  In this paper, we introduce a delayed Mittag-Leffler type function. With the
help of the delayed Mittag-Leffler type functions, we give an explicit formula
of solutions to linear nonhomogeneous fractional time-delay Langevin equations
involving two Riemann-Liouville fractional derivatives. The existence and
uniqueness of solutions are obtained by using an estimation of delayed
Mittag-Leffler type functions in terms of exponential functions and a weighted
norm via fixed point theorems. Further, we present Ulam--Hyers stability
results.
