Digital Signal Processing Reference
In-Depth Information
the message will always be correctly received by the cooperating users as long as the
transmission is collision-free. This basically assumes that the users are sufficiently close
to each other.
Suppose that user 1 is transmitting a certain packet for the first time from its source
buffer while user 2 remains silent. During this time slot, user 2 will be able to over-
hear the message transmitted by user 1 and record the message in its cooperative buffer
(i.e.,
CB
2
). If an
e
feedback is received from the access point, the packet will remain at
the head of line (HOL) and both users will be allowed to retransmit the packet in later
time slots until one of them succeeds. When a transmission is eventually successful, i.e.,
a
1
feedback is received from the access point, the packet will be dropped from both
users. Since no new packet is transmitted by the source user until the HOL packet is
successfully transmitted, the length of the cooperative queue cannot exceed 1 and, thus,
the stability of this queue will not be a concern. Let
CQ
1
[
m
],
CQ
2
[
m
] ∈{0, 1} and
Q
1
[
m
],
Q
2
[
m
] ∈{0, 1, 2, …} represent the number of packets stored in the buffers
CB
1
,
CB
2
,
B
1
,
and
B
2
, respectively, at the beginning of the
m
th
time slot.
Let
p
i
be the transmission probability of user
i
and let
A
i
[
m
],
V
i
[
m
],
~
i
[
m
], and
H
i
[
m
]
be as defined in the previous subsection. If user
i
decides to transmit in the current slot,
and both
B
i
and
CB
i
are nonempty, user
i
will choose the packet from
B
i
with probability
q
i
, and
CB
i
with probability
-
i
= 1 -
q
i
. To incorporate this into the queue evolution equa-
tion, we introduce the new Bernoulli random process {
U
i
[
m
]}
m
=0
, where Pr{
U
i
[
m
] = 1} =
q
i
and
U
i
[
m
] = 1 indicate the event that user
i
transmits a packet from
B
i
instead of
CB
i
when both the buffers are nonempty. A packet from the source buffer
B
i
will be trans-
mitted either if (1)
CB
i
is empty and
B
i
is not or if (2)
B
i
and
CB
i
are both nonempty but
user
i
chooses to transmit from
B
i
, i.e.,
U
i
[
m
] = 1. Therefore, the event that a transmission
attempt is made by a packet in
B
i
is represented with
coop
VmVm
[]
= ⋅
[]
χ
(
CQ mVmUm
[
])
+ ⋅
[] []
⋅
χ
N
(
CQ m
i
[
]),
(11. 5)
i
i
{}
0
i
i
i
where
V
i
[
m
] =
~
i
[
m
] · χ
(
Q
i
[
m
]). Notice that
V
i
coop
[
m
] = 1 if a packet is transmitted from
B
i
, and
V
i
coop
[
m
] = 0 otherwise. On the other hand, a packet from the cooperative buffer
CB
i
is transmitted if user
i
attempts a transmission and the packet transmitted is not
from
B
i
. Therefore, we have
1
coop
coop
CV
[]
mVm
=
[ [
−
χ
(
Q mCQm
[ )
⋅
χ
(
[]
)]
−
Vm
i
[]
.
(11.6)
i
i
{}
0
i
{}
0
i
More specifically, a packet from the cooperative buffer is transmitted if
CV
i
coop
[
m
] = 1;
otherwise,
CV
i
coop
[
m
] = 0. Since the successful transmission of a packet can be achieved
either by the source itself or by the cooperating user, the departure process of user
i
is
Dm VmHm
[]
=
coop
[] [ (
1
−
VmCV
coop
[
]
−
coop
[
m
]
)
i
i
i
j
j
(11.7)
coop
coop
coop
+
CV
[] [ (
mH mV
1
−
[
mmCVm
]
−
[
])
,
for
i
=, ≠.
12
and
i
j
j
j
i
i
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