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Little proved L=λW for consistently defined queue occupancy, arrival rate and sojourn time under finite-mean strict stationarity and an ergodic interarrival process, extending earlier heuristic and special-model uses of the relation. How many requests are usually inside a system? How long do they stay? Little’s relation connects these quantities: average occupancy equals the arrival rate multiplied by average time in the system. His1961 article proves this under explicit stationary stochastic-process assumptions, without requiring Poisson arrivals, a particular service distribution or first-come-first-served order. The key counting picture is the area under occupancy over time: it equals customers’ accumulated time inside, with carry-over corrections at the ends of an observation window. Ergodic arguments control the long-run averages and corrections. The same system boundary must be used for all three quantities. If some offered arrivals never enter, combining the offered arrival rate with the mean stay of admitted customers gives the wrong result. One can instead use the admitted rate, or include zero stays for rejected arrivals. The formula was already known heuristically and for particular models; this record recognizes the broader proof. Little explicitly credited H.Newton Garber with the key occupancy definition and its implications.
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Scientific scope
The identity relates long-run occupancy, arrival rate and time in system under the paper’s stationarity/existence and accounting assumptions. It is not a pointwise identity for every finite interval or a stability proof when averages do not exist. System boundaries and customer accounting must agree. Later transient and generalized formulations are separate.
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- Little: a general stationary-queue proof linking occupancy, arrivals and time in system — complete assessed article text
- INFORMS original publication metadata and abstract: A Proof for the Queuing Formula
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People and systems
John D. C. Little
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English catalogue narrative
Little: a general stationary-queue proof linking occupancy, arrivals and time in system
Little proved L=λW for consistently defined queue occupancy, arrival rate and sojourn time under finite-mean strict stationarity and an ergodic interarrival process, extending earlier heuristic and special-model uses of the relation.
How many requests are usually inside a system? How long do they stay? Little’s relation connects these quantities: average occupancy equals the arrival rate multiplied by average time in the system. His1961 article proves this under explicit stationary stochastic-process assumptions, without requiring Poisson arrivals, a particular service distribution or first-come-first-served order. The key counting picture is the area under occupancy over time: it equals customers’ accumulated time inside, with carry-over corrections at the ends of an observation window. Ergodic arguments control the long-run averages and corrections. The same system boundary must be used for all three quantities. If some offered arrivals never enter, combining the offered arrival rate with the mean stay of admitted customers gives the wrong result. One can instead use the admitted rate, or include zero stays for rejected arrivals. The formula was already known heuristically and for particular models; this record recognizes the broader proof. Little explicitly credited H.Newton Garber with the key occupancy definition and its implications.
Scientific scope
The identity relates long-run occupancy, arrival rate and time in system under the paper’s stationarity/existence and accounting assumptions. It is not a pointwise identity for every finite interval or a stability proof when averages do not exist. System boundaries and customer accounting must agree. Later transient and generalized formulations are separate.
People and systems
John D. C. Little