Cooperative Communications in Random Access Networks - Adaptive Signal Processing in Wireless Communications - page 316

the noncooperative and cooperative cases to illustrate the cooperative advantages in the
slotted ALOHA system.

11.3.2 Stability Region of the Noncooperative Slotted
ALOHA System
To derive the stability region of the slotted ALOHA system, Rao and Ephremides [27]
proposed the use of an auxiliary hypothetical system that has the same arrival process
and follows the same transmission policy as the original system, but is less likely to be
stable. This system is called a dominant system . The stability region of the dominant sys-
tem may be easier to derive if it is set up properly, but is only guaranteed to be an inner
bound of the true stability region. However, in certain cases, the stability region of both
systems may actually coincide, as is the case in the following example. Even if they do
not coincide, the dominant system still provides useful insights on the behavior of the
original system.
A dominant system can be constructed by assuming that a subset of users in the net-
work are fully loaded , i.e., they always have a packet to transmit regardless of the actual
queue state in the original system. Under this assumption, the departure rate of each
user will be no larger than that in the original system since contention between users
increases.
Interestingly, following the same procedures as in [27], we can show that the systems
considered in the following satisfy the conditions of Loynes' formulation, and thus, by
applying Loynes' theorem [29], we can say that the system is stable when the arrival rates
λ 1 , … , λ N are smaller than the service rates μ 1 , …, μ N , for a given set of transmission
probabilities p 1 , … , p N , i.e., λ i < μ i for all i . The service rate of user i refers to the average
number of user i 's packets that can be served in each time slot. On the other hand, the
system is unstable if there exists i such that λ i > μ i . In this case, our task reduces to find-
ing the departure rate of each system.
To derive the stability region of the two-user system, let us first consider the case where
user 1 is fully loaded. In this case, a packet from user 2 can be successfully received at the
base station if and only if the reception is successful, which occurs with probability ψ 2 ,
and user 1 does not transmit. Therefore, the service rate of user 2, which is a function of
p 1 and p 2 , is equal to μ 1 2 ( p 1 , p 2 ) = ψ 2 p 2 (1 - p 1 ). If user 2 transmits at a rate λ 2 strictly less
than the service rate μ 1 2 ( p 1 , p 2 ), it follows from Little's theorem [30] that the probability
the buffer of user 2 is non-empty is equal to

λ

2
22

) , ψ p ( 1 − p

1

and the service rate of user 1 is equal to


 


 

λ

∗

1

µ ( pp , = − ) ψ p 1

2

.

1 12 11

ψ ( 1 − p )

2 1