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arxiv: 2607.00230 · v1 · pith:IKKCIDJHnew · submitted 2026-06-30 · 📊 stat.AP · math.PR

Waiting time analysis in a finite-capacity multi-server systems with dynamic priorities, dynamically evolving customer types, and abandonment

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

Pith reviewed 2026-07-02 16:47 UTC · model grok-4.3

classification 📊 stat.AP math.PR
keywords waiting time distributionscontinuous-time Markov chainKrylov subspace methodsquasi-birth-and-death processmatrix-analytic methodsdynamic prioritiescustomer abandonmentfinite capacity queues
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The pith

A CTMC with Krylov approximation and a QBD process with matrix-analytic recursions compute conditional and long-run waiting time distributions in finite multi-server systems with dynamic priorities, evolving types, and abandonment.

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

The paper establishes that waiting times in service systems with two customer types, dynamic admission priorities, type evolution, abandonment, finite servers, and finite capacity can be analyzed by modeling the system as a continuous-time Markov chain. Krylov subspace methods provide a scalable numerical way to obtain the conditional waiting time distributions for each type. The same system is also modeled as a quasi-birth-and-death process whose matrix-analytic structure yields exact recursive expressions for both conditional and steady-state waiting time distributions. The approach is demonstrated on hospital data with complex and other patients.

Core claim

The CTMC-Krylov methodology offers a scalable computational approach well suited for analysing large complex systems; analytical expressions are derived via matrix-analytic methods to evaluate conditional and long-run waiting time distributions.

What carries the argument

The CTMC-Krylov methodology that approximates the transient behavior of the continuous-time Markov chain model of the queue.

If this is right

  • Conditional waiting time distributions for Type 1 and Type 2 customers become numerically evaluable for systems of practical size.
  • Long-run waiting time distributions follow directly from the matrix-analytic recursion on the QBD representation.
  • The same framework applies to other finite multi-server systems whose state can be tracked by a CTMC or QBD with similar structure.
  • Numerical examples on real hospital data illustrate that the computed distributions can inform resource allocation decisions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Krylov approach may remain feasible even when the number of customer types increases, provided the resulting state space does not explode.
  • The matrix-analytic recursions could be combined with sensitivity analysis to identify which priority rules most affect waiting times.
  • Direct comparison of the two methods on the same hospital instances would quantify the accuracy-cost trade-off between approximation and exact recursion.

Load-bearing premise

The system dynamics of dynamic admission priorities, evolving customer types, and abandonment behavior can be accurately captured by a continuous-time Markov chain whose state space remains tractable under the finite-capacity and finite-server assumptions.

What would settle it

If the waiting time distributions obtained from the CTMC-Krylov or QBD recursions do not match the empirical distributions computed from the Australian hospital dataset, the modeling assumptions would be falsified.

Figures

Figures reproduced from arXiv: 2607.00230 by M. Abdullah Khokhar, Malgorzata M. O'Reilly, Richard Turner.

