Simple Correlated Flow and Its Application

Abstract

A correlated flow of arrivals is considered in a case, where interarrival times {X _n } correspond to the Markov process with the continuous state space R ₊ = (0, ∞ ). The conditional probability density function of X _n + 1 given {X _n  = z} is determined by means of

$$ q(x|z)=q(x|X_n =z)=\sum_{i=1}^k p_i(z)h_i(x), \;\;\;\; z,x\in R_+, $$

where {p ₁(z),...,p _k (z)} is a probability distribution, p ₁(z) + ... + p _k (z) = 1 for all z ∈ R ₊; {h ₁(x),...,h _k (x)} is a family of probability density functions on R ₊.

This flow is investigated with respect to stationary case. One is considered as the Semi-Markov process J(t) on the state set {1, ..., k}. Main characteristics are considered: stationary distribution of J and interarrival times X, correlation and Kendall tau (τ) for adjacent intervals, and so on. Further one is considered a Markovian system on which the described flow arrives. Numerical results show that the dependence between interarrival times of the flow exercises greatly influences the efficiency characteristics of considered systems.