 This can be computed exactly using the Pollaczek–Khinchine formula as (the ruin function here is equivalent to the tail function of the stationary distribution of waiting time in an M/G/1 queue)
where formula_6 is the transform of the tail distribution of formula_7,
and formula_9 denotes the formula_10-fold convolution.In the case where the claim sizes are exponentially distributed, this simplifies to

E. Sparre Andersen extended the classical model in 1957 by allowing claim inter-arrival times to have arbitrary distribution functions.where the claim number process formula_13 is a renewal process and formula_14 are independent and identically distributed random variables.