BRAVO for many-server QED systems with finite buffers

This paper demonstrates the occurrence of the feature called BRAVO (Balancing Reduces Asymptotic Variance of Output) for the departure process of a finite-buffer Markovian many-server system in the QED (Quality and Efficiency-Driven) heavy-traffic regime. The results are based on evaluating the limit of a formula for the asymptotic variance of death counts in finite birth--death processes

Asymptotic analysis by the saddle point method of the Anick-Mitra-Sondhi model

We consider a fluid queue where the input process consists of N identical sources that turn on and off at exponential waiting times. The server works at the constant rate c and an on source generates fluid at unit rate. This model was first formulated and analyzed by Anick, Mitra and Sondhi. We obtain an alternate representation of the joint steady state distribution of the buffer content and the number of on sources. This is given as a contour integral that we then analyze for large N. We give detailed asymptotic results for the joint distribution, as well as the associated marginal and conditional distributions. In particular, simple conditional limits laws are obtained. These shows how the buffer content behaves conditioned on the number of active sources and vice versa. Numerical comparisons show that our asymptotic results are very accurate even for N=20

The asymptotic variance of departures in critically loaded queues

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case that the system load rho equals 1, and prove that the asymptotic variance rate satisfies lim_t Var D(t)/t = lambda (1 - 2/pi) (c_a2 + c_s2) where lambda is the arrival rate and c_a2 and c_s2 are squared coefficients of variation of the inter-arrival and service times respectively. As a consequence, the departures variability has a remarkable singularity in case rho equals 1, in line with the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs) which was previously encountered in the finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multi-server queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue we present an explicit expression of the variance of D(t) for any t

Finite Block-Length Achievable Rates for Queuing Timing Channels

The exponential server timing channel is known to be the simplest, and in some sense canonical, queuing timing channel. The capacity of this infinite-memory channel is known. Here, we discuss practical finite-length restrictions on the codewords and attempt to understand the amount of maximal rate that can be achieved for a target error probability. By using Markov chain analysis, we prove a lower bound on the maximal channel coding rate achievable at blocklength $n$ and error probability $\epsilon$ is approximated by $C- n^{-1/2} \sigma Q^{-1}(\epsilon)$ where $Q$ denotes the Q-function and $\sigma^2$ is the asymptotic variance of the underlying Markov chain. A closed form expression for $\sigma^2$ is given