The Spatial Multiplexing based MIMO-OFDM System with COUS . 119

With taking multiuser scheduling into consideration, COUS selects the target user according to (7.4). Denote the average effective output SNR of user k as Ω_k = _M^ρλ^ave_k,1, the CDF of Ω_k will be equal to (7.13) for each user k. The corresponding CDF of COUS is hence given by

F_Ωk∗(γ) = P_r

½

maxk∈K{Ω_k} ≤ γ

¾

= [F_Ωk(γ)]^K , (7.16)

where k^∗ represents the scheduled user. The link outage of COUS can be expressed as P_out = P_r(Ω_k^∗ ≤ γ_th) = F_Ωk∗(γ)

The corresponding closed-form expression of the reliable coverage is given by

RCOUS =

Similarly, by substituting N = 1 into (7.17) and (7.18), the equivalent results can be found in [30, Section IV. C] for the SWNSF scheduling.

In the Special Case K = 1

As the degrees of freedom on user domain is merely one, the performance metrics of COUS (7.17) and (7.18) are straightforwardly same to that of random scheduling in Section 7.3.1.

7.3.3 The Spatial Multiplexing based MIMO-OFDM System with COSA

With COSA scheduling algorithm, each user k competes with the other (K − 1) users for winning the service on each subchannel n according to (7.6). We label the selected user with a superscript k^∗. Then the effective output SNR of the selected user’s weakest substream in subchannel n is now denoted as γ_k^∗_,n and its corresponding CDF is given by

F_γk∗,n(γ) = P_r

where we omit the suffix index “1” which indicates the minimal eigenmode to ease the notations. For brevity, we use the symbol F_k^∗_,n instead of F_γk∗,n for the CDF of γ_k^∗_,n in the following. With the help of order statistics method [120], we reorder a given sample of the N variables {γ_k^∗_,1, ...γ_k^∗_,N} as γ_k^∗_,(1) ≤ γ_k^∗_,(2) ≤ ... ≤ γ_k^∗_{,(N )}. Then γ_k^∗_,(i) is called the i-th order statistic and F_k^∗_,(i)(·) is the corresponding CDF. The link outage probability of the spatial multiplexing based MIMO-OFDM systems with COSA can be written as

P_out = P_r

where we use Glivenko-Cantelli Theorem to get the approximation (a) with the assump-tion that γ_k^∗_,(˜ω) is the value closest to _N¹ P_N

n=1γ_k^∗_,n. This approximation can help us to facilitate deriving an approximated analytical closed form for reliable coverage.

*,(1)

J

J

J

J



J

1 N 2 N N Z 1

N

( ) F J

Figure 7.3: Illustration of the empirical distribution for an i.i.d. sequence.

Find the Approximate Value ˜ω

To find ˜ω, we introduce a function called the empirical distribution [121]. Assume that X₁, X₂, . . . are i.i.d. random variables in R with common CDF F (x). The empirical distri-bution for {X₁, X₂, . . .} is a random variable defined as

Fm(x) = 1 m

Xm i=1

I(−∞,x](Xi) , (7.22)

where I_(−∞,x](·) is the indicator function, i.e.

I_(−∞,x](y) =







1, if y ≤ x 0, if y > x .

(7.23)

The Glivenko-Cantelli Theorem [121] says that sup

x∈R

|F_m(x) − F (x)| (7.24)

converges to 0 with probability 1, where “sup” means the supremum. In other words, the theorem implies the strong statement

P_r where (a) comes from the Glivenko-Cantelli Theorem and (b) satisfies the assumption γ_k^∗_,(˜ω) is the value closest to _N¹ P_N

n=1γk^∗,n. The notation E[γk^∗,n] is the expectation value of γk^∗,n. With the help of the order statistics, we have the PDF of f_k^∗_,n(γ) as follows

f_k^∗_,n(γ) = KM²

By substituting (7.29) into (7.27), we obtain

˜

where (a) comes from (7.19) and N_ω˜ is an approximation integer value of ˜ω by rounding ˜ω to the nearest integer number and is a function of N and K.

Approximated Analytical Formula by Order Statistics

In (7.20), the link outage is transformed into another form of probability, which means at least N_ω˜ of the {γ_k^∗_,n}^N_n=1 are less than or equal to γ_th. By applying the theories of order is the regularized incomplete beta function. The result (a) comes from a well-known result for the regularized incomplete beta function Ip(l, L − l + 1) = P_L

i=l

¡_L

i

¢pⁱ(1 − p)^L−i [122].

The equalities in (b) and (c) are based on (7.19). Now we have the analytical closed-form approximation for the link outage.

The reliable coverage RCOSA can be derived from this approximate formula (7.31). We first introduce the inverse regularized incomplete beta function which is shown as follows

z = Ip(a, b) ⇒ p = I_z⁻¹(a, b) . (7.33) By substituting (7.30) and (7.33) into (7.31), we can obtain the reliable coverage as

RCOSA '

where the value Nω˜ is shown in (7.30). Now we have an analytical closed-form approximation of the reliable coverage for the spatial multiplexing based MIMO-OFDM systems with COSA, and it is a function composed of the given parameters {M, N, K, Pt, σ², µ, γth} and required P_out.

Annotation of the Approximated Formula

For the spatial multiplexing based MIMO-OFDM systems with COSA, the provided approx-imated form will be more accurate as the number of subchannels N is large according to the Glivenko-Cantelli Theorem. In the special case N = 1 and K = 1, the integer N_ω˜ = 1 due to ˜ω ∼= 0.63. In this case of no degrees of freedom on both user and frequency domains, the approximation (7.34) of COSA have the same value as (7.15) of random scheduling. As for the special case N = 1 and user population K which lacks degrees of freedom on frequency domain, the approximation (7.34) of COSA have the same value as (7.18) of COUS. It is because COUS only taking advantage of multiuser diversity.

7.4 Discussion

7.4.1 Fairness Issue

Compared with the spatial multiplexing based MIMO-OFDM systems using FOSA, the average subchannels for each user assigned using COSA is N/K. As for COUS, each user has 1/K probability selected in each time slot and then assigned N subchannels so that also N/K assigned subchannels on average. Thus, COSA and COUS are fair in the sense of equally resource allocation. To examine long-term fairness performance, we define the fairness index F in the multiuser spatial multiplexing based MIMO-OFDM systems according to Jain’s fairness index [123]

where T_i is the cumulated number of subchannels allocated to the i-th user. For F = 1, it is the fairest condition between users, and it is not fair as F ¿ 1. Figure 7.4 shows the fairness performance comparison for FOSA, COSA and COUS with system scheduling time slots when K = 10 and M = 3. We find that COSA achieves almost the same fairness performance as FOSA when the number of subchannels is larger and the scheduling time index is increasing. However, the fairness performance of COUS is poorer than FOSA

5 10 15 20 25 30 0.8

0.85 0.9 0.95 1

Scheduling time index

Fairness index

5 10 15 20 25 30

0 0.2 0.4 0.6 0.8

Scheduling time index

Fairness index

COUS FOSA COSA N =32

N =128

N =128 N =32

Figure 7.4: Fairness performance comparison for K = 10 and M = 3.

and COSA even if each user of COUS can get the same number of subchannels as COSA theoretically. In fact, the fairness performance of COUS can also approach that of FOSA and COSA as scheduling time slots are extremely large.