Absrracr-In this correspondence we prove the following properties of the stability of the finite population model of slotted ALOHA systems.1) The input-output packet flow balance principle [l] and the concept of expected drift [2] used for the stability analysis of the slotted ALOHA system'are mathemtaically equivalent.2 ) The slotted ALOHA system can only have either one stable equilibrium point or three equilibrium points, with the first one and the third one stable and the second one unstable.3) If the retransmission probability (which is assumed to be greater than the probability of new packet generation) is less than or equal to 2/N, where N is the total number of active users in the system, then the system has only one stable equilibrium point.
.I. INTRODUCTIONThe stability of the slotted ALOHA system has been of considerable interest to many researchers [ 11-[4]. Kleinrock and Lam [ 11 used a fluid approximation to study an infiiite population model. By using the principle of input-output packet flow balance, they demonstrated the three possible system behaviors: stable, bistable, and saturation. Carleial and Hellman [2] studied the dynamics of the finite population model' of the slotted ALOHA scheme by introducing the concept of the expected drift. Those states with zero expected drift are identified as equilibrium states. Some numerical examples were presented to illustrate the same three possible system behaviosrs. However, the question of how many stable equilibrium points are theoretically possible remained unanswered.In this correspondence, we prove the properties listed in the above Abstract.
SOME ANALYTICAL RESULTS
Following the model considered in[ 11 and [2], let us consider a slotted ALOHA channel with N users accessing a central station. Each user can be in either the originating mode or the retransmission mode. A user in the originating mode will transmit a new packet in any given time slot with probability p o , while a user in the retransmission mode will try a Paper approved by the Editor for Computer Communication of the IEEE Communications Society for publication without oral presentation.1 In some degenerated cases, the fiist and 'the second or the second slot with probability pr. In a real system p o is determined by the average behavior of a user, while pr can be controlled by the central station. Usually pr is much larger than p o ; hence, we will always assume that pr > p o in this note. Let x(t), t = 1, 2, 3, ... be the number of users in the r e transmission mode at the beginning of time slot. The quantity x ( t ) can assume one of the (N + 1) possible values ( 0 , 1 , 2 , '", N } , and can be considered as the state variable of the system. Because of the memoryless assumption, the process x ( t ) is a finite-state Markov chain with the state-transition probability= n ] is given by Let d(n) be the expected drift of the system at state n, that is [21, and let a(. ) be the system's expected packet flow per time slot at state n; then [ 21 We now show that the expected system drift ...
