# Engset 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. https://doi.org/10.25972/WUP-978-3-95826-153-2

The Engset model considers a finite population. In contrast to models with Poisson arrival processes, where arriving traffic flows are thought of to be generated by an infinite number of customers, we consider here -- more realistically -- a finite number $m$ of customers that generate the arrival traffic ($m > n$). Note for $m<n$, a customer is always served and no blocking occurs.

After a service period or after an unsuccessful attempt where the customer is blocked, the customer returns to the state idle. The service stage consists of $n$ servers with (negative) exponentially distributed service time.

## State Probability¶

The offered traffic of an idle customer is $a^*=\frac{\alpha}{\mu}$. The state probabilities are:

$x(i) = P(X=i)= \frac{\displaystyle{\binom{m}{i}} \, {a^{*^i}}} {\displaystyle\sum_{k=0}^n \binom{m}{k} \, {a^{*^k}}}$ for $i = 0,1,\ldots, n$ in statistical equilibrium.

## Arrival Theorem¶

In general, the system state $i$ as seen by an incoming customer has a different distribution than the state seen by a random observer. In queueing theory, the arrival theorem (also referred to as the random observer property) states that upon arrival at a server, a customer observes the system as if in steady state at an arbitrary instant for the system without that customer.

For closed networks, i.e. with a finite number of $m$ customers, the state probabilities $x^m_A(i)$ seen by a customer entering a state $i$ are the same as the arbitrary-time probabilities $x^{m-1}(i)$ in a system with $m - 1$ customers.

## State Probability for Arriving Customer¶

The arrival theorem is now applied to the state probabilities.

$x_A(i) = \frac{\displaystyle{\binom{m-1}{i}} \, {a^*}^i} {\displaystyle\sum_{k=0}^n {\binom{m-1}{k}} \, {a^*}^k} \, .$

## Engset Formula¶

A blocking event occurs in case the test customer finds the system state [$X_A = n$]. We finally obtain the following relationship called the Engset formula:

$p_B = x_A(n)$