  The recently developed technique of DOC kernels has been a great success in
the stability and convergence analysis for BDF2 scheme with variable time
steps. However, such an analysis technique seems not directly applicable to
problems with initial singularity. In the numerical simulations of solutions
with initial singularity, variable time-steps schemes like the graded mesh are
always adopted to achieve the optimal convergence, whose first adjacent
time-step ratio may become pretty large so that the acquired restriction is not
satisfied. In this paper, we revisit the variable time-step implicit-explicit
two-step backward differentiation formula (IMEX BDF2) scheme presented in [W.
Wang, Y. Chen and H. Fang, \emph{SIAM J. Numer. Anal.}, 57 (2019), pp.
1289-1317] to compute the partial integro-differential equations (PIDEs) with
initial singularity. We obtain the sharp error estimate under a mild
restriction condition of adjacent time-step ratios $r_{k}: =\tau_{k}/\tau_{k-1}
\; (k\geq 3) < r_{\max} = 4.8645 $ and a much mild requirement on the first
ratio, i.e., $r_2>0$. This leads to the validation of our analysis of the
variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a
simple strategy, i.e., the graded mesh $t_k=T(k/N)^{\gamma}$. In this
situation, the convergence of order $\mathcal{O}(N^{-\min\{2,\gamma \alpha\}})$
is achieved with $N$ and $\alpha$ respectively representing the total mesh
points and indicating the regularity of the exact solution. This is, the
optical convergence will be achieved by taking $\gamma_{\text{opt}}=2/\alpha$.
Numerical examples are provided to demonstrate our theoretical analysis.
