Alternate Wars gives this formula for computing the length of a rocket burn:
$$\Delta T = \frac {M_L E_V} {F} (1 - e ^ {-\frac {\Delta V } {E_V}}) $$
Where:
$\Delta T$: Length of burn in seconds
$M_L$: Total mass of the rocket at the beginning of the burn (often written $m_0$)
$E_V$ = Exhaust Velocity in meters/second (often written as $v_e$).
$F$: Thrust of the rocket in Newtons.
$\Delta V$ = Delta-V of burn in meters/second.
Your $M_0$ is this equation's $M_L$. Exhaust velocity $E_V$ is equivalent to thrust divided by mass flow rate (that's your $F$ and $M_1$).
Exhaust velocity is one of two standard forms for representing mass-specific impulse. More often you'll see specific impulse called $I_{sp}$ and measured in seconds (but really meaning pounds-of-force-seconds-per-pound-of-mass); $I_{sp}$ in seconds times gravity at Earth's surface ($\approx 9.81 \frac {m} {s^2}$) yields exhaust velocity.