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Following up on a recently renewed interest in computational methods for M/G/1-type processes, this paper considers an M/G/1-like system in which the service time distribution is represented by a Coxian series of memoryless stages. We present a novel approach to the solution of such systems. Our method is based on conditional probabilities, and provides a simple, computationally efficient and stable approach to the evaluation of the steady-state queue length distribution. We provide a proof of the numerical stability of our method. Without explicit use of matrix-geometric techniques or stochastic complementation, we are able to handle systems with state-dependent service and arrival rates. The proposed approach can be used to compute the queue length distribution for both finite and infinite M/G/1-like queues. In the case of an infinite, state-independent queue, our method allows us to show using elementary tools that the queue length distribution is asymptotically geometric. The parameter of the asymptotic geometric can be expressed through a simple set of equations, easily solved using fixed point iteration. Our approach is very thrifty in terms of memory requirements, easy to implement, and generally fast. Numerical examples illustrate the performance of the proposed method.