M/M/1 and M/M/c queue: utilisation, probability of waiting (Erlang C), queue len...

{ "method": "GET", "queryParams": { "arrivalRate": 8, "servers": 1, "serviceRate": 10, "waitThreshold": 0.5 }, "type": "http" }

{ "example": { "asOf": "2026-10-11T20:19:39.979Z", "degraded": false, "disclaimer": "Screening signal computed from the named public source. Not financial, legal or investment advice; provided as is. Verify at the source before acting.", "meanInSystem": 4.000000000000002, "meanQueueLength": 3.2000000000000015, "meanServiceTime": 0.1, "meanTimeInSystem": 0.5000000000000002, "meanWaitGivenWaiting": 0.5, "meanWaitInQueue": 0.4000000000000002, "model": "M/M/1", "notChecked": [ "non-Poisson arrivals and non-exponential service times (real service times with higher variance wait longer: Pollaczek-Khinchine for M/G/1)", "limited queue capacity, abandonment and retries", "…2 more" ], "offeredLoadErlangs": 0.8, "probabilityEmptySystem": 0.19999999999999996, "probabilityNoWait": 0.19999999999999996, "probabilityOfWaiting": 0.8, "probabilityWaitExceedsThreshold": { "probability": 0.2943035529371539, "t": 0.5 }, "reasons": [ "Waiting time explodes as utilisation approaches 100%: for M/M/1 the mean wait is rho / (mu (1 - rho)).", "Erlang C gives the probability that an arrival must queue; pooling servers (one queue, c servers) is much better than c separate queues.", "…1 more" ], "source": "computed locally (M/M/c queueing theory)", "summary": "M/M/1: utilisation 80.0%, P(wait) 80.00%, mean wait 0.4000, response 0.5000 (service alone 0.1000), 3.200 waiting on average. P(wait > 0.5) = 29.43%.", "utilization": 0.8, "verdict": "BUSY" }, "type": "json" }