Wireless Network Design: Optimization Models and Solution ...

6 Optimization Based WLAN Modeling and Design 139
ws binary ∀s ∈ S (6.13)
ys f binary ∀s ∈ S, f ∈ F (6.14)
zsm binary ∀s ∈ S,m ∈ M (6.15)
xs 1s 2 f binary ∀s1,s2 ∈ S, f ∈ F|s1 < s2 (6.16)
NS integer (6.17)
Ps ∈ P ∀s ∈ S (6.18)
Since we seek a maximum capacity, the objective function is
Maximize ∑ log
m∈M
1 + ∑s∈S
vm
tsm
usm
(6.19)
The capacity maximization WLAN design model defined by (1)-(19) is an integer
nonlinear mathematical program. The objective function is nonlinear and constraints
of type (9) and (10) are nonlinear. If the transmit power (Ps), for each
AP is fixed, then the constraints are linear. Design under a fixed budget constraint
may be achieved by fixing NS, the number of APs. A channel assignment
model is obtained by fixing the ws variables for all APs. When the transmit
power is fixed, the objective function is composed of the sum of terms of the form
g(a,b,c) = [log(1 +a/b)]/c. It can be shown that the Hessian of g(a,b,c) is neither
positive nor negative semi-definite, and therefore the function is of undetermined
concavity. Hence, (19) is neither concave nor convex. A plot of g(.) along a straight
line from (2.0e−4,1.0e−6,9) to (1.0e−6,1.0e−6,24) is illustrated in Figure 6.4.
Of all the models in the literature, the model of Bahri and Chamberland [2] is most
Fig. 6.4 The Function g(a,b,c) Plotted Along a Straight Line
closely related to (1)-(19). The objective of their model is simply minimize NS. In
addition, they assume that the data rate between each AP and the associated MD is