AMR Fleet Sizing

Every AMR proposal shows peak throughput at some utilization percentage — 80%, 85% — with fleet size derived from dividing demand by that rate. What those slides never show is what happens to throughput when utilization crosses 70%.

Little’s Law Fleet Sizing Formula

AMR fleet sizing lives in queueing theory (open-shop queueing model), not simple division. Little’s Law — derived by John Little, 1961 — gives the fundamental relationship:

$$N = \lambda \times W$$

Applied to AMR fleet sizing with utilization:

$$\text{Fleet Size} = \frac{\lambda \times T_{\text{cycle}}}{U}$$

• λ = arrival rate of transport missions (missions/hr)
• T_cycle = average time to complete one full mission cycle (hr)
• U = target vehicle utilization (decimal)

Cycle Time Decomposition

T_cycle is not a single number. It is the sum of every component from task assignment to ready-for-next:

$$T_{\text{cycle}} = t_{\text{empty travel}} + t_{\text{load}} + t_{\text{loaded travel}} + t_{\text{unload}} + t_{\text{return/reposition}} + t_{\text{queue wait}}$$

Queue wait is the component that kills most analyses — it is not zero, and it grows nonlinearly with utilization (see below).

For systems with scheduled charging, adjust effective cycle time:

$$T_{\text{cycle, effective}} = T_{\text{cycle}} + \frac{T_{\text{charge}} \times f_{\text{charge}}}{1 - f_{\text{charge}}}$$

Worked Example: 480 Missions/Hr GTP System

Six goods-to-person ports, each requiring one tote every 45 seconds. AS/RS buffer zone ~180 ft from port center.

• Task rate: λ = 6 × (3600/45) = 480 missions/hr
• Loaded travel (port → buffer): 3 min
• Empty travel (return): 1.5 min
• Load + unload: 0.5 + 0.5 = 1.0 min
• Queue wait (well-designed system): 0.5 min
• Total T_cycle = 6.0 min = 0.10 hr

At 80% utilization (nominal design point before safety margin):

$$\text{Fleet (operational)} = \frac{480 \times 0.10}{0.80} = 60 \text{ AMRs}$$

Add 15% for scheduled maintenance, in-shop repairs, and spares:

$$N_{\text{specified}} = 60 \times 1.15 = \mathbf{69 \text{ AMRs}}$$

The 9-robot maintenance float is not pad. Without it, the first two robots entering scheduled maintenance push below the throughput commitment.

Why Throughput Collapses Past 70% Utilization

This is the single most misrepresented aspect of AMR fleet sizing.

The foundation is the M/G/1 queueing model. In any system with random service times, queue waiting time as utilization ρ approaches 1:

$$W_{\text{queue}} = \frac{\rho}{1 - \rho} \times \frac{C_s^2 + 1}{2} \times \bar{S}$$

The utilization multiplier alone:

Utilization (ρ)	Queue Wait Multiplier
50%	1.0×
60%	1.5×
70%	2.33×
80%	4.0×
85%	5.67×
90%	9.0×

This is before path congestion — when multiple robots compete for the same intersection, collision-avoidance delays compound on top of queueing.

Simulation studies consistently show effective throughput peaking at 65–75% utilization and declining past that point as rerouting and queue backup dominate.

Design rule: ≤70% utilization at peak hour. If sizing math produces 85% utilization at peak, you have an undersized fleet. Adding robots to ≤70% is not overbuilding — it is what delivers the promised throughput.

Charging Strategy

Opportunity Charging

FMS routes vehicles to chargers during idle periods. Works cleanly when:

• Fleet utilization stays below 70% (enough natural idle time)
• Battery discharge allows meaningful 5–15 min top-ups
• Charger count ≥ 20–30% of total fleet

Failure mode: if utilization spikes and the system stops producing idle windows, vehicles run out of charge without opportunity to top up. Degrades silently.

Scheduled Rotation Charging

Define the fraction charging at any moment:

$$f_c = \frac{T_{\text{charge}}}{T_{\text{battery}} + T_{\text{charge}}}$$

Example: Battery life = 4 hr. Charge time = 45 min. f_c = 0.75 / (4 + 0.75) = 15.8%

Fleet requirement to maintain 60 operational robots: N = 60 / (1 - 0.158) = 72 AMRs

Verify consistency: 72 × 84.2% available = 60.6 operational ✓

Battery Swap

Swappable battery packs eliminate charging from the operational cycle (60–90 second swap vs. 30–60 min charge).

• Need 1 charged spare battery per 8 hr of operation per vehicle
• AMR capital cost 40–60% higher for swap-capable platforms
• Justified at sustained 80%+ utilization on three-shift operations where vehicles cannot get off the floor long enough to charge
• Almost never justifiable at ≤70% utilization

Path Design and Fleet Size Are Coupled

Path design is as important as fleet size. A bi-directional main aisle with overtake bays every 15 meters increases effective throughput 25–40% versus a single-direction aisle at the same fleet size.

Narrow intersections — single-vehicle-width choke points — are hard throughput constraints that no fleet size increase can overcome. They serialize robot passage by geometry.

Critical sequence: define aisle width and intersection count in the floor layout before finalizing the fleet size model. Solving fleet size first, then leaving path design to the architect, produces a correctly-sized fleet delivering incorrect throughput.

Robotic Picking: Net Pick Rate

For goods-to-robot or goods-to-person with robotic arm:

$$\text{Net picks/hr} = \frac{3600 \times P_{\text{success}}}{T_{\text{cycle}} + (1 - P_{\text{success}}) \times T_{\text{exception}}}$$

T_exception is the full elapsed time per failed pick: robot stop + alert + human response + manual intervention + reset + restart. In a real deployment with a handler working multiple cells, T_exception = 30–45 seconds — not 5 seconds.

99% vs. 95% Success Rate Impact

Success Rate	Net Picks/hr	Exceptions/hr	Handler Coverage
99%	~829	8.3	1 handler per 10–15 cells
95%	~622	31	1 handler per 2–3 cells

At 95% success, the robot doesn’t eliminate labor — it moves it from picking to exception handling. The net labor saving is negligible. The business case evaporates.

Current benchmarks: commercial robotic piece-picking achieves 90–98% on well-defined rigid items. Soft goods, polybag items, and items with transparent packaging challenge current vision systems significantly. Your SKU profile determines which end of that range you actually get.

Minimum ROI threshold: approximately 600 picks/hr. Below that, human picking with put-to-light assist delivers better economics.

Vision System Cycle Time Budget

For a 4-second target pick cycle:

• Robot motion budget (travel + place): ~3,100 ms
• Vision budget (3D acquisition + processing): 900 ms maximum

Asynchronous triggering: fire the camera during the deposit motion of the previous cycle. Vision processing occurs while the robot is moving, hiding vision latency inside travel time. Without this, a 500 ms point-cloud acquisition eats 12.5% of a 4-second cycle. With it, the marginal cycle time cost of 3D vision approaches zero.

Mixed-Case Palletizing

Robot arm throughput is rarely the constraint — infeed sequencing is.

A palletizing robot at 800 cases/hr achieves that only if cases arrive in the sequence the algorithm planned. If the infeed conveyor delivers out of sequence, the robot either waits (losing throughput) or deviates from the planned stack pattern (degrading cube utilization and stability).

Cube utilization benchmarks:

• Manual mixed-case: 65–75%
• Rule-based robotic (pattern layers): 70–80%
• AI-optimized 3D bin-packing: 75–88%
• Theoretical maximum: 92–95% (not achievable in practice)

Design the infeed sortation and sequencing buffer before sizing the robot arm.

Source: 2.6-advanced-automation-design