual gain achieved by these users is high (Figure 8) but overall, as their number is small, the total gain is limited, and ii. when the number of users of type 2 is higher (scenario 1), the gain is limited as the number of users of type 1 that can take advantage of their cooperation is limited and hence the gain that would in turn come back to them is limited too (Figure 9). Figure 10 represents the maximum capacity of the cell obtained by the calculation of the following harmonic mean: Pr pj Prj=1 pj j=1 cj
30 Scenario 1 Scenario 2 Scenario 3
25
20 Gain (%)
Mean Transfer Time (s)
0.07
15
10
5
0 0.8
Fig. 8.
1
1.2
1.4 1.6 1.8 Offered traffic (bps)
Gain for users of type 1
2
2.2
2.4 6
x 10
6
10
12 Scenario 1 Scenario 2 Scenario 3
9 8
All users Group 1 Group 2
10
7
8 Gain (%)
Gain (%)
6 5
6
4 4
3 2
2
1 0 0.8
Fig. 9.
1
1.2
1.4 1.6 1.8 Offered traffic (bps)
2
2.2
Fig. 11.
where pj is the proportion of users of type j in the system and cj is their service rate, with and without HM. 6
x 10
Without HM With HM
4
Cell capacity (bps)
3.6
3.4
3.2
3
Fig. 10.
0.3
0.4 0.5 0.6 Percentage of users of type 1
0.5
1
1.5 2 Offered traffic (bps)
2.5
3 6
x 10
Gain
the proportion of users of each type: the optimum being for the case where both types of users are of equal proportions. We eventually extended HM to the case where users with bad radio conditions are also superposed on the those with good ones and showed that the gain is even bigger in this case. R EFERENCES
3.8
2.8
0
6
Gain for users of type 2
4.2
0
2.4 x 10
0.7
Maximal traffic that can be carried by the cell
We observe, in accordance with our previous results, that the maximum increase with HM takes place when the percentage of users of each type is equal to 50%. We eventually show, in Figure 11 the gain for all the users in the system for the case of classical HM and our proposal in which users of type 1 help alos those of type 2. We observe that there is indeed a gain, and it now profits more to users of type 2. VI. C ONCLUSION We considered in this paper Hierarchical Modulation (HM), a physical layer technique wherein users of good radio conditions are superposed on subcarriers of ones with lower radio conditions, by means of constellation embedding. We modeled such a system at the flow level taking into account the user dynamics in terms of arrivals and departures after service. We quantified the gain obtained by such a technique, at the individual and global levels and showed that it does vary with
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