Virtual Waiting Room Calculator

This calculator applies queuing theory to a telehealth virtual waiting room, estimating how long patients wait and how likely they are to wait at all, using the M/M/c queuing model with an Erlang-C formula for probability of waiting. M/M/c assumes patients arrive following a Poisson process (random, independent arrivals at a known average rate) and that each visit's duration follows an exponential distribution, with c identical providers (servers) working in parallel -- the standard model used across service-operations and call- center capacity planning, applied here to virtual care. The calculator first converts visit duration plus a charting/transition buffer into a per-provider service rate, then computes offered load (arrival rate divided by service rate, in "Erlangs") and provider utilization (offered load divided by provider count). When utilization reaches or exceeds 100% -- meaning patients are arriving faster than the providers can possibly see them -- the queue is mathematically unstable and wait time grows without bound; the calculator flags this saturated state rather than reporting a meaningless finite number.

Below that threshold, it computes probability of waiting, expected wait time, and average queue length (via Little's Law: queue length equals arrival rate times average wait). It also reports the maximum patient arrival rate the current staffing can sustain at a comfortable 85% utilization, and searches upward from the current provider count to recommend the minimum staffing that keeps average wait under 5 minutes. This model assumes arrivals and service times behave like the idealized Poisson/ exponential processes it's built on; real clinic no-show rates, visit-length variability, and non-random patient arrival patterns (everyone booking the top of the hour) will all cause real wait times to diverge from this idealized estimate to some degree.