Operations Research & Service Parts Logistics
Palm’s Theorem (S-1, S) Spare-Parts Base-Stock Formula
Size repairable spare-parts base stock S using Palm’s Theorem Poisson pipeline mean mu_p = lambda * T_rep, exact Fill Rate P(X <= S - 1), and Expected Backorders EBO(S).
Use Palm’s Theorem + finite on-hand identity to compute EXACT Expected Backorders!
High-cost repairable modules (jet-engine line-replaceable units, MRI coils, GPU chassis) use one-for-one (S - 1, S) base-stock replenishment: whenever a module fails (Poisson rate lambda) and enters an ample-capacity M/G/infinity repair depot with mean turnaround T_rep, a spare is pulled immediately from stock S. By Palm’s Theorem (1938) — cornerstone of Sherbrooke’s USAF METRIC model — steady-state units in repair X follow an EXACT Poisson(mu_p = lambda * T_rep) distribution regardless of the repair-time distribution shape! Moreover, conservation of inventory E[On-Hand] - EBO(S) = S - mu_p lets you compute exact Expected Backorders from a finite sum over 0..S-1: EBO(S) = (mu_p - S) + SUM_{x=0}^{S-1} (S - x) * P(X = x)!
Verify on failure rate lambda = 0.50 / wk (G2), depot turnaround T_rep = 4.0 wks (G3 -> pipeline mean mu_p = 2.00 units), and spare stock S = 4 (G4) across states x = 0..5 (A2:C7): immediate Fill Rate P(X <= S - 1) = P(X <= 3) is (19/3) * EXP(-2) = 0.8571 (85.71%), expected on-hand inventory is (46/3) * EXP(-2) = 2.0751 spares, and exact Expected Backorders is (2 - 4) + 2.0751 = 0.0751 grounded units! Test S = 0 to confirm Fill Rate = 0 and EBO(0) = mu_p = 2.0000.
| Base-Stock Metric (lambda=0.50, T_rep=4.0 -> mu_p=2.0) | Formula | Verified Output (S = 4) |
|---|---|---|
| Pipeline Mean in Repair (mu_p) | lambda * T_rep | 2.0000 units |
| Immediate Fill Rate P(X <= S - 1) | SUM_{x=0}^{S-1} (mu_p^x * EXP(-mu_p) / FACT(x)) | 0.8571 (85.71%) |
| Expected On-Hand Spares E[OH] | SUM_{x=0}^{S-1} (S - x) * P(X = x) | 2.0751 units |
| Exact Expected Backorders EBO(S) | (mu_p - S) + E[OH] | 0.0751 units |
Palm’s Theorem Immediate Fill Rate P(X <= S - 1) Formula
Sum exact Poisson state probabilities p(x) = POWER(mu_p, x) * EXP(-mu_p) / FACT(x) for x <= S - 1 (returning 0 when S = 0).
=IF(OR(G2 <= 0, G3 <= 0, G4 < 0, G4 > 6), "Require S in 0..6 and Positive Rates", IF(G4 = 0, 0, ROUND(SUMIF(A2:A7, "<=" & (G4 - 1), B2:B7), 4)))Exact Finite-Sum Expected Backorders EBO(S) = (mu_p - S) + E[On-Hand] Formula
Because MAX(0, S - x) = 0 for all x >= S, summing (S - x) * p(x) over x = 0..5 evaluates exact infinite-tail EBO(S) for any integer S in 0..6 with zero truncation error!
=IF(OR(G2 <= 0, G3 <= 0, G4 < 0, G4 > 6), "Require S in 0..6 and Positive Rates", ROUND((G2 * G3 - G4) + SUM(C2:C7), 4))Try the example
Use the original values or change them to test your understanding.