On A Single Server Queue with Two-Stage First Essential Service Followed by One of the Two Types of Additional Optional Service and Optional Deterministic Server Vacations

We study the steady state behavior of a batch arrival single server queue in which the first service consisting of two stages with general service times G1 and G2 is compulsory. After completion of the two stages of the first essential service, a customer has the option of choosing one of the two types of additional service with respective general service times G1 and G2 . Just after completing both stages of first essential service with or without one of the two types of additional optional service, the server has the choice of taking an optional deterministic vacation of fixed (constant) length of time. We obtain steady state probability generating functions for the queue size for various states of the system at a random epoch of time in explicit and closed forms. The steady state results of some interesting special cases have been derived from the main results.

Estimating Smooth and Convex Functions

We propose a new method for estimating an unknown regression function $f_*:[\alpha, \beta] \rightarrow \mathbb{R}$ from a dataset $(X_1, Y_1), \dots, (X_n,$ $Y_n)$ when the only information available on $f_*$ is the fact that $f_*$ is convex and twice differentiable. In the proposed method, we fit a convex function to the dataset that minimizes the sum of the roughness of the fitted function and the average squared differences between the fitted function and $f_*$. We prove that the proposed estimator can be computed by solving a convex quadratic programming problem with linear constraints. Numerical results illustrate the superior performance of the proposed estimator compared to existing methods when i) $f_*$ is the price of a stock option as a function of the strike price and ii) $f_*$ is the steady-state mean waiting time of a customer in a single server queue.

BROWNIAN MOTION MINUS THE INDEPENDENT INCREMENTS: REPRESENTATION AND QUEUING APPLICATION

This paper relaxes assumptions defining multivariate Brownian motion (BM) to construct processes with dependent increments as tractable models for problems in engineering and management science. We show that any Gaussian Markov process starting at zero and possessing stationary increments and a symmetric smooth kernel has a parametric kernel of a particular form, and we derive the unique unbiased, jointly sufficient, maximum-likelihood estimators of those parameters. As an application, we model a single-server queue driven by such a process and derive its transient distribution conditional on its history.