Birth-death processes with mass annihilation and state-dependent immigration

A birth-death process is subject to mass annihilation at rate β with subsequent mass immigration occurring into state j at rateα j . This structure enables the process to jump from one sector of state space to another one (via state 0) with transition rate independent of population size. First, we highlight the difficulties encountered when using standard techniques to construct both time-dependent and equilibrium probabilities. Then we show how to overcome such analytic difficulties by means of a tool developed in Chen and Renshaw (1990, 1993b); this approach is applicable to many processes whose underlying generator on E\{0} has known probability structure. Here we demonstrate the technique through application to the linear birth-death generator on which is superimposed an annihilation/immigration process.

Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration

Attention has recently focussed on stochastic population processes that can undergo total annihilation followed by immigration into state j at rate αj. The investigation of such models, called Markov branching processes with instantaneous immigration (MBPII), involves the study of existence and recurrence properties. However, results developed to date are generally opaque, and so the primary motivation of this paper is to construct conditions that are far easier to apply in practice. These turn out to be identical to the conditions for positive recurrence, which are very easy to check. We obtain, as a consequence, the surprising result that any MBPII that exists is ergodic, and so must possess an equilibrium distribution. These results are then extended to more general MBPII, and we show how to construct the associated equilibrium distributions.

The M/M/1 queue with mass exodus and mass arrivals when empty

An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate {αr; r≥ 1} when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.

Rats, floods and flowers: London's Gothicized nature

AUTHOR's OVERVIEW This chapter attempts a definition of London eco-gothic, beginning with an ecocritical reading of the ubiquitous London rat. Following Dracula, popular London gothic has been overrun, from the blunt horror-schlock of James Herbert’s 1970s Rats series to China Miéville’s King Rat. Maud Ellman’s elegant discussion of the modernist rat as a protean figure associated with a ‘panoply of fears and fetishes’, underlines how the rat has always featured in anti-urban discourse: as part of racist representations of immigration; as an expression of fear of disease and poverty; or through a quasi-supernatural anxiety about their indestructible and illimitable nature which makes them a staple feature of post-apocalyptic landscapes. Even so, the London rat is a rather more mundane manifestation of urban eco-gothic than the ‘city wilderness’ metaphors common to representations of New York or Los Angles as identified by eco-critic Andrew White. London’s gothic noses its way out through cracks in the pavements, grows from seeds in suburban gardens or accumulates through the steady drip of rainwater. However, I will suggest, in texts such as Maggie Gee’s The Flood and P. D. James’ Children of Men, London eco-gothic becomes less local and familiar as it responds to global environmental crisis with more dramatic tales of minatorial nature.