{"uid":"cap_M3KXcLEiNhWKZEluoGJ_H","slug":"m-m-1-and-m-m-c-queue-theory-calculator-9e983dfc","name":"M/M/1 and M/M/c Queue Theory Calculator","description":"M/M/1 and M/M/c queue: utilisation, probability of waiting (Erlang C), queue length, wait and response time, tail odds.","url":"https://data-calc.x402tookit.com/queue-model?utm_source=zero.xyz","method":"GET","headers":{},"bodySchema":{"type":"object","$schema":"https://json-schema.org/draft/2020-12/schema","required":["input"],"properties":{"input":{"type":"object","required":["type","method"],"properties":{"type":{"type":"string","const":"http"},"method":{"enum":["GET"],"type":"string"},"queryParams":{"type":"object","required":["arrivalRate","serviceRate"],"properties":{"servers":{"type":"integer","description":"Number of servers c (default 1 = M/M/1), 1 to 1000"},"arrivalRate":{"type":"number","description":"Arrival rate lambda per time unit (e.g. per second)"},"serviceRate":{"type":"number","description":"Service rate mu of ONE server per the same time unit (1 / mean service time)"},"waitThreshold":{"type":"number","description":"Optional time t (same unit as the rates inverse): returns P(wait in queue > t)"}}}},"additionalProperties":false},"output":{"type":"object","required":["type"],"properties":{"type":{"type":"string"},"example":{"type":"object","properties":{"asOf":{"type":["null","string"]},"model":{"type":["null","string"]},"source":{"type":["null","string"]},"reasons":{"type":["array","null"],"items":{"type":"string"}},"summary":{"type":["null","string"]},"verdict":{"type":["null","string"]},"degraded":{"type":["boolean","null"]},"disclaimer":{"type":["null","string"]},"notChecked":{"type":["array","null"],"items":{"type":"string"}},"utilization":{"type":["null","number"]},"meanInSystem":{"type":["null","number"]},"meanQueueLength":{"type":["null","number"]},"meanServiceTime":{"type":["null","number"]},"meanWaitInQueue":{"type":["null","number"]},"meanTimeInSystem":{"type":["null","number"]},"probabilityNoWait":{"type":["null","number"]},"offeredLoadErlangs":{"type":["null","number"]},"meanWaitGivenWaiting":{"type":["null","number"]},"probabilityOfWaiting":{"type":["null","number"]},"probabilityEmptySystem":{"type":["null","number"]},"probabilityWaitExceedsThreshold":{"type":["null","object"],"properties":{"t":{"type":["null","number"]},"probability":{"type":["null","number"]}}}}}}}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.02","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.02/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.02","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.02","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_NaiNsEe7MIbRVDolNDMgx","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.02","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes queueing model metrics (utilisation, Erlang C probability, queue length, wait times, tail probabilities) for M/M/1 and M/M/c systems given arrival and service rates.","exampleAgentPrompt":"My web service gets 80 requests per second and each server handles 100 requests per second — model the M/M/1 queue and tell me the average wait time, queue length, utilisation, and the probability that a request has to wait at all.","exampleUseCases":[{"title":"Call center agent capacity planning","prompt":"My call center receives 120 calls per hour and each agent handles 15 calls per hour — model the M/M/c queue with 10 agents and give me the Erlang C probability, average wait time, and utilisation."},{"title":"Web API tail latency estimation","prompt":"My API gets 50 requests per second and each instance serves 60 requests per second with 3 servers — what fraction of requests will wait longer than 0.1 seconds in the queue?"},{"title":"Single-server queue bottleneck check","prompt":"I have a background job processor with an arrival rate of 5 tasks per minute and a processing rate of 8 tasks per minute — run the M/M/1 model and tell me the mean number of jobs waiting and the average time a job spends in the system."}],"resultDescription":"Returns a JSON object with queueing metrics including: utilization (rho), mean queue length (Lq), mean number in system (L), mean wait in queue (Wq), mean time in system (W), mean service time, probability of waiting (Erlang C), probability of no wait, probability the system is empty, offered load in Erlangs, and optionally the probability that wait exceeds a specified threshold (t). Also includes a model identifier, summary, and any degraded/disclaimer flags.","failureModes":["Arrival rate >= service rate times number of servers (unstable system, rho >= 1) — system is overloaded and queue grows without bound, endpoint may return error or degraded flag","Missing required arrivalRate or serviceRate parameters — returns validation error","servers parameter out of range (less than 1 or greater than 1000) — returns validation error","Non-positive or zero rates — undefined behaviour or error response","waitThreshold provided but system is unstable — tail probability computation may fail"],"whenToPreferThis":"Choose this endpoint when you need rigorous closed-form M/M/1 or M/M/c queueing theory metrics — utilisation, Erlang C waiting probability, mean queue length, wait times, or tail probabilities — for capacity planning, infrastructure sizing, or performance modelling. It is ideal when you have a stable system (lambda < mu * c) and want analytic results rather than simulation. Prefer it over simulation-based tools when you need fast, exact steady-state answers. Use the sibling Erlang B/C endpoint if you specifically need trunk or agent sizing to hit a blocking or service-level target rather than raw queue statistics.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-10-30T00:34:29.754Z","isFirstParty":false,"canonicalSlug":"m-m-1-and-m-m-c-queue-theory-calculator-9e983dfc"}