Queue and Loss Distributions in Finite-Buffer Queues

We derive simple bounds on the queue distribution in finite-buffer queues with Markovian arrivals. The bounds capture a truncated exponential behavior, involving joint horizontal and vertical shifts of an exponential function; this is fundamentally different than existing results capturing horizontal shifts only. We also obtain similar bounds on the loss distribution, which is a key metric to understand the impact of finite-buffer queues on real-time applications. Simulations show that the bounds are accurate in heavy-traffic regimes, and improve existing ones by orders of magnitude.

[1]  A. Zwart,et al.  ON AN EQUIVALENCE BETWEEN LOSS RATES AND CYCLE MAXIMA IN QUEUES AND DAMS , 2005, Probability in the Engineering and Informational Sciences.

[2]  Søren Asmussen,et al.  Monotone Stochastic Recursions and their Duals , 1996, Probability in the Engineering and Informational Sciences.

[3]  Predrag R. Jelenkovic,et al.  Subexponential loss rates in a GI/GI/1 queue with applications , 1999, Queueing Syst. Theory Appl..

[4]  Tatsuya Suda,et al.  Cell loss and output process analyses of a finite-buffer discrete-time ATM queueing system with correlated arrivals , 1995, IEEE Trans. Commun..

[5]  Tao Yang,et al.  A novel approach to estimating the cell loss probability in an ATM multiplexer loaded with homogeneous on-off sources , 1995, IEEE Trans. Commun..

[6]  A. Zwart.  A fluid queue with a finite buffer and subexponential input , 2000, Advances in Applied Probability.

[7]  Wenyu Jiang,et al.  Modeling of Packet Loss and Delay and Their Effect on Real-Time Multimedia Service Quality , 2000 .

[8]  Israel Cidon,et al.  Analysis of packet loss processes in high-speed networks , 1993, IEEE Trans. Inf. Theory.

[9]  San-Qi Li,et al.  Study of information loss in packet voice systems , 1989, IEEE Trans. Commun..

[10]  Bert Zwart,et al.  Loss rates in the single-server queue with complete rejection , 2015, Math. Methods Oper. Res..

[11]  Bruno Sericola.  A Finite Buffer Fluid Queue Driven by a Markovian Queue , 2001, Queueing Syst. Theory Appl..

[12]  Aurel A. Lazar,et al.  Monitoring the packet gap of real-time packet traffic , 1992, Queueing Syst. Theory Appl..

[13]  Chris Blondia,et al.  Statistical Multiplexing of VBR Sources: A Matrix-Analytic Approach , 1992, Perform. Evaluation.

[14]  Don Towsley,et al.  Packet loss correlation in the MBone multicast network , 1996, Proceedings of GLOBECOM'96. 1996 IEEE Global Telecommunications Conference.

[15]  Donald F. Towsley,et al.  Approximation Techniques for Computing Packet Loss in Finite-Buffered Voice Multiplexers , 1991, IEEE J. Sel. Areas Commun..

[16]  Tetsuya Takine,et al.  Loss probability in a finite discrete-time queue in terms of the steady state distribution of an infinite queue , 1999, Queueing Syst. Theory Appl..

[17]  László Pap,et al.  An Efficient Bandwidth Assignment Algorithm for Real-Time Traffic in ATM Networks , 1997, Modelling and Evaluation of ATM Networks.

[18]  F. Baccelli,et al.  Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences , 2010 .

[19]  Rhonda Righter.  A NOTE ON LOSSES IN M/GI/1/n QUEUES , 1999 .

[20]  Jean-Chrysostome Bolot,et al.  End-to-end packet delay and loss behavior in the internet , 1993, SIGCOMM '93.

[21]  J. Kingman.  A martingale inequality in the theory of queues , 1964 .

[22]  Wojciech M. Kempa,et al.  A comprehensive study on the queue-size distribution in a finite-buffer system with a general independent input flow , 2017, Perform. Evaluation.

[23]  Peter W. Glynn,et al.  Lévy Matters V , 2015 .

[24]  M. Handley.  An Examination of MBone Performance , 1997 .

[25]  P. Taylor.  Insensitivity in Stochastic Models , 2011 .

[26]  Kyung C. Chae,et al.  Transform-free analysis of the GI/G/1/K queue through the decomposed Little's formula , 2003, Comput. Oper. Res..

[27]  Roger C. F. Tucker,et al.  Accurate method for analysis of a packet-speech multiplexer with limited delay , 1988, IEEE Trans. Commun..

[28]  N. Duffield,et al.  Exponential upper bounds via martingales for multiplexers with Markovian arrivals , 1994 .

[29]  De Kok,et al.  Asymptotic results for buffer systems under heavy load , 1987 .

