Abstract
A simple bound in GI/G/1 queues was obtained by Kingman using a discrete martingale transform. We extend this technique to 1) multiclass Σ\textrmGI/G/1 $ queues and 2) Markov Additive Processes (MAPs) whose background processes can be time-inhomogeneous or have an uncountable state-space. Both extensions are facilitated by a necessary and sufficient ordinary differential equation (ODE) condition for MAPs to admit continuous martingale transforms. Simulations show that the bounds on waiting time distributions are almost exact in heavy-traffic, including the cases of 1) heterogeneous input, e.g., mixing Weibull and Erlang-k classes and 2) Generalized Markovian Arrival Processes, a new class extending the Batch Markovian Arrival Processes to continuous batch sizes.
- Nail Akar and Khosrow Sohraby. 2004. Infinite- and Finite-Buffer Markov Fluid Queues: A Unified Analysis. Journal of Applied Probability , Vol. 41, 2 (2004), 557--569.Google Scholar
Cross Ref
- David Anick, Debasis Mitra, and Man M. Sondhi. 1982. Stochastic Theory of a Data-Handling System with Multiple Sources. Bell Systems Technical Journal , Vol. 61, 8 (Oct. 1982), 1871--1894.Google Scholar
Cross Ref
- Søren Asmussen. 1995. Stationary Distributions via First Passage Times. In Advances in Queueing Theory: Theory, Methods and Open Problems (Editor J. H. Dshalalow). CRC Press, Inc., 79--102.Google Scholar
- Søren Asmussen. 2003. Applied Probability and Queues. Springer.Google Scholar
- Søren Asmussen and Tomasz Rolski. 1994. Risk Theory in a Periodic Environment: The Cramér-Lundberg Approximation and Lundberg's Inequality. Mathematics of Operations Research , Vol. 19, 2 (May 1994), 410--433. Google Scholar
Digital Library
- Demitris Bertsimas and Georgia Mourtzinou. 1997. Multiclass Queueing Systems in Heavy Traffic: An Asymptotic Approach Based on Distributional and Conservation Laws. Operations Research , Vol. 45, 3 (June 1997), 470--487. Google Scholar
Digital Library
- Jean-Yves noopsortBoudec Le Boudec and Patrick Thiran. 2001. Network Calculus. Springer Verlag, Lecture Notes in Computer Science, LNCS 2050.Google Scholar
- Emmanuel Buffet and Nick G. Duffield. 1994. Exponential Upper Bounds via Martingales for Multiplexers with Markovian Arrivals. Journal of Applied Probability , Vol. 31, 4 (Dec. 1994), 1049--1060.Google Scholar
Cross Ref
- Cheng-Shang Chang. 2000. Performance Guarantees in Communication Networks .Springer Verlag. Google Scholar
Digital Library
- Cheng-Shang Chang and Jay Cheng. 1995. Computable Exponential Bounds for Intree Networks with Routing. In Proceedings of IEEE Infocom . 197--204. Google Scholar
Digital Library
- Joseph T. Chang. 1994. Inequalities for the overshoot. The Annals of Applied Probability , Vol. 4, 4 (Nov. 1994), 1223--1233.Google Scholar
Cross Ref
- Julian Cheng, Chintha Tellambura, and Norman C. Beaulieu. 2004. Performance of Digital Linear Modulations on Weibull Slow-Fading Channels. IEEE Transactions on Communications , Vol. 52, 8 (Aug 2004), 1265--1268.Google Scholar
Cross Ref
- Gagan L. Choudhury, David M. Lucantoni, and Ward Whitt. 1996. Squeezing the Most out of textATM . IEEE Transactions on Communications , Vol. 44, 2 (Feb. 1996), 203--217.Google Scholar
Cross Ref
- Erhan cC inlar. 1972. Markov Additive Processes. I. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete , Vol. 24, 2 (1972), 85--93.Google Scholar
Cross Ref
- Florin Ciucu, Felix Poloczek, and Jens Schmitt. 2014. Sharp Per-Flow Delay Bounds for Bursty Arrivals: The Case of FIFO, SP, and EDF Scheduling. In IEEE Infocom. 1896--1904.Google Scholar
- Florin Ciucu, Felix Poloczek, and Jens Schmitt. 2016. Stochastic Upper and Lower Bounds for General Markov Fluids. In International Teletraffic Congress (ITC) .Google Scholar
