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Loss asymptotics in large buffers fed by heterogeneous long-tailed sources

Part of: Special processes Operations research and management science Computer system organization

Published online by Cambridge University Press:  01 July 2016

Nikolay Likhanov  and
Ravi R. Mazumdar
Nikolay Likhanov*
Affiliation:
Institute for Problems of Information Transmission, Moscow
Ravi R. Mazumdar*
Affiliation:
Purdue University
*
∗ Postal address: Institute for Problems of Information Transmission, 19 Bolshoi Karetnyi, GSP-4, Moscow 101447, Russia.
∗∗ Postal address: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285, USA. Email address: mazum@ecn.purdue.edu
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Abstract

This paper presents the large-buffer asymptotics for a multiplexer which serves N types of heterogeneous sessions which have long-tailed session lengths. Specifically, the model considered is that sessions of type i ∈ {1,…,N} arrive as a Poisson process with rate λi. Each type of session (independently) remains active for a random duration, say τi, where P(τi > x) ~ αix-(1 + βi) for positive numbers αi and βi. While active, a session transmits at a rate ri. Under the assumption that the average load ρ = ∑Ni=1riλiE[τi] < C, where C denotes the server capacity, we show that both the tail distribution of the stationary buffer content and the loss asymptotics in finite buffers of size z behave approximately as z-κJ0, where κJ0 depends not only on the βi but also on the transmission rates ri; it is the ratio of βi to ri which determines κJ0. When specialized to the homogeneous case, i.e., when ri=r and βi = β for all i, the result coincides with results reported in the literature which have been shown under more restrictive hypotheses. Finally, it is a simple observation that light-tailed sessions only have the effect of reducing the available capacity for long-tailed sessions, but do not contribute otherwise to the definition of κJ0.

Keywords

Fluid queueslong-tailed inputsstationaryasymptotics

MSC classification

Primary: 60K25: Queueing theory
Secondary: 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 1168 - 1189
Copyright
Copyright © Applied Probability Trust 2000 

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