A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio ‘age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csörg et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals.

The paper obtains the exact distribution of Bookstein's shape variables under his plausible model for landmark data. We consider its properties including invariances, marginal distributions and the relationship with Kendall's uniform measure. Particular cases for triangles and quadrilaterals are highlighted. A normal approximation to the distribution is obtained, extending Bookstein's result for three landmarks. The adequacy of these approximations is also studied.

A general study of the asymptotic law of a skew-product of two real diffusions is proposed; a complete and elementary form of the Rosenkrantz theorem is given; the results are applied to three examples: the Brownian motions on SO3, on S3, and on SH3.

In this paper we compare the distribution of the supremum of the Gaussian random fields Z(P) = ∫CpG(P, P′) dW(P′) and U(P) = ∫CpdW(P'), where CP are circles of fixed radius, dW is a white noise field and G are special deterministic response functions.

The results obtained permit us to establish upper bounds for the distribution of the supremum of Z(P) by applying some well-known inequalities on U(P).

The comparison of the suprema is carried out also, when CP = ℝ2, between fields with different response functions.

A max-autoregressive moving average (MARMA(p, q)) process {Xt} satisfies the recursion for all t where φ i, , and {Zt} is i.i.d. with common distribution function Φ1,σ (X): = exp {–σ x–1} for . Such processes have finite-dimensional distributions which are max-stable and hence are examples of max-stable processes. We provide necessary and sufficient conditions for existence of a stationary solution to the MARMA recursion and we examine the reducibility of the process to a MARMA(p′, q′) with p′ <p or q′ < q. After introducing a natural metric between two jointly max-stable random variables, we consider the prediction problem for MARMA processes. Assuming that X1, …, Xn have been observed, we restrict our class of predictors to be max-linear, i.e. of the form , and find b1, …, bn to minimize the distance between this predictor and Xn+k for k 1. The optimality criterion is designed to minimize the probability of large errors and is similar in spirit to the dispersion criterion adopted in Cline and Brockwell (Stoch. Proc. Appl. 19 (1985), 281-296) for the prediction of ARMA processes with stable noise. Most of our results remain valid for the case when the distribution of Z1 is only in the domain of attraction of Φ1,σ. In addition, we give a naive estimation procedure for the φ 's and the θ 's which, with probability 1, identifies the true parameter values exactly for n sufficiently large.

Loss networks have long been of interest to telephone engineers, and are becoming increasingly important as models of computer and information systems. In this paper we present an asymptotic analysis of loss networks exhibiting various forms of symmetry. Our aim is to better understand the behaviour of networks involving very large numbers of links and routes, where an exact analysis is not possible and approximations are required.

This paper studies the behaviour of large loss networks, paying particular attention to links at a certain critical loading where the load offered very nearly matches capacity. We correct and extend an earlier central limit theorem for the stationary distribution of a loss network with critically loaded links. We then use this result to show that acceptance probabilities have a limiting product-form decomposition, despite marked dependencies between the occupancies of critically loaded links.

We prove a theorem which can be used to show that the expectation of a non-negative function of the state of a time-homogeneous Markov process is uniformly bounded in time. This is reminiscent of the classical theory of non-negative supermartingales, except that our analog of the supermartingale inequality need not hold almost surely. Consequently, the theorem is suitable for establishing the stability of systems that evolve in a stabilizing mode in most states, though from certain states they may jump to a less stable state. We use this theorem to show that ‘joining the shortest queue' can bound the expected sum of the squares of the differences between all pairs among N queues, even under arbitrarily heavy traffic.

In this paper we consider a queueing system having n exponential servers, each with its own queue and service rate. Customers arrive according to a Poisson process with rate λ, and upon arrival each customer must be routed to some server's queue. No jockeying amongst the queues is allowed and each server serves its queue according to a first-come-first-served discipline.

Each server i, 1 ≦ i ≦ n, provides service with a state-dependent rate μ(i)(k), k = 0, 1, …. In addition, at every queue i, there is a deterministic holding cost which occurs at rate h(i)(k) while there are k customers at the queue.

An admissible routing policy is a policy that assigns each arriving customer to one of the queues. A decision at time t may be randomized and dependent on the queue lengths and decisions till time t. An optimal routing policy is an admissible policy that minimizes the long-run average holding cost.

