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We propose a new model for the evolution of a star cluster's System MassFunction (SMF). The model involves both turbulent fragmentation and competitiveaccretion. Turbulent fragmentation creates low-mass seed proto-systems (i.e.single and multiple protostars). Some of these low-mass seed proto-systems thengrow by competitive accretion to produce the high-mass power-law tail of theSMF. Turbulent fragmentation is relatively inefficient, in the sense that thecreation of low-mass seed proto-systems only consumes a fraction, $\sim 23\%$(at most $\sim 50\%$), of the mass available for star formation. The remainingmass is consumed by competitive accretion. Provided the accretion rate onto aproto-system is approximately proportional to its mass ($dm/dt \propto m$), theSMF develops a power-law tail at high masses with the Salpeter slope ($\sim-2.3$). If the rate of supply of mass accelerates, the rate of proto-systemformation also accelerates, as appears to be observed in many clusters.However, even if the rate of supply of mass decreases, or ceases and thenresumes, the SMF evolves homologously, retaining the same overall shape, andthe high-mass power-law tail simply extends to ever higher masses until thesupply of gas runs out completely. The Chabrier SMF can be reproduced veryaccurately if the seed proto-systems have an approximately log-normal massdistribution with median mass $\sim 0.11 {\rm M}_{_\odot}$ and logarithmicstandard deviation $\sigma_{\log_{10}(M/{\rm M}_\odot)}\sim 0.47$).

On the emergent System Mass Function: the contest between accretion and fragmentation