In this paper, we present monotonicity properties of the Leaky Bucket (LB) input rate regulation scheme. We show that the asymptotic version of the inter-departure time decreases in the convex ordering as the size of the token pool decreases or as the token generation period increase. When measured by the coefficients of variations of the inter-departure times, the burstiness of the output traffic from the LB increases with the sizes of the token pool, and decreases with the token generation period. These results may prove useful in designing the LB.