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

Published online by Cambridge University Press:  14 July 2016

W. J. Padgett  and
C. P. Tsokos
W. J. Padgett
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia
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Extract

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:

  1. (i) ω∊Ω, where Ω is the supporting set of the probability measure space (Ω,B,P);

  2. (ii) x(t; ω) is the unknown random variable for t ∊ R+, where R+ = [0, ∞);

  3. (iii) y(t; ω) is the stochastic free term or free random variable for t ∊ R+;

  4. (iv) k(t, τ; ω) is the stochastic kernel, defined for 0 ≦ τ ≦ t < ∞; and

  5. (v) f(t, x) is a scalar function defined for t ∊ R+ and x ∊ R, where R is the real line.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 2 , June 1971 , pp. 269 - 275
Copyright
Copyright © Applied Probability Trust 1971 

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References

[1] Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and Their Applications. McGraw-Hill, New York.Google Scholar
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