Yaglom limit for stochastic fluid models

In this paper we analyse the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of models in the theory of matrix-analytic methods. So far, only transient and stationary analyses of SFMs have been considered in the literature. The limiting conditional distribution gives useful insights into what happens when the process has been evolving for a long time, given that its busy period has not ended yet. We derive expressions for the Yaglom limit in terms of the singularity˜$s^*$ such that the key matrix of the SFM, ${\boldsymbol{\Psi}}(s)$, is finite (exists) for all $s\geq s^*$ and infinite for $s<s^*$. We show the uniqueness of the Yaglom limit and illustrate the application of the theory with simple examples.

The Sudan Years

Chapter 4 explores the years after 1991, when Bin Laden was given sanctuary in Sudan. Five years later he and his followers were expelled. The Sudan years were an exceptionally busy period of building alliances and laying the groundwork for attacks on U.S. embassies and military installations in Saudi Arabia, Yemen, Nairobi, and Tanzania. At the same time, Bin Laden was supporting militants across the Muslim world and funding military instructors in training camps in Afghanistan, Southeast Asia, and East Africa. He was also seeking out terrorist expertise wherever he could find it. One of those places was Iran, which for a period became a sleeping partner in Bin Laden’s terrorism business. Above all, he was working to integrate Arab and North African exiled militants into a coherent, global organization. From Sudan, Bin Laden assisted his Egyptian allies in their terrorist campaign in Egypt, funded experimentation with the use of liquids bombs to blow up international commercial airliners, made plans to set up a base in the Balkans for Al Qaeda, set up a business office in London, and initiated preparations for the 1998 bombings of the U.S. embassies in Kenya and Tanzania.

An Uncertain Single-Server Queueing Model

In the queueing theory, the interarrival times between customers and the service times for customers are usually regarded as random variables. This paper considers human uncertainty in a queueing system, and proposes an uncertain queueing model in which the interarrival times and the service times are regarded as uncertain variables. The busyness index is derived analytically which indicates the service efficiency of a queueing system. Besides, the uncertainty distribution of the busy period is obtained.

Frequency scaling in multilevel queues

In this paper, we study a variant of PS+PS multilevel scheduling, which we call the PS+IS queue. Specifically, we use Processor Sharing (PS) at both queues, but with linear frequency scaling on the second queue, so that the latter behaves like an Infinite Server (IS) queue. The goals of the system are low response times for small jobs in the first queue, and reduced power consumption for large jobs in the second queue. The novelty of our model includes the frequency scaling at the second queue, and the batch arrival process at the second queue induced by the busy period structure of the first queue which has strictly higher priority. We derive a numerical solution for the PS+IS queueing system in steady-state, and then study its properties under workloads obtained from fitting of TCP flow traces. The simulation results confirm the e

Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue

<p style='text-indent:20px;'>We give analytically explicit solutions for the distribution of the number of customers served during a busy period for the <inline-formula><tex-math id="M1">\begin{document}$ M^X/PH/1 $\end{document}</tex-math></inline-formula> queues when initiated with <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> customers. When customers arrive in batches, we present the functional equation for the Laplace transform of the number of customers served during a busy period. Applying the Lagrange inversion theorem, we provide a refined result to this functional equation. From a phase-type service distribution, we obtain the distribution of the number of customers served during a busy period for various special cases such as exponential, Erlang-k, generalized Erlang, hyperexponential, Coxian, and interrupted Poisson process. The results are exact, rapid and vigorous, owing to the clarity of the expressions. Moreover, we also consider computational results for several service-time distributions using our method. Phase-type distributions can approximate any non-negative valued distribution arbitrarily close, making them a useful practical stochastic modelling tool. These distributions have eloquent properties which make them beneficial in the computation of performance models.</p>

A study on some infinite server queues in discrete-time

This work analysis some discrete-time queueing mechanisms with infinitely many servers.By using a shot noise process, general results on the system size in discrete-time are given both in transient state and in steady state. For this we use the classical differentiation formula of F´a di Bruno. First two moments of the system size and distribution of the busy period of the system are also computed.