Little's law relates inventory, throughput rate, and flow time as which of the following formulas?

Question

BlackTom AI · Accepted Answer

Little's Law states that Inventory equals Throughput rate multiplied by Flow time: Inventory = Throughput rate × Flow time.

Relevant concepts:
- Inventory (also called work-in-process) is the average number of items in a system (units).
- Throughput rate (or arrival/completion rate) is the average rate items leave the system (units per time).
- Flow time (or cycle time) is the average time an item spends in the system (time).
Little’s Law links these steady-state averages.

Step-by-step reasoning and algebra:
1. Let I = average inventory (units), R = throughput rate (units/time), and T = average flow time (time).
2. Consider observing the system for a long period Δt. About R × Δt items will be completed during Δt.
3. Each completed item on average spends T time in the system, so the total item-time consumed during Δt is approximately (R × Δt) × T = R × T × Δt.
4. The average inventory I is the total item-time divided by Δt, so I = (R × T × Δt) / Δt = R × T.
Thus I = R × T, i.e., Inventory = Throughput rate × Flow time.

Numerical example (intermediate steps):
- If throughput R = 5 units/hour and flow time T = 2 hours:
  Inventory I = R × T = 5 units/hour × 2 hours = 10 units.
This shows units cancel correctly: (units/hour)×(hour)=units.

Why the correct choice is correct:
- Dimensionally consistent and derived from counting items over time; it gives correct rearrangements: T = I / R and R = I / T.

Why other options are incorrect:
- "Flow time = Inventory + Throughput rate": Adds incompatible units (time ≠ units + units/time); nonsense dimensionally.
- "Inventory = Throughput rate + Flow time": Again mismatched units and wrong operation (addition instead of multiplication).
- "Inventory × Throughput rate = Flow time": Multiplying inventory by throughput yields units^2/time, not time; incorrect algebraic relation.

Conclusion: The correct formula is Inventory = Throughput rate × Flow time, matching the provided answer.

Little's law relates inventory, throughput rate, and flow time as which of the following formulas?Single choice

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