  In this paper we make a systematic study of the multiplicity of the jumping
points associated to the mixed multiplier ideals of a family of ideals in a
complex surface with rational singularities. In particular we study the
behaviour of the multiplicity by small perturbations of the jumping points. We
also introduce a Poincar\'e series for mixed multiplier ideals and prove its
rationality. Finally, we study the set of divisors that contribute to the
log-canonical wall.
