transmitter will only occur at the start instant of the packet
transmission, whereas collisions induced by the jammers
hidden to the transmitter may occur during the whole
packet transmission period.
Accordingly, in this paper we introduce the concepts of
the instantaneous collision zone and the persistent colli-
sion zone of receivers. We suggest that packet collisions
induced by the hidden jammers in the persistent collision
zone of receivers are the primary causes for the weighted
fairness problem of WLAN mesh networks. We put forward
a three-dimensional Markov chain model to determine
how to precisely tune the backoff persistence factors to
achieve weighted fairness for flows with diverse quality
of service (QoS) demands. Our analytical model introduces
the pseudo states into the Markov chain to effectively dis-
tinguish the different backoff procedures induced by the
RTS collisions and the data collisions in multi-hop environ-
ments. By means of the proposed model, the bandwidth
allocation ratio between two classes of flows can be
expressed as a function of the station numbers and the
BPFs of the flows. According to this function, we determine
how to properly tune the BPFs to achieve the predefined
weighted fairness goal between the two classes of flows
under different network conditions. Finally, we validate
the accuracy of our model by comparing the analytical
results with that obtained by means of simulations.
The remainder of the paper is organized as follows. Sec-
tion 2 gives a detailed analysis of the weighted fairness
problem in multi-hop WLAN mesh networks. Section 3
presents our analytical model and derives the expression
of the bandwidth allocation ratio between different classes
of flows. Validations of the analytical model are carried out
in Section 4. Finally, we conclude this paper in Section 5.
2. Weighted fairness problem
In this section, we introduce the concepts of the instan-
taneous collision zone and the persistent collision zone of
receivers, and analyze the unfairness behavior of the
tagged nodes through extensive simulations. Our objective
is to reveal the primary cause of the MAC-layer weighted
fairness problem in multi-hop WLAN mesh networks. Con-
sidering the function of the MAC protocol is to coordinate
the use of the communication medium, and to establish
reliable one-hop communication links for data transfer,
the packet destination is restricted to one hop from the
source in our paper.
2.1. Collision zones
We depict a simple multi-hop mesh topology with two
flows in Fig. 1. We neglect the physical layer issues that
would complicate the MAC-layer throughput unfairness
analysis and assume idea channel conditions, i.e., the chan-
nel is error-free and there is no capture effect. All nodes
have the same type of radio and the identical transmission
power. The transmission range (r
tx
) and the carrier sense
range (r
cs
) of each node are fixed, which are denoted by
the solid line circle and the dotted line circle in Fig. 1,
respectively. The carrier sense range is typically two times
larger than the transmission range [31]. Nodes located in
the carrier sense range of a transmitter can detect a busy
channel by the carrier sense, but only nodes located in
the transmission range of the transmitter can successfully
decode its packets.
We consider the two-ray ground propagation model,
the relation between the transmission power P
tx
and the
received power P
rx
is
P
rx
¼
P
tx
h
2
t
h
2
r
G
t
G
r
d
4
ð2Þ
where G
t
and G
r
are the antenna gains of the transmitter
and the receiver, respectively; h
t
and h
r
are the antenna
heights of the transmitter and the receiver, respectively;
and d is the distance between the transmitter and the
receiver. To successfully receive the packets from the
transmitter, the signal-to-interference-plus-noise ratio
(SINR) at the receiver should be greater than a threshold
SINR
rx
. Therefore, the transmission of a jammer located
within a distance of
r
co
¼ d
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
SINR
rx
4
p
ð3Þ
from the receiver can collide with the current transmis-
sion. As shown in Fig. 1, the radius r
co
denoted by the
dashed line circle is often referred to as the collision range
of receivers. In practice, SINR
rx
is usually set to 10 dB. Thus,
the value of r
co
is approximate to 1.78d.
When the transmitter of Flow 1 (Node A) starts to trans-
mit packets to its receiver (Node B), the jammer (Node C)
within the collision range of Node B may simultaneously
transmit packets and leads to the collisions. However, dur-
ing the packet transmission period of Node A, the jammers
within both the carrier sense range of Node A and the col-
lision range of Node B can defer their transmissions by the
physical carrier sense mechanism and avoid collisions. In
other words, the collisions induced by the jammers within
the above mentioned area will only occur at the start
instant of the packet transmission of Node A. Therefore,
we define the instantaneous collision zone as the area
within both the physical carrier sense range of the trans-
mitter and the collision range of the receiver, which is
shown as the shaded area in Fig. 1.
d
r
co
r
cs
d
r
tx
AB CD1
Instantaneous
collision zone
Persistent
collision zone
r
tx
d
if
2
Fig. 1. Collision zones of receiver.
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