powers of all APs are set to the maximum value at the
beginning, and are gradually reduced based on the pro-
posed algorithms for load-balancing. A simulation study on
a 20-AP network with 10 power levels was reported in [12]
and the results indicated that the total number of power
levels to be decreased to achieve the resultant power setting
was about 100, which makes the beacon powers of many
APs drop to low levels. This implies that in sparse AP
deployments, the problem of service loophole could be
hardly avoided.
Given the topology of the APs and the clients, an optimal
mapping of clients to APs can be found through linear pro-
gramming techniques [13]. In this study [13], once the number
of clients in the network exceeds the maximum limitation of
an AP, some clients might not be served, also leading to the AP
service loophole problem. These observations are verified by
our simulation study reported in Sect. 5.2.
This paper proposes a variable polyhedron GA to
compute near-optimal beacon ranges that guarantee AP
service availability. This algorithm does not make any
assumption on the AP deployment density. Moreover, it
does not place any constraint on the number of clients each
AP can serve. Nevertheless, it yields very good perfor-
mance in terms of load-balancing and network throughput.
3 Problem definition
3.1 Network model
In this paper, a WLAN consisting of K APs is deployed in a
region R. Each AP is equipped with an omnidirectional
antenna that can transmit data and beacons at different power
levels, resulting in different disk-shaped
2
data coverage
areas and beacon coverage areas. When the (data or beacon)
power of an AP increases, its (data or beacon) coverage area
enlarges. Two APs are neighbors if and only if their data
coverage areas overlap. For simplicity, we assume that the
region R is a rectangle
3
and that all the data powers jointly
provide a full network coverage, i.e., each point in R is
located within the data coverage area of at least one AP.
This paper also assumes that a region controller is
deployed for load-balancing, which takes charge of the
optimal beacon power computation based on our proposed
algorithm. Each AP reports its load information to the
controller periodically or when its load changes abruptly.
Upon receiving the load information, the region controller
updates/recomputes the load density distribution through
existing methods [21, 22].
The utilization of the load density function instead of the
load directly measured at each AP can be justified as fol-
lows. An iterative algorithm such as the proposed variable
polyhedron GA requires the network to measure the AP
loads at each iteration before the final beacon ranges are
obtained. If instant load information is employed, some
intermediate iterative results will lead to a large number of
unnecessary handoffs before the final result is obtained. A
similar consideration is also taken in [12] for the same
reason. On the other hand, a load density function derived
from the real AP loads can provide a good load estimation
for each iteration before the algorithm converges.
3.2 Problem formulation
In a WLAN, a client selects the AP with the strongest RSSI
to associate with. Two critical parameters are involved in
this process:
Definition 3.1 (Beacon Range (BR)) The BR of AP
i (BR
i
) is the range within which the beacon power of AP
i can be sensed.
Definition 3.2 (Transmission Range (TR)) The TR of AP
i (TR
i
) is the range within which an associated client can
transmit and receive data through AP i.
According to these definitions, only when a client is
located within an AP’s BR can it associate with the AP;
and only when it also is located within the AP’s TR can it
communicate to the AP.
When a client actively probes the existence of APs in its
neighborhood via probe requests, the AP whose probe
response has the strongest RSSI is selected. In such a case,
the BR of the AP is determined by the power of its probe
responses. For simplicity, this paper assumes that the
power of the probe responses of an AP is the same as that
of its beacons. The goal of our design on coverage
adjustment is to balance the loads of all APs. Load bal-
ancing is formally defined as follows:
Definition 3.3 (Load Balancing) A network realizes load-
balancing if the load variance of its APs, denoted by d
L
2
,is
minimized, where d
L
2
is represented by
d
2
L
¼
P
K
i¼1
LðiÞ
LðÞ
2
K
; ð1Þ
with L(i) being the load of AP i;
L being the average load of
all APs, i.e.,
L ¼
P
K
i¼1
LðiÞ=K; and K being the total
number of APs.
2
In this study, we choose to employ a disk-shaped coverage for each
AP because it is a good approximation of an AP’s coverage: WiFi
signals can penetrate walls and other barriers without much difficulty,
which justifies why many relevant research adopts the same
assumption of disk-shaped coverage [7, 20].
3
Actually, our algorithm is applicable to a more general case where
R could have any shape.
478 Wireless Netw (2014) 20:475–491
123
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