The authors describe and solve a Markov model of the leaky bucket ATM generic flow control mechanism. The model has a space cardinality which grows quickly with its parameters and is challenging to solve. Exploiting the cyclic nature of the model, the authors develop a methodology which allows them to efficiently solve instances of the model with 3,905,134 states and 53,869,532 transitions using 29.8 Mbytes of memory and 222 Mbytes of disc storage. The CPU utilisation is high (between 70% and 90%). The methodology is new and can be easily extended to any kind of finite cyclic Markov models.