The queueing system studied in this paper is the one in which

1. (i) there are an infinite number of servers,

2. (ii) initially (at t = 0) all the servers are idle,

3. (iii) one server serves only one customer at a time and the service times are independent and identically distributed with distribution function B(t) (t > 0) and mean β(< ∞),

4. (iv) the arrivals are in batches such that a batch arrives during (t, t + δt) with probability λ(t)δt + o(δt) (λ(t) > 0) and no arrival takes place during (t, t + δt) with the probability 1 –λ(t)δt + o(δt),

5. (v) the batch sizes are independent and identically distributed with mean α(< ∞), and the probability that a batch size equals r is given by ar(r ≧ 1),

6. (vi) the batch sizes, the service times and the arrivals are independent.

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).

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.

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.

This paper is a sequel to [1] and considers a more realistic formulation of the same question: that of finding an optimal policy for controlling the path of a space-ship as it moves towards its target. The difference here is that we no longer suppose there is an infinite quantity of fuel, always available at a fixed price, for modifying the current direction of motion. This complicates the problem of reducing the final miss distance, by introducing an extra variable. As before, we shall be particularly concerned to find a control procedure which always minimizes the mean square terminal miss. From the theoretical point of view we are also interested to see whether the techniques used to approximate the optimal policy can be extended, and how far we shall be forced to adopt a new approach. Results are derived which provide bounds on the form of the optimal policy. The derivation depends on a comparison technique whose validity is intuitively obvious, but which is still only a conjecture. However, further confirmation is obtained in the quadratic case from asymptotic expansions giving the form of the solution both when the space-ship is far away from its target and during its final approach.

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.

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.

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.

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.

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.

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.

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.

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.

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x≧0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is ≦x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random.

Conditions are given on a process so that a shock model governed by the process will have nbue survival time. A related condition is given so that survival will be NBU and it is shown that a pure birth process satisfies this condition. A result of A-Hameed and Proschan is obtained as a corollary.

We introduce a new class of Monte Carlo methods, which we call exact estimation algorithms. Such algorithms provide unbiased estimators for equilibrium expectations associated with real-valued functionals defined on a Markov chain. We provide easily implemented algorithms for the class of positive Harris recurrent Markov chains, and for chains that are contracting on average. We further argue that exact estimation in the Markov chain setting provides a significant theoretical relaxation relative to exact simulation methods.

A generalisation of multistate coherent structures is proposed where the state of each component in a binary coherent structure can take any value in the unit interval, as can the structure function. The notions of duality, critical elements and strong coherency for such a structure are discussed and the functional form of the structure function is analysed. An expression is derived for the distribution function of the state of the system, given the distributions of the states of the components, and generalisations of the Moore–Shannon and IFRA and NBU closure theorems are proved. The states of the components are then permitted to vary with time and a first-passage-time distribution is discussed. A simple model for such a process, based on the concept of partial availability, is then proposed. Lastly, an alternative continuum structure function is introduced and discussed.

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.