The Large Deviations of Estimating Rate Functions

Meyn

2011

Queueing Syst

Loynes' distribution, which characterizes the one dimensional marginal of the stationary solution to Lindley's recursion, possesses an ultimately exponential tail for a large class of increment processes. If one can observe increments but does not know their probabilistic properties, what are the statistical limits of estimating the tail exponent of Loynes' distribution? We conjecture that in broad generality a consistent sequence of non-parametric estimators can be constructed that satisfies a large deviation principle. We present rigorous support for this conjecture under restrictive assumptions and simulation evidence indicating why we believe it to be true in greater generality.

Section: Rigorous Evidence In Support Of the Conjecturementioning

confidence: 68%

“…As a corollary to a result regarding a related estimation problem, [9,Theorem 2] proves that the sequence of estimates {θ * (n)} satisfy the LDP under less restrictive conditions than those of the following theorem, but does not establish its consistency. …”

Section: Rigorous Evidence In Support Of the Conjecturementioning

confidence: 99%

Estimating Loynes’ exponent

Meyn

2011

Queueing Syst

“…Note that when ν = L n , the empirical law of (7), then J ν = J n , which is defined in (5). For a one-dimensional process, assuming that condition (S) holds, Duffy and Metcalfe [10] proved that rate function estimates satisfy the LDP. The arguments in [10]…”

Section: Resultsmentioning

confidence: 99%

How to estimate the rate function of a cumulative process

Metcalfe

2005

J. Appl. Probab.

Self Cite

Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate function J(⋅,⋅) and whose cumulative process satisfies the LDP with rate function I(⋅). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of a cumulative renewal process it is demonstrated that this approach is favourable to a more direct method, as it ensures that the laws of the estimates converge weakly to a Dirac measure at I.

“…The rate function for the associated random walk can be calculated using methods described in [7]. For example, from [10] we have that…”

Section: 2mentioning

confidence: 99%

Large Deviation Asymptotics for Busy Periods

Meyn

2014

Stochastic Systems

The busy period for a queue is cast as the area swept under the random walk until it first returns to zero. Encompassing non-i.i.d. increments, the large-deviations asymptotics of the busy period B is addressed, under the assumption that the increments satisfy standard conditions, including a negative drift. The main conclusions provide insight on the probability of a large busy period, and the manner in which this occurs. The scaled probability of a large busy period has the asymptote, for any b > 0, [Formula: see text], where [Formula: see text], with [Formula: see text], and with Λ denoting the scaled cumulant generating function of the increments process. The most likely path to a large swept area is found to be a simple rescaling of the path on [0, 1] given by [Formula: see text]. In contrast to the piecewise linear most likely path leading the random walk to hit a high level, this is strictly concave in general. While these two most likely paths have distinctly different forms, their derivatives coincide at the start of their trajectories, and at their first return to zero. These results partially answer an open problem of Kulick and Palmowski [18] regarding the tail of the work done during a busy period at a single server queue. The paper concludes with applications of these results to the estimation of the busy period statistics (λ*, K) based on observations of the increments, offering the possibility of estimating the likelihood of a large busy period in advance of observing one.