Wireless Network Design: Optimization Models and Solution ...

14 Cross Layer Scheduling in Wireless Networks 323
(termed as slot in this chapter) boundaries. This model is termed as Block Fading
Model.
Under this model, if a user transmits a signal χn in slot n, then the received signal
Yn is given by:
Yn = Hnχn + Zn, (14.1)
where Hn corresponds to the time-varying channel (tap) gain due to fading and Zn is
the complex Additive White Gaussian Noise (AWGN) (with zero mean and variance
N0). Usually Hn is modeled as a zero mean complex Gaussian random variable.
Let σ 2 denote the variance of Hn. Then |Hn| is a Rayleigh random variable and
Xn = |Hn| 2 is an exponentially distributed random variable with probability density
function expressed as:
fX(x) = 1
σ
2 exp(−x2
), x ≥ 0. (14.2)
2σ 2
This model is called Rayleigh fading model.
We refer to Xn as channel state in slot n. Note that the channel state Xn may
change from slot to slot either in an independent and identically distributed (i.i.d.)
fashion or in a correlated fashion (e.g. may follow a Markov model). Moreover, the
channel state Xn is a continuous (exponential) random variable. However, for the
scheduling problems considered later in this chapter, we assume that the channel
state Xn takes values from a finite discrete set X. This discretization can be achieved
by partitioning the channel state into equal probability bins with preselected thresholds.
For example, let x(1) < ... < x(L) be these thresholds. Then the channel is
said to be in state xk if x ∈ [x(k),x(k + 1)), k = 1,...,L. The channel state space X
can be represented as X = {x1,...,xL}.
For a wireless channel, capacity analysis can be performed both in the presence
as well as absence of Channel State Information (CSI) at the transmitter. Throughout
this chapter, we assume that the transmitter has the knowledge of perfect CSI. In a
Time Division Duplex (TDD) system, due to channel reciprocity, it may be possible
for the transmitter to obtain the CSI through channel estimation based on the signal
received on the opposite link. In a Frequency Division Duplex (FDD) system, the
receiver has to estimate the CSI and feed this information back to the transmitter. In
practice, e.g., in IEEE 802.16 [26], the channel related information can be conveyed
using ranging request (RNG-REQ) messages. In this chapter, we do not take into
account the specific feedback mechanisms; rather, we assume that the transmitter
has the knowledge of perfect CSI.
Different notions of capacity of fading channels have been defined in the literature.
The classical notion of Shannon capacity defines the maximum information
rate that can be achieved over the channel with zero probability of error. This notion
involves a coding theorem and its converse, i.e., that there exists a code that achieves
the capacity (information can be reliably transmitted using this code at a rate less
than or equal to the capacity) and that reliable communication is not possible if
information is transmitted at a rate higher than the capacity.