Chapter 5.4

GI/M/1 Delay System with Geometric Approach

(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.

The geometric approach is used to analyze the condition. The state probability is

$x(j)=(1-\sigma)\sigma^j, j>=0, \rho<1$

The parameter $ \sigma $ can be determined numerically by solving the following equation. The random variable $ \Gamma $ is the number of requests that can be served during an interarrival time $A$. While $ A $ follows a general distribution, the service time $B$ is described by an exponential distribution with rate $ \mu $, i.e. $ B \sim \mathrm{EXP}(\mu) $.

$ z = \Gamma_{GF}(z)$

In the following we consider a uniform distribution in the interval $[0,2/\lambda]$ for the interarrival time $A$ with the mean value $E[A]=1/\lambda$. It is $A \sim U(0,2/\lambda)$. The Laplace transform of the continuous uniform distribution is

$ \Phi_A(s) = \frac{e^{-sa}-e^{-sb}}{s(b-a)} = \frac{1-e^{-sb}}{s\cdot b} $

with $a=0$ and $b = 2/\lambda$.

The generating function is obtained with the help of the Laplace transform of the uniform distribution.

$ \Gamma_{GF}(z) = \phi_A(\mu(1-s))$

Now we need to solve

$ z = \Gamma_{EF}(z)$.

To do this, we calculate the solution of $\Gamma_{EF}(z)-z=0$. There are numerical methods such as fsolve.

Waiting Time Distribution

The waiting time distribution function $W(t)$ for customers in a GI/M/1 delay system with the waiting probability $\sigma$ is:

$W(t) = 1 \: - \: \sigma \; e^{\displaystyle -(1\,-\, \sigma) \, \mu \, t}\,. $

The waiting probability is

$ p_W = \sigma \quad$ (for GI/M/1) and therefore depends on the type of distribution of the interarrival time $A$.

In contrast, for an M/GI/1 system, the waiting probability follows from the utilization $\rho$ of the system and is independent of the type of distribution of the service time $B$:

$p_W = \rho \quad$ (for M/GI/1).