Chapter 4.2

M/M/n Delay System

(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 M/M/n delay system represents a fundamental queueing system. The steady state probabilities $x(i)$ for $i=0,...,n$ can be derived based on the analysis of Markov systems. The system consists of $n$ parallel servers and an infinite waiting room. Customers arrive according to a Poisson process with rate $\lambda$. The service time $B$ is exponentially distributed with mean $E[B]=1/\mu$.

State Probability

The offered load is $a=\lambda E[B]=\frac{\lambda}{\mu}$. The utilization is identical to the normalized offered traffic $\rho=a/n$. The state probabilities are:

$ x(i) = \begin{cases} x(0) \displaystyle\frac{a^i}{i!} \; , & i=0,1,...,n\\ \\ x(0) \displaystyle\frac{a^n}{n!} \left({\frac{a}{n}} \right)^{i-n} = x(n) {\rho}^{i-n} \; , & i>n \end{cases} \\ \Big(x(0)\Big)^{-1} = \displaystyle\sum\limits_{k=0}^{n-1}\frac{a^k}{k!} + \frac{a^n}{n!} \frac{1}{1-\rho}\;. $

Waiting Probability

An arriving customer has to enter the queue when all servers are occupied at the time of arrival. The corresponding waiting probability $p_W$ is thus

$ p_W = \sum\limits_{i=n}^{\infty} x(i) \, = \, x(n)\sum\limits_{i=0}^{\infty} {\rho}^i \, = \, x(n) \; \frac{1}{1-\rho} $

Mean Waiting Time

The mean waiting time of all customers $E[W]$ and of waiting customers $E[W_1]$ can be derived using Little's law.

All Customers: $E[W]$

The mean queue length is $ \Omega = E[X_W] =p_W \cdot \frac{\rho}{1-\rho}\,. $

The server utilization is $ \rho = E[X_B] = \lambda/\mu = \lambda E[B]$

The mean number of customers in the system is $E[X]=E[X_W]+E[X_B]=\Omega+\rho$.

According to Little: $\Omega+\rho = \lambda \cdot (E[W]+E[B]) = \lambda E[W] + \rho$

Hence, the mean waiting time of all customers follows: $E[W] = \frac{\Omega}{\lambda}$

Waiting Customers: $E[W_1]$

The waiting probability is $p_W$. The mean waiting time of waiting customers follows from Little: $\Omega = p_W \cdot \lambda \cdot E[W_1]$

$E[W_1] = \displaystyle\frac{\Omega}{\lambda \cdot p_W} = \frac{1}{\lambda} \cdot \frac{\rho}{1-\rho} = \frac{E[W]}{p_W} $