Capacity of time-slotted ALOHA systems

t,r;insrnit.. Depending on the presence or absence of ot.lier users, each user will be able to achieve different rates. In particular. certain bit,s n-ill be .received in the absence of other users but not in t,he presence of other users. Since some transniitted bits mill be lost owing to collisions. the capacit.y lve consider is maxiinurn rel iablyreceived rate rather than niasiiiium rel iably t7~ms7nitted rat,e. The difference betn-een these tu-o r:ates has been explored iii [7]. but the basicidea is as folloms. Shannon's definition of capacitjrequires t,liat all transmitted bits are correctly decoded at, the receiver. It is shon-n in [7] that by allowing soiiie bits to be received--in error. the rate of bit.s correctly received (the reliably received rate) caii be increased above Shannon capacity (the reliably transniitted rate). Reliably received rate is of interest for'applications -where it is -desirable t o iiiasiiiiize received rate n-ithout iiec. eisarily kiion-iiig at the transiiiitter n-liich rate is received. Sincc iiiost -4LOH-i sj-stenis 1iAl-e a fredliack ~iiecliaiiism that allon-s the receiver to tell the traiimiitter n-hich bits to reseiid. reliably received rate is the.appropriate capacity iiieasure for. these s\-stems. The organization of our paper &.as follon-s. In . Section 11. n-e preseiit the simplified case of tn'o iisers. earl1 of n-liicii seiids a t iliost one packet. The users share a single channel with additil-p ivliite Gaussiaii iioise (.A\\-GS). \Ye coiisit1t.r that. th r duratioii of a packet is long cnougli to guar;mtw tliat capacitjn-ill. be achieved 01-er a single packet traiisiiiissioii. The purpose of the sectio!i is :to establish a niodel wliicli. ca!i thcn he ex. tiwitxi. iii Svctioii III. . t i ' t ~ i e iiiorc iiitwestiiig '( '~IS( ' of s c i ~ ~ a l IISWS comiiiuiiic.atiii~ o l ~ r arhitrarily loiig tiniiis. 111 Section I\-. n-c presentour roiicliisioiis and diwctioiis for fiirtliw r r s t ~ ~ c h .