  In this paper, we consider a type of time-changed Markov process, where the
time-change is an inverse killed subordinator. This can be seen as an extension
of Chen (Chen, Z., Time fractional equations and probabilistic representation,
Chaos Solitons and Fractals, 168-174, 2017). As a result, it constructs a
one-to-one correspondence between general Bernstein functions (with infinite
L\'{e}vy measure) and a class of generalized time-fractional partial
differential equations.
