  Lifetimes of the doubly heavy baryons ${\cal B}_{bb}$ and ${\cal B}_{bc}$ are
analyzed within the framework of the heavy quark expansion (HQE). Lifetime
differences arise from the spectator effects such as $W$-exchange and Pauli
interference. For doubly bottom baryons, the lifetime pattern is
$\tau(\Omega_{bb}^-)\sim \tau(\Xi_{bb}^{-})>\tau(\Xi_{bb}^0)$. The
$\Xi_{bb}^{0}$ baryon is shortest-lived owing to the $W$-exchange contribution,
while $\Xi_{bb}^{-}$ and $\Omega_{bb}^{-}$ have similar lifetimes as they both
receive contributions from destructive Pauli interference. We find the lifetime
ratio $\tau(\Xi_{bb}^{-})/\tau(\Xi_{bb}^0)=1.26$\,. The large $W$-exchange
contribution to $\Xi_{bc}^0$ through the subprocess $cd\to us\to cd$ and the
sizable destructive Pauli interference contribution to $\Xi_{bc}^+$ imply a
substantial lifetime difference between $\Xi_{bc}^+$ and $\Xi_{bc}^0$. In the
presence of subleading $1/m_c$ and $1/m_b$ corrections to the spectator
effects, we find that $\tau(\Omega_{bc}^0)$ becomes longest-lived. This is
because $\Gamma^{\rm int}_+$ and $\Gamma^{\rm semi}$ for $\Omega_{bc}^0$ are
subject to large cancellation between dimension-6 and -7 operators. This
implies that the subleading corrections are too large to justify the validity
of the HQE. Demanding that $\Gamma^{cs}_{{\rm int+}}(\Omega_{bc}^0)$,
$\Gamma^{{\rm SL},cs}_{\rm int}(\Omega_{bc}^0)$ be positive and
$\Gamma^{cu}_{{\rm int-}}(\Xi^+_{bc})$ be negative, we conjecture that
$1.68\times 10^{-13}s<\tau(\Omega_{bc}^0)< 3.70\times 10^{-13}s$ , $4.09\times
10^{-13}s<\tau(\Xi_{bc}^+)< 6.07\times 10^{-13}s$ and $0.93\times
10^{-13}s<\tau(\Xi_{bc}^0)< 1.18\times 10^{-13}s$. Hence, the lifetime
hierarchy of ${\cal B}_{bc}$ baryons is expected to be the pattern
$\tau(\Xi_{bc}^{+})>\tau(\Omega_{bc}^0)>\tau(\Xi_{bc}^0)$.
