The increase in strangeness production with charged particle multiplicity, as seen by the ALICE collaboration at CERN in p-p, p-Pb and Pb-Pb collisions, is investigated in the hadron resonance gas model taking into account interactions among hadrons using S-matrix corrections based on known phase shift analyses. Strangeness conservation is taken into account in the framework of the canonical strangeness ensemble. A very good description is obtained for the variation of the strangeness content in the final state as a function of the number of charged hadrons in the mid-rapidity region using the same fixed temperature value as obtained in the most central Pb-Pb collisions. It is shown that the number of charged hadrons is linearly proportional to the volume of the system. For small multiplicities the canonical ensemble with local strangeness conservation restricted to mid-rapidity leads to a stronger suppression of (multi-)strange baryons than seen in the data. This is compensated by introducing a global conservation of strangeness in the whole phase-space which is parameterized by the canonical correlation volume larger than the fireball volume at the mid-rapidity. The results on comparing the hadron resonance gas model with and without S-matrix corrections, are presented in detail. It is shown that the interactions introduced by the phase shift analysis via the S-matrix formalism are essential for a better description of the yields data.
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