- Jim G. Dai and Thomas G. Kurtz. 1995. A multiclass Station with Markovian Feedback in Heavy Traffic. Mathematics of Operations Research , Vol. 20, 3 (Aug. 1995), 721--742.Google Scholar
Cross Ref
- Nick G. Duffield. 1994. Exponential Bounds for Queues with Markovian Arrivals. Queueing Systems , Vol. 17, 3--4 (Sept. 1994), 413--430.Google Scholar
Cross Ref
- Anwar I. Elwalid and Debasis Mitra. 1993. Effective bandwidth of General Markovian Traffic Sources and Admission Control of High Speed Networks. IEEE/ACM Transactions on Networking , Vol. 1, 3 (June 1993), 329--343. Google Scholar
Digital Library
- Anwar I. Elwalid, Debasis Mitra, and Thomas E. Stern. 1991. Statistical Multiplexing of Markov-Modulated Sources: Theory and Computational Algorithms. In International Teletraffic Congress (ITC) .Google Scholar
- Stewart N. Ethier and Thomas G. Kurtz. 1986. Markov processes -- Characterization and Convergence .John Wiley & Sons Inc.Google Scholar
- Youjian Fang, Michael Devetsikiotis, Ioannis Lambadaris, and A. Roger Kaye. 1995. Exponential Bounds for the Waiting Time Distribution in Markovian Queues, with Applications to TES/GI/1 Systems. In Proceedings of the 1995 ACM SIGMETRICS Joint International Conference on Measurement and Modeling of Computer Systems (SIGMETRICS '95/PERFORMANCE '95). 108--115. Google Scholar
Digital Library
- Wolfgang Fischer and Kathleen Meier-Hellstern. 1993. The Markov-Modulated Poisson Process (MMPP) Cookbook. Performance Evaluation , Vol. 18, 2 (1993), 149 -- 171. Google Scholar
Digital Library
- Qi-Ming He and Attahiru A. Alfa. 2018. Space Reduction for a Class of Multidimensional Markov Chains: A Summary and Some Applications. INFORMS Journal on Computing , Vol. 30, 1 (2018), 1--10.Google Scholar
Digital Library
- Harry Heffes and David M. Lucantoni. 1986. A Markov Modulated Characterization of Packetized Voice and Data Traffic and Related Statistical Multiplexer Performance. IEEE Journal on Selected Areas in Communications , Vol. 4, 6 (Sept. 1986), 856--867. Google Scholar
Digital Library
- Roger Horn and Charles R. Johnson. 1991. Topics in Matrix Analysis .Cambridge University Press. Google Scholar
Digital Library
- Donald L. Iglehart and Ward Whitt. 1970. Multiple Channel Queues in Heavy Traffic. I . Advances in Applied Probability , Vol. 2, 1 (Spring 1970), 150--177.Google Scholar
Cross Ref
- Yuming Jiang and Yong Liu. 2008. Stochastic Network Calculus .Springer. Google Scholar
Digital Library
- Yuming Jiang and Vishal Misra. 2017. Delay Bounds for Multiclass FIFO . CoRR , Vol. abs/1605.05753v2 (2017). https://arxiv.org/abs/1605.05753v2Google Scholar
- John F. C. Kingman. 1964. A Martingale Inequality in the Theory of Queues. Cambridge Philosophical Society , Vol. 60, 2 (April 1964), 359--361.Google Scholar
Cross Ref
- John F. C. Kingman. 1970. Inequalities in the Theory of Queues. Journal of the Royal Statistical Society, Series B , Vol. 32, 1 (1970), 102--110.Google Scholar
- Leendert Kosten. 1974. Stochastic Theory of a Multi-Entry Buffer I . Delft Progress Report, Series F. University of Delft. 10--18 pages.Google Scholar
- Vidyadhar G. Kulkarni. 1997. Fluid Models for Single Buffer Systems. In Frontiers in Queueing: Models and Applications in Science and Engineering. (Editor J. H. Dshalalow). CRC Press, Inc., 321--338. Google Scholar
Digital Library
- Chengzhi Li, Almut Burchard, and Jörg Liebeherr. 2007. A Network Calculus with Effective Bandwidth. IEEE/ACM Transactions on Networking , Vol. 15, 6 (Dec. 2007), 1442--1453. Google Scholar
Digital Library
- Zhen Liu, Philippe Nain, and Don Towsley. 1994. On a Generalization of Kingman's Bounds. Technical Report No. 2423, INRIA.Google Scholar
- Zhen Liu, Philippe Nain, and Don Towsley. 1997. Exponential Bounds with Applications to Call Admission. J. ACM , Vol. 44, 3 (May 1997), 366--394. Google Scholar
Digital Library
