The Single Server Queue in Heavy Traffic.

Kingman, J. F. C. (1961). The Single Server Queue in Heavy Traffic. Mathematical Proceedings of the Cambridge Philosophical Society, 57(4), 902-904.

Type

Journal article

Intellectual base

Primary research

Year

1961

DOI

10.1017/s0305004100036094

Link

https://doi.org/10.1017/S0305004100036094

Cited from

chemistry_materials, computer_science, operations_research

Cited by

5 citations across 5 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Buffering
  • Engineers reason about buffer utilization (what fraction of capacity is in use during typical operation?) and buffer latency (how long does material spend in the buffer?), a trade-off Kingman (1961) formalized in his heavy-traffic approximation showing that mean queue waiting time grows roughly as ρ/(1−ρ) times a variability term.
• Interference and Contention
  • A core function of "interference and contention" is to distinguish between overload and contention, a separation Kingman (1961) made tractable through his analysis of waiting times in single-server queues with arbitrary arrival processes.
• Queueing
  • Not always Markovian:

    This sourceHeavy-traffic approximation for the GI/G/1 queue, giving bounds on mean wait when exact (Markovian) solutions are unavailable.

• Unevenness Waste
  • The underlying regularity is queueing-theoretic: delay scales roughly as \(\rho\sigma^2/(1-\rho)\) for utilization \(\rho\) and variability \(\sigma\), a formula that has been independently re-derived across substrates.

    This sourceDerives the heavy-traffic approximation in which mean delay scales with the variability of arrivals and service over (1−ρ), pricing variance as a cost additive to the mean.

Mechanisms

• Staffing Level Adjustment
  • People take time to redeploy and to ramp, so a manager who chases every spike whipsaws staff between stations and exhausts them, while a loop tuned to average load saturates the moment demand is genuinely heavy — the queueing math is unforgiving, since service degrades sharply as utilization approaches capacity.

Verification

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