research-article

How Hard are Steady-State Queueing Simulations?

Authors:

Shane G. HendersonAuthors Info & Claims

ACM Transactions on Modeling and Computer Simulation (TOMACS), Volume 25, Issue 4

Article No.: 27, Pages 1 - 21

https://doi.org/10.1145/2749460

Published: 16 November 2015 Publication History

Abstract

Some queueing systems require tremendously long simulation runlengths to obtain accurate estimators of certain steady-state performance measures when the servers are heavily utilized. However, this is not uniformly the case. We analyze a number of single-station Markovian queueing models, demonstrating that several steady-state performance measures can be accurately estimated with modest runlengths. Our analysis reinforces the meta result that if the queue is “well dimensioned,” then simulation runlengths will be modest. Queueing systems can be well dimensioned because customers abandon if they are forced to wait in line too long, or because the queue is operated in the “quality- and efficiency-driven regime” in which servers are heavily utilized but wait times are short. The results are based on computing or bounding the asymptotic variance and bias for several standard single-station queueing models and performance measures.

References

[1]

S. Asmussen. 1992. Queueing simulation in heavy traffic. Mathematics of Operations Research 17, 84--111.

Digital Library

[2]

H. P. Awad and P. W. Glynn. 2007. On the theoretical comparison of low-bias steady-state estimators. ACM Transactions on Modeling and Computer Simulation 17, 1, Article 4.

Digital Library

[3]

P. Billingsley. 1968. Convergence of Probability Measures. Wiley, New York, NY.

[4]

S. Borst, A. Mandelbaum, and M. I. Reiman. 2004. Dimensioning large call centers. Operations Research 52, 17--34.

Digital Library

[5]

E. Cinlar. 1972. Superposition of point processes. In Stochastic Point Processes: Statistical Analysis, Theory, and Applications, P. A. W. Lewis (Ed.). Wiley Interscience, New York, 549--606.

[6]

Michael A. Crane and Donald L. Iglehart. 1974a. Simulating stable stochastic systems, I : General multiserver queues. Journal of the ACM 21, 1, 103--113.

Digital Library

[7]

Michael A. Crane and Donald L. Iglehart. 1974b. Simulating stable stochastic systems, II: Markov chains. Journal of the ACM 21, 1, 114--123.

Digital Library

[8]

Michael A. Crane and Donald L. Iglehart. 1975. Simulating stable stochastic systems: III. Regenerative processes and discrete-event simulations. Operations Research 23, 1, 33--45.

Digital Library

[9]

J. G. Dai and Shuangchi He. 2011. Queues in service systems: Customer abandonment and diffusion approximations. In Tutorials in Operations Research: Transforming Research into Action, Joseph Geunes (Ed.). INFORMS, Hanover MD, 31--59.

[10]

J. G. Dai and Shuangchi He. 2012. Many-server queues with customer abandonment: A survey of diffusion and fluid approximations. Journal of Systems Science and Systems Engineering 21, 1--36.

[11]

D. Down, S. P. Meyn, and R. L. Tweedie. 1995. Exponential and uniform ergodicity of Markov processes. Annals of Probability 23, 4, 1671--1691.

[12]

O. Garnett, A. Mandelbaum, and M. Reiman. 2002. Designing a call center with impatient customers. Manufacturing & Service Operations Management 4, 3, 208--227.

Digital Library

[13]

P. W. Glynn and S. P. Meyn. 1996. A Liapounov bound for solutions of the Poisson equation. Annals of Probability 24, 916--931.

[14]

Shlomo Halfin and Ward Whitt. 1981. Heavy-traffic limits for queues with many exponential servers. Operations Research 29, 3, 567--588.

Digital Library

[15]

J. M. Harrison. 1990. Brownian Motion and Stochastic Flow Systems (2nd ed.). Krieger, Malabar, FL.

[16]

D. L. Iglehart and W. Whitt. 1970a. Multichannel queues in heavy traffic I. Advances in Applied Probability 2 (1970), 150--177.

[17]

D. L. Iglehart and W. Whitt. 1970b. Multichannel queues in heavy traffic II: sequences, networks, and batches. Advances in Applied Probability 355--369.

[18]

S. Karlin and H. M. Taylor. 1975. A First Course in Stochastic Processes (2nd ed.). Academic Press, Boston, MA.

[19]

S. Karlin and H. M. Taylor. 1981. A Second Course in Stochastic Processes. Academic Press, Boston, MA.

[20]

P. Mandl. 1968. Analytical Treatment of One-Dimensional Markov Processes. Springer-Verlag, New York, NY.

[21]

S. P. Meyn and R. L. Tweedie. 1993a. Markov Chains and Stochastic Stability. Springer-Verlag, London.

[22]

S. P. Meyn and R. L. Tweedie. 1993b. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Advances in Applied Probability 25, 518--548.

[23]

B. L. Nelson. 2013. Foundations and Methods of Stochastic Simulation. International Series in Operations Research & Management Science, Vol. 187. Springer, New York, NY.

Digital Library

[24]

Sidney I. Resnick. 1992. Adventures in Stochastic Processes. Birkhäuser, Boston, MA.

Digital Library

[25]

Rayadurgam Srikant and Ward Whitt. 1996. Simulation run lengths to estimate blocking probabilities. ACM Transactions on Modeling and Computer Simulation 6, 1, 7--52.

