In this article we investigate a class of superprocess with cut-off branching, studying the long-time behavior of the occupation time process. Persistence of the process holds in all dimensions. Central-limit-type theorems are obtained, and the scales are dimension dependent. The Gaussian limit holds only when d ≤ 4. In dimension one, a full large deviation principle is established and the rate function is identified explicitly. Our result shows that the super-Brownian motion with cut-off branching in dimension one has many features that are similar to super-Brownian motion in dimension three.

We consider a stochastic model for the spread of a carrier-borne epidemic amongst a closed homogeneously mixing population, in which a proportion 1 − π of infected susceptibles are directly removed and play no part in spreading the infection. The remaining proportion π become carriers, with an infectious period that follows an arbitrary but specified distribution. We give a construction of the epidemic process which directly exploits its probabilistic structure and use it to derive the exact joint distribution of the final size and severity of the carrier-borne epidemic, distinguishing between removed carriers and directly removed individuals. We express these results in terms of Gontcharoff polynomials. When the infectious period follows an exponential distribution, our model reduces to that of Downton (1968), for which we use our construction to derive an explicit expression for the time-dependent state probabilities.

In this paper we study the assignment of packets to M parallel heterogeneous servers with no buffers. The controller does not have knowledge on the state of the servers and bases all decisions on the number of time slots ago that packets have been sent to the different servers. The objective of the controller is to minimize the expected average cost. We consider a general stationary arrival process, with no independence assumptions. We show that the problem can be transformed into a Markov Decision Process (MDP). We further show under quite general conditions on cost functions that only a finite number of states have to be considered in this MDP, which then implies the optimality of periodic policies. For the case of two servers we obtain a more detailed structure of the cost and optimal policies. In particular we show that the average cost function is multimodular, and we obtain expressions for the cost for MMPP and MAP processes. Finally we present an application to optimal robot scheduling for Web search engines.

Let {X(n): n = 0, 1, 2, …} be a Markov chain with state space {0, 1, 2, …, N} and transition probability matrix P = (pij).

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

Let X1, X2, · ··, Xn, · ·· be independent and identically distributed non-negative integer-valued random variables with finite mean and variance. For any positive integer n and m we consider the random vector i.e., L has the same distribution as the conditional distribution of (X1, · ··, Xm) given the condition It is easy to see that our model includes the classical urn model, the Bose–Einstein urn model and the Pólya urn model as special cases. For any non-negative integer s define G(s) = the number of Li′s such that Li = s, and U = the number of Li′s such that Li is an even number; in this paper we study the asymptotic behaviour of the random variables considered above. Some central limit theorems and a multinormal local limit theorem are proved.

In this paper the distributions of the number of intersections of three, four and five random chords to a circle are obtained by a reduction technique employing Ptolemy's theorem. These results are then used to obtain the skewness and kurtosis of the number of intersections of n random chords to a circle.

Necessary and sufficient conditions for the finiteness of moments of virtual work for GI/G/c queues are presented.

The object of this paper is to study the stochastic asymptotic exponential stability of a stochastic integral equation of the form

A random solution of the stochastic integral equation is considered to be a second order stochastic process satisfying the equation almost surely. The random solution, y(t, ω) is said to be. stochastically asymptotically exponentially stable if there exist some β > 0 and a γ > 0 such that for t∈ R+.

The results of the paper extend the results of Tsokos' generalization of the classical stability theorem of Poincaré-Lyapunov.

Emitted particles arrive at the counter with prolonging dead time so that the interarrival times and the lengths of impulses in any dead time are independent but not necessarily identically distributed random variables, and whenever the counter is idle then the following evolution starts from the beginning. For this class of counters we derive the probability laws of the numbers of particles arriving at the counters during their dead times, and the Laplace transform of the cycle, respectively.

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk}and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.

A bivariate point process consisting of an original stationary point process and its random translation is considered. Westcott's method is applied to show that if the original point process is ergodic then the bivariate point process is also ergodic. This result is applied to an identification problem of the displacement distribution. It is shown that if the spectrum of the original process is the real line then the displacement distribution is identifiable from almost every sample realisation of the bivariate process.

Let Tx be the length of time to accumulate x units of a resource. In queueing, the resource could be service. We derive a sufficient condition for the process to have stationary increments where Tx is an additive functional of a Markov process. This condition is satisfied in symmetric queues and generalized semi-Markov schemes with insensitive components. As a corollary, we show that the conditional expected response time in a symmetric queue is linear in the service requirement. A similar result holds for the conditional average residence time of an insensitive component in a GSMS.

Let X(t), N(t) respectively denote the number of cells alive at t and the total number of cells born by t in a critical age-dependent Bellman-Harris branching process.

The asymptotic behavior of the conditional moments, for 0 < α ≦ 1, E(Nn(αt) | X(t) > 0), E(Nn(t) |X(αt) > 0), is obtained.

We develop shock model theory in different scenarios for the ℳ-class of life distributions introduced by Klar and Müller (2003). We also study the cumulative damage model of A-Hameed and Proschan (1975) in the context of ℳ-class and establish analogous results. We obtain moment bounds and explore weak convergence issues within the ℳ-class of life distributions.

We consider a Markov chain of point processes such that each state is a superposition of an independent cluster process with the previous state as its centre process together with some independent noise process and a thinned version of the previous state. The model extends earlier work by Felsenstein (1975) and Shimatani (2010) describing a reproducing population. We discuss when closed-form expressions of the first- and second-order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

The forward equations for the death process in ℕ2 with Q-matrix determined by q(i, j; i − 1, j) = αij, i ≧ 1, q(i, j; i, j − 1) = μj, j ≧ 1 are solved by giving a special interpretation to this Q-matrix.

Asymptotic normality and the weak invariance principle are studied for an iterated renewal process of order A martingale decomposition is used in this context to provide a simple proof.

We consider the problem of reorganizing a linear list, when the individual queries consist of accesses to a subset of the elements stored, as opposed to the individual elements themselves. In this paper, which to our knowledge represents the first reported result in this model of query processing, we first propose a simple model for a query generator which emits set queries. Subsequently, we present extensions to the classical move-to-front (MTF) and transposition (TR) rules under this generalized query generation mechanism and analyze their performance.

A subfair discounted red-and-black game with a house limit is introduced and the optimality of the modified bold strategy is shown.