Tail Asymptotics for HOL Priority Queues Handling a Large Number of Independent Stationary Sources

In this paper we study the asymptotics of the tail of the buffer occupancy distribution in buffers accessed by a large number of stationary independent sources and which are served according to a strict HOL priority rule. As in the case of single buffers, the results are valid for a very general class of sources which include long-range dependent sources with bounded instantaneous rates. We first consider the case of two buffers with one of them having strict priority over the other and we obtain asymptotic upper bound for the buffer tail probability for lower priority buffer. We discuss the conditions to have asymptotic equivalents. The asymptotics are studied in terms of a scaling parameter which reflects the server speed, buffer level and the number of sources in such a way that the ratios remain constant. The results are then generalized to the case of M buffers which leads to the source pooling idea. We conclude with numerical validation of our formulae against simulations which show that the asymptotic bounds are tight. We also show that the commonly suggested reduced service rate approximation can give extremely low estimates.

[1]  Jean C. Walrand,et al.  Effective bandwidths for multiclass Markov fluids and other ATM sources , 1993, TNET.

[2]  Alain Simonian,et al.  Large Deviations Approximation for Fluid Queues Fed by a Large Number of on/off Sources , 1994 .

[3]  V. V. Petrov.  Sums of Independent Random Variables , 1975 .

[4]  Walter Willinger,et al.  On the self-similar nature of Ethernet traffic , 1993, SIGCOMM '93.

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

[6]  Nick G. Duffield,et al.  Large deviations, the shape of the loss curve, and economies of scale in large multiplexers , 1995, Queueing Syst. Theory Appl..

[7]  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.

[8]  David L. Black,et al.  An Architecture for Differentiated Service , 1998 .

[9]  R. Weber,et al.  Buffer overflow asymptotics for a buffer handling many traffic sources , 1996, Journal of Applied Probability.

[10]  Ward Whitt,et al.  Effective bandwidths with priorities , 1998, TNET.

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

[12]  Joseph Y. Hui.  Resource allocation for broadband networks , 1988, IEEE J. Sel. Areas Commun..

[13]  Onno J. Boxma,et al.  Fluid queues with long-tailed activity period distributions , 1997, Comput. Commun..

[14]  R. R. Bahadur,et al.  On Deviations of the Sample Mean , 1960 .

[15]  Roderick Wong,et al.  Asymptotic approximations of integrals , 1989, Classics in applied mathematics.

[16]  Frank Kelly,et al.  Notes on effective bandwidths , 1994 .

[17]  Stephen R. E. Turner,et al.  Large Deviations for Join the Shorter Queue , 1999 .

[18]  Neil O'Connell.  Large deviations for queue lengths at a multi-buffered resource , 1998 .

[19]  John N. Tsitsiklis,et al.  Asymptotic buffer overflow probabilities in multiclass multiplexers: an optimal control approach , 1998, IEEE Trans. Autom. Control..

[20]  R. Srikant,et al.  Many-Sources Delay Asymptotics with Applications to Priority Queues , 2001, Queueing Syst. Theory Appl..