  Analytic scaling relations are derived for a phenomenological model of the
plasmoid instability in an evolving current sheet, including the effects of
reconnection outflow. Two scenarios are considered, where the plasmoid
instability can be triggered either by an injected initial perturbation or by
the natural noise of the system (here referred to as the system noise). The two
scenarios lead to different scaling relations because the initial noise decays
when the linear growth of the plasmoid instability is not sufficiently fast to
overcome the advection loss caused by the reconnection outflow, whereas the
system noise represents the lowest level of fluctuations in the system. The
leading order approximation for the current sheet width at disruption takes the
form of a power-law multiplied by a logarithmic factor, and from that, the
scaling relations for the wavenumber and the linear growth rate of the dominant
mode are obtained. When the effects of the outflow are neglected, the scaling
relations agree, up to the leading order approximation, with previously derived
scaling relations based on a principle of least time. The analytic scaling
relations are validated with numerical solutions of the model.
