Poisson Arrivals See Time Averages

1982/04/01 by Ronald W. Wolff · 1,250 citations
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #Algorithm #Applied mathematics #Compound Poisson process #Computer science #Fraction (chemistry) #Markovian arrival process #Mathematical economics #Mathematics #Poisson distribution #Poisson process #Process (computing) #Queueing theory #Random Matrices and Applications #Renewal theory #State (computer science) #Statistics #Stochastic process #Stochastic processes and statistical mechanics

paper · doi:10.1287/opre.30.2.223

published in Operations Research 30(2), 223-231 (Institute for Operations Research and the Management Sciences)

openalex publication_date 1982/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In many stochastic models, particularly in queueing theory, Poisson arrivals both observe (see) a stochastic process and interact with it. In particular cases and/or under restrictive assumptions it has been shown that the fraction of arrivals that see the process in some state is equal to the fraction of time the process is in that state. In this paper, we present a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process.

Cited by

• Beyond Accuracy and Cost: Latency-Aware LLM Query Routing for Dynamic Workloads
• Analytical Modeling of Asynchronous Event-Driven Readout Architectures Using Queueing Theory
• Traffic-aware Two-stage Queueing Communication Networks: Queue Analysis and Energy Saving
• Load Balancing Guardrails: Keeping Your Heavy Traffic on the Road to Low Response Times
• Economies-of-scale in resource sharing systems: tutorial and partial review of the QED heavy-traffic regime
• Non-Clairvoyant Scheduling with Progress Bars
• Measuring Pulsed Interference in 802.11 Links
• The analysis of batch sojourn-times in polling systems
• Asymptotic dimensioning of stochastic service systems
• Optimal Pricing for Tandem Queues: Does It Have to Be Dynamic Pricing to Earn the Most?
• Large Deviations Optimal Scheduling of Closed Queueing Networks
• Losses in M/GI/m/n Queues
• Greedy Dynamic Matching
• Fleet Sizing and Charger Allocation in Electric Vehicle Sharing Systems
• Stability and Instability Conditions for Slotted Aloha with Exponential Backoff
• Flexibility in an asymmetric system with prolonged service time at non-dedicated servers
• Score-Aware Policy-Gradient and Performance Guarantees using Local Lyapunov Stability
• SOAP: One Clean Analysis of All Age-Based Scheduling Policies
• Zero-rating of Content and its Effect on the Quality of Service in the Internet
• Arrival Control in Quasi-Reversible Queueing Systems: Optimization and Reinforcement Learning
• On the Steady State of Continuous Time Stochastic Opinion Dynamics with Power Law Confidence
• Telephone Call Centers: Tutorial, Review, and Research Prospects
• Performance analysis of a call center with interactive voice response units
• Online Distributed Queue Length Estimation
• Emergency Department Patient Flow Optimization with an Alternative Care Threshold Policy
• A jump persistent turning walker to model zebrafish locomotion. [europepmc]
• Flexible bed allocations for hospital wards. [europepmc]
• Queues with the dropping function and general service time. [europepmc]

Related

• Approximating Systems Fed by Poisson Processes with Rapidly Changing Arrival Rates
• First Steps Towards an Imprecise Poisson Process
• Queueing systems with pre-scheduled random arrivals
• Superposition of time-changed Poisson processes and their hitting times
• Poisson Hypothesis for Information Networks (A study in non-linear Markov processes) I. Domain of Validity