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We develop explicit bounds for the tail of the distribution for the all-time supremum of a random walk with negative drift, where the increments have a truncated heavy-tailed distribution. As an application, we consider a ruin problem in the presence of re-insurance.

Motivated by the needs in the theory of large deviations and in the theory of Lundberg's equation with heavy-tailed distribution functions, we study for $n=0,1,...$ the maximization of $S:~\Bigl(1-e^{-s}\Bigl(1+\frac{s^1}{1!}+...+\frac{s^n}{n!}\Bigr)\Bigr)/s^{\delta} = E_{n,\delta}(s)$ over $s\geq0$, with $\delta\in(0,n+1)$, $U:~({-}1)^{n+1}\Bigl(e^{-u}-\Bigl(1-\frac{u^1}{1!}+...+({-}1)^n\,\frac{u^n}{n!} \Bigr)\Bigr)/u^{\delta}=G_{n,\delta}(u)$ over $u\geq0$ with $\delta\in(n,n+1)$. We show that $E_{n,\delta}(s)$ and $G_{n,\delta}(u)$ have a unique maximizer $s=s_n(\delta)>0$ and $u=u_n(\delta)>0$ that decrease strictly from $+\infty$ at $\delta=0$ and $\delta=n$, respectively, to 0 at $\delta=n+1$. We use Taylor's formula for truncated series with remainder in integral form to develop a criterion to decide whether a particular smooth function $S(\delta)$, $\delta\in(0,n+1)$, or $U(\delta)$, $\delta\in(n,n+1)$, respectively, is a lower/upper bound for $s_n(\delta)$ and $u_n(\delta)$, respectively. This criterion allows us to find lower and upper bounds for $s_n$ and $u_n$ that are reasonably tight and simple at the same time. Furthermore, as a consequence of the identities $\frac{d}{d\delta}\,[{\rm ln}\,ME_{n,\delta}] ={-}{\rm ln}\,s_n(\delta)$ and $\frac{d}{d\delta}\,[{\rm ln}\,MG_{n,\delta}]={-}{\rm ln}\,u_n(\delta)$, we show that $ME_{n,\delta}$ and $MG_{n,\delta}$ are log-convex functions of $\delta\in(0,n+1)$ and $\delta\in(n+1,n)$, respectively, with limiting values 1 ($\delta\downarrow0$) and $1/(n+1)!$ ($\delta\uparrow n+1$) for $E$, and $1/n!\,(\delta\downarrow n)$ and $1/(n+1)!\,(\delta\uparrow n+1)$ for $G$. The minimal values $\hat{E}_n$ and $\hat{G}_n$ of $ME_{n,\delta}$ and $MG_{n,\delta}$, respectively, as a function of $\delta$, as well as the minimum locations $\delta_{n,E}$ and $\delta_{n,G}$ are determined in closed form.

Emden-Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden-Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden-Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden-Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden-Fowler type equation that we consider., Comment: 33 pages, 3 figures

Heavy-traffic limit theory deals with queues that operate close to criticality and face severe queueing times. Let $W$ denote the steady-state waiting time in the ${\rm GI}/{\rm G}/1$ queue. Kingman (1961) showed that $W$, when appropriately scaled, converges in distribution to an exponential random variable as the system's load approaches 1. The original proof of this famous result uses the transform method. Starting from the Laplace transform of the pdf of $W$ (Pollaczek's contour integral representation), Kingman showed convergence of transforms and hence weak convergence of the involved random variables. We apply and extend this transform method to obtain convergence of moments with error assessment. We also demonstrate how the transform method can be applied to so-called nearly deterministic queues in a Kingman-type and a Gaussian heavy-traffic regime. We demonstrate numerically the accuracy of the various heavy-traffic approximations.