  We introduce for a general correlation scenario a new simulation model, a
local quasi hidden variable (LqHV) model, where locality and the
measure-theoretic structure inherent to an LHV model are preserved but
positivity of a simulation measure is dropped. We specify a necessary and
sufficient condition for LqHV modelling and, based on this, prove that every
quantum correlation scenario admits an LqHV simulation. Via the LqHV approach,
we construct analogs of Bell-type inequalities for an N-partite quantum state
and find a new analytical upper bound on the maximal violation by an N-partite
quantum state of S_{1}x...xS_{N}-setting Bell-type inequalities - either on
correlation functions or on joint probabilities and for outcomes of an
arbitrary spectral type, discrete or continuous. This general analytical upper
bound is expressed in terms of the new state dilation characteristics
introduced in the present paper and not only traces quantum states admitting an
S_{1}x...xS_{N}-setting LHV description but also leads to the new exact
numerical upper estimates on the maximal Bell violations for concrete N-partite
quantum states used in quantum information processing and for an arbitrary
N-partite quantum state. We, in particular, prove that violation by an
N-partite quantum state of an arbitrary Bell-type inequality (either on
correlation functions or on joint probabilities) for S settings per site cannot
exceed (2S-1)^{N-1} even in case of an infinite dimensional quantum state and
infinitely many outcomes.
