Work out queue length estimate instantly with clear inputs, formula shown and shareable results.
This uses the M/M/1 result with the servers pooled into a single service rate: utilisation is arrivals over capacity and the average queue is utilisation squared over one minus utilisation. Queue length explodes as utilisation approaches one, which is why a till running at 95% capacity has a queue nineteen times longer than one at 50%.
Utilisation
rho = arrival rate / (servers x service rate per server)
Queue length
Lq = rho^2 / (1 - rho)
rho = arrival rate / (servers x service rate per server). This uses the M/M/1 result with the servers pooled into a single service rate: utilisation is arrivals over capacity and the average queue is utilisation squared over one minus utilisation.
Queue length explodes as utilisation approaches one, which is why a till running at 95% capacity has a queue nineteen times longer than one at 50%.
Enter arrivals per hour, customers one server handles per hour, number of servers. The defaults shown are a realistic worked example — swap in your own figures to get a result you can use.