Chapter 6.1¶

Recurrence Time Distribution of Discrete-Time Renewal Processes¶

(c) Tobias Hossfeld (Aug 2021)

This script and the figures are part of the following book. The book is to be cited whenever the script is used (copyright CC BY-SA 4.0):
Tran-Gia, P. & Hossfeld, T. (2021). Performance Modeling and Analysis of Communication Networks - A Lecture Note. Würzburg University Press. https://doi.org/10.25972/WUP-978-3-95826-153-2

The forward recurrence time $ R $ of discrete-time renewal processes is defined as the time interval from a random observation time $ t^* $ to the next arrival. The observation instant lies also at the equidistant time instants on the discretized time axis. Here we distinguish between two cases: the observation instant is considered to be immediately before or after a discretized time instant.

Observation prior to discretized time instants¶

A discrete-time renewal process is viewed from a random time $t^\ast$ by an independent outside observer. It is assumed that the observation time $ t^\ast$ is immediately before a time instant of the discretized time axis. If an arrival occurs at the same time as the observation, the forward recurrence time is zero. The distribution of the discrete-time forward recurrence time is:

$ \displaystyle r(k) = \frac{1}{E[A]} \Big( 1 - \sum_{i=0}^{k} a(i)\Big) \; , \quad k =0,1,\dots \;, $

for the interarrival time $A$ being a discrete random variable.

In [9]:

ks = np.arange(A.xmax+1) # value range of the recurrence time
rk = np.zeros(len(ks)) # probabilities

for k in ks:
    rk[k] = (1-A.cdf(k))/A.mean()

R = DiscreteDistribution(ks, rk)    

A.plotPMF(label='interarrival time')
R.plotPMF(label='recurrence time')
plt.xlabel('i')
plt.ylabel('PMF')
plt.legend();
plt.grid(which='major');

Observation just after discretized time instants¶

Now the observation time $ t^\ast$ is located immediately after a time instant of the discretized time axis. The distribution of the forward recurrence time $R$ is

$\displaystyle r(k) = \frac{1}{E[A]} (1 - \sum_{i=0}^{k-1} a(i)) \;, \quad k =1,2,\dots \;,$ for the interarrival time $A$.

In [10]:

ks = np.arange(1,A.xmax+1) # value range of the recurrence time
rk = np.zeros(len(ks)+1) # probabilities

for k in ks:
    rk[k] = (1-A.cdf(k-1))/A.mean()

R_after = DiscreteDistribution(ks, rk[1:])    

#A.plotPMF(label='interarrival time')
R.plotPMF(label='before')
R_after.plotPMF(label='after')
plt.xlabel('i')
plt.ylabel('PMF')
plt.legend();
plt.grid(which='major');