Support Queue: Staffing & Wait-Time Simulator

Desk Configuration

Ticket arrival rate λ 2.0 / min

Support agents c 3

Avg handle time 1/μ 2.0 min

SLA target 60 s

Live Queue Stats

Waiting now

Utilization ρ

Avg wait (sim)

– s

Avg wait (Erlang-C)

– s

Within SLA

– %

Resolved

How it works

This is an M/M/c queue: tickets arrive one at a time as a Poisson process with rate λ (per minute), and c identical agents each resolve tickets at rate μ = 1/(handle time), with exponentially distributed service durations. A ticket that arrives when every agent is busy waits in a single FIFO line until an agent frees up.

offered load   a  = λ / μ
utilization    ρ  = a / c            (stable only if ρ < 1)

Erlang-C wait probability:
P_wait = [ aᶜ / (c!·(1-ρ)) ] / [ Σ_{k=0}^{c-1} aᵏ/k! + aᶜ/(c!·(1-ρ)) ]

Expected wait (Erlang-C):
Wq = P_wait / (c·μ − λ)

Arrival rate λ — how fast new tickets arrive; raising it increases offered load a = λ/μ.
Agents c — headcount answering the queue; more agents lower ρ and the wait, at diminishing returns.
Handle time 1/μ — average minutes an agent spends per ticket (triage + reply).
SLA target — the wait threshold used for the "within SLA" readout (e.g. 60s = answered inside a minute).

The "Avg wait (sim)" readout is measured directly from the running discrete-event simulation (a real running average over resolved tickets); "Avg wait (Erlang-C)" is the closed-form theoretical prediction from the formula above for the same λ, c, μ. They converge as more tickets are resolved — this is the same math contact centres and mobile-app support desks use to size headcount against a wait-time SLA.

Inbox / queue Idle agent Busy agent Resolved

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