Use Little’s law and a capacity screen to relate arrival rate, service time, parallel servers, WIP, and utilization.
Parameters
Arrival rate lambda
1/h
Input Arrival rate lambda.
Service time
min
Input Service time.
Servers c
count
Input Servers c.
Target lead time
min
Input Target lead time.
While paused, move the sliders to update the result instantly.
Live metrics
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Arrival rate λ [items/h]
—
Avg time in system W [min]
—
Number in system L (measured)
—
λ·W (theory)
Real-time queue visualization
Poisson arrivals line up in the queue, are processed by a free server (service time), then depart. The time-average number in system L converges to λ·W, confirming Little's Law L = λW.
Model and equations
$$L=\lambda W,\quad \rho=\frac{\lambda}{c\mu}$$
This simplified model captures the main relationship only. Boundary conditions, losses, nonlinear effects, and code-specific corrections still need separate checks.
How to read it
Use the main plot to read the controlling trend, including break points that a single result card can hide.
Use the sensitivity view to find input combinations where margin collapses quickly.
For early design, focus on which input controls margin before trusting the absolute value.
Learn Capacity Planning Little Law by dialogue
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When reading Capacity Planning Little Law, where should I look first? Moving Arrival rate lambda changes both the plots and the result cards.
🎓
Start with Average WIP L, but do not treat the number as the whole answer. Use Flow, WIP, and lead time to confirm the assumed state, then read Capacity and load breakdown for the distribution or trend. Use the main plot to read the controlling trend, including break points that a single result card can hide.
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I can see why Arrival rate lambda changes Average WIP L. How should I judge the influence of Service time?
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Move Service time in small steps and watch Capacity. That reveals which term is controlling the result. This simplified model captures the main relationship only. Boundary conditions, losses, nonlinear effects, and code-specific corrections still need separate checks. A single operating point is not enough; sweep the realistic scatter range.
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What is Arrival-service margin map for? It feels like the ordinary curve already tells the story.
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Arrival-service margin map is for finding boundaries where the condition becomes risky or margin collapses quickly. Use the sensitivity view to find input combinations where margin collapses quickly. In First-pass comparison of design options before review, the important question is often what happens after a small change, not only the nominal value.
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So if Average WIP L is within the target, can I accept the condition?
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Treat this as a first-pass review. It helps with Narrowing controlling factors and worst-side conditions before detailed analysis and Teaching or explaining the equation, numbers, and visualization under the same inputs, but final decisions still need standards, measured data, detailed analysis, and vendor limits. For early design, focus on which input controls margin before trusting the absolute value.
Practical use
First-pass comparison of design options before review.
Narrowing controlling factors and worst-side conditions before detailed analysis.
Teaching or explaining the equation, numbers, and visualization under the same inputs.
FAQ
Start with Average WIP L and Capacity. Then use Flow, WIP, and lead time to confirm the assumed state and Capacity and load breakdown to read distribution or bias. Use the main plot to read the controlling trend, including break points that a single result card can hide
Move Arrival rate lambda alone, then move Service time by a comparable amount and compare the change in Average WIP L. Arrival-service margin map shows combinations where margin or performance changes quickly.
Use it for First-pass comparison of design options before review. Instead of trusting a single point, widen the input range and check whether Average WIP L keeps enough margin before moving to detailed analysis.
This simplified model captures the main relationship only. Boundary conditions, losses, nonlinear effects, and code-specific corrections still need separate checks. Final decisions still require standards, measured data, detailed analysis, and vendor limits.
How to Use
Enter arrival rate (lambda) in jobs/hour—for example, 15 parts/hour arriving at a CNC machining center
Set average service time in minutes per job—typical values: 8 minutes for drilling operations, 12 minutes for milling
Specify number of parallel servers (machines)—input 2 for a dual-spindle lathe setup or 4 for a job shop with four workstations
The simulator calculates average WIP (Work-In-Process), system capacity, utilization percentage, and queue growth index using Little's Law (L = λ × W)
Adjust parameters interactively to observe how bottlenecks form when utilization exceeds 85%
Worked Example
A semiconductor wafer fab receives 20 wafers/hour (λ=20) at an etching station with average service time of 18 minutes (μ=3.33 wafers/hour). With 3 parallel etchers, system capacity = 3 × 3.33 = 10 wafers/hour per server. Utilization = 20/(3×3.33) = 200%, indicating severe overload. Reducing arrival rate to 8 wafers/hour yields utilization = 80%, average WIP = 8 × 0.3 hours = 2.4 wafers in system, with queue growth index stable at 0.2. Adding a fourth etcher reduces utilization to 60% and WIP to 1.6 wafers.
Practical Notes
Target utilization 75-85% for manufacturing systems—above 90% causes exponential queue buildup; below 60% wastes capital equipment
Service time variability (standard deviation) amplifies queueing delays more than arrival rate changes; batch similar parts to reduce variation
For automotive assembly lines with 45-second takt time and 8 workers in series, calculate effective λ and μ per station to identify the bottleneck constraint
Use queue growth index >0.5 as a trigger to increase servers, reduce arrival rate, or implement priority dispatch rules