On the asymptotic relationship between the overflow probability and the loss ratio

In this paper we study the asymptotic relationship between the loss ratio in a finite buffer system and the overflow probability (the tail of the queue length distribution) in the corresponding infinite buffer system. We model the system by a fluid queue which consists of a server with constant rate c and a fluid input. We provide asymptotic upper and lower bounds on the difference between log P{Q > x} and logP L (x) under different conditions. The conditions for the upper bound are simple and are satisfied by a very large class of input processes. The conditions on the lower bound are more complex but we show that various classes of processes such as Markov modulated and ARMA type Gaussian input processes satisfy them.

[1]  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.

[2]  Vincent Hodgson,et al.  The Single Server Queue. , 1972 .

[3]  D. Mitra,et al.  Stochastic theory of a data-handling system with multiple sources , 1982, The Bell System Technical Journal.

[4]  Ward Whitt,et al.  Characterizing Superposition Arrival Processes in Packet Multiplexers for Voice and Data , 1986, IEEE J. Sel. Areas Commun..

[5]  Joseph D. Langford,et al.  Models for analysis of packet voice communications systems , 1986, IEEE J. Sel. Areas Commun..

[6]  Ward Whitt,et al.  Limits for queues as the waiting room grows , 1989, Queueing Syst. Theory Appl..

[7]  W. D. Ray.  Stationary Stochastic Models , 1991 .

[8]  Anwar I. Elwalid.  Markov-modulated rate processes for modeling, analysis and control of communication networks , 1992 .

[9]  P. Franken,et al.  Stationary Stochastic Models. , 1992 .

[10]  P. Glynn,et al.  Logarithmic asymptotics for steady-state tail probabilities in a single-server queue , 1994, Journal of Applied Probability.

[11]  W. Whitt,et al.  Asymptotics for steady-state tail probabilities in structured markov queueing models , 1994 .

[12]  An approximation for performance evaluation of stationary single server queues , 1994, IEEE Trans. Commun..

[13]  Cheng-Shang Chang,et al.  Stability, queue length, and delay of deterministic and stochastic queueing networks , 1994, IEEE Trans. Autom. Control..

[14]  Nick Duffield,et al.  Large deviations and overflow probabilities for the general single-server queue, with applications , 1995 .

[15]  Ravi Mazumdar,et al.  Cell loss asymptotics in buffers fed with a large number of independent stationary sources , 1998, Proceedings. IEEE INFOCOM '98, the Conference on Computer Communications. Seventeenth Annual Joint Conference of the IEEE Computer and Communications Societies. Gateway to the 21st Century (Cat. No.98.

[16]  Ness B. Shroff,et al.  Improved loss calculations at an ATM multiplexer , 1998, TNET.

[17]  R. Mazumdar,et al.  Cell loss asymptotics for buffers fed with a large number of independent stationary sources , 1999 .

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

[19]  Ness B. Shroff,et al.  On the supremum distribution of integrated stationary Gaussian processes with negative linear drift , 1999, Advances in Applied Probability.

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

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