On stability of slotted-ALOHA systems with delay constraints and geometric transmission probability

It has been well-known that slotted ALOHA systems exhibit unfavorable bistable behavior, since 1970s. It was proved that we can eliminate bistable region by limiting the number of allowable transmission attempts per packet to eight or less. In this paper, we investigate bistable behavior of slotted ALOHA systems with delay constraints and geometric transmission probability in terms of the catastrophe theory. A lifetime of a packet is confined within D time-slots and packet transmission probability is determined based on geometric series depending on the number of elapsed time-slots after its generation. Let p
_d
= pα
^d
be the transmission probability in a slot for a packet with d-time-slot expiration after generation, d = 0, 1,
_⋯
, D - 1. Then, we numerically indicate that bistable region may exist, if the sum p
₀
+ p
₁
+
_⋯
+ p
_D-1
is greater than approximately 8.30, and that the system can operate with a unique stable equilibrium point, if the sum is less than approximately 8.30.