On a stochastic integral equation of the Volterra type in telephone traffic theory

In mathematical models of phenomena occurring in the general areas of the engineering, biological, and physical sciences, random or stochastic equations appear frequently. In this paper we shall formulate a problem in telephone traffic theory which leads to a stochastic integral equation which is a special case of the Volterra type of the form where: (i)ω∊Ω, where Ω is the supporting set of the probability measure space (Ω,B,P);(ii)x(t; ω) is the unknown random variable for t ∊ R+, where R+ = [0, ∞);(iii)y(t; ω) is the stochastic free term or free random variable for t ∊ R+;(iv)k(t, τ; ω) is the stochastic kernel, defined for 0 ≦ τ ≦ t < ∞; and(v)f(t, x) is a scalar function defined for t ∊ R+ and x ∊ R, where R is the real line.