Age of Information in Multi-Source Broadcast Channel

We consider a time slotted status update system with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(N+1)$</tex> sources, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> users and one base station (BS). User <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$i$</tex> aims to receive fresh information from the stochastic processes at source <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$i$</tex> and source <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(N+1)$</tex>. We measure the freshness of the system with the age of information metric. Thus, for user <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$i$</tex>, there are two notions of age of information, one corresponding to the age of source <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$i$</tex>, and another corresponding to the age of source <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(N+1)$</tex>. The BS aims to employ an optimal scheduling policy to minimize the total average age of information. When the BS samples an update packet from the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$i$</tex>th source, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1\leq i\leq N$</tex>, the BS transmits that sample only to the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$i$</tex>th user. However, when the BS samples an update packet from the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(N+1)$</tex>th source, the BS transmits that sample to all the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> users. Due to the energy constraint of the BS, we assume that when the BS transmits an update packet to a dedicated user, the probability of successful transmission is higher compared to the probability of successful transmission to the same user, when the BS transmits an update packet to all the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex> users. Due to this, there is a trade-off between the age of information and the probability of successful transmission. We find several structures for the optimal scheduling policy and exploit those properties to design a low-complexity relative value iteration-based scheduler. We propose a further lower complexity scheduling policy which is a mixture of the optimal scheduling policy and the max-weight policy.