Some properties of variable length packet shapers

The min-plus theory of greedy shapers has been developed from R.L. Cruz\´s results (1991) on the calculus of network delays. The theory of greedy shapers establishes a number of properties such as the series decomposition of shapers or the conservation of arrival constraints by reshaping. It applies either to fluid systems or to packets of constant size such as ATM. For variable length packets, due to the distortion introduced by packetization, the theory is no longer valid. We elucidate the relationship between shaping and packetization effects. We show a central result, the min-plus representation of a packetized greedy shaper. We find a sufficient condition under which series decomposition of shapers and conservation of arrival constraints still holds in the presence of packetization effects. This allows us to demonstrate the equivalence of implementing a buffered leaky bucket controller based on either virtual finish times or on bucket replenishment. However, in some examples, if the condition is not satisfied, then the property may no longer hold. Thus, for variable size packets, there is a fundamental difference between constraints based on leaky buckets and constraints based on general arrival curves, such as spacing constraints. The latter are used in the context of ATM to obtain tight end-to-end delay bounds. We use a min-plus theory and obtain results on greedy shapers for variable length packets which are not readily explained with the max-plus theory of C.S. Chang (see "Performance Guarantees in Communication Networks", Springer-Verlag, 2000)