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This paper addresses the problem of estimating the worst-case end-to-end delay for a flow in a tandem of FIFO multiplexing nodes, following up our previous work [12]. We show that, contrary to the expectations, the state-of-the-art method for computing delay bounds, i.e. upper bounds on the worst-case delay, called the Least Upper Delay Bound (LUDB) methodology, may actually be larger than the worst-case delay even in simple cases. Thus, we first devise a method to compute improved delay bounds. Then, in order to assess how close the derived bounds are to the actual, still unknown, worst-case delays, we devise a method to compute lower bounds on the worst-case delay. Our analysis shows that the gap between the upper and lower bounds is quite small in many practical cases, which implicitly validates the upper bounds themselves.