Asymptotics of palm-stationary buffer content distributions in fluid flow queues

We study a fluid flow queueing system with m independent sources alternating between periods of silence and activity; m ≥ 2. The distribution function of the activity periods of one source, is supposed to be intermediate regular varying. We show that the distribution of the net increment of the buffer during an aggregate activity period (i.e. when at least one source is active) is asymptotically tail-equivalent to the distribution of the net input during a single activity period with intermediate regular varying distribution function. In this way, we arrive at an asymptotic representation of the Palm-stationary tail-function of the buffer content at the beginning of aggregate activity periods. Our approach is probabilistic and extends recent results of Boxma (1996; 1997) who considered the special case of regular variation.

[1]  A. Lazar,et al.  Asymptotic results for multiplexing subexponential on-off processes , 1999, Advances in Applied Probability.

[2]  E. Pitman Subexponential distribution functions , 1980 .

[3]  Onno Boxma Regular variation in a multi-source fluid queue , 1996 .

[4]  Onno J. Boxma Fluid Queues and Regular Variation , 1996, Perform. Evaluation.

[5]  Jacob Cohen,et al.  On regenerative processes in queueing theory , 1976 .

[6]  A. Pakes ON THE TAILS OF WAITING-TIME DISTRIBUTIONS , 1975 .

[7]  Charles M. Goldie,et al.  Subexponentiality and infinite divisibility , 1979 .

[8]  Jacob Cohen On the effective bandwidth in buffer design for the multi-server channels , 1994 .

[9]  T. V. Lakshman,et al.  Source models for VBR broadcast-video traffic , 1996, TNET.

[10]  Gennady Samorodnitsky,et al.  Heavy Tails and Long Range Dependence in On/Off Processes and Associated Fluid Models , 1998, Math. Oper. Res..

[11]  P. Embrechts,et al.  Estimates for the probability of ruin with special emphasis on the possibility of large claims , 1982 .

[12]  Gennady Samorodnitsky,et al.  Patterns of buffer overflow in a class of queues with long memory in the input stream , 1997 .

[13]  Zbigniew Palmowski,et al.  A note on martingale inequalities for fluid models , 1996 .

[14]  Ward Whitt,et al.  Long-Tail Buffer-Content Distributions in Broadband Networks , 1997, Perform. Evaluation.

[15]  Charles M. Goldie,et al.  On convolution tails , 1982 .

[16]  D. B. Cline,et al.  Intermediate Regular and Π Variation , 1994 .

[17]  Ward Whitt,et al.  Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues , 1993, Telecommun. Syst..

[18]  Daren B. H. Cline,et al.  Convolution tails, product tails and domains of attraction , 1986 .