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  "slug": "sharp-multiserver-pricing-guarantees",
  "title": "Sharp multiserver pricing guarantees and the worst congestion load",
  "shortTitle": "A fixed price with a cap: the published guarantees cannot be improved",
  "url": "https://evidencepress.org/releases/sharp-multiserver-pricing-guarantees/",
  "oneLine": "The published worst-case guarantees for a throughput-matched static price with a capacity cap are exact for every number of servers, with an explicit worst congestion load and an exact revenue-congestion curve.",
  "abstract": "A simple static price with a capacity cap has published worst-case guarantees for profit, revenue and congestion, and Bergquist and Elmachtoub conjectured that their multiserver bounds cannot be improved. This candidate proves all three sharpness claims for every number of servers and every capacity, with uniform valuations. It finds the offered load that maximises the congestion ratio: an explicit quartic decides between an interior maximiser and critical load, and also decides whether the worst revenue and congestion ratios can be approached together. Its waiting-capacity threshold is asymptotic to 1.7882539728 times the square root of the number of servers. The paper also proves the many-server limit of the maximal ratio, the exact worst-case revenue-congestion frontier, and sharpness of the companion sojourn-time guarantees. With five servers and capacity seven the construction keeps at least 81.85% of optimal revenue and raises occupancy by at most a factor 1.01003, and the congestion figure occurs at a load of about 2.59 rather than 5. These results concern the specified throughput-matched construction, not the best static policy, which stays open. The optimal throughput is a proof device, and the occupancy ratio can hide queues. The frontier and sojourn proofs are elementary with close antecedents. Pacheco (1994) has the same threshold for one server, and the two differ at two servers. This is not a priority claim.",
  "datePublished": "2026-11-04",
  "dateModified": "2026-11-04",
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  "doi": "10.5281/zenodo.23953351",
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    "Anonymous"
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  "keywords": [
    "dynamic pricing",
    "static pricing",
    "queueing",
    "Erlang loss",
    "finite buffer",
    "sharp approximation guarantees",
    "congestion",
    "many-server asymptotics"
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    "Sharpness (Theorem 2.1, Proposition 3.4, proof): the profit, revenue and congestion constants are sharp for every C and N, even for uniform valuations; optimal throughput runs through all of (0,C) in the linear-demand family.",
    "Worst load (Theorem 2.2, proof): an explicit quartic Q_C decides whether the congestion ratio peaks below critical load; the threshold is asymptotic to 1.7882539728√C.",
    "Joint attainability (Corollary 4.1, proof): the worst revenue and congestion ratios can be approached together exactly when d ≥ d_crit(C).",
    "Exact frontier (Theorem 4.3, Proposition 4.2, proof): the load-dependent bounds are tight at every load, giving the whole worst-case revenue-congestion curve.",
    "Many servers and rates (Theorems 6.1, 7.1, Corollary 6.2, proof): limits of the maximal ratio and its maximiser with c√C waiting places, and finite-parameter bounds on the profit deficit.",
    "Sojourn time (Theorem 8.1, proof): the companion sojourn-time constants are sharp and always approached together."
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  "relatedWorks": [
    {
      "citation": "Balseiro, Ma and Zhang (2026). Dynamic Pricing for Reusable Resources: The Power of Two Prices. Operations Research. The occupation programme that Section 3 specialises.",
      "url": "https://doi.org/10.1287/opre.2023.0475",
      "doi": "10.1287/opre.2023.0475"
    },
    {
      "citation": "Bergquist and Elmachtoub (2026). Static Pricing Guarantees for Queueing Systems. Stochastic Systems. The guarantees and the conjecture proved here.",
      "url": "https://doi.org/10.1287/stsy.2023.0057",
      "doi": "10.1287/stsy.2023.0057"
    },
    {
      "citation": "Besbes, Elmachtoub and Sun (2022). Technical Note—Static Pricing: Universal Guarantees for Reusable Resources. Operations Research. Averaged arrival rate and Jensen for static pricing of loss systems.",
      "url": "https://doi.org/10.1287/opre.2020.2054",
      "doi": "10.1287/opre.2020.2054"
    },
    {
      "citation": "Dupont et al. (2026). Dynamic Parking Pricing for Electric Vehicles in a Loss Queue. IEEE Transactions on Intelligent Transportation Systems. Computational study of electric-vehicle parking prices.",
      "url": "https://doi.org/10.1109/TITS.2026.3702272",
      "doi": "10.1109/TITS.2026.3702272"
    },
    {
      "citation": "Daley, van Leeuwaarden and Nazarathy (2015). BRAVO for Many-Server QED Systems with Finite Buffers. Advances in Applied Probability. Equation (2.17) gives the first limit of Corollary 6.2.",
      "url": "https://doi.org/10.1239/aap/1427814589",
      "doi": "10.1239/aap/1427814589"
    },
    {
      "citation": "Karp, Vishnyakova and Zhang (2025). Unimodality Preservation by Ratios of Functional Series and Integral Transforms. Results in Mathematics. Coefficient-ratio monotonicity, which gives the peak shape.",
      "url": "https://doi.org/10.1007/s00025-025-02431-4",
      "doi": "10.1007/s00025-025-02431-4"
    },
    {
      "citation": "Pacheco (1994). Second-order properties of the loss probability inM/M/s/s +c systems. Queueing Systems. The same threshold for one server; the transitions differ for two.",
      "url": "https://doi.org/10.1007/BF01189242",
      "doi": "10.1007/BF01189242"
    },
    {
      "citation": "Whitt (2004). A Diffusion Approximation for the G/GI/n/m Queue. Operations Research. The stationary density of Section 6.",
      "url": "https://doi.org/10.1287/opre.1040.0136",
      "doi": "10.1287/opre.1040.0136"
    },
    {
      "citation": "Elmachtoub and Shi (2025). The Power of Static Pricing for Reusable Resources. EC 2025. Extended abstract on static pricing for reusable resources.",
      "url": "https://doi.org/10.1145/3736252.3742552",
      "doi": "10.1145/3736252.3742552"
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