Let F be a distribution function on [0,∞) with finite expectation. In terms of the hazard rate of F several conditions are given which simultaneously imply subexponentiality of F and of its integrated tail distribution F1. These conditions apply to a wide class of longtailed distributions, and they can also be used in connection with certain random walks which occur in risk theory and queueing theory.

A motorist drives his car toward his destination along a street and looks for a motor pool. Motor pools are assumed to occur independently, with probability p. Observing whether there exists a motor pool or not, the driver decides either to stop (i.e., return to the latest motor pool observed so far and park there) or continue driving. Once the driver stops, he walks the remaining distance to his destination. Let r, 0 < r < 1, be the relative speed of driving a car compared with that on foot. Then the time duration required to reach the destination is measured by r · (distance driven) + (distance on foot) and the objective of the driver is to find a parking policy which minimizes the expected time duration. It is shown that, under an optimal policy, a U-turn never occurs before the destination, but may occur beyond the destination. Moreover, the expected time is computed and some comparisons are made between our problem and the classical parking problem.

Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions. A general technique is sketched, whereby the basic ingredients necessary for the application of Stein's method may be derived, and this is applied to a simple problem in Poisson point process approximation.

General exact formulas are established to calculate the reliability of a consecutive-k-out-of-n: F system. The optimality of the system, under permutation of the components, is examined too.

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

A characterization of the gamma distribution is considered which arises from a random difference equation. A proof without characteristic functions is given that if V and Y are independent random variables, then the independence of V · Y and (1 – V) · Y results in a characterization of the gamma distribution (after excluding the trivial cases).

For a strong Markov process on the line with continuous paths the Karlin–McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations. The extended formula is applied to a variety of combinatorial problems including counts of non-crossing paths in the plane with variable start and end points, dominance orderings, numbers of dominated majorization orderings, and time-inhomogeneous random walks.

For independent random lifelengths of the units in use stochastic comparisons of the number of failures and removal in [0,s] under age and block replacement policies are performed. A new concept of NBU (NWU) in sequence is introduced.

A system is subject to shocks; each shock at time t increases the cumulative damage λ (t) by a constant amount, while the system is subject to repair in between the shocks which brings down λ (t) at a constant rate. The shock arrival process is an inhomogeneous Poisson process with intensity function λ (t) and each shock weakens the system making it more expensive to run. The long-run expected cost per unit time of running the system is obtained as well as the variance of the cost which are used to get optimal times of replacement of the system.

Finiteness of mean extinction times for certain predator-prey models has been established by Hitchcock (1986) with the aid of a criterion of Reuter (1957). Using this criterion and a ‘minimisation' lemma this note shows that the mean extinction times tend to zero as the combined initial population of predators and prey becomes large.

A random stream of convex particles in , all moving uniformly in the same direction, is given. A convex test body is brought into the stream by means of a random rigid motion, independent of the stream. The number of particles that hit the test body at a preassigned boundary set within unit time is a random variable. Its expected value is determined under different model assumptions.

If T = (T1, · ··, Tn) is a vector of random lifetimes then its distribution can be determined by a set λof multivariate conditional hazard rates. In this paper, sufficient conditions on λare found which imply that T is Block–Savits MIFRA (multivariate increasing failure rate average) or Savits MIFR (multivariate increasing failure rate). Applications for a multivariate reliability model of Ross and for load-sharing models are given. The relationship between Shaked and Shanthikumar model of multivariate imperfect repair and the MIFRA property is also discussed.

The general cost structure of a unit on line is assumed to be governed by a stochastic process , where R(t) denotes the operating cost on [0, t)and ζ denotes the time of an unscheduled (or unplanned) replacement by a new unit at a cost c1. For an age replacement maintenance policy, scheduled (or planned) replacements occur whenever an operating unit reaches age T, whereas in the block replacement case, scheduled replacements occur every T units of time. Such scheduled replacements cost c2. The expected long-run cost per unit time can then be expressed in the form A(T)/Ε [min(ζ, T)] and B(T)/T respectively. Our main result shows that where U is the associated renewal function generated by ζ.

Results showing the weak convergence of certain stochastic processes are used to derive both known and new (asymptotic) properties of signs of residuals from regression; other weak convergence results are derived, and used to determine the behaviour of runs of residuals.

A non-homogeneous Poisson shock model has a continuous mean function Λ(t). The kth shock Sk causes simultaneous failure of the components j ∊ J ∊ {1, ···, n} with probability pJ(Sk). If Tj is the lifetime of component j, it is shown that (T1, · ··, Tn) belongs to various multivariate non-parametric life classes depending on the life class of .

Consider an ‘experiment' which can be repeated indefinitely often resulting in independent random outcomes. Fix attention on a finite number of possible (sets of) outcomes E1, E2, … and define W = W(N1, N2, …) to be the expected number of repetitions needed to ensure that E1 has occurred (at least) N1 times, E2 has occurred (at least) N2 times, etc. This article examines the asymptotic behavior of W as a function of the sum ΣjNj, as the latter grows without bound.

A tandem queue with a FIFO multiserver system at each stage, i.i.d. service times and a renewal process of external arrivals is shown to be regenerative by modeling it as a Harris-ergodic Markov chain. In addition, some explicit regeneration points are found. This generalizes the results of Nummelin (1981) in which a single server system is at each stage and the result of Charlot et al. (1978) in which the FIFO GI/GI/c queue is modeled as a Harris chain. In preparing for our result, we study the random assignment queue and use it to give a new proof of Harris ergodicity of the FIFO queue.

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.

Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

Slotted ALOHA with a finite number of users, each with infinite buffer, is considered. For stationary, metrically transitive, non-independent input, the problem of existence of stationary queue length distributions is solved. Results are obtained for zero and arbitrary (finite a.s.) initial conditions. Continuity, in probability, of queue lengths with respect to input sequence is proved.