A proof for the queueing formula¶
Little, J. D. C. (1961). A proof for the queueing formula: L = λW. Operations Research, 9(3), 383-387.
Cited by¶
8 citations across 8 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Buffering
- Stability emerges when average inflow equals average outflow, the steady-state relationship Little (1961) formalized as L = λW connecting average inventory, arrival rate, and time-in-system; perturbation is absorbed when spare capacity exists.
This sourceFoundational result of queueing theory: in any stable queueing system the mean number of items in the system equals the arrival rate times the mean residence time (L = λW), giving the steady-state relation among average inventory, arrival rate, and time-in-system.
- Stability emerges when average inflow equals average outflow, the steady-state relationship Little (1961) formalized as L = λW connecting average inventory, arrival rate, and time-in-system; perturbation is absorbed when spare capacity exists.
- Interference and Contention
- The pattern captures a fundamental trade-off: shared resources improve overall utilization (the resource is rarely idle) but introduce contention risk (concurrent demand causes interference and performance loss), the utilization–contention coupling Little (1961) made precise via his foundational queueing-system law.
This sourceFoundational result of queueing theory: in any stable queueing system, the mean number of items in the system equals arrival rate times mean residence time, providing the substrate-independent law that governs throughput-based liquidity in trading, networking, and operations.
- The pattern captures a fundamental trade-off: shared resources improve overall utilization (the resource is rarely idle) but introduce contention risk (concurrent demand causes interference and performance loss), the utilization–contention coupling Little (1961) made precise via his foundational queueing-system law.
- Liquidity
- The structural analogy is exact at the queueing level: Little (1961) proved that the mean number of items in any stable queueing system equals arrival rate times mean residence time (L = λW), so a system's effective liquidity—throughput per unit residence time—is governed by the same law whether the items are trades, packets, queries, or approval requests.
This sourceFoundational result of queueing theory: in any stable queueing system, the mean number of items in the system equals arrival rate times mean residence time, providing the substrate-independent law that governs throughput-based liquidity in trading, networking, and operations.
- The structural analogy is exact at the queueing level: Little (1961) proved that the mean number of items in any stable queueing system equals arrival rate times mean residence time (L = λW), so a system's effective liquidity—throughput per unit residence time—is governed by the same law whether the items are trades, packets, queries, or approval requests.
- Queueing
- Not identical to scheduling:
This sourceFoundational result: in any stable queueing system the mean number in system equals arrival rate times mean residence time, a relation that holds regardless of queue discipline.
- Not identical to scheduling:
- Reaction Intermediate
- This is a deep and portable identity: it generalizes Little's Law, in which queue depth equals arrival rate times sojourn time, and it connects to the Michaelis-Menten analysis in enzymology, to work-in-process accounting in operations, and to bottleneck detection in dataflow analysis.
This sourceEstablishes Little's Law relating standing population to arrival rate times sojourn time, generalizing the formation/consumption steady-state of an intermediate.
- This is a deep and portable identity: it generalizes Little's Law, in which queue depth equals arrival rate times sojourn time, and it connects to the Michaelis-Menten analysis in enzymology, to work-in-process accounting in operations, and to bottleneck detection in dataflow analysis.
- Receptor Saturation
- Following the capacity-utilization framework Little (1961) established in his foundational queueing-theory result L = λW,
This sourceFoundational result of queueing theory: in any stable queueing system, the mean number of items in the system equals arrival rate times mean residence time, providing the substrate-independent law that governs throughput-based liquidity in trading, networking, and operations.
- Following the capacity-utilization framework Little (1961) established in his foundational queueing-theory result L = λW,
- Unevenness Waste
- The steady-state mean number in system is \(L = \rho / (1 - \rho)\), and by Little's law the mean delay is \(W = 1 / (\mu - \lambda)\) — both containing the \(1/(1-\rho)\) term that explodes as \(\rho \to 1\).
This sourceProves Little's law relating mean number in system, arrival rate, and mean delay, used to convert queue length into waiting time.
- The steady-state mean number in system is \(L = \rho / (1 - \rho)\), and by Little's law the mean delay is \(W = 1 / (\mu - \lambda)\) — both containing the \(1/(1-\rho)\) term that explodes as \(\rho \to 1\).
Mechanisms¶
- Active-Site Capacity Dashboard
- A real anchor makes the point precise: Little's Law relates average queue length, arrival rate, and wait time (L = λW), which is why a capacity board must show queue and cycle time, not just how busy the machines look.
This sourceStates Little's Law as L = λW, relating mean number in a system, arrival rate, and time in the system.
- A real anchor makes the point precise: Little's Law relates average queue length, arrival rate, and wait time (L = λW), which is why a capacity board must show queue and cycle time, not just how busy the machines look.
Verification¶
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