# M/M/1 Queue Calculator (L, Lq, W, Wq) Steady-state M/M/1 metrics: ρ, P0, L, Lq, W, Wq, state probabilities, and Little's-law cross-checks; unstable λ ≥ μ inputs are rejected. > Canonical page: https://elysiatools.com/en/tools/queuing-theory-mm1 - **Category:** Math & Numbers - **Keywords:** queuing theory, mm1, m/m/1, queue, waiting line, little's law, utilization, poisson arrivals, exponential service, operations research ## Overview The M/M/1 Queue Calculator computes steady-state performance metrics for single-server queuing systems with Poisson arrivals and exponentially distributed service times. It delivers instant calculations for traffic intensity, idle probability, queue length, waiting times, and state distributions verified by Little's law. ## Inputs - **Arrival rate λ (per time unit)** (number): Average arrivals per time unit (Poisson process). Must be below the service rate μ. - **Service rate μ (per time unit)** (number): Average services per time unit (exponential distribution). Must exceed the arrival rate λ. - **Decimal Places** (number) ## When to use - Analyzing single-server service desks, ticket counters, or technical support lines under steady-state conditions. - Evaluating queue latency, system occupancy, and server utilization for capacity planning. - Validating operations research assignments and queuing theory models using Little's law formulas. ## How it works - Enter the arrival rate (λ) representing the average number of incoming requests or customers per time unit. - Enter the service rate (μ) representing the average number of requests or customers the server can handle per time unit (μ must exceed λ for stability). - Specify the desired decimal precision for the output metrics. - Generate steady-state metrics including server utilization (ρ), idle probability (P0), mean customers in system (L), mean queue length (Lq), average system time (W), average waiting time (Wq), and discrete state probabilities. ## Use cases - Capacity planning for single-lane drive-thrus or service booths to keep wait times within target thresholds. - Sizing single-threaded background job processors or single-socket network buffers to prevent buffer overflow. - Solving and cross-checking operations research problem sets involving Poisson arrival processes and exponential service distributions. ## Frequently asked questions ### What does an M/M/1 queue represent? An M/M/1 queue models a single-server system with Poisson arrivals (Markovian), exponentially distributed service durations (Markovian), and one server operating under first-come, first-served discipline. ### Why must the service rate (μ) be greater than the arrival rate (λ)? If arrival rate equals or exceeds service rate (λ ≥ μ), the queue grows indefinitely, making the system unstable and preventing steady-state equilibrium. ### What is the difference between W and Wq? W represents the total expected time spent in the system (waiting plus service), while Wq represents only the expected time spent waiting in the queue. ### What is Little's law in this calculator? Little's law provides mathematical validation where L = λ × W and Lq = λ × Wq, relating average quantities to average times. ### What does the utilization metric (ρ) signify? 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