  A $q$-analogue of the tau function of the modified KP hierarchy is defined by
a change of independent variables. This tau function satisfies a system of
bilinear $q$-difference equations. These bilinear equations are translated to
the language of wave functions, which turn out to satisfy a system of linear
$q$-difference equations. These linear $q$-difference equations are used to
formulate the Lax formalism and the description of quasi-classical limit. These
results can be generalized to a $q$-analogue of the Toda hierarchy. The results
on the $q$-analogue of the Toda hierarchy might have an application to the
random partition calculus in gauge theories and topological strings.
