Standard models in stochastic resource allocation concern the economic processing of all jobs in some set J. We consider a set up in which tasks in various subsets of J are deemed to be alternative to one another, in that only one member of such a subset of alternative tasks will be completed during the evolution of the process. Existing stochastic scheduling methodology for single-machine problems is developed and extended to this novel class of models. A major area of application is in research planning.

We consider the G/G/1 queue as an operator that maps inter-arrival times to inter-departure times of points, given the service times. For arbitrarily fixed statistics of service times, we are interested in the existence of distributions of inter-arrival times that induce identical distributions on the inter-departure times. In this note we prove, by construction, the existence of one of such distribution.

We prove that the frequencies of the oldest, second-oldest, third-oldest, … alleles in the stationary infinitely-many-neutral-alleles diffusion model are distributed as X1, (1 − X1)X2, (1 − X1)(1 − X2)X3, …, where X1, X2,X3, … are independent beta (1, θ) random variables, θ being twice the mutation intensity; that is, the frequencies of age-ordered alleles have the so-called Griffiths–Engen–McCloskey, or GEM, distribution. In fact, two proofs are given, the first involving reversibility and the size-biased Poisson–Dirichlet distribution, and the second relying on a result of Donnelly and Tavaré relating their age-ordered sampling formula to the GEM distribution.

This paper deals with continuous-time Markov decision processes in Polish spaces, under the discounted and average cost criteria. All underlying Markov processes are determined by given transition rates which are allowed to be unbounded, and the costs are assumed to be bounded below. By introducing an occupation measure of a randomized Markov policy and analyzing properties of occupation measures, we first show that the family of all randomized stationary policies is ‘sufficient’ within the class of all randomized Markov policies. Then, under the semicontinuity and compactness conditions, we prove the existence of a discounted cost optimal stationary policy by providing a value iteration technique. Moreover, by developing a new average cost, minimum nonnegative solution method, we prove the existence of an average cost optimal stationary policy under some reasonably mild conditions. Finally, we use some examples to illustrate applications of our results. Except that the costs are assumed to be bounded below, the conditions for the existence of discounted cost (or average cost) optimal policies are much weaker than those in the previous literature, and the minimum nonnegative solution approach is new.

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C, the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.

In the setting of incomplete markets, this paper presents a general result of convergence for derivative assets prices. It is proved that the minimal martingale measure first introduced by Föllmer and Schweizer is a convenient tool for the stability under convergence. This extends previous well-known results when the markets are complete both in discrete time and continuous time. Taking into account the structure of stock prices, a mild assumption is made. It implies the joint convergence of the sequences of stock prices and of the Radon-Nikodym derivative of the minimal measure. The convergence of the derivatives prices follows.

This property is illustrated in the main classes of financial market models.

A threshold AR(1) process with boundary width 2δ > 0 was defined by Brockwell and Hyndman [5] in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold AR(1) process with δ = 0 in terms of the weak solution of a certain stochastic differential equation. Two characterizations of the distributions of the process are investigated. Both express the characteristic function of the transition probability distribution as an explicit functional of standard Brownian motion. It is shown that the joint distributions of this solution with δ = 0 are the weak limits as δ ↓ 0 of the distributions of the solution with δ > 0. The sense in which an approximating sequence of processes used by Brockwell and Hyndman [5] converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron–Martin–Girsanov formula and results of Engelbert and Schmidt [9]. We also derive the stationary distribution (under appropriate assumptions) and investigate stability of these processes.

The stationary characteristics of an n-component periodic review system which is subject to stochastic deterioration (but not to failure) are investigated. When the n-component vector which expresses the state of deterioration of the system pierces a certain surface the entire multicomponent system is replaced by items of identical cost structure at the time of the next review. In the absence of this situation nothing is replaced. It is assumed that there is a fixed cost associated with each replacement and that the operating cost of each item is a strictly increasing function of its state of deterioration. The conditions for minimizing the long-term cost of maintaining a system which operates under the stated policy were found through solution of a problem in variational calculus. Two examples are worked. A useful graph which aids in the solution of such problems is provided.

