# The Mysterious τ

There seems to be a lot of confusion as the parameters of the production line models. I'll try to clear it up without creating too much more.

## 1. Continuous Time, Continuous Material Models

There should be no confusion about this case. $$r \delta t$$ is the probability of a repair during an interval of length $$\delta t$$ and is therefore dimensionless. $$p\delta$$t is the probability of a failure during an interval of length $$\delta t$$ and is also dimensionless. Since $$\delta t$$ is in units of time, $$r$$ and $$p$$ are in units of $$\frac{1}{time}$$. The time unit could be seconds or minutes or hours, years, or centuries, whichever is more convenient. $$\frac{1}{r}$$ is the MTTR of a machine and $$\frac{1}{p}$$ is the MTTF of a machine, and they have units of time.

$$\mu$$ is in units of stuff / time. ("Stuff" could be parts, it could be a weight unit, it could be a volume unit, etc.)

Just be sure that the time units are consistent for all parameters and that the stuff units of $$\mu$$ is the same as the stuff units for the buffer sizes $$N$$.

## 2. Discrete Time, Discrete Material (AKA Deterministic Processing Time) Models

$$τ$$ is the common operation time of all the machines in the system being studied. It is a time and is expressed in natural time units.

However, the time unit for this model is the number of operation times. Events only occur at integer multiples of $$τ$$. In this model, $$r$$ and $$p$$ are probabilities and therefore dimensionless. $$r$$ is the probability of a repair during an operation time; $$p$$ is the probability of a failure during an operation time. $$\frac {1}{r}$$ is the MTTR of the machine expressed in units of operation times.

That is, MTTR is the mean number of operation times until a machine is repaired. We treat MTTR as dimensionless. Similarly, $$\frac{1}{p}$$ is the MTTF of the machine, also expressed in units of operation times.

The stuff unit is a part, and is treated as dimensionless. The buffer sizes are therefore also dimensionless.

The confusion arises when I present problems in natural time units (seconds, minutes, etc.). For example, suppose I tell you that the operation time $$τ$$ of a machine is 15 seconds and that its MTTR is 30 minutes. In order to use this information in the deterministic processing time model, we have to express MTTR as a multiple of $$τ$$. The relationship is

${MTTR\,in\,seconds} = [MTTR\,in\,operation\,times] \times [the\,number\,of\,seconds\,in\,an\,operation\,time]$

or

${MTTR\,in\,seconds} = [MTTR\,in\,operation\,times] \times \,{τ \;in \,seconds}$

so in this case,

$30 \times 60 = [MTTR\,in\,operation\,times] \times 15$

or

$[MTTR\,in\,operation\,times] = \frac{30 \times 60}{15} = 120$

so

$r = \frac{1}{MTTR} = 0.0083333$

and similarly for MTTF and $$p$$.