research-article

Exact Sampling of Stationary and Time-Reversed Queues

Authors:

Aya WallwaterAuthors Info & Claims

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

Article No.: 26, Pages 1 - 27

https://doi.org/10.1145/2822892

Published: 16 November 2015 Publication History

Abstract

We provide the first algorithm that, under minimal assumptions, allows simulation of the stationary waiting-time sequence of a single-server queue backward in time, jointly with the input processes of the queue (interarrival and service times). The single-server queue is useful in applications of Dominated Coupling from the Past (DCFTP), which is a well-known protocol for simulation without bias from steady-state distributions. Our algorithm terminates in finite time, assuming only finite mean of the interarrival and service times. To simulate the single-server queue in stationarity until the first idle period in finite expected termination time, we require the existence of finite variance. This requirement is also necessary for such idle time (which is a natural coalescence time in DCFTP applications) to have finite mean. Thus, in this sense, our algorithm is applicable under minimal assumptions.

References

[1]

S. Asmussen. 2003. Applied Probability and Queues (2nd Ed.). Springer, New York, NY.

[2]

J. Blanchet and X. Chen. 2012. Steady-State Simulation for Reflected Brownian Motion and Related Networks. Retrieved October 5, 2015, from http://arxiv.org/pdf/1202.2062.pdf.

[3]

J. Blanchet and J. Dong. 2012. Sampling point processes on stable unbounded regions and exact simulation of queues. In Proceedings of the 2012 Winter Simulation Conferrence.

Digital Library

[4]

J. Blanchet and K. Sigman. 2011. On exact sampling of stochastic perpetuities. Journal of Applied Probability 48A, 165--182.

[5]

S. Connor and S. Kendall. 2014. Perfect Simulation of M/G/c Queues. Retrieved October 5, 2015, from http://arxiv.org/abs/1402.7248.

[6]

R. Durrett. 2005. Probability: Theory and Examples. Duxbury Advanced Series.

Digital Library

[7]

K. B. Ensor and P. W. Glynn. 2000. Simulating the maximum of a random walk. Journal of Statistical Planning and Inference 85, 127--135.

[8]

S. Foss and A. Sapozhnikov. 2006. Convergence rates in monotone separable stochastic networks. Queueing Systems 52, 2, 125--137.

Digital Library

[9]

J. Blanchet, J. Dong, and Y. Pei. 2015. Perfect Sampling of GI/G/c Queues. Retrieved October 5, 2015, from http://www.columbia.edu/&sim;jb2814/papers/exact_multiserver_v12.pdf.

[10]

W. S. Kendall. 1998. Perfect simulation for the area-interaction point process. In Probability Towards 2000. Lecture Notes in Statistics, Vol. 128. Springer, 218--234.

[11]

J. F. C. Kingman and M. F. Atiyah. 1961. The single server queue in heavy traffic. Proceedings of the Cambridge Philosophical Society 57, 902.

[12]

K. Murthy, S. Juneja, and J. Blanchet. 2013. State-Independent Importance Sampling for Random Walks with Regularly Varying Increments. Retrieved October 5, 2015, from http://arxiv.org/pdf/1206.3390.pdf.

[13]

J. G. Propp and D. B. Wilson. 1996. Exact sampling with coupled Markov chains and applications to statistical mechanics. In Proceedings of the 7th International Conference on Random Structures and Algorithms. 223--252.

Digital Library

[14]

K. Sigman. 2011. Exact simulation of the stationary distribution of the FIFO M/G/c queue. Journal of Applied Probability 48A, 209--216.

[15]

K. Sigman. 2012. Exact simulation of the stationary distribution of the FIFO M/G/c queue: The general case for ρ < c. Queueing Systems 70, 37--43.

