We develop an algorithm for numerically inverting multidimensional transforms. Our algorithm applies to any number of continuous variables (Laplace transforms) and discrete variables (generating functions). We use the Fourier-series method; that is, the inversion formula is the Fourier series of a periodic function constructed by aliasing. This amounts to an application of the Poisson summation formula. By appropriately exponentially damping the given function, we control the aliasing error. We choose the periods of the multidimensional periodic function so that each infinite series is a finite sum of nearly alternating infinite series. Then we apply the Euler transformation to compute the infinite series from finitely many terms. The multidimensional inversion algorithm enables us, evidently for the first time, to calculate probability distributions quickly and accurately from several classical transforms in queueing theory. For example, we apply our algorithm to invert the two-dimensional transforms of the joint distribution of the duration of a busy period and the number served in that busy period, and the time-dependent transient queue-length and workload distributions in the M/G/1 queue. In other related work, we have applied the inversion algorithms here to calculate time-dependent distributions in the transient BMAP/G/1 queue (with a batch Markovian arrival process) and the piecewise-stationary $M_t/G_t/1$ queue.
"Multidimensional Transform Inversion with Applications to the Transient M/G/1 Queue." Ann. Appl. Probab. 4 (3) 719 - 740, August, 1994. https://doi.org/10.1214/aoap/1177004968