Chapter 4.4

Retrial Model

(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.

In communication networks, especially for systems operating in loss mode, rejected customer requests sooner or later lead to new attempts - this happens rather sooner. Especially when looking at systems with a finite number of sources, this effect has a significant impact on the overall range of traffic and on the performance of the system. We are looking at an extension of the loss system with finite number of sources (Engset model), to a more realistic model considering this retrial or repeated-attempt modeling approach.

A detailed description of the customer retrial model is provided in Chapter 4.4. This script provides an implementation of the analysis in which a two-dimensional Markov state process is used and solved with a recursive algorithm.

The script maps the two-dimensional Markov states $[X=i,Z=j]$ to a single dimension using the function c2i(i,j) (and vec2matrix as inverse mapping). This allows to use the previously implemented methods for analyzing a continuous-time Markov chain.

Transition Rate Matrix

The rate transition matrix is defined now rateMatrix(). Thereby, we consider the transitions $[X=i_0,Z=j_0] \to [X=i_1,Z=j_1]$. The two-dimensional index is mapped to an one-dimensional index with the mapping helper functions above.

The function printMatrix provides a visualization of the matrix.

Example: Parameter Setting

A particular system and its parameter is considered. The corresponding rate matrix is plotted.

Solve the Steady State Equation System

The steady state equation system needs to be solved (as in the one-dimensional case):

$ X \cdot Q = 0$ and $X \cdot e = 1$

Alternatively, the recursion algorithm in the book in Chapter 4.4 can be implemented.

Calculation of Traffic Flows

To calculate system characteristics, we determine the rates of traffic processes in the model:

The indices of the traffic flows are defined as follows.

According to this indexing scheme, $\lambda_{FS}$ is e.g. the rate (in calls per unit of time) of fresh calls that are accepted and immediately served.

A characteristic performance measure for the retrial effect is the average number of calls -- including the initiating fresh calls -- that a call has to make

$ \displaystyle \eta = \frac{\lambda_{R}+\lambda_{F}}{\lambda_F} = 1 + \frac{\lambda_R}{\lambda_F} $

The blocking probability experienced by a user initiating a fresh call is given as

$ \displaystyle p_{B_F} = \frac{\lambda_{FU}+\lambda_{FB}}{\lambda_F} $

Parameter Study

A parameter sensitivity study is conducted by varying the retrial probability $\Theta$ and the offered load of idle customers.

Note: It will take some time until the computation is done.