  The L\'evy walk process for the lower interval of the time of flight
distribution ($\alpha<1$) and with finite resting time between consecutive
flights is discussed. The motion is restricted to a region bounded by two
absorbing barriers and the escape process is analysed. By means of a Poisson
equation, the total density, which includes both flying and resting phase, is
derived and the first passage time properties determined: the mean first
passage time appears proportional to the barrier position; moreover, the
dependence of that quantity on $\alpha$ is established. Two limits emerge from
the model: of short waiting time, that corresponds to L\'evy walks without
rests, and long waiting time which exhibits properties of a L\'evy flights
model. The similar quantities are derived for the case of a position-dependent
waiting time. Then the mean first passage time rises with barrier position
faster than for L\'evy flights model. The analytical results are compared with
Monte Carlo trajectory simulations.