[30]  U. C. Gupta,et al.  Computing Steady State Probabilities in Lambda(n)/G/1/K Queue , 1996, Perform. Evaluation.

[31]  Dieter Fiems,et al.  Packet loss characteristics for M/G/1/N queueing systems , 2009, Ann. Oper. Res..

[32]  L. Brown,et al.  Interval Estimation for a Binomial Proportion , 2001 .

[33]  R. M. Loynes,et al.  The stability of a queue with non-independent inter-arrival and service times , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[34]  Catherine Rosenberg,et al.  A survey of straightforward statistical multiplexing models for ATM networks , 1996, Telecommun. Syst..

[35]  Felix Poloczek,et al.  Scheduling analysis with martingales , 2014, Perform. Evaluation.

[36]  Rajeev Koodli,et al.  One-way Loss Pattern Sample Metrics , 2002, RFC.

[37]  K. Arrow.  The Demand for Information and the Distribution of Income , 1987, Probability in the Engineering and Informational Sciences.

[38]  Peter W. Glynn,et al.  Levy Processes with Two-Sided Reflection , 2015 .

[39]  Henning Schulzrinne,et al.  Loss correlation for queues with bursty input streams , 1992, [Conference Record] SUPERCOMM/ICC '92 Discovering a New World of Communications.

[40]  D. Siegmund.  The Equivalence of Absorbing and Reflecting Barrier Problems for Stochastically Monotone Markov Processes , 1976 .

[41]  Israel Cidon,et al.  The ballot theorem strikes again: Packet loss process distribution , 2000, IEEE Trans. Inf. Theory.

[42]  David M. Lucantoni,et al.  The BMAP/G/1 QUEUE: A Tutorial , 1993, Performance/SIGMETRICS Tutorials.

[43]  Yashar Ghiassi-Farrokhfal,et al.  On the impact of finite buffers on per-flow delays in FIFO queues , 2012, 2012 24th International Teletraffic Congress (ITC 24).

[44]  Henk C. Tijms,et al.  Computing Loss Probabilities in Discrete-Time Queues , 1998, Oper. Res..

[45]  C. Blondia.  The n/g/l finite capacity queue , 1989 .

[46]  Nick G. Duffield,et al.  Exponential bounds for queues with Markovian arrivals , 1994, Queueing Syst. Theory Appl..

[47]  Marco Listanti,et al.  Loss Performance Analysis of an ATM Multiplexer Loaded with High-Speed ON-OFF Sources , 1991, IEEE J. Sel. Areas Commun..

[48]  Harshinder Singh,et al.  Reliability Properties of Reversed Residual Lifetime , 2003 .

[49]  Ness B. Shroff,et al.  Loss probability calculations and asymptotic analysis for finite buffer multiplexers , 2001, TNET.

[50]  Jean-Yves Le Boudec,et al.  Network Calculus: A Theory of Deterministic Queuing Systems for the Internet , 2001 .

[51]  Ward Whitt,et al.  A Diffusion Approximation for the G/GI/n/m Queue , 2004, Oper. Res..

[52]  M. Pihlsgård.  Loss Rate Asymptotics in a GI/G/1 Queue with Finite Buffer , 2005 .

[53]  W. Stadje,et al.  A new look at the Moran dam , 1993, Journal of Applied Probability.

[54]  S. Ross.  Bounds on the delay distribution in GI/G/1 queues , 1974, Journal of Applied Probability.

[55]  David Cronshaw,et al.  On using a hardware monitor as an intelligent peripheral , 1973, PERV.

[56]  Felix Poloczek,et al.  Sharp per-flow delay bounds for bursty arrivals: The case of FIFO, SP, and EDF scheduling , 2014, IEEE INFOCOM 2014 - IEEE Conference on Computer Communications.

[57]  J. Keilson.  The Ergodic Queue Length Distribution for Queueing Systems with Finite Capacity , 1966 .

[58]  On a property of a refusals stream , 1997 .

[59]  Jay Cheng,et al.  Computable exponential bounds for intree networks with routing , 1995, Proceedings of INFOCOM'95.

[60]  Majid Raeis,et al.  Analysis of the Leakage Queue: A Queueing Model for Energy Storage Systems with Self-discharge , 2017, ArXiv.

[61]  Haining Liu.  Buffer size and packet loss in a tandem queueing network , 1993 .

[62]  Richard A. Davis,et al.  The convex hull of a random sample in , 1987 .

[63]  Felix Poloczek,et al.  Two Extensions of Kingman's GI/G/1 Bound , 2018, Proc. ACM Meas. Anal. Comput. Syst..

[64]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[65]  Mohan L. Chaudhry,et al.  On exact computational analysis of distributions of numbers in systems for M/G/1/N + 1 and GI/M/1/N + 1 queues using roots , 1991, Comput. Oper. Res..