We present a model of one-dimensional asymmetric random walks. Random walkers alternate between flights (steps of constant velocity) and sticking (pauses). The sticking time probability distribution function (PDF) decays as P(t)∼t-ν. Previous work considered the case of a flight PDF decaying as P(t)∼t-μ [Weeks et al., Physica D 97, 291 (1996)]; leftward and rightward flights occurred with differing probabilities and velocities. In addition to these asymmetries, the present strongly asymmetric model uses distinct flight PDFs for leftward and rightward flights: PL(t)∼t-μ and PR(t)∼t-η, with μ≠η. We calculate the dependence of the variance exponent γ (σ2∼tγ) on the PDF exponents ν, μ, and η. We find that γ is determined by the two smaller of the three PDF exponents, and in some cases by only the smallest. A PDF with decay exponent less than 3 has a divergent second moment, and thus is a Lévy distribution. When the smallest decay exponent is between 3/2 and 3, the motion is superdiffusive (1<γ<2). When the smallest exponent is between 1 and 3/2, the motion can be subdiffusive (γ<1); this is in contrast with the case with μ=η.
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