As it has been known since its introduction Network Calculus can be used to obtain achievable delay bounds for a single First In First Out (FIFO) queue whose inputs are constrained by deterministic bounds known as "input envelopes." Using these results it's possible to obtain a bound on the endtoend delay for two FIFO queues in tandem but whether this bound was achievable or not was an open problem. In order to find a better bound, we define a general service abstraction, which is defined in terms of a ``service mapping," which is a monotone operator that maps an arrival process to a lower bound on the corresponding departure process. For a network element with a shift invariant service mapping, we obtain bounds on the maximum delay, maximum backlog, and a traffic envelope for the departure process, assuming that the arrival process to the network element conforms to a traffic envelope. Using the service mapping abstraction to analyze two FIFO queues in tandem we can obtain an achievable delay that is smaller than those that can be obtained with previously proposed methods. In the second part of the presentation we address the problem of worst case average delay for a single FIFO queue with constrained inputs, where the average is over time. If a flow has a piecewise linear and concave envelope and shares a single FIFO queue with another flow that has a concave envelope it's possible to obtain a tight bound on the worst case average delay.





