Mathematical expression of a component availability with shared warehouse

[Saladin]

Hello,

I would like to estimate mathematically (i.e. without petri nets or other simulation) the availability of a component in the following situation:
- System with N components of the same type, same MTTF and same MTTR (constants)
- Warehouse with 1 spare component
- When a component fails, a spare is directly shipped from the warehouse. It takes LDT (Logistics delay time) to arrive next to the failed component to be replaced
- When the component is replaced, the failed one is shipped to a lab (LDT), the lab needs a time T to repair it and when repaired, the component goes back in the warehouse (takes another LDT).
- The component can then be shipped again.

Is there any mathematical way to express the availability of one component of the system?

I have tried by expressing the probability of an empty warehouse multiplied by the time needed by a repaired component to come back in the warehouse and augment it by the MTTR and LDT to calculate a kind of "real MTTR as seen by the component", but the results obtained don't seem to match with results obtained by simulations.

Many thanks in advance for any help,
Saladin.

[Adam]

I don't know of any way you can come up with a closed-form solution for this problem (if it exists). Maybe if you use Markov chains and solve the partial differential equations.