# The Azimuth Project Queueing theory (Rev #1)

## Idea

The simplest sort of queuing model describes (for example) customers entering a line, being dealt with by one or more ‘servers’, and then leaving. Such queuing models can be represented using Kendall’s notation:

$A/B/S/K/N/D$

where:

• $A$ is the interarrival time distribution: the probability distribution of times for the next customer to enter the queue.
• $B$ is the service time distribution: the probability distribution of times for a customer being service to have their service completed.
• $S$ is the number of servers
• $K$ is the system capacity, meaning the maximum number of customers who can be ‘waiting in line’.
• $N$ is the calling population, meaning the maximum number of customers who enter.
• D is the service discipline assumed: for example, ‘first in first out’ ($FIFO$), ‘last in first out’ ($LIFO$), etcetera.

Often the last members are omitted, so the notation becomes $A/B/S$ and it is assumed that $K = \infty$, $N = \infty$ and $D = FIFO$.

Some of the most popular probability distributions for $A$ and $B$ are:

• $M$: Markovian, meaning a Poisson process.

• $E^k$: an Erlang $k$-process, which is the convolution of $k$ identical Poisson processes.

• $D$: a degenerate distribution, meaning a delta function at some fixed waiting time.

category: methods