The authors describe and solve a Markov model of the leaky bucket ATM generic now 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 29.8Mbyte of memory and 222Mbyte 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.