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# Utilization

[yoot-l-ahyz]
In queueing theory, utilisation is the proportion of the system's resources which is used by the traffic which arrives at it. It should be strictly less than one for the system to function well. It is usually represented by the symbol $rho$. If $rho geq 1$ then the queue will continue to grow as time goes on. In the simplest case of an M/M/1 queue (Poisson arrivals and a single Poisson server) then it is given by the mean arrival rate over the mean service rate, that is,
$rho=frac\left\{lambda\right\}\left\{mu\right\}$
where $lambda$ is the mean arrival rate and $mu$ is the mean service rate.

More generally:

$rho=frac\left\{lambda\right\}\left\{mutimes M\right\}$
where $lambda$ is the mean arrival rate, $mu$ is the mean service rate, and M is the number of servers.

In most cases a lower utilization will mean less queuing for customers but will mean that the system is idle more (which could be considered inefficient).

In production, an increase of utilization will require more preventative maintenance and more wear and tear on the machine. Sometimes it is more economical to buy new equipment, but increasing the utilization of the current equipment is usually the most practical. In the case of cellular manufacturing, each product is produced in one area or cell, sometimes requiring more machines and lower utilizations.

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