In this paper a unified derivation of the upper and lower bounds (in terms of the mean) of an IFR, DFR, IFRA, DFRA, NBU or NWU reliability function is presented. The method of proof provides a simpler alternative to the various proofs known in the literature. Moreover, this method can be used to generalize the existing results in two ways, as demonstrated here. First, the bounds for the reliability function are obtained in terms of any finite moment in all these cases. Subsequently we provide a moment bound on a reliability function which ages faster or slower than a known one in some sense. The existing literature does not offer any of these generalizations, except for a few results which are available in an unnecessarily complicated form.

The theory of piecewise-deterministic Markov processes is used in order to investigate insurance risk models where borrowing, investment and inflation are present.

In this paper we obtain the Beňes equation for the evolution of the probability distribution of the excursion process associated with the level crossings of a general storage process. We then show that under stationarity and ergodicity assumptions on the process we can recover the well-known rate conservation law (RCL). Using the stationary solution we then show that the existence of an invariant solution can be studied in terms of an operator equation and we show how this characterization leads to a very simple explicit computation of the stationary distribution.

Perturbation analysis estimators for expectations of possibly discontinuous functions of the time-stationary workload were derived in [2]. The expressions obtained may, however, not be valid if the customer-stationary distribution of the workload has atoms (at points other than zero). This was pointed out by Brémaud and Lasgouttes in [1]. In this note we clearly state the additional condition required for the validity of the expressions in [2]. We furthermore show how our approximation scheme can also be used to obtain the correct expressions for the right and left derivatives given in [1].

Existence and finiteness of the sample-mean limit of sojourn times of jobs in a queueing system are investigated. The queueing system operates under rather general multiprocessor disciplines allowing job classes and priorities. The input stream of jobs consisting of job classes and interarrival and processing times is stationary and ergodic and may contain batch arrivals. Existence of the sample-mean limit is proved by means of the superadditive ergodic theorem, and its finiteness is controlled by uniform mixing of the input stream.

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

This paper examines the availability of a maintained system where the rate of deterioration is governed by an exogenous random environment. We provide a qualitative result that exposes the relationship between remaining lifetime, environment, and repairs. This result leads to simple bounds that can be used to choose inspection rates that guarantee a specified level of availability. The principal result requires no specific distributional assumptions, is intuitively appealing and can be directly applied by practitioners. Our development employs techniques from stochastic calculus.

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.

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network.

This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p , where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

The double-stranded molecule, DNA, has the unique property of replication and, because of this, it is the central molecule of life. The mechanism of replication for each single strand is intricate, involving enzymes which move along each of the single strands building a complementary copy. At the frontier of this action, the events have a strong stochastic character due to the random location on the DNA of key ‘sites' where copying commences. A model of this process is analysed. The central problem of interest is the mean length of certain ‘islands' of newly replicated DNA developed at the randomly located ‘sites'. These islands, which have been observed experimentally, are called Okazaki fragments.

Consider GI/G/1 processor sharing queues with traffic intensity tending to 1. Using the theory of random measures and the theory of branching processes we investigate the limiting behaviour of the queue length, sojourn time and random measures describing attained and residual processing times of customers present.

The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times are shown.

We provide the explicit expression for the mean coverage function of a generalized voter model on a regular lattice and establish a characterization of the class of the above processes. As a result, we derive the exact rate of convergence of the considered processes to the steady state. We also prove the existence of different processes with the same mean coverage function on a given lattice.

A Markov-modulated Poisson process (MMPP) is a Poisson process whose rate is a finite Markov chain. The Poisson process is a simple MMPP. An MMPP/M/1 queue is a queue with MMPP arrivals, an infinite capacity, and a single exponential server. We prove that the output of an MMPP/M/1 queue is not an MMPP process unless the input is Poisson. We derive this result by analyzing the structure of the non-linear filter of the state given the departure process of the queue. The practical relevance of the result is that it rules out the existence of simple finite descriptions of queueing networks with MMPP inputs.

This paper examines the infinitely high dam with seasonal (periodic) Lévy input under the unit release rule. We show that a periodic limiting distribution of dam content exists whenever the mean input over a seasonal cycle is less than 1. The Laplace transform of dam content at a finite time and the Laplace transform of the periodic limiting distribution are derived in terms of the probability of an empty dam. Necessary and sufficient conditions for the periodic limiting distribution to have finite moments are given. Convergence rates to the periodic limiting distribution are obtained from the moment results. Our methods of analysis lean heavily on the coupling method and a stochastic monotonicity result.

In this paper a particular loss network consisting of two links with C 1 and C 2 circuits, respectively, and two fixed routes, is investigated. A call on route 1 uses a circuit from both links, and a call on route 2 uses a circuit from only the second link. Calls requesting routes 1 and 2 arrive as independent Poisson streams. A call requesting route 1 is blocked and lost if there are no free circuits on either link, and a call requesting route 2 is blocked and lost if there is no free circuit on the second link. Otherwise the call is connected and holds a circuit from each link on its route for the holding period of the call.

The case in which the capacities C 1, and C 2, and the traffic intensities v 1, and v 2, all become large of O(N) where N » 1, but with their ratios fixed, is considered. The loss probabilities L 1 and L 2 for calls requesting routes 1 and 2, respectively, are investigated. The asymptotic behavior of L 1 and L 2 as N→ ∞ is determined with the help of double contour integral representations and saddlepoint approximations. The results differ in various regions of the parameter space (C 1, C 2, v 1, v 2). In some of these results the loss probabilities are given in terms of the Erlang loss function, with appropriate arguments, to within an exponentially small relative error. The results provide new information when the loss probabilities are exponentially small in N. This situation is of practical interest, e.g. in cellular systems, and in asynchronous transfer mode networks, where very small loss probabilities are desired.

The accuracy of the Erlang fixed-point approximations to the loss probabilities is also investigated. In particular, it is shown that the fixed-point approximation E 2 to L 2 is inaccurate in a certain region of the parameter space, since L 2 « E 2 there. On the other hand, in some regions of the parameter space the fixed-point approximations to both L 1 and L 2 are accurate to within an exponentially small relative error.

We consider an Mx /G/1 queueing system with N-policy and multiple vacations. As soon as the system empties, the server leaves for a vacation of random length V. When he returns, if the queue length is greater than or equal to a predetermined value N(threshold), the server immediately begins to serve the customers. If he finds less than N customers, he leaves for another vacation and so on until he finally finds at least N customers. We obtain the system size distribution and show that the system size decomposes into three random variables one of which is the system size of ordinary Mx /G/1 queue. The interpretation of the other random variables will be provided. We also derive the queue waiting time distribution and other performance measures. Finally we derive a condition under which the optimal stationary operating policy is achieved under a linear cost structure.

A one-dimensional packing approach is used to obtain limiting results for inter-crack distances after multiple fracture of a long brittle-matrix composite with continuous aligned fibres. The results may also be appropriate for applications of the Rényi car-parking model in which there is a reduced probability of cars parking bumper to bumper.

In this paper we consider the problem of routing customers to identical servers, each with its own infinite-capacity queue. Under the assumptions that (i) the service times form a sequence of independent and identically distributed random variables with increasing failure rate distribution and (ii) state information is not available, we establish that the round-robin policy minimizes, in the sense of a separable increasing convex ordering, the customer response times and the numbers of customers in the queues.