relife.renewal_process.renewal_equation_solver
relife.renewal_process.renewal_equation_solver¶
- relife.renewal_process.renewal_equation_solver(F: numpy.ndarray, Fm: numpy.ndarray, y: numpy.ndarray, D: Optional[numpy.ndarray] = None) numpy.ndarray[source]¶
Renewal equation solver.
- Parameters
F (ndarray) – The cumulative distribution function evaluated at each point of the timeline.
Fm (ndarray) – The cumulative distribution function evaluated at each midpoint of the timeline.
y (ndarray) – A given function evaluated at each point of the timeline.
D (ndarray, optional) – Discount function value at each point of the timeline, by default None.
- Returns
Renewal function evaluated at each point of the timeline.
- Return type
ndarray
Notes
The timeline must start at 0.
Solve the renewal equation for \(z\), such that for \(t\geq 0\).
\[\begin{split}\begin{aligned} z(t) & = y(t) + \int_0^t z(t-x) D(x) \mathrm{d}F(x) \\ & = y(t) + z \ast F(t) \end{aligned}\end{split}\]where \(F\) is the cumulative distribution function, \(y\) a function and \(D\) the exponential discounting factor.
References
- 1
Dragomir, S. S. (2011). Approximating the Riemann–Stieltjes integral by a trapezoidal quadrature rule with applications. Mathematical and computer modelling, 54(1-2), 243-260.
- 2
Tortorella, M. (2005). Numerical solutions of renewal-type integral equations. INFORMS Journal on Computing, 17(1), 66-74.