Figure 1
Figure 1. Figure 1: Structure of the system under consideration. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Stationary distribution of the QBD model assuming [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Stationary distribution of the QBD model assuming [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Stationary distribution of the QBD model assuming [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Stationary distribution of the QBD model assuming [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Stationary distribution of the QBD model assuming [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Stationary distribution of the QBD model assuming [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Probability densities f1(n, b1, w1)(t), with n = B + w1, w1 = 5, 10, 15, 20, of the waiting times of Type 1 patient from the moment they arrive in the system, and so they are 5th (top left figure), 10th (top right figure), 15th (bottom left figure), and 20th tagged patient at the end of Queue 1 in the waiting area, respectively, given b1 = 30, b2 = 50, w2 = 0, and assuming parameters λ1 = 0, λ2 = 0, µ1 = 0… view at source ↗
Figure 9
Figure 9. Figure 9: Probability densities f2(n, b1, w1, w2)(t), with n = B + w1 + w2, w1 = 2, w2 = 3, 8, 13, 18, of the waiting times of Type 2 patient from the moment they arrive in the system, and so they are 5th (top left figure), 10th (top right figure), 15th (bottom left figure), and 20th tagged patient in the waiting area, respectively, given b1 = 30, b2 = 50, and assuming parameters λ1 = 5.7961, λ2 = 0, µ1 = 0.1517, µ2… view at source ↗
Figure 10
Figure 10. Figure 10: Probability densities f2(n, b1, w1, w2)(t), with n = B + w1 + w2, w1 = 2, w2 = 3, 8, 13, 18, of the waiting times of Type 2 patient from the moment they arrive in the system, and so they are 5th (top left figure), 10th (top right figure), 15th (bottom left figure), and 20th tagged patient in the waiting area, respectively, given b1 = 30, b2 = 50, and assuming parameters λ1 = 5.7961, λ2 = 17.9039, µ1 = 0.1… view at source ↗
Figure 11
Figure 11. Figure 11: Probability densities f1(n, b1, w1)(t), with n = B + w1, b1 = 30, w1 = 20, and w2 = 0, of the waiting times for a Type 1 patient arriving in a congested system with N = 100, B = 80, so that they are the 20th tagged Type 1 patient in the waiting area. These densities are computed assuming the parameters λ1 = 5.7961, λ2 = 17.9039, µ1 = 0.1517, µ2 = 0.4113, and β2→1 = 3 (see [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 12
Figure 12. Figure 12: Probability densities f2(n, b1, w1, w2)(t), with n = B + w1 + w2, b1 = 30, w1 = 2, and w2 = 18, of the waiting times for a Type 2 patient arriving in a congested system with N = 100, B = 80, so that they are the 20th tagged Type 2 patient in the waiting area. These densities are computed assuming the parameters λ1 = 5.7961, λ2 = 17.9039, µ1 = 0.1517, µ2 = 0.4113, and β2→1 = 3 (see [PITH_FULL_IMAGE:figure… view at source ↗
Figure 13
Figure 13. Figure 13: Long-run probability densities fi(t) of the waiting time for Type 1 (left figure) and Type 2 (right figure) patients, evaluated using the CTMC, at the moment when they arrive in the system of capacity of N = 100 and B = 80, under the priority policy r1 = 1, r2 = 0. The parameters used are given by λ1 = 5.7961, λ2 = 17.9039, µ1 = 0.1517, µ2 = 0.4113, and β2→1 = 0. The corresponding long-run mean waiting ti… view at source ↗
Figure 14
Figure 14. Figure 14: Long-run probability densities fi(t) of the waiting time for Type 1 (left figure) and Type 2 (right figure) patients, evaluated using the CTMC, at the moment when they arrive in the system of capacity of N = 100 and B = 80, under the priority policy r1 = 1, r2 = 0. The parameters used are given by λ1 = 5.7961, λ2 = 17.9039, µ1 = 0.1517, µ2 = 0.4113, and β2→1 = 3. The corresponding long-run mean waiting ti… view at source ↗
read the original abstract

In many service systems, an estimation of customers' waiting times for the service can assist in decision making focused on enhancing the operational efficiency, improving the customers' experience, and ensuring efficient resource allocation. In this paper, we study the customers' waiting times in a finite-capacity service system with a finite number of parallel servers and a shared waiting area. We consider two customer types, Type 1 and Type 2, with dynamic admission priorities, dynamically evolving customer type, and abandonment. We model the system under such assumptions using a continuous-time Markov chain (CTMC) and develop a methodology based on Krylov subspace approximation methods to evaluate the conditional waiting time distributions of Type 1 and Type 2 customers in the system. This methodology (CTMC-Krylov) offers a scalable computational approach that is well suited for analysing large complex systems. Next, we model this system using a quasi-birth-and-death (QBD) process and derive analytical expressions building on matrix-analytic methods to evaluate the conditional and long-run waiting time distributions using recursion. We illustrate the practical applicability of our models in a hospital system through a suite of numerical examples based on a large dataset obtained from a tertiary referral hospital in Australia, considering two types of patients, complex (Type 1) and other (Type 2).

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 / 2 minor

Summary. The paper models a finite-capacity multi-server system with two customer types, dynamic admission priorities, type evolution, and abandonment as a CTMC. It develops a CTMC-Krylov subspace method to compute conditional waiting time distributions and represents the system as a QBD process to derive analytical expressions for conditional and long-run waiting time distributions via matrix-analytic methods and recursion. The approach is illustrated with numerical examples from a large Australian hospital dataset distinguishing complex (Type 1) and other (Type 2) patients.

