Queue Wait Time (Little's Law)
Calculate queue wait times and required capacity using queueing theory. Enter values for instant results with step-by-step formulas.
Formula
L = λW; ρ = λ/(cμ); Wait explodes as ρ → 1
Worked Examples
Example 1: Call Center Staffing
Problem: Call center receives 50 calls/hour. Each call takes 5 minutes average. How many agents needed for average wait under 1 minute?
Solution: Parameters:\nλ = 50 calls/hour\nμ = 12 calls/hour per agent (60 min / 5 min = 12)\n\nRequired capacity: λ/μ = 50/12 = 4.17 agents (minimum)\n\nWith 5 agents:\nρ = 50 / (5 × 12) = 0.833 (83.3% utilization)\nAverage wait ≈ 2-3 minutes (too long)\n\nWith 6 agents:\nρ = 50 / (6 × 12) = 0.694 (69.4% utilization)\nAverage wait ≈ 30-45 seconds (acceptable)\n\nWith 7 agents:\nρ = 50 / (7 × 12) = 0.595 (59.5% utilization)\nAverage wait ≈ 15-20 seconds (good)\n\nRecommendation: 6-7 agents for <1 min average wait.\nNote: Peak hours may need more; use 8 for safety.
Result: 6-7 agents needed | 60-70% utilization target | 30-45 sec average wait
Example 2: Web Server Capacity
Problem: E-commerce site: 100 requests/second average, server handles 25 req/sec. Current: 5 servers. Analyze capacity.
Solution: Parameters:\nλ = 100 requests/second\nμ = 25 requests/second per server\nc = 5 servers\n\nUtilization:\nρ = 100 / (5 × 25) = 0.80 (80%)\n\nThis is at the edge of acceptable!\n\nUsing M/M/c approximation:\nAverage in queue: ~3-4 requests\nAverage wait: ~30-40ms\n\nDuring traffic spikes (+20%):\nλ = 120 req/s\nρ = 120 / 125 = 0.96 (96%)\nWait times explode to 200-500ms+\n\nRecommendation:\nAdd 2 servers (7 total)\nNew ρ = 100 / 175 = 0.57 (57%)\nHandles 40% traffic surge safely\n\nLittle's Law verification:\nL = λW = 100 × 0.04 = 4 requests in system ✓
Result: 80% utilization (risky) | Add 2 servers | Target 60% for surge capacity
Example 3: Retail Checkout Lines
Problem: Store: 120 customers/hour at peak. Checkout takes 3 minutes average. 4 registers. What's expected wait?
Solution: Parameters:\nλ = 120 customers/hour\nμ = 20 customers/hour per register (60/3 = 20)\nc = 4 registers\n\nUtilization:\nρ = 120 / (4 × 20) = 0.75 (75%)\n\nM/M/4 queue analysis:\nAverage in queue: ~1.5 customers per line\nAverage in system: ~2.5 customers per line\nTotal in store checkout: ~10 customers\n\nWait time:\nQueue wait: ~45 seconds average\nCheckout time: 3 minutes\nTotal time: ~3:45 average\n\nLittle's Law:\nL = 120/60 × (3.75/60) = 2 × 0.0625 = ... \nL = 120 customers/hr × (3.75 min / 60 min/hr) = 7.5 customers\n\nPeak periods (150 customers/hour):\nρ = 150 / 80 = 0.94 (danger!)\nWait explodes to 5+ minutes\nOpen 5th register for peaks.
Result: 75% utilization | ~45 sec wait normally | Open 5th register for peaks
Frequently Asked Questions
What is Little's Law?
Little's Law states: L = λW, where L = average number in system, λ = arrival rate, W = average time in system. It's remarkably simple yet universally applicable—works for any stable queue regardless of arrival or service distributions. Named after John Little (1961), though the relationship was known earlier.
Why do wait times explode near 100% utilization?
At high utilization, any random variation in arrivals or service creates backlogs that don't clear quickly. At 90% utilization, small surges cause significant queues. At 99%, even tiny variations create massive waits. This non-linear relationship is why capacity planning targets 70-80% utilization, not 95%+.
What's the difference between wait time and system time?
Wait time = time in queue before service starts. System time = wait time + service time. A customer arriving at empty queue has 0 wait but still has service time. System time is what customers experience end-to-end; wait time is the 'wasted' portion.
How does adding servers affect wait time?
Adding servers reduces wait time non-linearly. Going from 1 to 2 servers at 80% total utilization dramatically reduces wait. Additional servers beyond 2-3 have diminishing marginal impact. The key is keeping per-server utilization reasonable (< 80-85%).
What is M/M/c queue notation?
Kendall notation describes queue characteristics: M/M/c = Markovian arrivals (Poisson), Markovian service (exponential), c servers. M/M/1 = single server, M/M/c = c servers. Other notation: G (general distribution), D (deterministic). Most real systems approximate M/M/c.
How do I reduce wait times without adding capacity?
Options: 1) Reduce service time (faster processing), 2) Reduce arrival rate (manage demand, pricing), 3) Buffer demand (appointments vs walk-ins), 4) Improve service variability (consistent process), 5) Priority queues (serve urgent first). All affect wait without adding servers.