Work out queueing delay (m/m/1) instantly with clear inputs, formula shown and shareable results.
For an M/M/1 queue with arrival rate lambda and service rate mu, total time in system is 1/(mu - lambda) and queueing delay is rho/(mu - lambda). The key behaviour is the denominator: as lambda approaches mu the delay diverges, so a system at 80 percent utilisation waits four times its service time and at 95 percent nineteen times.
M/M/1 results
rho = lambda/mu; W = 1/(mu - lambda); Wq = rho/(mu - lambda); L = rho/(1 - rho); p-th percentile of W = ln(1/(1-p))/(mu - lambda)
It is a simplification — real service times are not exponential and there is usually more than one server — but it captures the crucial non-linear relationship between utilisation and delay.
Because response time in M/M/1 is exponentially distributed with rate mu - lambda, so every percentile scales with 1/(mu - lambda).