Ideas and results published in two papers by R. L. Cruz in IEEE Transactions on Information Theory in 1991 gave rise to what is called now network calculus. A key role in it plays a certain inequality characterizing the behaviour of cumulative traffic curves. It defines the so-called burstiness constraint by which many kinds of traffics can be described, as for example those occurring in computer networks. Interpretation of this constraint, which can be expressed in two equivalent forms: with and without the use of min-plus convolution, can be found in papers of R. L. Cruz. Nothing however was said about how to obtain it practically, for example, for each of representatives of a family of measured cumulative traffic curves being upperbounded. This problem is tackled in this paper, and as a result, a relation between the Cruz’s constraining function and an upper-bounding function of measured traffic curves is found. The relation obtained is quite general and valid also for the case of non-fulfilment of the so-called sub-additivity property by traffic curves. For the purpose of its derivation, a notion of sub-additivity property with some tolerance Δ was introduced, and the corresponding theorem exploiting it formulated and proved. Further, to complement discussion of the above relation, a minimal burstiness constraint was added to the original Cruz’s inequality and related with a lower bound of a family of measured cumulative traffic curves. The derivations presented in this paper are illustrated by examples.
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