relife.renewal_process.delayed_renewal
relife.renewal_process.delayed_renewal¶
- relife.renewal_process.delayed_renewal(z: numpy.ndarray, F1: numpy.ndarray, F1m: numpy.ndarray, y1: numpy.ndarray, D: Optional[numpy.ndarray] = None) numpy.ndarray[source]¶
Add delay for the first renewal to a solution of a renewal equation.
- Parameters
z (ndarray) – The solution of a renewal equation at each point of the timeline.
F1 (ndarray) – The cumulative distribution function of the first renewal evaluated at each point of the timeline.
F1m (ndarray) – The cumulative distribution function of the first renewal evaluated at each midpoint of the timeline.
y (ndarray) – A given function related to the first renewal evaluated at each point of the timeline.
D (ndarray, optional) – Discount function value at each point of the timeline, by default None.
- Returns
Delayed solution of the renewal equation at each point of the timeline.
- Return type
ndarray
Notes
The solution of the renewal equation is delayed by computing:
\[z_1(t) = y_1(t) + \int_0^t z(t-x) D(x) \mathrm{d}F_1(x)\]where \(z\) is a solution of a renewal equation, \(D\) the exponential discounting factor, \(F_1\) is the cumulative distribution function of the first renwal and \(y_1\) a function related to the first renewal.