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It is now a widely held view that, in their formation and early evolution, stars build up mass in bursts. The burst mode of star formation scenario proposes that the stars grow in mass via episodic accretion of fragments migrating from their gravitationally-unstable circumstellar discs and it naturally explains the existence of observed pre-main-sequence bursts from high mass protostars. We present a parameter study of hydrodynamical models of massive young stellar objects (MYSOs) that explores the initial masses of the collapsing clouds (Mc = 60-200Mo) and ratio of rotational-to-gravitational energies (beta = 0:005-0:33). An increase in Mc and/or beta produces protostellar accretion discs that are more prone to develop gravitational instability and to experience bursts. We find that all MYSOs have bursts even if their pre-stellar core is such that beta <= 0.01. Within our assumptions, the lack of stable discs is therefore a major difference between low- and high-mass star formation mechanisms. All our disc masses and disk-to-star mass ratios Md=M* > 1 scale as a power-law with the stellar mass. Our results confirm that massive protostars accrete about 40-60% of their mass in the burst mode. The distribution of time periods between two consecutive bursts is bimodal: there is a short duration (~ 1-10 yr) peak corresponding to the short, faintest bursts and a long duration peak (at ~ 10^3-10^4 yr) corresponding to the long, FU-Orionis-type bursts appearing in later disc evolution, i.e., around 30 kyr after disc formation. We discuss this bimodality in the context of the structure of massive protostellar jets as potential signatures of accretion burst history.