The practical applications of the systems of Josephson devices meet a number of difficulties associated with the lack of understanding of the dynamic processes occurring in them. The accurate consideration of this problem requires the solution of the unequilibrium equations of the microscopic theory of superconductivity and it is very difficult task [1] On the other hand there are simplified models concentrated on four physical mechanisms of coupling between Josephson contacts. They are inductive interaction between adjacent junctions [2], modification of pair potential due to proximity effect [3], a charge accumulation of condensate [4] and a quasiparticle accumulation [5]. In this work we study analytically and numerically the influence of the quasiparticle charge imbalance on the dynamics of the asymmetric Josephson stack formed by two inequivalent junctions: the fast capacitive junction JJ1 and slow non-capacitive junction JJ2 [6]. We find, that the switching of the fast junction into resistive state leads to significant increase of the effective critical current of the slow junction. At the same time, the initial switching of the slow junction may either increase or decrease the effective critical current of the fast junction, depending on ratio of their resistances and the value of the capacitance. Finally, we have found that the slow quasiparticle relaxation (in comparison with Josephson times) leads to appearance of the additional hysteresis on current-voltage characteristics.