- David M. Lucantoni. 1993. The BMAP/G/1 Queue: A Tutorial. In Performance Evaluation of Computer and Communication Systems, Lorenzo Donatiello and Randolph Nelson (Eds.). Springer, 330--358. Google Scholar
Digital Library
- Edward W. Ng and Murray Geller. 1969. A Table of Integrals of the Error Functions. J. Res. Natl. Bur. Stand., Sec. B: Math. Sci. , Vol. 73B, 1 (Jan.-Mar. 1969), 1--20.Google Scholar
Cross Ref
- António Pacheco and Narahari U. Prabhu. 1995. Markov-Additive Processes of Arrivals. In Advances in Queueing Theory: Theory, Methods and Open Problems (Editor J. H. Dshalalow). CRC Press, Inc., 167--194.Google Scholar
- Zbigniew Palmowski and Tomasz Rolski. 1996. A Note on Martingale Inequalities for Fluid Models. Statistics & Probability Letters , Vol. 31, 1 (Dec. 1996), 13--21.Google Scholar
Cross Ref
- Felix Poloczek and Florin Ciucu. 2014. Scheduling Analysis with Martingales. Performance Evaluation (Special Issue: IFIP Performance 2014) , Vol. 79 (Sept. 2014), 56 -- 72.Google Scholar
Cross Ref
- Felix Poloczek and Florin Ciucu. 2015. Service-Martingales: Theory and Applications to the Delay Analysis of Random Access Protocols. In IEEE Infocom. 945--953.Google Scholar
- Vaidyanathan Ramaswami. 1985. Independent Markov Processes in Parallel. Communications in Statistics. Stochastic Models , Vol. 1, 3 (1985), 419--432.Google Scholar
Cross Ref
- Philippe Robert. 2003. Stochastic Networks and Queues .Springer.Google Scholar
- Leonard C. G. Rogers. 1994. Fluid Models in Queueing Theory and Wiener-Hopf Factorization of Markov Chains. The Annals of Applied Probability , Vol. 4, 2 (May 1994), 390--413.Google Scholar
Cross Ref
- Sheldon M. Ross. 1974. Bounds on the Delay Distribution in GI/G/1 queues. Journal of Applied Probability , Vol. 11, 2 (June 1974), 417--421.Google Scholar
Cross Ref
- Sanjay Shakkottai and Rayadurgam Srikant. 2000. Delay Asymptotics for a Priority Queueing System. In ACM Sigmetrics. 188--195. Google Scholar
Digital Library
- Ness B. Shroff and Mischa Schwartz. 1998. Improved Loss Calculations at an textA™ Multiplexer. IEEE/ACM Transactions on Networking , Vol. 6, 4 (Aug. 1998), 411--421. Google Scholar
Digital Library
- Stanislaw J. Szarek and Elisabeth Werner. 1999. A Nonsymmetric Correlation Inequality for Gaussian Measure. Journal of Multivariate Analysis , Vol. 68, 2 (Feb 1999), 193--211. Google Scholar
Digital Library
- David Williams. 1991. Probability with Martingales .Cambridge University Press.Google Scholar
- Dapeng Wu and Rohit Negi. 2003. Effective Capacity: A Wireless Link Model for Support of Quality of Service. IEEE Transactions on Wireless Communication , Vol. 2, 4 (July 2003), 630--643. Google Scholar
Digital Library
Index Terms
Two Extensions of Kingman's GI/G/1 Bound
Recommendations
Queue and Loss Distributions in Finite-Buffer Queues
We derive simple bounds on the queue distribution in finite-buffer queues with Markovian arrivals. Our technique relies on a subtle equivalence between tail events and stopping times orderings. The bounds capture a truncated exponential behavior, ...
Two Extensions of Kingman's GI/G/1 Bound
SIGMETRICS '19: Abstracts of the 2019 SIGMETRICS/Performance Joint International Conference on Measurement and Modeling of Computer SystemsA simple bound in GI/G/1 queues was obtained by Kingman using a discrete martingale transform. We extend this technique to 1) multiclass ΣGI/G/1 queues and 2) Markov Additive Processes (MAPs) whose background processes can be time-inhomogeneous or have ...
Two Extensions of Kingman's GI/G/1 Bound
A simple bound in GI/G/1 queues was obtained by Kingman using a discrete martingale transform [5]. We extend this technique to 1) multiclass (GI/G/1 queues and 2) Markov Additive Processes (MAPs) whose background processes can be time-inhomogeneous or ...






Comments