Queueing and capacity

M/M/1 formulasqueue length and waiting timehow many serverstraffic engineering

Utilisation, queue length, waiting time, blocking and Little's law: how much load a server takes before the line stops moving.

M/M/1 Queue Utilisation

\rho = \frac{\lambda}{\mu}

Fraction of time a single server is busy: the arrival rate divided by the service rate. Below 1 the queue is stable; at 1 or above it grows without limit.

M/M/1 Average Number in System

L = \frac{\lambda}{\mu - \lambda}

Average number of jobs in a single-server queue, waiting plus in service. It rises gently until the server is about 70% busy and then goes vertical.

M/M/1 Average Time in System

W = \frac{1}{\mu - \lambda}

Average time a job spends in a single-server queue, from arrival to departure. It depends on the difference between the two rates, not on their ratio.

Little's Law (Work in Process)

\mathit{WIP} = \mathit{TH} \times \mathit{CT}

The most general law in operations: the work sitting in a system equals the rate it flows through times how long each job stays. True for any stable system, whatever the arrival pattern.

Erlang B Blocking Probability

B = \frac{A^{N}/N!}{\displaystyle\sum_{k=0}^{N} A^{k}/k!}

Fraction of calls turned away by N circuits carrying A erlangs of offered traffic, on the assumption that a blocked call simply goes away. The standard sizing tool for trunks, agents and connection pools.

Poisson Probability

P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}

Chance of exactly k events in a fixed interval when events occur independently at a constant average rate lambda.

How they fit together

Everyone who has sized a call centre, a checkout bank, a thread pool or a machine shop has met the same unpleasant surprise: capacity does not degrade gently. Start with utilisation, which is only arrivals divided by service rate, and read the other formulas as answers to what that single ratio does. At 50% busy the queue holds one job. At 90% busy it holds nine. At 95% it holds nineteen, and the last five points of load doubled the wait without adding much work.

Number in system and time in system are the same curve asked two ways, and the difference matters when you are choosing what to fix. Queue length depends on the ratio of the two rates, so it is unchanged if you double both. Waiting time depends on their difference, so doubling both halves it. That is the whole argument for many small fast servers over one slow big one.

Little's law is the one to trust when the assumptions look shaky. It needs no distribution at all, only a stable system, so it survives arrival patterns that break the M/M/1 model outright. Measure the work sitting in the system and the rate it leaves, and it hands you the lead time you are actually quoting, whatever the schedule claims. Erlang B answers the other version of the question, the one where a blocked caller hangs up rather than waits, which is how trunks, connection pools and hold-free phone lines are sized. Poisson sits underneath all of it as the arrival model these formulas assume.