Digital Library

Cited By

Blanchet JChen LDong J(2022)Exact Sampling for the Maximum of Infinite Memory Gaussian ProcessesAdvances in Modeling and Simulation10.1007/978-3-031-10193-9_3(41-63)Online publication date: 1-Dec-2022
https://doi.org/10.1007/978-3-031-10193-9_3
Liu ZBlanchet JDieker AMikosch T(2019)On logarithmically optimal exact simulation of max-stable and related random fields on a compact setBernoulli10.3150/18-BEJ107625:4AOnline publication date: 1-Nov-2019
https://doi.org/10.3150/18-BEJ1076
Blanchet JChen X(2019)Perfect Sampling of Generalized Jackson NetworksMathematics of Operations Research10.1287/moor.2018.0941Online publication date: 19-Apr-2019
https://doi.org/10.1287/moor.2018.0941
Show More Cited By

Index Terms

Exact Sampling of Stationary and Time-Reversed Queues
1. Mathematics of computing
  1. Probability and statistics
2. Theory of computation
  1. Design and analysis of algorithms
    1. Algorithm design techniques
      1. Preconditioning
  2. Randomness, geometry and discrete structures
    1. Random walks and Markov chains

Recommendations

Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a ...
Discrete-time single-server finite-buffer queues under discrete Markovian arrival process with vacations

This paper treats a discrete-time single-server finite-buffer exhaustive (single- and multiple-) vacation queueing system with discrete-time Markovian arrival process (D-MAP). The service and vacation times are generally distributed random variables and ...
On the exact asymptotics for the stationary sojourn time distribution in a tandem of queues with light-tailed service times

We study the asymptotics of the stationary sojourn time Z of a "typical customer" in a tandem of single-server queues. It is shown that in a certain "intermediate" region of light-tailed service time distributions, Z may take a large value mostly ...

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 September 2015

Revised: 01 June 2015

Received: 01 August 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

NSF

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

10
Total Citations
View Citations
101
Total Downloads

Downloads (Last 12 months)5
Downloads (Last 6 weeks)1

Reflects downloads up to 12 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Blanchet JChen LDong J(2022)Exact Sampling for the Maximum of Infinite Memory Gaussian ProcessesAdvances in Modeling and Simulation10.1007/978-3-031-10193-9_3(41-63)Online publication date: 1-Dec-2022
https://doi.org/10.1007/978-3-031-10193-9_3
Liu ZBlanchet JDieker AMikosch T(2019)On logarithmically optimal exact simulation of max-stable and related random fields on a compact setBernoulli10.3150/18-BEJ107625:4AOnline publication date: 1-Nov-2019
https://doi.org/10.3150/18-BEJ1076
Blanchet JChen X(2019)Perfect Sampling of Generalized Jackson NetworksMathematics of Operations Research10.1287/moor.2018.0941Online publication date: 19-Apr-2019
https://doi.org/10.1287/moor.2018.0941
Blanchet JDong JLiu Z(2019)Exact sampling of the infinite horizon maximum of a random walk over a nonlinear boundaryJournal of Applied Probability10.1017/jpr.2019.956:01(116-138)Online publication date: 12-Jul-2019
https://doi.org/10.1017/jpr.2019.9
Blanchet JPei YSigman K(2019)Exact sampling for some multi-dimensional queueing models with renewal inputAdvances in Applied Probability10.1017/apr.2019.4551:4(1179-1208)Online publication date: 15-Nov-2019
https://doi.org/10.1017/apr.2019.45
Blanchet JDong JPei Y(2018)Perfect sampling of GI/GI/c queuesQueueing Systems: Theory and Applications10.1007/s11134-018-9573-290:1-2(1-33)Online publication date: 1-Oct-2018
https://dl.acm.org/doi/10.1007/s11134-018-9573-2
Dieker ALagos G(2017)A dichotomy for sampling barrier-crossing events of random walks with regularly varying tailsJournal of Applied Probability10.1017/jpr.2017.6054:04(1213-1232)Online publication date: 30-Nov-2017
https://doi.org/10.1017/jpr.2017.60
Glynn PHuschka TChick S(2016)Exact simulation vs exact estimationProceedings of the 2016 Winter Simulation Conference10.5555/3042094.3042130(193-205)Online publication date: 11-Dec-2016
https://dl.acm.org/doi/10.5555/3042094.3042130
Glynn P(2016)Exact simulation vs exact estimation2016 Winter Simulation Conference (WSC)10.1109/WSC.2016.7822089(193-205)Online publication date: Dec-2016
https://doi.org/10.1109/WSC.2016.7822089
Bouillard ABušić ARovetta C(2014)Perfect sampling for closed queueing networksPerformance Evaluation10.1016/j.peva.2014.07.01079(146-159)Online publication date: Sep-2014
https://doi.org/10.1016/j.peva.2014.07.010

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