Concave iteration semigroups of linear continuous set-valued functions

Let {F t: t ≥ 0} be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) − F t (x) exist for x ∈ K and t > 0, then D t F t (x) = (−1)F t ((−1)G(x)) for x ∈ K and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and $$G(x) = \mathop {\lim }\limits_{s \to 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord{\left/ {\vphantom {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \kern-\nulldelimiterspace} {\left( { - s} \right)}}$$ for x ∈ K.

On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel

We analyse a stack protocol of the Capetanakis–Tsybakov–Mikhailov type for resolving collisions in a random multiple-access channel. We obtain a functional equation for the generating function of the expected collision resolution interval (CRI) durations, which is non-local with a non-commutative iteration semigroup. Using Mellin transform techniques and geometric properties of the iteration semigroup we show that for arrival rates smaller than a fixed threshold, the mean CRI duration for n initial colliders is asymptotically proportional to n. Ergodicity conditions are also demonstrated.

On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel

We analyse a stack protocol of the Capetanakis–Tsybakov–Mikhailov type for resolving collisions in a random multiple-access channel. We obtain a functional equation for the generating function of the expected collision resolution interval (CRI) durations, which is non-local with a non-commutative iteration semigroup. Using Mellin transform techniques and geometric properties of the iteration semigroup we show that for arrival rates smaller than a fixed threshold, the mean CRI duration for n initial colliders is asymptotically proportional to n. Ergodicity conditions are also demonstrated.