Perfect Simulation of M/G/cM/G/c Queues

In this paper we describe a perfect simulation algorithm for the stable $M/G/c$ queue. Sigman (2011: Exact Simulation of the Stationary Distribution of the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209--213) showed how to build a dominated CFTP algorithm for perfect simulation of the super-stable $M/G/c$ queue operating under First Come First Served discipline, with dominating process provided by the corresponding $M/G/1$ queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service), and exploiting the fact that the workload process for the $M/G/1$ queue remains the same under different queueing disciplines, in particular under the Processor Sharing discipline, for which a dynamic reversibility property holds. We generalize Sigman's construction to the stable case by comparing the $M/G/c$ queue to a copy run under Random Assignment. This allows us to produce a naive perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled $M/G/c$ queues started respectively from the empty state and the state of the $M/G/c$ queue under Random Assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarize statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.Comment: 28 pages, 5 figure

Optimal co-adapted coupling for the symmetric random walk on the hypercube

Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z2n. We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class

Metalliferous Biosignatures for Deep Subsurface Microbial Activity

Acknowledgments We thank the British Geological Survey (BGS) for the provision of samples and the Science & Technology Facilities Council (STFC) grant (ST/L001233/1) for PhD funding which aided this project. Research on selenium in reduction spheroids was also supported by NERC grants (NE/L001764/1 and NE/ M010953/1). The University of Aberdeen Raman facility was funded by the BBSRC. We also thank John Still for invaluable technical assistance.Peer reviewedPublisher PD

Optimal co-adapted coupling for a random walk on the hyper-complete-graph

The problem of constructing an optimal co-adapted coupling for a pair of symmetric random walks on $Z_2^d$ was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such co-adapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal co-adapted coupling for the continuous-time symmetric random walk on $K_n^d$, where $K_n$ is the complete graph with $n$ vertices. Moreover, we show that although this coupling is not maximal for any $n$ (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as $n\to\infty$.Comment: 20 pages, 1 figur

An Examination of Effective Length in Moment Frames

The honors thesis project I have been working on is called An Examination of Effective Length in Moment Frame. When loaded axially in compression columns experience a failure mode in compression that axially loaded members don’t experience in tension. This failure mode, elastic column buckling, doesn’t involve yielding or rupture; the column changes shape and deforms to the side. In order to come up with a solution to this failure mode, Leonhard Euler developed the critical buckling load equation. However, this equation uses an effective length of columns, which is the distance between two points of zero moment, or inflection points. This is found by multiplying an effective length factor, K, by the column length. The K factor has been found using a nomograph that is strictly based off the columns section properties and only accounts for axial loads. In a moment frame however, patterned live loads can cause joint rotation, which the nomograph does not account for. Increased joint rotation would theoretically change what the effective length of the column is

Institutional change and learning for sustainable development

This research project has been commissioned by Land and Water Australia,and focuses on lessons drawing for Australia from selected international examples of institutional change for sustainable development in the past decade. An interim product of the project is now available as a CRES Working Paper, setting out the conceptual basis of the research and the proposed case studies

State-dependent Foster-Lyapunov criteria for subgeometric convergence of Markov chains

We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster-Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed by Connor & Kendall (2007) concerning the existence of so-called 'tame' Markov chains. Furthermore, we show that our 'subsampled drift condition' implies the existence of finite moments for the return time to a small set.Comment: 20 pages, LaTeX: paper reduced in lengt