Approximations for the Long-Term Behavior of an Open-Population Epidemic Model

A simple stochastic epidemic model incorporating births into the susceptible class is considered. An approximation is derived for the mean duration of the epidemic. It is proved that the epidemic ultimately dies out with probability 1. The limiting behavior of the epidemic conditional on non-extinction is studied using approximation methods. Two different diffusion approximations are described and compared.     

Translating IΔ0 + exp Proofs into Weaker Systems

The purpose of this paper is to explore the relationship between IΔ₀ + exp and its weaker subtheories. We give a method of translating certain classes of IΔ₀ + exp proofs into weaker systems of arithmetic such as Buss' systems S₂. We show if IE_i (exp) ⊢ A with a proof P of expind‐rank(P) ≤ n + 1where all (∀ ≤: right) or (∃ ≤: left) have bounding terms not containing function symbols, then Sⁱ ₂ ⊇ IE_i,2 ⊢ Aⁿ. Here A is not necessarily a bounded formula. For IOpen(exp) we prove a similar result. Using our translations we show IOpen(exp) ⊊ IΔ₀(exp). Here IΔ₀(exp) is a conservative extension of IΔ₀ + exp obtained by adding to IΔ₀ a symbol for 2 ^x to the language as well as defining axioms for it.     

Markovian bulk-arrival and bulk-service queues with state-dependent control

We study a modified Markovian bulk-arrival and bulk-service queue incorporating state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated and the relationship with our queueing model is examined and exploited. Equilibrium behaviour is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting time and busy period distributions are answered in detail and the Laplace transforms of these distributions are presented. Further properties including expectations of hitting times and busy period are also explored.     

On the problem of evaluating quasistationary distributions for open reaction schemes

A number of recent papers have been concerned with the stochastic modeling of autocatalytic reactions. In some instances the birth and death model has been criticized for its apparent inadequacy in being able to describe the long-term behavior of the catalyst, in particular the fluctuations in the concentration of the catalyst about its macroscopically stable state. This criticism has been answered, to some extent, with the introduction of the notion of a quasistationary distribution; a number of authors have established the existence of limiting conditional distributions that can adequately describe these fluctuations. However, much of the work appears only to be appropriate for dealing with closed systems, for attention is usually restricted to finite-state birth and death processes. For open systems it is more appropriate to consider infinite-state processes and, from the point of view of establishing conditions for the existence of quasistationary distributions, extending the results for closed systems is far from straightforward. Here, simple conditions are given for the existence of quasistationary distributions for Markov processes with a denumerable infinity of states. These can be applied to any open autocatalytic system. The results also extend to explosive processes and to processes that terminate with probability less than 1.     

Strengths and Weaknesses of LH Arithmetic

In this paper we provide a new arithmetic characterization of the levels of the og‐time hierarchy (LH). We define arithmetic classes and that correspond to ‐LOGTIME and ‐LOGTIME, respectively. We break and into natural hierarchies of subclasses and . We then define bounded arithmetic deduction systems ′ whose ‐definable functions are precisely B( ‐LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that , (2) TAC, a theory that is slightly stronger than ′ whose (LH)‐definable functions are LH, proves LH is not equal to ‐TIME(s) for any m> 0, where 2^s ∈L, s(n) ∈ ω(log n), and (3) TAC proves LH ≠ for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.     

A Reduced Load Approximation Accounting for Link Interactions in a Loss Network

This paper is concerned with evaluating the performance of loss networks. Accurate determination of loss network performance can assist in the design and dimensioning of telecommunications networks. However, exact determination can be diffcult and generally cannot be done in reasonable time. For these reasons there is much interest in developing fast and accurate approximations. We develop a reduced load approximation that improves on the famous Erlang fixed point approximation (EFPA) in a variety of circumstances. We illustrate our results with reference to a range of networks for which the EFPA may be expected to perform badly.     

Quasistationary distributions for autocatalytic reactions

We provide simple conditions for the existence of quasistationary distributions that can be used to describe the long-term behaviour ofopen autocatalytic reaction systems. We illustrate with reference to a particular example that the quasistationary distribution is close to the usual stationary diffusion approximation.     

Nature of strain recovery in polyvinyl chloride following steady laminar shearing

Summary It is well known that viscoelastic materials, subjected to steady laminar shearing, develop a tension in the direction of shear, and it is of both practical and theoretical, interest to determine whether, this can give rise to normal components of strain recovery parallel and perpendicular to this direction, in addition to the tangential component corresponding to the applied shear. This has been investigated for the condition where there is opportunity for stress relaxation after shearing, plasticised polyvinyl chloride being used for the purpose. The hot material was sheared in a cone- and-plate rheometer at 1.0 sec^–1, and then allowed to cool without recovery being permitted. It was then removed, and suitable specimens were cut and re-heated above the shearing temperature, the resultant recovery being measured. In addition to partial shear recovery there was considerable retraction in the direction of shear. Moreover the accompanying expansion in the transverse plane was greater perpendicular to the shear planes than parallel to them. The strain recovery increased with applied shear till the latter reached 10–15, and then diminished, in agreement with previous tests on shear recovery immediately after shearing.

---

Zusammenfassung Bekanntlich entwickelt viskoelastisches Material, einer stationären laminaren Scherung unterworfen, eine Spannung in Scherrichtung, und es ist von Interesse, ob bei Verschwinden der Zugspannung diese Anlaß zu Normalkomponenten der Dehnungserholung führen kann, zusätzlich zur Tangentialkomponente. Dies wurde unter Bedingungen untersucht, bei denen nach der Scherung Gelegenheit für Spannungsrelaxation gegeben war, bei plastifiziertem PC. Das heiße Material wurde in einen Konsus-Platten Rheometer bei 1.0 sec^–1 geschert und konnte dann ohne Rückfederung abkühlen. Es wurden geeignete Proben ausgeschnitten und über die Temperatur bei der Scherung hinauf aufgeheizt und anschließend die Rückfederung gemessen. Zusätzlich zu einer teilweisen Rückbildung der Scherung traten beträchtliche Schrumpfungen in der Scherrichtung auf. Außerdem war die Verdickung senkrecht zu den Scherebenen stärker als parallel. Die Deformationserholung stieg bis zu 10–15 an und verminderte sich dann in Übereinstimmung mit früheren Versuchen über Schererholung unmittelbar nach dem Scheren.

---

---

The author wishes to record his thanks to Mr.W. C. Barry B. Sc. (Eng.) M.I.E.E., Manager of the Research Laboratories, to Mr.S. E. Goodall M. Sc. (Eng.) M. I. E. E., Chief Engineer of W. T. Henley's Telegraph Works Co., Ltd., and to the Company, for permission to publish this paper.