Digital Library

[26]

Samuel G. Steckley and Shane G. Henderson. 2006. The error in steady-state approximations for the time-dependent waiting time distribution. Stochastic Models 23, 307--332.

[27]

N. M. Steiger, E. K. Lada, J. R. Wilson, J. A. Joines, C. Alexopoulos, and D. Goldsman. 2005. ASAP3: A batch means procedure for steady-state simulation analysis. ACM Transactions on Modeling and Computer Simulation 15, 1, 39--73.

Digital Library

[28]

R. J. Wang and P. W. Glynn. 2014. On the Marginal Standard Error Rule and the testing of initial transient deletion methods. Submitted for publication 2014.

[29]

Ward Whitt. 1989. Planning queueing simulations. Management Science 35, 1341--1366.

Digital Library

[30]

Ward Whitt. 1992. Asymptotic formulas for Markov processes with applications to simulation. Operations Research 40, 2, 279--291.

Digital Library

[31]

Ward Whitt. 2002. Stochastic-Process Limits. Springer, New York, NY.

[32]

Ward Whitt. 2006. Analysis for design. In Handbook of Simulation, S. G. Henderson and B. L. Nelson (Eds.). Elsevier, Amsterdam, The Netherlands, 381--413.

[33]

R. W. Wolff. 1989. Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.

[34]

Sergey Zeltyn and Avishai Mandelbaum. 2005. Call centers with impatient customers: Many-server asymptotics of the M/M/n + G queue. Queueing Systems: Theory and Applications 51, 3--4, 361--402.

Digital Library

Cited By

Liu WSun X(2022)Energy-Aware and Delay-Sensitive Management of a Drone Delivery SystemManufacturing & Service Operations Management10.1287/msom.2021.105624:3(1294-1310)Online publication date: May-2022
https://doi.org/10.1287/msom.2021.1056
Niño-Mora J(2020)Solving Poisson’s equation for birth-death chains: Structure, instability, and accurate approximationPerformance Evaluation10.1016/j.peva.2020.102163(102163)Online publication date: Oct-2020
https://doi.org/10.1016/j.peva.2020.102163
Sun XWhitt W(2018)Creating Work Breaks from Available IdlenessManufacturing & Service Operations Management10.1287/msom.2017.068220:4(721-736)Online publication date: 1-Oct-2018
https://dl.acm.org/doi/10.1287/msom.2017.0682
Show More Cited By

Index Terms

How Hard are Steady-State Queueing Simulations?
1. Computing methodologies
  1. Modeling and simulation
    1. Simulation evaluation
2. Mathematics of computing
  1. Probability and statistics

Recommendations

Validity of heavy-traffic steady-state approximations in many-server queues with abandonment

We consider $$GI/Ph/n+M$$ G I / P h / n + M parallel-server systems with a renewal arrival process, a phase-type service time distribution, $$n$$ n homogenous servers, and an exponential patience time distribution with positive rate. We show that in the Halfin---Whitt regime, the sequence of ...
Diffusion approximations for double-ended queues with reneging in heavy traffic

We study a double-ended queue consisting of two classes of customers. Whenever there is a pair of customers from both classes, they are matched and leave the system. The matching is instantaneous following the first-come---first-match principle. If a ...
On a Preemptive Markovian Queue with Multiple Servers and Two Priority Classes

We consider a queueing system with multiple servers and two classes of customers operating under a preemptive resume priority rule. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Modeling and Computer Simulation

ACM Transactions on Modeling and Computer Simulation Volume 25, Issue 4

Special Issue on Don Iglehart

November 2015

139 pages

ISSN:1049-3301

EISSN:1558-1195

DOI:10.1145/2774955

Editor:
Adelinde M. Uhrmacher
University of Rostock, Germany

Issue’s Table of Contents

Copyright © 2015 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 16 November 2015

Accepted: 01 March 2015

Revised: 01 January 2015

Received: 01 July 2013

Published in TOMACS Volume 25, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

National Science Foundation

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
124
Total Downloads

Downloads (Last 12 months)11
Downloads (Last 6 weeks)0

Reflects downloads up to 28 Jul 2024

Other Metrics

View Author Metrics

Citations

Cited By

Liu WSun X(2022)Energy-Aware and Delay-Sensitive Management of a Drone Delivery SystemManufacturing & Service Operations Management10.1287/msom.2021.105624:3(1294-1310)Online publication date: May-2022
https://doi.org/10.1287/msom.2021.1056
Niño-Mora J(2020)Solving Poisson’s equation for birth-death chains: Structure, instability, and accurate approximationPerformance Evaluation10.1016/j.peva.2020.102163(102163)Online publication date: Oct-2020
https://doi.org/10.1016/j.peva.2020.102163
Sun XWhitt W(2018)Creating Work Breaks from Available IdlenessManufacturing & Service Operations Management10.1287/msom.2017.068220:4(721-736)Online publication date: 1-Oct-2018
https://dl.acm.org/doi/10.1287/msom.2017.0682
Liu WSun X(undefined)Dynamic Scheduling of a Battery-Operated QueueSSRN Electronic Journal10.2139/ssrn.3542316
https://doi.org/10.2139/ssrn.3542316

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents