Queues with superposition arrival processes in heavy traffic

AbstractTo help provide a theoretical basis for approximating queues with superposition arrival processes, we prove limit theorems for the queue-length process in a Σ GIi/G/s model, in which the arrival process is the superposition of n independent and identically distributed stationary renewal processes each with rate n−1. The traffic intensity ρ is allowed to approach the critical value one as n increases. If n(1−ρ)2 → c, 0 < c < ∞, then a limit is obtained that depends on c. The two iterated limits involving ρ and n, which do not agree, are obtained as c → 0 and c → ∞

Large Deviations in Renewal Models of Statistical Mechanics

In Ref. [1] the author has recently established sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a separable Banach space. The renewal model has been there identified with constrained and non-constrained pinning models of polymers, which amount to Gibbs changes of measure of a classical renewal process. In this paper we show that the constrained pinning model is the common mathematical structure to the Poland-Scheraga model of DNA denaturation and to some relevant one-dimensional lattice models of Statistical Mechanics, such as the Fisher-Felderhof model of fluids, the Wako-Sait\^o-Mu\~noz-Eaton model of protein folding, and the Tokar-Dreyss\'e model of strained epitaxy. Then, in the framework of the constrained pinning model, we develop an analytical characterization of the large deviation principles for cumulative rewards corresponding to multivariate deterministic rewards that are uniquely determined by, and at most of the order of magnitude of, the time elapsed between consecutive renewals. In particular, we outline the explicit calculation of the rate functions and successively we identify the conditions that prevent them from being analytic and that underlie affine stretches in their graphs. Finally, we apply the general theory to the number of renewals. From the point of view of Equilibrium Statistical Physics and Statistical Mechanics, cumulative rewards of the above type are the extensive observables that enter the thermodynamic description of the system. The number of renewals, which turns out to be the commonly adopted order parameter for the Poland-Scheraga model and for also the renewal models of Statistical Mechanics, is one of these observables

On the area between a L\'evy process with secondary jump inputs and its reflected version

We study the the stochastic properties of the area under some function of the difference between (i) a spectrally positive L\'evy process $W_t^x$ that jumps to a level $x>0$ whenever it hits zero, and (ii) its reflected version $W_t$. Remarkably, even though the analysis of each of these processes is challenging, we succeed in attaining explicit expressions for their difference. The main result concerns the Laplace-Stieltjes transform of the integral $A_x$ of (a function of) the distance between $W_t^x$ and $W_t$ until $W_t^x$ hits zero. This result is extended in a number of directions, including the area between $A_x$ and $A_y$ and a Gaussian limit theorem. We conclude the paper with an inventory problem for which our results are particularly useful