Functional equations with multiple recursive terms

Abstract

In this paper, we study a functional equation for generating functions of the form $$f(z) = g(z) \sum _{i=1}^M p_i f(\alpha _i(z)) + K(z)$$ f ( z ) = g ( z ) ∑ i = 1 M p i f ( α i ( z ) ) + K ( z ) , viz. a recursion with multiple recursive terms. We derive and analyze the solution of this equation for the case that the $$\alpha _i(z)$$ α i ( z ) are commutative contraction mappings. The results are applied to a wide range of queueing, autoregressive and branching processes.

Logistics Infrastructure and Supply Efficiency of Petroleum Products in Kenya: A Survey of Oil Marketing Companies in Kenya

“…According to Adan and Resing (2015), a queuing system will ensure that customers will receive input service and that they will be required to wait for it in the event that it is unavailable. The optimization of transit and waiting times, which in turn leads to a reduction in fuel consumption, can be achieved by setting priorities for tasks that belong to any aspect of supply chain management.…”

Untitled

“…Of course, because the system contains a finite buffer, regardless of the intensity of customer input and the speed of their service, the steady state exists (see e.g. Adan, Resing, 2015;Tijms, 2003;Heyman, Sobel, 1982 for basics of stochastic modeling in this area).…”

Queueing and risk models with dependencies

“…In [1], recursions of exactly this type are treated. The commutativity of ζ i (s) := s+μ i c and ζ j (s) := s + μ j c, i.e., ζ i (ζ j (s)) = ζ j (ζ i (s)), makes the recursion (54) relatively easy even though in each iteration step a term φ(s + K i=1 b i μ i c) gives rise to K new terms.…”