2 The paradigm of the repairman

2.1 Description of the M/GI repairman

The "M/GI repairman" system does not involve an M/GI queue, this expression being used to shorten: "all of the "next times to failure" are i.i.d. M, and all of the "next times to repair" are i.i.d. G". More precisely, the aim is to model a workshop of identical machines, maintained by a single repairman. The next times to failure (nttf) are assumed to be i.i.d. exponential variables, with expectation (mttf) $1/\lambda$.

When the $n$-th failure takes place (at $T_{a}\left( n \right)$), the broken machine enters in the repair cycle: waiting in a fifo file until the previous failures are cleared (at $T_{s}\left( n \right)$), then being repaired and coming back to activity (at $T_{q}\left( n \right)$). The next times to repair (nttr) are assumed to be i.i.d. variables, with pdf $b\left( \tau \right)$ and expectation (mttr) $1/\mu$.

It is well known that the ratio $\rho \doteq \lambda /\rho$ is the paramount parameter of the system: when $\rho \ll 1$, failures are 'not too frequent' and a complete failure of the workshop is a rare event. On the contrary, i.e. when $\rho \approx 1$ failures are 'frequent' and the full availability is now the rare event.

2.2 The M/M repairman and the availability chain

The M/M repairman's problem is usually solved [2] by breaking the continuous time in intervals where the number $Y\in \left[ 0, M\right]$ of broken machines remains constant. In other words, the states of that embedded discrete chain are beginning just after each failure or repair, i.e. at each of the $T_{a}\left( n \right)$, $T_{s}\left( n \right)$ and $T_{q}\left( n \right)$ (some of these events being the same). Let us call this chain the availability chain and $\mathrm{M}_{a}$ its matrix (to avoid confusion with the later defined repairing chain, with matrix $\mathrm{M}_{r}$)

In the special case of the M/M repairman, the availability chain is a Markov chain, whose solution is easy to obtain since the probability that state $y>0$ ends by a repair instead of a new failure is: $\mu /\left( \mu +y \lambda \right)$. FIG. 1 exemplifies this availability chain with $M=4, \lambda =1/2, \mu =1$.

FIG. 1: The M/M availability chain.

The stationary vector $\mathrm{V}_{a}$ of matrix $\mathrm{M}_{a}$ can be found as any column of the Cesaro's limit of $\left( \mathrm{M}_{a}\right) ^{n}$ for increasing $n$ or (since the process is a birth and death one) by a decomposition of $\mathrm{M}_{a}^{2}$ in two matrices acting over, respectively, the even states and the odd states. An element of $\mathrm{V}_{a}$ gives the individual probability $iPr\left( y \right)$ of state $Y=y$, i.e., the probability conditioned by an uniform sorting over the discrete events. To obtain the temporal probabilities $tPr\left( y \right)$, i.e. the probabilities conditioned by an uniform sorting over the instants of the continuous time, these $iPr\left( y \right)$ have to be weighted by the average duration of each discrete event.

These durations are known to be $1/\left( M \lambda \right)$ for $y=0$ and $1/\left( \mu +\lambda \left( M-y\right) \right)$ for the other states, leading to:

$$\begin{array}{cccccc} y & 0 & 1 & 2 & 3 & 4\\ iPr\left( y ... ...\left( y \right) & 2/21 & 4/21 & 6/21 & 4/21 & 3/21 \end{array}$$

Other quantities of interest are the stationary probabilities $iPr\left( a\mapsto b \right)$ of the edges of the chain, obtained by uniform sorting over the discrete chain. We obviously have the conditioning: $iPr\left( a\mapsto b \right) =iPr\left( a \right) \times \mathrm{Pr}\left( a \vert b \right)$ and, for example, the lower $2/19$ in FIG. 2 is $iPr\left( 1 \right) =5/19$ (from $\mathrm{V}_{a}$) times $\mathrm{Pr}\left( 0 \vert 1 \right) =2/5$ (from FIG. 1).

FIG. 2: Stationary probabilities of the M/M availability edges.

2.3 The impossible generalization

These results relative to the M/M repairman seem to be clear and sound but they can't contain all the truth about the problem since they can't be generalized in any way to the M/G repairman. In such a case, the availability chain is no more a Markov chain and another solution is to be rebuilt from scratch.