Stochastic-service

427. 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).

427.1. 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.

427.2. The trade-off

	Guaranteed-service	Stochastic-service
Service times	deterministic / committed	random waiting times
Demand model	bounded ( $𝑧$ -quantile)	full distribution
Tractability	easy (DP on trees, MILP for nets)	hard — queueing-style analysis
Safety stock	closed form $𝑧 𝜎 \sqrt{𝜏}$	must solve queueing model at each stage

427.3. 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 $𝑖$ , the effective lead time downstream sees is:

𝐿_{𝑖}^{eff} = 𝐿_{𝑖}^{nominal} + 𝑊_{𝑖}^{upstream stockout wait}

where $𝑊_{𝑖}$ is random. Compute $𝐸 [𝑊_{𝑖}]$ and $Var (𝑊_{𝑖})$ via queueing approximations (typically Poisson-process / $\frac{\frac{𝑀}{𝑀}}{1}$ -style models on the order arrival stream).

427.4. 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.

427.5. 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

427.6. 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

427.7. See also

Graves-Willems — guaranteed-service
METRIC / VARI-METRIC — Poisson demand repairables
Multi-Echelon overview