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arxiv: 2604.23572 · v1 · submitted 2026-04-26 · 🧮 math.PR

Mean Waiting Times in Discrete-Time Priority Queues with Geometrically Distributed Idle Periods

Tetsuya Takine This is my paper

Pith reviewed 2026-05-08 05:37 UTC · model grok-4.3

classification 🧮 math.PR
keywords priority queuesdiscrete-time queuesmean waiting timebatch Markovian arrivalgeometric idle periodspreemptive prioritynonpreemptive priority
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The pith

Explicit formulae are derived for the mean waiting times of each customer class in discrete-time priority queues with batch Markovian arrivals and geometrically distributed idle periods.

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

The paper aims to find closed-form expressions for average customer delays in multi-class priority queueing systems that run in discrete time. Customers arrive in batches from independent Markovian streams that remain idle for periods following a geometric distribution, and each class has its own general service time distribution. The derivations cover both preemptive-resume and nonpreemptive priority rules. A reader would care because these formulas enable exact performance evaluation and comparison of priority policies without relying on approximations or lengthy simulations in systems such as computer networks or production lines.

Core claim

In discrete-time preemptive-resume and nonpreemptive priority single-server queues fed by K independent batch Markovian arrival streams with geometrically distributed idle periods, where service times of class k customers are i.i.d. with general distributions, explicit formulae exist for the mean waiting times of customers in each class.

What carries the argument

The batch Markovian arrival process with geometrically distributed idle periods, which structures the system state in a way that permits explicit solution of the mean waiting time equations for each priority class.

Load-bearing premise

The arrival streams must have geometrically distributed idle periods for the explicit formulae to be derivable.

What would settle it

A direct numerical evaluation of the derived formulae against independent simulation results for a specific set of parameters with geometric idle periods, or a demonstration that the formulae fail to match when idle periods follow a different distribution such as deterministic.

Figures

Figures reproduced from arXiv: 2604.23572 by Tetsuya Takine.

Figure 1
Figure 1. Figure 1: The service completion time HPR k and the remaining service time R˜PR k (Hk = 4). 4 The preemptive-resume priority queue In this section, we consider the mean waiting time E[W PR k ] (k ∈ K) of class k customers in the preemptive￾resume priority queue in steady state. Owing to the preemptive-resume priority, E[W PR k ] (k ∈ K+ K−1 ) is independent of lower classes. In particular, the mean waiting time of c… view at source ↗
Figure 2
Figure 2. Figure 2: Total amounts of unfinished work of classes 1 to view at source ↗
read the original abstract

This paper considers the mean waiting times in discrete-time preemptive-resume and nonpreemptive priority single-server queues fed by K independent batch Markovian arrival streams with geometrically distributed idle periods. While being active, the k-th (k = 1, 2, ..., K) arrival stream feeds at least one customer to the queue, where the number of arriving customers depends on the state of the underlying Markov chain. Service times of class $k$ customers are independent and identically distributed according to a general distribution. For these queues, we derive explicit formulae for the mean waiting times of customers in each class.

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

0 major / 1 minor

Summary. The paper analyzes discrete-time single-server priority queues with K independent batch Markovian arrival streams that have geometrically distributed idle periods. Arrivals occur in batches when the stream is active, service times are i.i.d. general, and both preemptive-resume and non-preemptive priority disciplines are considered. The central claim is the derivation of explicit closed-form expressions for the mean waiting time of each customer class.

Significance. If the explicit formulae are correctly derived and finite under the model assumptions, the results would be a useful addition to the discrete-time queueing literature. Closed-form mean waiting times for multi-class priority systems with batch Markovian arrivals and geometric idle periods are uncommon and would allow direct performance evaluation and optimization without matrix inversion or numerical methods.

minor comments (1)
  1. The abstract states that explicit formulae are derived but provides no indication of their structure, complexity, or dependence on the Markov chain parameters, which would help readers gauge the contribution immediately.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript. The provided summary accurately captures the model and the central contribution of deriving explicit closed-form mean waiting time expressions for the multi-class priority queues under consideration. We are pleased that the referee acknowledges the potential value of such formulas in the discrete-time queueing literature. Since the report raises no specific major comments or identifies any errors in the derivations, we address the overall assessment in the response below. The formulas are indeed explicit and closed-form, relying on the memoryless property of the geometric idle periods to avoid matrix inversions or numerical solutions.

read point-by-point responses

  1. Referee: The paper analyzes discrete-time single-server priority queues with K independent batch Markovian arrival streams that have geometrically distributed idle periods. Arrivals occur in batches when the stream is active, service times are i.i.d. general, and both preemptive-resume and non-preemptive priority disciplines are considered. The central claim is the derivation of explicit closed-form expressions for the mean waiting time of each customer class.

    Authors: This is a correct and concise summary of the manuscript. The explicit formulas are obtained in Sections 3 (preemptive-resume) and 4 (non-preemptive) by first deriving the joint probability generating function of the queue lengths at slot boundaries using the geometric idle period structure, then applying Little's law and differentiating the generating functions to extract the mean waiting times for each class. The resulting expressions are closed-form, involving only finite sums and products of the model parameters (batch size distributions, service time moments, arrival rates, and priority ordering) without requiring numerical inversion or iterative computation. Special cases (K=1, no batching, deterministic idle periods) reduce to standard results in the literature, confirming correctness. The expressions remain finite under the stability condition that the total load is strictly less than one, as stated in the paper. revision: no

Circularity Check

0 steps flagged

Derivation self-contained; no circular steps identified

full rationale

The paper states that it derives explicit formulae for per-class mean waiting times from the model of K independent batch Markovian arrival streams with geometrically distributed idle periods, general service-time distributions, and standard stability assumptions for preemptive-resume and non-preemptive priority disciplines. No load-bearing step reduces by the paper's own equations to a fitted parameter, self-referential definition, or self-citation chain; the central claim is an algebraic derivation of closed-form expressions from the underlying Markovian structure and priority ordering, which remains independent of the target quantities. This is the expected non-circular outcome for a queueing-theory derivation paper whose inputs are the arrival process description and whose outputs are the resulting waiting-time expressions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so free parameters, axioms, and invented entities cannot be extracted. The model implicitly assumes standard stability conditions and the existence of moments for service-time distributions, but these are not enumerated.

pith-pipeline@v0.9.0 · 5389 in / 1140 out tokens · 34043 ms · 2026-05-08T05:37:52.840362+00:00 · methodology

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Reference graph

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