**Overview**

This example shows how to model a single-queue single-server system with a single traffic source and an infinite storage capacity. In the notation, the M stands for Markovian; M/M/1 means that the system has a Poisson arrival process, an exponential service time distribution, and one server. Queuing theory provides exact theoretical results for some performance measures of an M/M/1 queuing system and this model makes it easy to compare empirical results with the corresponding theoretical results.

**Structure**

The model includes the components listed below:

**Time Based Entity Generator block:**It models a Poisson arrival process by generating entities (also known as "customers" in queuing theory).

**Exponential Interarrival Time Distribution subsystem:**It creates a signal representing the interarrival times for the generated entities. The interarrival time of a Poisson arrival process is an exponential random variable.

**FIFO Queue block:**It stores entities that have yet to be served.

**Single Server block:**It models a server whose service time has an exponential distribution.

**Results and Displays**

The model includes these visual ways to understand its performance:

Display blocks that show the waiting time in the queue and the server utilization

A scope showing the number of entities (customers) in the queue at any given time

A scope showing the theoretical and empirical values of the waiting time in the queue, on a single set of axes. You can use this plot to see how the empirical values evolve during the simulation and compare them with the theoretical value.

**Theoretical Results**

Queuing theory provides the following theoretical results for an M/M/1 queue with an arrival rate of and a service rate of :

Mean waiting time in the queue =

The first term is the mean total waiting time in the combined queue-server system and the second term is the mean service time.

Utilization of the server =

**Experimenting with the Model**

Move the Arrival Rate Gain slider during the simulation and observe the change in the queue content, shown in the Q Content Scope.

**Related Examples**

Generating Entities as a Markov-Modulated Poisson ProcessGenerating Entities as a Markov-Modulated Poisson Process

G/G/1 Queuing System and Little's LawG/G/1 Queuing System and Little's Law

Converting Between Time-Based and Event-Based SignalsConverting Between Time-Based and Event-Based Signals

**References**

[1] Kleinrock, Leonard, Queueing Systems, Volume I: Theory, New York, Wiley, 1975.

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