Importance sampling is a widely used variance reduction technique to compute sample quantiles such as value at risk. The variance of the weighted sample quantile estimator is usually a difficult quantity to compute. In this paper we present the exact convergence rate and asymptotic distributions of the bootstrap variance estimators for quantiles of weighted empirical distributions. Under regularity conditions, we show that the bootstrap variance estimator is asymptotically normal and has relative standard deviation of order O(n−1/4).

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

Let X(t) be a real-valued stochastic process with sample functions which are almost surely continuously differentiable. We say that X(t) has an upcrossing at level u at time t if X(t) = u, X′(t) > 0.

Equipping the edges of a finite rooted tree with independent resistances that are inverse Gaussian for interior edges and reciprocal inverse Gaussian for terminal edges makes it possible, for suitable constellations of the parameters, to show that the total resistance is reciprocal inverse Gaussian (Barndorff-Nielsen 1994). This result is extended to infinite trees. Also, a connection to Brownian diffusion is established and, for the case of finite trees, an exact distributional and independence result is derived for the conditional model given the total resistance.

Techniques currently available in the literature in dealing with problems in geometric probabilities seem to rely heavily on results from differential and integral geometry. This paper provides a radical departure in this respect. By using purely algebraic procedures and making use of some properties of Jacobians of matrix transformations and functions of matrix argument, the distributional aspects of the random p-content of a p-parallelotope in Euclidean n-space are studied. The common assumptions of independence and rotational invariance of the random points are relaxed and the exact distributions and arbitrary moments, not just integer moments, are derived in this article. General real matrix-variate families of distributions, whose special cases include the mulivariate Gaussian, a multivariate type-1 beta, a multivariate type-2 beta and spherically symmetric distributions, are considered.

The concept of a characterization and its stability for queueing models is introduced. The principle of two stages in the study of the stability property is formulated. A series of results concerning the G/G/1 model is obtained.

For continuous-time Markov chains with semigroups P, P' taking values in a partially ordered set, such that P ≤ stP', we show the existence of an order-preserving Markovian coupling and give a way to construct it. From our proof, we also obtain the conditions of Brandt and Last for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the existence of Markovian couplings between two Jackson networks.

We review characteristics of data traffic which we term stylized facts: burstiness, long-range dependence, heavy tails, bursty behavior determined by high-bandwidth users, and dependence determined by users without high transmission rates. We propose an infinite-source Poisson input model which supplies traffic in adjacent time slots. We study properties of the model as slot width decreases and traffic intensity increases. This model has the ability to account for many of the stylized facts.

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

In this paper we identify three questions concerning the management of risk networks with a central branch, which may be solved using the extensive machinery available for one-dimensional risk models. First, we propose a criterion for judging whether a subsidiary is viable by its readiness to pay dividends to the central branch, as reflected by the optimality of the zero-level dividend barrier. Next, for a deterministic central branch which must bailout a single subsidiary each time its surplus becomes negative, we determine the optimal bailout policy, as well as the ruin probability and other risk measures, in closed form. Moreover, we extend these results to the case of hierarchical networks. Finally, for nondeterministic central branches with one subsidiary, we compute approximate risk measures by applying rational approximations, and by using the recently developed matrix scale methodology.

Spatial point processes are considered whose points are subjected to certain classes of affine transformations indexed by some variable, T. Under some hypotheses, for large T integrals with respect to such a point process behave approximately as if the process were Poisson. Under stronger hypotheses, the transformed process converges as a process to a Poisson process. The result gives the asymptotic distribution of certain density estimates.

We consider d-dimensional lattice systems of bounded real-valued spins with ferromagnetic random interaction between nearest neighbours. We establish an outer and, in two dimensions, an inner bound of the parameter region where spontaneous magnetization occurs. These bounds provide an estimate of the singularity of the critical temperature at the threshold for percolation through active bonds. We derive a relationship between the ferromagnetic region and the percolative region for a correlated site–bond percolation problem, and we investigate the latter. Bounds for the level sets of the expected magnetization are also obtained.