Machine Repair Model

Example

Consider $M$ machines and one serviceman. Each machine runs for an $\text{Exp}(\lambda )$ time before breaking down, and repair takes $\text{Exp}(\mu )$ time.

Let $X(t)$ be the number of machines not in use (down). This is a birth and death process with:

${\lambda }_n=\{\begin{array}{ll}(M-n)\lambda ,& n\le M\\ 0,& n>M\end{array}$ ${\mu }_n=\mu ,n\ge 1$

A machine failing is a “birth” (increases down count), and a repaired machine is a “death” (decreases down count).

Limiting Probabilities

From the D balance equation solution:

$P_0=\frac{1}{1+{\sum }_{n=1}^M(\lambda /\mu {)}^n\frac{M!}{(M-n)!}}$ $P_n=\frac{(\lambda /\mu {)}^n\frac{M!}{(M-n)!}}{1+{\sum }_{k=1}^M(\lambda /\mu {)}^k\frac{M!}{(M-k)!}},n=0,1,\dots ,M$

Key Quantities

Average number of machines not in use:

${\sum }_{n=0}^{M}n{P}_n=\frac{{\sum }_{n=0}^{M}n(\lambda /\mu {)}^n\frac{M!}{(M-n)!}}{1+{\sum }_{n=1}^M(\lambda /\mu {)}^n\frac{M!}{(M-n)!}}$

Proportion of time a machine is working:

$P\{\text{machine working}\}={\sum }_{n=0}^M\frac{M-n}{M}{P}_n=1-\frac{{\sum }_{n=0}^{M}n{P}_n}{M}$

Interpretation

The machine repair model is the prototypical finite-source queue. Instead of customers arriving from an infinite population, jobs (machine failures) come from a finite set of $M$ sources. The arrival rate decreases as more machines are already down.

Special Case: $M=N=R$

When every machine’s operator is also the repairman ($M=N=R$, equivalent to $M$ independent two-state Markov chains), the stationary distribution is binomial:

$P_n=\left(\genfrac{}{}{0ex}{}{M}{n}\right){\left(\frac{\lambda }{\lambda +\mu }\right)}^n{\left(\frac{\mu }{\lambda +\mu }\right)}^{M-n}$

Related

• Birth and Death Process
• Solving BD Process Balance Equations
• Machine Repair Model