A number of papers have recently appeared in the literature in which customers, in moving from node to node in the network arrive as either a positive customer or as a batch of negative customers. A positive customer joining its queue increases the number of customers at the queue by 1 and each negative customer decreases the queue length by 1, if possible. It has been shown that the equilibrium distribution for these networks assumes a geometric product form, that certain partial balance equations prevail and that the parameters of the geometric distributions are, as in Jackson networks, the service facility throughputs of customers. In this paper the previous work is generalised by allowing state dependence into both the service and routing intensities and by allowing the possibility, although not the necessity, for negative customers to build up at the nodes.

We study a one-dimensional telegraph process (M t ) t≥0 describing the position of a particle moving at constant speed between Poisson times at which new velocities are chosen randomly. The exact distribution of M t and its first two moments are derived. We characterize the level hitting times of M t in terms of integro-differential equations which can be solved in special cases.

The time until extinction for the closed SIS stochastic logistic epidemic model is investigated. We derive the asymptotic distribution for the extinction time as the population grows to infinity, under different initial conditions and for different values of the infection rate.

Recently defined classes of life distributions are considered, and some relationships among them are proposed. The life distribution H of a device subject to shocks occurring randomly according to a Poisson process is also considered, and sufficient conditions for H to belong to these classes are discussed.

An exponential server splits its service capacity between two independent Poisson streams of customers, unless one queue is empty, in which case the full service capacity is granted to the other queue. We present the stationary joint generating function of the two queue sizes explicitly in the form of an elliptic integral.

Consider a branching process in which each individual reproduces independently of all others and has probability aj (j = 0, 1, ···) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where, with probability bj (j = 0, 1, ···) j objects enter the population at each generation. Then letting Xn (n = 0, 1, ···) be the population size of the nth generation, it is known (Heathcote (1965), (1966)) that {Xn} defines a Markov chain on the non-negative integers and it is called a branching process with immigration (b.p.i.). We shall call the process sub-critical or super-critical according as the mean number of offspring of an individual, , satisfies α < 1 or α > 1, respectively. Unless stated specifically to the contrary, we assume that the following condition holds.

We consider records from the sequence Yj = Xj, + cj, j ≧ 1 where c > 0 if upper records are of interest and c < 0 if lower records are studied. If {Xj} is stationary the record rate is asymptotically constant and if {Xj} is i.i.d. the record rate is, in addition, asymptotically normal. These results are illustrated by analysis of the times in the mile run.

A definition of the Markovian property is given for a lattice process and a Gaussian Markovian lattice process is constructed on a torus lattice. From this a Gaussian Markovian process is constructed for a lattice in the plane and its properties are studied.

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n2).

A ‘Secretary Problem’ with no recall but which allows the applicant to refuse an offer of employment with a fixed probability 1 – p, (0 < p < 1), is considered. The optimal stopping rule and the maximum probability of employing the best applicant are derived.

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.

Let X1, X2,· ·· be a (linear or circular) sequence of trials with three possible outcomes (say S, S∗ or F) in each trial. In this paper, the waiting time for the first appearance of an S-run of length k or an S∗-run of length r is systematically investigated. Exact formulae and Chen-Stein approximations are derived for the distribution of the waiting times in both linear and circular problems and their asymptotic behaviour is illustrated. Probability generating functions are also obtained when the trials are identical. Finally, practical applications of these results are discussed in some detail.

In this paper we consider an independent and identically distributed sequence {Yn} with common distribution function F(x) and a random variable X0, independent of the Yi's, and define a Markovian sequence {Xn} as Xi = X0, if i = 0, Xi = k max{Xi− 1, Yi}, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.

In this paper two burn-in procedures for a general failure model are considered. There are two types of failure in the general failure model. One is Type I failure (minor failure) which can be removed by a minimal repair or a complete repair and the other is Type II failure (catastrophic failure) which can be removed only by a complete repair. During a burn-in process, with burn-in Procedure I, the failed component is repaired completely regardless of the type of failure, whereas, with burn-in Procedure II, only minimal repair is done for the Type I failure and a complete repair is performed for the Type II failure. In field use, the component is replaced by a new burned-in component at the ‘field use age’ T or at the time of the first Type II failure, whichever occurs first. Under the model, the problems of determining optimal burn-in time and optimal replacement policy are considered. The two burn-in procedures are compared in cases when both the procedures are applicable.

The Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance, and many other fields. In this paper we obtain a functional central limit theorem for the nonlinear Hawkes process. Under the same assumptions, we also obtain a Strassen's invariance principle, i.e. a functional law of the iterated logarithm.

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by Barndorff-Nielsen and Shephard (2001) as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes, we take advantage of the nonnegativity of the increments of the driving Lévy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0,h,…,Nh. We also show how to reconstruct the background driving Lévy process from a continuously observed realization of the process and use this result to estimate the increments of the Lévy process itself when h is small. Asymptotic properties of the coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process.

We study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on the borrowing and short-selling of a ‘hedger’ in this market, and can be described by a closed convex set K. We find explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or infinity, and also, on if we have linear dynamics for the stock price process, and whether volatility process depends on the stock price. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.

Wright (1921) computed various correlations of relatives by a rather cumbersome procedure called the “method of path coefficients”. Wright's method is basically a disguised form of the use of Bayes' rule and the law of total probabilities. Malecot (1948) reorganized Wright's calculations by introducing the fundamental concept of identity by descent and exploiting its properties. The method of identity by descent has been perfected and developed by Malécot and his students, especially Gillois, Jauquard and Bouffette. Kempthorne (1957) has applied the concept of identity by descent to the study of quantitative inheritance. Kimura (1963) elegantly employed the ideas of identity by descent in determining rates of approach to homozygosity in certain mating situations with finite population size. Later in this chapter we will extend and refine the results of Kimura (1963) to give a more complete study of rates of approach to homozygosity. Ellison (1966) established several important limit theorems corresponding to polyploid, multi-locus random mating infinite populations by judicious enlargement of the concepts of identity by descent. Kesten (unpublished)) has recently refined the technique of Ellison.

Random processes of the diffusion type have the property that microscopic fluctuations of the trajectory make possible the identification of certain statistical parameters from one continuous observation. The paper deals with the construction of parameter estimates when observations are made at discrete but very dense time points.

A distribution F of a non-negative random variable belongs to the subexponential family of distributions S if 1 – F(2)(x) ~ 2(1 – F(x)) as x →∞. This family is of considerable interest in branching processes, queueing theory, transient renewal theory and infinite divisibility theory. Much is known about the kind of distributions that belong to S but the question of whether S is closed under convolution has remained unresolved for some time. This paper contains an example which demonstrates that S is not in fact closed.