Digital Signal Processing Reference
In-Depth Information
Hence, given
p
1
and
p
2
, the stability region obtained by assuming that user 1 is fully
loaded is given by
{
}
.
∗
∗
∗
R
1
1
1
(
pp
, =, : <
)
(
λλ λµ
)
(
pp
, , <
)
λµ
112
(
p
,
p
)
(11. 2)
12
12
1
1
12
2
2
For λ
2
< ψ
2
p
2
(1 -
p
1
), the bounds for λ
1
are not only inner bounds, but in fact coincide
with that of the true system. More specifically, when λ
1
is greater than μ
1
∗
1
(
p
1
,
p
2
), user 1
will be unstable in the dominant system and the queue length will go to infinity without
emptying with finite probability, which means that there are sample paths that do not
return to zero infinitely often. For such sample paths, the queue state will not return to
zero after a certain time, in which case the dominant system and the original system
will be identical. This shows that the original system will be unstable for these values of
λ
1
as well.
Similarly, by assuming that user 2 is fully loaded, the stability region can be expressed as
{
}
,
∗
∗
∗
R
2
2
2
(
pp
, =, : <
)
(
λλ λµ
)
(
pp
, , <
)
λµ
112
(
p
,
p
)
(11. 3)
12
12
1
1
12
2
2
where μ
2
1
(
p
1
,
p
2
) = ψ
1
p
1
(1 -
p
2
) and
λ
∗
2
µ
(
pp
, = −
)
ψ
p
1
1
.
2
12
22
ψ
(
1
−
p
)
1
2
By taking the union of R
1
∗
(
p
1
,
p
2
) and R
2
∗
(
p
1
,
p
2
) for all possible transmission probabili-
ties, the stability region of the two-user slotted ALOHA system is given by
λ
ψ
λ
ψ
∪
∪
∗
∗
1
2
(11.4)
R
=
R
(
pp
,
)
R
(
pp
,
)
=, : + <
(
λλ
)
1
1
2
2
1
.
12
12
12
2
(
pp
, ∈,
)
[]
01
12
0.75 and ψ
2
= 0.5 (solid curve) and ψ
1
= ψ
2
= 0.25 (dash-dotted curve). The case with no
channel errors (dotted curve), ψ
1
= ψ
2
= 1, is also plotted as a reference.
11.3.3 Cooperation in a Two-User Slotted ALOHA System
In the conventional slotted ALOHA system, each user transmits independently of oth-
ers, and thus, the stable throughput of each user is limited by its local channel quality.
It is possible that the users experiencing bad channels on average will be seriously back-
logged, while the others, i.e., the ones that experience good channels, have nothing to
transmit. Intuitively, if the users are willing to cooperate, those that are more capable
can help by relaying the messages from the less capable users and thereby increase their
stable throughput.
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