Significance. If the QBD reduction and matrix-analytic derivations hold, the work supplies both scalable numerical tools and exact recursive expressions for waiting times in queueing systems with dynamic features, which is valuable for operational decision-making in healthcare. The application to real hospital data and the emphasis on finite capacity and abandonment are practical strengths.

major comments (2)
  1. [QBD modeling and matrix-analytic derivations] The abstract and modeling description assert that the system with dynamic type evolution and admission priorities can be represented as a standard QBD process admitting matrix-geometric solutions. However, state-dependent rate changes from type transitions (Type 1/2) and priority rules typically destroy the level-independent repeating blocks (A0, A1, A2) required for the matrix-analytic method. The manuscript must exhibit the explicit infinitesimal generator or the repeating blocks in the QBD section to substantiate this reduction; without it the claimed analytical expressions rest on an unverified structural assumption.
  2. [Numerical examples and CTMC-Krylov section] The CTMC-Krylov methodology is presented as scalable for large systems, yet the abstract and numerical examples provide no error bounds, convergence rates, or direct validation against discrete-event simulation. In the hospital case-study section, add quantitative accuracy metrics (e.g., Kolmogorov-Smirnov distances or tail-probability errors) between the Krylov approximations and simulation benchmarks.
minor comments (2)
  1. [Model description] Clarify the precise definition of 'dynamic admission priorities' and the transition rates between customer types; the current description leaves the phase space and rate dependencies ambiguous for replication.
  2. [Analytical expressions] The abstract states that recursion is used to evaluate long-run distributions; supply the explicit recursive formulas or the matrix-geometric rate matrix R in an appendix or dedicated subsection.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the two major comments point by point below, agreeing to revisions that strengthen the presentation while clarifying the modeling choices.

read point-by-point responses

  1. Referee: [QBD modeling and matrix-analytic derivations] The abstract and modeling description assert that the system with dynamic type evolution and admission priorities can be represented as a standard QBD process admitting matrix-geometric solutions. However, state-dependent rate changes from type transitions (Type 1/2) and priority rules typically destroy the level-independent repeating blocks (A0, A1, A2) required for the matrix-analytic method. The manuscript must exhibit the explicit infinitesimal generator or the repeating blocks in the QBD section to substantiate this reduction; without it the claimed analytical expressions rest on an unverified structural assumption.

    Authors: We appreciate this observation. The state space is defined with levels as the total number of customers present and phases encoding the customer-type configuration, server assignments, and abandonment status. Because type-evolution and priority rules act only on the phase variables and the per-customer transition rates remain constant across interior levels, the blocks A0, A1, A2 are level-independent. The current manuscript describes this structure in Section 4 but does not display the explicit generator matrices. We will add the full infinitesimal generator together with the repeating blocks A0, A1, A2 in the revised version so that the matrix-analytic derivations are fully substantiated. revision: yes

  2. Referee: [Numerical examples and CTMC-Krylov section] The CTMC-Krylov methodology is presented as scalable for large systems, yet the abstract and numerical examples provide no error bounds, convergence rates, or direct validation against discrete-event simulation. In the hospital case-study section, add quantitative accuracy metrics (e.g., Kolmogorov-Smirnov distances or tail-probability errors) between the Krylov approximations and simulation benchmarks.

    Authors: We agree that quantitative validation will improve the numerical section. In the revised manuscript we will augment the hospital case-study results with direct comparisons against discrete-event simulation, reporting Kolmogorov-Smirnov distances, maximum tail-probability errors, and observed convergence behavior of the Krylov subspace method. A brief discussion of error bounds will also be added to the CTMC-Krylov methodology subsection. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations apply standard matrix-analytic methods to explicitly modeled CTMC/QBD

full rationale

The paper explicitly constructs a CTMC for the system with dynamic priorities, type evolution, and abandonment, then represents it as a QBD to which established matrix-analytic techniques are applied for waiting-time distributions. No step reduces a claimed prediction or analytical expression to a fitted parameter, self-citation, or definitional tautology; the results are obtained by direct application of known recursion and Krylov methods to the constructed state space. The derivation chain is therefore self-contained and independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; the ledger cannot be populated with specific free parameters, axioms, or invented entities because the full derivations, state-space definitions, and any fitted quantities are not provided.

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