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Density dependent Markov population processes in large populations of size Nwere shown by Kurtz (1970), (1971) to be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, and to exhibit stochastic fluctuations about this deterministic solution on the scale vNthat can be approximated by a diffusion process. Here, motivated by an example from evolutionary biology, we are concerned with describing how such a process leaves an absorbing boundary. Initially, one or more of the populations is of size much smaller than N, and the length of time taken until all populations have sizes comparable to Nthen becomes infinite as N? 8. Under suitable assumptions, we show that in the early stages of development, up to the time when all populations have sizes at least N1-afor 1/3 < a < 1, the process can be accurately approximated in total variation by a Markov branching process. Thereafter, it is well approximated by the deterministic solution starting from the original initial point, but with a random time delay. Analogous behaviour is also established for a Markov process approaching an equilibrium on a boundary, where one or more of the populations become extinct.

We consider a storage model that can be on or off. When on, the content increases at some state-dependent rate and the system can switch to the off state at a state-dependent rate as well. When off, the content decreases at some state-dependent rate (unless it is at zero) and the system can switch to the on position at a state-dependent rate. This process is a special case of a piecewise deterministic Markov process. We identify the stationary distribution and conditions for its existence and uniqueness.

We consider a dam process with a general (state dependent) release rule and a pure jump input process, where the jump sizes are state dependent. We give sufficient conditions under which the process has a stationary version in the case where the jump times and sizes are governed by a marked point process which is point (Palm) stationary and ergodic. We give special attention to the Markov and Markov regenerative cases for which the main stability condition is weakened. We then study an intermittent production process with state dependent rates. We provide sufficient conditions for stability for this process and show that if these conditions are satisfied, then an interesting new relationship exists between the stationary distribution of this process and a dam process of the type we explore here.

Consider a closed queueing network of single-server stations, as introduced by Jackson, Gordon-Newell and Whittle, but with service times that are arbitrarily distributed with finite means. We verify that such a network has a unique stationary distribution if one of its service times is unbounded. The proof is carried out by modeling the network as a piecewise-deterministic Markov process which is shown to be regenerative. If, furthermore, the unbounded service-time distribution is non-lattice, then the stationary distribution is also the unique equilibrium. When the service-times enjoy finite second moments, and the number of customers in the network grows indefinitely, this equilibrium (properly normalized) converges to the equilibrium of a reflected Brownian motion on a simplex

There are k(≧2) competing normal populations with common known variance and unknown means [image omitted] . Let [image omitted]  denote the ordered values of the {&z.Theta;i}. Nothing is known concerning the pairing of the {z.Θi} and the {z.Θ[i]}. In the location invariant identification problem, the differences [image omitted]  are given and it is desired to select the population associated with {z.Θ[k]}. Of particular interest is the slippage configuration [image omitted]  where δ*>0 is given. We restrict attention to procedures that guarantee that the population with the largest mean is correctly selected with probability at least P* where [image omitted]  is preassigned. Essentially, this requirement is satisfied by the stopping rule of Bechhofer, Kiefer and Sobel (Sequential Identification and Ranking Procedures, Univ. of Chicago Press 1968, Chap. 3) independently of the sampling rule and thus data dependent allocation rules can be considered. Unlike the case k=2 (see Robbins and Siegmund ( 1974) J. Am. Statist. Assoc.69, 132-139), when k≧3,substantial savings in expected total sample size can be obtained when adaptive sampling is used instead of the equal allocation rule ("vector at a time" sampling). Several procedures are proposed and investigated with respect to this criterion. Also their performance with regard to the alternative criterion of minimizing expected number of observations on the inferior populations is studied. Both theoretical and simulation results are presented. Comparisons are made with the performance of elimination-type procedures, such as that of Paulson (1964)Ann. Math. Statist.35, 174-180.Finally, for the ranking problem where the differences [image omitted]  are now unknown, it is shown that, when adaptive sampling rules are used, the slippage configuration is no longer necessarily least favorable for the usual indifference zone approach. This poses some interesting problems for future research.

We consider a storage/production process where the production rate is state dependent, the demand arrival is a renewal process, and the sizes of the demands are i.i.d exponentially distributed random variables. The resulting content process is non-Markovian but regenerative. We construct a dual Markovian dam process with drift, jump rate and jump sizes that are state dependent and use it to compute the limiting one-dimensional distribution of the content process.

We consider a stochastic fluid network with independent subordinator inputs to the various stations and deterministic internal flow which is of feed-forward type. We show that under suitable conditions the process of fluid contents in the station has a limiting distribution, where the limit holds in total variation and is independent of the initial condition. We also show that this limiting distribution is of product form only for trivial setups.