  Random walk has wide applications in many fields, such as machine learning,
biology, physics, and chemistry. Random walk can be discrete or continuous in
time and space. Asymmetric random walk could be described by drift-diffusion
equation. The discrete asymmetric random walk provides a basis for
understanding biased drift-diffusion. However, the current reported theoretical
results for discrete random walks do not give a general theoretical treatment
for the asymmetry due to unequal jump times. In this paper, the theoretical
expressions for asymmetric random walks with unequal jump probabilities, times,
and lengths are derived. The obtained theoretical results can be reduced to
reported results when jump times are equal. Additionally, discrete random walk
simulations are performed to verify the obtained theoretical results. There are
good agreements between the theoretical predictions and simulation results.
