423. Stochastic-service
A relaxation of Graves-Willems guaranteed-service that allows random wait times upstream when stockouts occur. Originated with Lee & Billington (1993) and Ettl et al. (2000).
423.1. Why relax guaranteed service?
Graves-Willems assumes each stage always meets its promised service time, achieved by holding “enough” safety stock for bounded demand. Real supply chains miss promises:
- Demand exceeds the bounded model occasionally
- Suppliers stock out and you wait
- Capacity constrained
Stochastic-service models accept that upstream stockouts cause downstream delays, then compute the resulting waiting-time distribution.
423.2. The trade-off
| Guaranteed-service | Stochastic-service | |
|---|---|---|
| Service times | deterministic / committed | random waiting times |
| Demand model | bounded (-quantile) | full distribution |
| Tractability | easy (DP on trees, MILP for nets) | hard — queueing-style analysis |
| Safety stock | closed form | must solve queueing model at each stage |
423.3. Modeling waiting times
At each upstream stage, when inventory is exhausted, downstream orders wait. The waiting-time distribution depends on:
- Replenishment lead time at the upstream stage
- Demand distribution (downstream orders arriving at upstream)
- Base-stock level at the upstream stage
For each stage , the effective lead time downstream sees is:
where is random. Compute and via queueing approximations (typically Poisson-process / -style models on the order arrival stream).
423.4. METRIC as a special case
For Poisson demand on repairable items, the METRIC model gives closed-form expressions for expected backorders at each echelon — see that page for the details. METRIC is the canonical example of a stochastic-service multi-echelon model.
423.5. When to use
- Service-parts networks — Poisson demand, repair pipelines, low-volume / high-value items
- Long-lead-time supply chains — upstream delays are material and frequent
- When guaranteed-service over-promises and you need realistic waiting-time estimates
423.6. When NOT to use
- Mass-market consumer goods where guaranteed-service is “close enough”
- Network design phase — guaranteed-service is much faster to optimize
- When you don’t have good lead-time / demand distribution data
423.7. See also
- Graves-Willems — guaranteed-service
- METRIC / VARI-METRIC — Poisson demand repairables
- Multi-Echelon overview