In this study, we bound the optimal cost from below, by considering an ideal system, where each server optimally selects a given proportion of customers, irrespective of other servers' selections. From this ideal system we construct a class of admissible routing policies, the overflow routing class, that approximates the ideal situation for each server. Finally, we evaluate the policies and compare them to the lower bound.

The queueing model known as precedence-based queueing discipline, proposed in [8] by Tsitsiklis et al. is addressed. In this queueing model, there are infinitely many servers and for any pair of customers i and j such that i arrived later than j, there is a fixed probability p that i will have to wait for the end of the execution of j before starting its execution. In the case where the customer service times are deterministic and the arrival process is Poisson, these authors have derived the stability condition which determines the maximum arrival rate that keeps the system stable. For more general statistics, they conjectured that the stability condition would possibly depend on the complete service time distribution functions but only on the first moment of the inter-arrival times.

In this paper, we consider this queueing model and relax the restrictive statistical assumptions mentioned above by only assuming that the service times and the inter-arrival times are stationary and ergodic sequences, so that these variables can receive general distribution functions and be correlated.

We derive a general expression for the stability condition which in turn proves the above conjectures. The results are obtained by establishing pathwise evolution equations for these queueing systems, and then a schema which, in certain sense, generalizes the schema of Loynes for the G/G/1 queues.

This paper presents a variation of a state-dependent reliability model first proposed in Lemoine and Wenocur [4], [5], and develops some of its corresponding analytical theory. In particular, we develop a reliability model in which system state is a random process satisfying a stochastic differential equation where the driving process is gamma distributed.

Structure functions relate the level of operations of a system as a function of the level of the operation of its components. In this paper structure functions are studied which have an intuitive property, called L-superadditive (L-subadditive). Such functions describe whether a system is more series-like or more parallel-like. L-superadditive functions are also known under the names supermodular, quasi-monotone and superadditive and have been studied by many authors. Basic properties of both discrete and continuous (i.e., taking a continuum of values) L-superadditive structure functions are studied. For binary structure functions of binary values, El-Neweihi (1980) showed that L-superadditive structure functions must be series. This continues to hold for binary-valued structure functions even if the component values are continuous (see Proposition 3.1). In the case of non-binary-valued structure functions this is no longer the case. We consider structure functions taking discrete values and obtain results in various cases. A conjecture concerning the general case is made.

A conjecture due to Block et al. (1989), concerning the number of critical connection vectors to the various performance levels of a discrete L-superadditive structure function, is proved. When the components of the discrete L-superadditive structure function are further assumed to satisfy a certain relevance condition due to Griffith (1980), it is shown that there is exactly one critical connection vector to each performance level.

We shall use three basic properties of Brownian motion to derive in an elegant and non-computational way the probability that standard Brownian motion, starting from 0, will ever cross the halflines t → αt + β or t → γt + δ where γ, δ < 0 < α, β.

In a given set, the elements are compared pairwise. The number W of similar pairs is studied, that is, the number of pairs with a certain property in common. Under certain conditions, W has, approximately, a Poisson distribution. Examples are considered connected with the birthday problem and with a circle problem involving DNA breakages.

Let dx(t) = y(t) dt, where y(t) is a one-dimensional Wiener process. In this note, we obtain a formula for the moment-generating function of y(T), where T is the 1/2-winding time about the origin of the integrated Wiener process x(t).

A system is repaired on failure. With probability p, it is returned to the ‘good as new' state (perfect repair) and with probability 1 – p, it is returned to the functioning state, but is only as good as a system of age equal to its age at failure (imperfect repair). In this article, we develop replacement policies for a deteriorating system with imperfect maintenance. The successive survival times and consecutive repair times form a geometric process which is stochastically non-increasing or non-decreasing respectively. Explicit expressions are obtained for the long-run expected cost under two kinds of replacement policies based on the working age of the system and the number of imperfect repairs before a replacement.

We consider the same circuit switching problem as in Mitra [1]. The calculation of the blocking probabilities is reduced to finding the partition function for a closed exponential pseudo-network with L−1 customers. This pseudo-network differs from that in [1] in one respect only: service rates at nodes 1, 2, …, p depend on the queue length. The asymptotic expansion developed in [1] follows from our exact expression for the partition function.