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Queueing Theory Calculator

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Queueing Theory Calculator
Erlang C, M/M/1/K queue metrics and Little's Law
Queue Model
Parameters
λ = customers arriving per hour/min/sec  |  μ = customers served per same unit  |  ρ = λ/μ(M/M/1) must be < 1 for stability
Little's Law

L = λW — the average number in the system equals the arrival rate times the average time in the system. This holds for any stable queue.

Kendall notation A/B/c/K/N — A = arrival distribution, B = service distribution, c = servers, K = capacity, N = population. M = Markovian (Poisson/exponential).

Applications: call centres, hospital ERs, manufacturing lines, network routers, software request queues, supermarket checkouts.

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What is Queueing Theory Calculator?

Queueing Theory Calculator analyzes waiting lines and service systems using mathematical queueing models, helping predict wait times, queue lengths, and system efficiency.

Queueing Theory Calculator Formula / Method

QueueingTheoryCalculator(M/M/1model)Queueing Theory Calculator (M/M/1 model)
ρ=λ/μρ = λ / μ
L=ρ/(1−ρ)L = ρ / (1 − ρ)
Lq=ρ2/(1−ρ)Lq = ρ² / (1 − ρ)
W=1/(μ−λ)W = 1 / (μ − λ)
Wq=λ/[μ(μ−λ)]Wq = λ / [μ(μ − λ)]

Queueing Theory Calculator Step-by-Step

  1. Enter all required inputs shown in the calculator.
  2. Check units/scales or grading format where applicable.
  3. Apply the calculator method for this tool.
  4. Review the computed output.
  5. Use the result for planning, verification, or comparison.

Queueing Theory Calculator Worked Example

QueueingTheoryCalculator(M/M/1)Queueing Theory Calculator (M/M/1)
Arrivalrateλ=8customers/hour,servicerateμ=10customers/hourArrival rate λ = 8 customers/hour, service rate μ = 10 customers/hour
ρ=8/10=0.8ρ = 8/10 = 0.8
AveragequeuelengthLq=(0.8)2/(1−0.8)=0.64/0.2=3.2customersAverage queue length Lq = (0.8)² / (1 − 0.8) = 0.64 / 0.2 = 3.2 customers
AveragewaitingtimeWq=8/[10×(10−8)]=8/20=0.4hours(24minutes)Average waiting time Wq = 8 / [10 × (10−8)] = 8/20 = 0.4 hours (24 minutes)

Where Queueing Theory Calculator Is Applied

Applied in operations management, call centers, computer networks, traffic engineering, healthcare (patient wait times), and service system optimization.

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