4338 IEEE TRANSACTIONS ON COMMUNICAT IONS, VOL. 62, NO. 12, DECEMBER 2014
Fig. 2. System Model of a MU-MIMO Downlink Scenario.
matrix, and s(t) denotes the i.i.d. user data streams. The signal-
to-interference-noise ratio (SINR) of user-n is written as,
γ
n
(t)=
ζ(t)
2
h
†
n
(t)w
n
(t)
2
∑
j=n
ζ(t)
2
h
†
n
(t)w
j
(t)
2
+ z
n
(t)
2
, (3)
where we write H(t)=[h
1
(t), h
2
(t),...,h
N
(t)]
†
and W (t)=
[w
1
(t), w
2
(t),...,w
N
(t)], and n is the user index. Furthermore,
let Q
n
(t) denote the queue length in bits of user n at the begin-
ning of t-th channel use, let a
n
(t) denote the number of arrival
bits from upper layer to the physical layer between (t − 1)-th
and t-th channel uses, and let µ
n
(t) denote the allocated number
of service bits to Queue-n, which equals the allocated service
rate (bits/channel use) in this case. Then the queuing dynamics
are written as
Q
n
(t + 1)=Q
n
(t) − ˜µ
n
(t)+a
n
(t), (4)
where ˜µ
n
(t)=min{Q
n
(t), µ
n
(t)} denotes the actual service
number of bits, considering the circumstances that sometimes
the queue is emptied given the amount of allocated service bits.
Definition 1: Queue-n is said to be strongly stable if [13]
limsup
T →∞
1
T
T
∑
t=1
E [Q
n
(t)] < ∞, (5)
when there is no bound on the buffer size for any n.
When all queues are strongly stable in the system, the time-
average actual service rate equals the arrival rate, i.e.,
lim
T →∞
1
T
T
∑
t=1
˜µ
n
(t)= lim
T →∞
1
T
T
∑
t=1
a
n
(t), ∀n. (6)
Notice t hat the left-right-side is the time average of the real-
izations of the actual service rate, thus we do not need the
expectation to hold (6).
The achievable ergodic rate region
R is defined as the convex
hull of all achievable rate points of n users. Denote all feasible
transmission schemes as
X and the transmission scheme π
s
∈
X , where s is the index for scheduling policies, is the user
scheduling scheme and the corresponding precoding scheme
with the rate of user-n at time t,
R
n
(H(t), π
s
(t)) = I (π
s
(t))log (1 +SINR (H(t), π
s
(t))), (7)
where I(π
s
(t)) is an indicator function which determines
whether user-n is scheduled, and SINR(H(t)) is the signal to
noise and interference ratio which is related to the channel
realization and the precoding scheme.
6
The user-n achievable
rate is defined as the time-average of user rate
¯
R
n
= lim
T →∞
1
T
T
∑
t=1
R
n
(H(t), π
s
(t)), ∀ n. (8)
Based on ergodicity, (8) equals
¯
R
n
= E
{
R
n
(H, π
s
)
}
, ∀ n, (9)
where the expectation is taken over all possible channel gain
H(t) and possibly π
s
(t) when a randomized control policy is
considered. The achievable ergodic rate region can be charac-
terized as
R = coh
π
s
∈X
{
¯
R :0≤
¯
R
n
≤ E [R
n
(H, π
s
)]
}
, (10)
where
¯
R is a N-dimensional region,
¯
R
n
is its n-th component,
and “coh” denotes the closure of a convex hull.
Definition 2 (Throughput-Optimality): A scheduling scheme
is throughput-optimal if for any arrival rate point inside the
achievable ergodic rate region, the system can be stabilized by
the scheduling scheme.
Note that the throughput in this paper refers to the downlink
throughput, not concerning uplink throughput. We consider a
generic scenario where each user has its distinct block length
which denotes the number of consecutive channel uses that
the user-channel stays static, or also referred to as channel
coherence time. The block fading channel model is adopted
in this paper, where every user’s channel stays constant for
T
n
consecutive channel uses, and changes to another constant
according to an i.i.d. (over time and users) random process.
Denote by T
n
the channel coherence time of user-n, and let
T = {T
1
, T
2
,...,T
N
}.
7
Notice that there are no units for both T
c
and T
k
, ∀k, since by definition, the block length in block fading
model equals channel coherence time multiplied by channel
coherence bandwidth. Therefore the block length describes
the channel coherence, both temporally and spectrally. Similar
notations have been used in [14], [2].
Here we assume that for every T
n
channel uses, the system
adopts one uplink channel training symbol (reciprocal channel
is considered here) to estimate the channel of user-n. By doing
this, we assume the BS obtains the perfect CSIT of user-n.
Hence, the number of concurrent scheduled users, denoted by
N
s
, equals the number of channel uses for channel estimation
numerically in each fading block. Note that this is actually
an optimistic assumption on the MU-MIMO system, since
normally we can only get a noisy version of the CSIT, and
the system has to do the channel training more than once to
refine the estimation. Nonetheless our results can be extended
6
Explicit expressions of R
n
(H(t), π
s
(t)) will be shown later in Section IV.
7
The distinction of user block lengths is due to the fact that there are several
factors that can affect the block length of each user, such as distinct user-
mobility, scattering environment nearby, frequency offset and etc. We assume
the BS knows the channel coherence time apriori, since channel coherence
time is a second-order channel statistics, which can be regarded to be static for
a relatively long period.
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