Among the fairness criteria for allocating indivisible resources to a group of agents, some are based on minimum utility levels. These levels can come from a specific allocation method, such as maximin fair-share criterion which is based on the cut-and-choose protocol. We propose to analyze criteria whose minimum utility levels are inspired by picking sequences, a well-established protocol for allocating indivisible resources. We study these criteria and investigate their connections with known fairness criteria, enriching the understanding of fair allocation of indivisible goods.