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arxiv: 2606.31133 · v1 · pith:WBHAIJRXnew · submitted 2026-06-30 · 🧮 math.PR

Level-dependent quasi-birth-and-death processes: Application to cost analysis of multi-server systems

M. Abdullah Khokhar , Malgorzata M. O'Reilly , Richard Turner This is my paper

Pith reviewed 2026-07-01 04:49 UTC · model grok-4.3

classification 🧮 math.PR
keywords level-dependent quasi-birth-and-death processesLaplace-Stieltjes transformscost distributionmulti-server queuesadmission policiessensitivity analysishospital bed management
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The pith

The Laplace-Stieltjes transforms of the distribution of total costs accumulated in specified level sets of level-dependent quasi-birth-and-death processes are derived analytically.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives analytical expressions for the Laplace-Stieltjes transforms of total cost distributions accumulated by level-dependent quasi-birth-and-death processes during times spent in given level sets. It supplies algorithms to evaluate the transforms numerically, including memory-efficient versions, and extends them to sensitivity analysis with respect to parameters. The expressions are applied to model finite-capacity multi-server queues under redirection, preemptive transfer, and guard-channel policies, using hospital data to obtain performance measures, times to bed availability, and cost distributions under each policy. A reader would care because the results supply exact tools for tracking and comparing operational costs in systems subject to stochastic arrivals and state-dependent transitions.

Core claim

Analytical expressions are derived for the Laplace-Stieltjes transforms of the distribution of total costs accumulated during the times the LD-QBD processes spend in a specified set of levels. Algorithms for the numerical evaluation of these LSTs, memory-efficient versions, and sensitivity analysis of the LSTs are presented. The results are illustrated on finite-capacity multi-server queueing systems with admission policies including redirection, preemptive transfer, and guard-channel threshold, using a large hospital dataset to compute long-run performance measures, the distribution of time until beds become available, and the associated cost distributions.

What carries the argument

Laplace-Stieltjes transform expressions for accumulated costs over LD-QBD level sets, obtained through matrix-analytic methods.

Load-bearing premise

Multi-server systems with the listed admission policies can be accurately represented as level-dependent quasi-birth-and-death processes whose cost accumulation depends only on time spent in specified level sets.

What would settle it

A direct simulation of costs in one of the LD-QBD hospital models whose empirical distribution, when transformed, deviates from the closed-form LST expression beyond numerical tolerance.

read the original abstract

Analysing costs is crucial for optimising the operational efficiency and resource allocation in systems evolving under uncertainty. In this paper, we study the distribution of costs associated with the evolution of level-dependent quasi-birth-and-death (LD-QBD) processes, which are useful in modelling many multi-server systems. We derive analytical expressions for the Laplace-Stieltjes transforms (LSTs) of the distribution of total costs accumulated during the times the LD-QBD processes spend in a specified set of levels. We present algorithms for the numerical evaluation of these LSTs. We also give memory efficient versions of the algorithms and discuss their algorithmic complexity. To assess the robustness of the distribution of costs with respect to model parameters, we develop algorithms for the sensitivity analysis of the corresponding LSTs. To illustrate the application potential of our results, we construct LD-QBD example models for a finite capacity multi-server queueing systems with admissions policies including redirection, preemptive transfer, and guard-channel threshold. The analysis is based on a large dataset obtained from a tertiary referral hospital in Australia. We compute the long-run performance measures, the distribution of time until some number of beds become available following congestion, and the distribution of the associated costs. We present valuable insights into how the system behaves under the various policies. We also perform the sensitivity analysis of the distribution of costs with respect to model parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript derives Laplace-Stieltjes transform (LST) expressions for the distribution of total costs accumulated by level-dependent quasi-birth-and-death (LD-QBD) processes while occupying specified level sets. It supplies recursive algorithms for LST evaluation (including memory-efficient variants and complexity bounds), sensitivity-analysis algorithms, and applies the framework to finite-capacity multi-server queues under redirection, preemptive-transfer, and guard-channel policies, using Australian tertiary-hospital data to obtain long-run measures, time-to-availability distributions, and cost distributions.

Significance. The work extends matrix-analytic methods to cost functionals on LD-QBDs, supplying explicit recursive schemes and real-data illustrations that are directly usable for operational analysis of multi-server systems. The combination of analytical LSTs, algorithmic evaluation, and sensitivity tools constitutes a coherent contribution to stochastic modeling of healthcare queues.

major comments (2)
  1. [§4.1] §4.1, the recursive scheme for the LST vector: the boundary equations for the finite-level truncation are stated without an explicit a-priori error bound on the tail probability; because the cost functional is an integral over occupation time, the truncation error propagates directly into the LST and should be quantified (e.g., via the spectral radius of the level-dependent rate matrix) to support the numerical claims.
  2. [§5.3] §5.3, the guard-channel example: the level-dependent birth rates under the threshold policy are defined piecewise, yet the LST derivation treats the cost rate as constant within each level set; the paper must verify that the piecewise definition does not introduce an additional state variable that would invalidate the standard LD-QBD generator used for the LST equations.
minor comments (3)
  1. Notation for the cost accumulation functional (integral of level-dependent rate over occupation time) is introduced without a dedicated display equation; a single numbered equation would improve readability when the LST is later expressed in terms of this functional.
  2. [§6] The hospital dataset is described only by source; adding the observation window length and number of patient records would allow readers to assess the statistical reliability of the fitted parameters used in the numerical examples.
  3. Algorithm 2 (memory-efficient LST evaluation) is presented in pseudocode; a short complexity table comparing it with the naïve recursion would make the claimed savings explicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive recommendation. We address each major comment below.

read point-by-point responses

  1. Referee: [§4.1] §4.1, the recursive scheme for the LST vector: the boundary equations for the finite-level truncation are stated without an explicit a-priori error bound on the tail probability; because the cost functional is an integral over occupation time, the truncation error propagates directly into the LST and should be quantified (e.g., via the spectral radius of the level-dependent rate matrix) to support the numerical claims.

    Authors: We agree that an explicit a-priori error bound on the truncation would strengthen the presentation. In the revised version we will add a short subsection deriving a bound on the tail probability using the spectral radius of the level-dependent rate matrices and showing how this controls the error in the LST of the cost functional. revision: yes

  2. Referee: [§5.3] §5.3, the guard-channel example: the level-dependent birth rates under the threshold policy are defined piecewise, yet the LST derivation treats the cost rate as constant within each level set; the paper must verify that the piecewise definition does not introduce an additional state variable that would invalidate the standard LD-QBD generator used for the LST equations.

    Authors: The guard-channel policy is encoded solely through piecewise level-dependent birth rates; the cost rate remains constant on each level by construction of the model. Because the policy depends only on the current level (already part of the state), no auxiliary variable is required and the standard LD-QBD generator remains valid. We will add one clarifying sentence in §5.3 to make this explicit. revision: partial

Circularity Check

0 steps flagged

No circularity: derivations are standard matrix-analytic LST constructions on LD-QBD rate matrices

full rationale

The paper states it derives LSTs of accumulated costs via level-dependent QBD theory and supplies recursive algorithms plus explicit rate matrices for the redirection/preemptive/guard-channel policies. These steps rest on the standard infinitesimal generator construction and the integral cost functional over occupation times; neither reduces to a fitted parameter renamed as prediction nor to a self-citation whose content is the target result. The hospital dataset is used only for numerical illustration after the expressions are obtained. No load-bearing self-citation, self-definitional step, or ansatz smuggling is exhibited in the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The modeling step that treats the hospital system as an LD-QBD process is implicit but not detailed.

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