M/G/1 queues with negative arrival: an iteration to solve a Fredholm integral equation of the first kind

The generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both positive and negative customers is investigated. The queueing discipline, first come first served, is chosen for the positive customers, together with a killing strategy in which the last customer in the queue is removed by a negative customer. This killing strategy leads to complex mathematical analysis and, ultimately, to a Fredholm integral equation of the first kind. This type of equation has no known reliable numerical solution method except in special cases. We derive an alternative, iterative method involving the numerical inversion of Laplace transforms