d
ij
denotes the Euclidean distance between node i and node
j and
D
P 0 is the guard factor to prevent interference. In a
time slot, the data that can be transmitted during a success-
ful transmission is normalized to one packet.
2.1.3. Traffic model
We consider the widely adopted permutation traffic
model [10,12,13], where there are n distinct traffic flows
in the network. Under such traffic model, each node acts
as the source of one traffic flow and at the same time the
destination of another traffic flow. The packet generating
process in each source node is assumed to be a Bernoulli
process, where a packet is generated by its source node
with probability k in a time slot [6]. We assume that each
source node has a first-come-first-serve queue (called
local-queue hereafter) with limited buffer size M > 0to
store its locally generated packets. Each locally generated
packet in a source node will be inserted into the end of its
local-queue if the queue is not full, and dropped otherwise.
2.2. MAC protocol
We adopt a commonly used MAC protocol based on the
concept of Equivalent-Class to address wireless medium
access issue in MANETs [10–12,15]. As illustrated in
Fig. 1b that an Equivalent-Class (EC) consists of a group
of cells with any two of them being separated by a horizon-
tal and vertical distance of some integer multiple of
a
ð1 6
a
6 mÞ cells. Under the EC-based MAC protocol
(MAC–EC), the whole network cells are divided into
a
2
ECs and ECs are then activated alternatively from time slot
to time slot. We call cells in an activated EC as active cells,
and only a node in an active cell could access the wireless
channel and do packet transmission. If there are multiple
nodes in an active cell, one of them is selected randomly
to have a fair access to wireless channel.
To avoid interference among concurrent transmissions
under the MAC–EC protocol, the parameter
a
should be
set properly. Suppose a node (say S in Fig. 1b) in an active
cell is transmitting to node R at the current time slot, and
another node W in one adjacent active cell is also transmit-
ting simultaneously. As required by the protocol model, the
distance d
WR
between W and R should satisfy the following
condition to guarantee successful transmission from S to R,
d
WR
P ð1 þ
D
Þr: ð1Þ
Notice that d
WR
P ð
a
2Þ=m, we have
ð
a
2Þ=m P ð1 þ
D
Þr: ð2Þ
Since
a
6 m and r ¼
ffiffiffi
8
p
=m;
a
should be set as
a
¼ min dð1 þ
D
Þ
ffiffiffi
8
p
þ 2e; m
no
; ð3Þ
where the function dxe returns the least integer value
greater than or equal to x.
Remark 1. Notice that for a time slot and an active cell, the
random selection of one node from multiple nodes to
access wireless channel can be implemented based on a
mechanism similar to DCF protocol [16,17]. At the begin-
ning of the time slot, each node in the active cell first
initiates a backoff timer with backoff period drawn
uniformly from ½0; CW (CW represents the contention
window size), all nodes then begin to count down. The
node whose timer reaches 0 first broadcasts a message
claiming its access to the wireless channel, and all other
nodes in the same active cell, after overhearing the
broadcasted message, stop their timers and remain silent
in the time slot.
2.3. PD-f scheme
Once a node (say S) got access to the wireless channel in
a time slot, it then executes the PD-f scheme (f P 1) sum-
marized in Algorithm 1 for packets dispatch.
Remark 2. The PD-f scheme is general and covers many
widely used packet dispatching schemes as special cases,
like the ones without packet redundancy [5,6,8] when
f ¼ 1 and only unicast transmission is allowed, the ones
with controllable packet redundancy [12,16,18] when
f > 1 and only unicast transmission is allowed, and the
ones with uncontrollable packet redundancy [7,19] when
f P 1 and broadcast transmission is allowed.
Algorithm 1. PD-f scheme.
1: if S has packets in its local-queue then
2: S checks whether its destination D is within its
transmission range;
3: if D is within its transmission range then
4: S transmits the head-of-line (HoL) packet in its
local-queue to D;
{source–destination transmission}
5: S removes the HoL packet from its local-queue;
6: S moves ahead the remaining packets in its
local-queue;
7: else
8: With probability q (0 < q < 1), S dispatches the
HoL packet;
9: if S conducts packet dispatch then
10: S dispatches the HoL packet for one time;
{packet-dispatch transmission}
11: if S has already dispatched the HoL packet
for f times then
12: S removes the HoL packet from its local-
queue;
13: S moves ahead the remaining packets in
its local-queue;
14: end if
15: end if
16: end if
17: else
18: S remains idle;
19: end if
3. QBD-based theoretical framework
In this section, a QBD-based theoretical framework is
developed to capture the packet dispatching process in a
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