Age of Information in a Cooperative Slotted Aloha Network: Marginal and Joint Distributions

In this paper, we investigate a slotted Aloha cooperative network where a source node and a relay node send status updates of two underlying stochastic processes to a common destination. Additionally, the relay node cooperates with the source by accepting its packets for further retransmissions, where the cooperation policy comprises acceptance and relaying probabilistic policies. Exact marginal steady state distributions of the source and relay Age of Information (AoI) and Peak AoI (PAoI) sequences are obtained using Quasi-Birth-Death (QBD) Markov chain models. Extending this approach, we also obtain the joint distribution of the source and relay AoI sequences out of which one can obtain the steady state distribution of the Squared Difference of the two AoI sequences (SDAoI), which finds applications in network scenarios where not only the timeliness of status updates of each process is desired but also their simultaneity is of crucial importance. In this regard, we numerically obtain the optimal cooperation policy in order to minimize the expected value of SDAoI subject to a constraint on the average PAoI of the relay. Finally, our proposed analytical approach is verified by simulations and the performance of the optimal policy is discussed based on the numerical results.