Quick guide
How to use this calculator
- Select the stated protocol, strategy, approximation, or conversion mode when one is available.
- Enter the labelled finite bound, probabilities, payoffs, sequence state, or special-function argument.
- Read the theoretical or deterministic result separately from any seeded empirical result.
Calculation method
Apply the stated mathematical model
ρ=λ/μ; L=λ/(μ−λ); Lq=ρL; W=1/(μ−λ); Wq=ρW.
The calculator validates model hypotheses, finite work limits, probability domains, singular cases, and numerical approximation ranges before returning a result.
Worked example
Worked example
λ=2,μ=3 in the same reciprocal-time unit gives utilization 2/3 and mean system size 2.
ρ=λ/μ; L=λ/(μ−λ); Lq=ρL; W=1/(μ−λ); Wq=ρW.
Supported inputs
Precision and limits
Model scope
Paradoxes depend on their information or randomization protocol. Queueing, games, fractals, and cellular automata use exactly the visible assumptions; changing those assumptions changes the answer.
Deterministic versus simulated
Exact combinatorial and theoretical probabilities are labelled separately from seeded pseudorandom simulations. Reusing a seed reproduces the same finite experiment.
Numerical methods
Finite double-precision arithmetic is used for probability models and special functions. Gamma uses a bounded Lanczos approximation, Bessel J uses a bounded convergent series, and erf/erfc disclose their approximation error.
Limits
Iteration, grid, candidate, trial, and series caps are visible in field labels or calculator-specific notes. A bounded trace never claims to prove an unresolved conjecture.
Calculator-specific rule
The M/M/1 model assumes Poisson arrivals, exponential service, one server, λ<μ, and matching reciprocal-time units for λ and μ. W and Wq use the corresponding time unit.