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This paper deals with the performance analysis of a system modeled by a queue. If we are interested in occupation problems and if we look at the transient phase, then it makes sense to study the maximum backlog observed in the queue over a finite period. This paper proposes an efficient algorithmic scheme to evaluate the distribution of this maximum backlog level, based on the uniformization technique. The approach is illustrated using the classical M/M/1 model, but it can be extended to more complex ones.