We investigate Leaky Bucket (LB) schemes with a threshold in the data buffer, where the leak rate changes depending on the contents of the data buffer. We use a Markov modulated Poisson process (MMPP) as a bursty input traffic. We obtain the limiting distributions of the system state at an embedded point and at an arbitrary time. As performance measures we obtain cell loss probability and mean cell delay. We present some numerical results to show the effects of the threshold value, the rate of token generation, the size of the token pool and the size of the data buffer on the performances of the LB scheme with a threshold. Numerical examples show that the LB scheme with a threshold improves the system performance in comparison with the LB scheme without a threshold. (C) 1997 Published by Elsevier Science B.V.