The limited tritium resources available for the first fusion power plants (FPPs) make fuel self-sufficiency and tritium inventory minimization leading issues in FPP design. This work builds on the model proposed by M. Abdou et al. \cite{abdou2020physics}, which analyzed the fuel cycle of a DEMO-class FPP with a time-dependent system-level model. Here, we use a modified version of their model to analyze the fuel cycle of an ARC-class tokamak and two versions of a STEP-class tokamak. The ARC-class tokamak breeds tritium in a FLiBe liquid immersion blanket (LIB), while the STEP-class tokamak breeds tritium utilizing either a liquid-lithium blanket design or an Encapsulated Breeding Blanket (EBB). A time-dependent system-level model is developed in Matlab Simulink~\textregistered\hspace{.2em} to simulate the evolution of tritium flows and tritium inventories in the fuel cycle. The main goals of this work are to assess tritium self-sufficiency of the ARC- and STEP-class designs and to determine quantitative design requirements that can be used to analyze the adequacy of a proposed fuel cycle system. These design requirements are aimed at achieving a low tritium inventory doubling time ($t_d$) and a low start-up inventory ($I_{\mathrm{startup}}$) while keeping the required tritium breeding ratio (TBR$_r$) as low as possible. We also consider how improvements in fuel cycle technology and plasma operations affect TBR$_r$ and $I_{\mathrm{startup}}$. The model results show that TBR$_r$ for ARC- and STEP-class FPPs should be achievable if the tritium burn efficiency (TBE) reaches 0.5-1\% (TBR$_r <$ 1.2). This assumes significant, but attainable, improvements over current abilities. However, the model results indicate that a FPP must achieve ambitious performance targets, including FPP availability $>$70\%, tritium processing time $<$4~h, and the implementation of direct internal recycling. If future research yields major improvements to achievable TBE, it may be possible to achieve tritium self-sufficiency while operating at lower availability and without implementing DIR.