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A group random-access (GILA) control discipline for a multi-access communication channel is presented and studied. A GILA scheme uses only certain channel time periods to allow some network terminals to transmit their information-bearing packets on a random-access basis. The channel can thus be utilized at other times to grant access to other terminals or other message types, by applying as appropriate group random-access, reservation, or fixed access-control procedures. GILA schemes could also be utilized to provide channel access to various network protocol packets. The average packet delay under a GRA discipline is evaluated by a Markov ratio limit theorem. To stabilize the channel, the GRA procedure is controlled dynamically by a control policy that rejects any newly arriving packets within certain time periods. Studying the associated Markov decision problem, the optimal control policy is characterized as yielding a minimal average packet delay under a prescribed maximal packet probability of rejection. This policy is shown to be represented by a single-threshold scheme. For such a scheme, a threshold value that attains the minimum probability of rejection is shown to exist and to yield a desirable control procedure. Performance curves are presented to demonstrate the delay-throughput characteristics induced by GRA procedures.