Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say is Laplace-smaller than for all s > 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.

Nash has extended Gittins' work to describe optimal strategies for a class of generalised bandit problems. Here we use a forwards induction argument to analyse ε -optimal strategies for generalised bandit problems. An evaluation procedure for such problems is described; this may be used to analyse models in research planning and stochastic scheduling.

We investigate a generalized variability ordering and its weaker versions among non-negative random variables (lifetimes of components). Our results include a necessary and sufficient condition which justifies the generalized variability interpretation of this dominance relation between life distributions, relationships to some weakly aging classes in reliability theory, closure properties and inequalities for the mean life of series and parallel systems under such ordering.

Analogues of some theorems of Brown (1980) concerning renewal measures with DFR or IMRL underlying distributions are proved for delayed renewal measures. Some related results, such as partial converses of the main theorems, are also presented.

A family of random variables {X(θ)} parameterized by the parameter θ satisfies stochastic convexity (SCX) if and only if for any increasing and convex function f(x), Ef[X(θ)] is convex in θ. This definition, however, has a major drawback for the lack of certain important closure properties. In this paper we establish the notion of strong stochastic convexity (SSCX), which implies SCX. We demonstrate that SSCX is a property enjoyed by a wide range of random variables. We also show that SSCX is preserved under random mixture, random summation, and any increasing and convex operations that are applied to a set of independent random variables. These closure properties greatly facilitate the study of parametric convexity of many stochastic systems. Applications to GI/G/1 queues, tandem and cyclic queues, and tree-like networks are discussed. We also demonstrate the application of SSCX in bounding the performance of certain systems.

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB(t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB(t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB(t) as t →∞.

In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied.

A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given.

Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.

In this study we examine repairable systems with random lifetime. Upon failure, a maintenance action, specifying the degree of repair, is taken by a controller. The objective is to determine an age-dependent maintenance strategy which minimizes the total expected discounted cost over an infinite planning horizon. Using several properties of the optimal policy which are derived in this study, we propose analytical and numerical methods for determining the optimal maintenance strategy. In order to obtain a better insight regarding the structure and nature of the optimal policy and to illustrate computational procedures, a numerical example is analysed. The proposed maintenance model outlines a new research channel in the area of reliability with interesting theoretical issues and a wide range of potential applications in various fields such as product design, inventory systems for spare parts, and management of maintenance crews.

We derive a rate conservation law for distribution densities which extends a result of Brill and Posner. Based on this conservation law, we obtain a generalized Takács equation for the G/G/m/B queueing system that only requires the existence of a stochastic intensity for the arrival process and the residual service time distribution density for the G/GI/1/B queue. Finally, we solve Takács' equation for the N/GI/1/∞ queueing system.

In this study an asymptotical analysis of the reliability of a complex renewable system with an unbounded number of repair units is provided. The system state is given by a binary vector e(t) = [e1(t), · ··, en(t)], ei(t) = 0(1), if at moment t the ith element is failure-free (failed). We assume that at the state e the ith element has failure intensity λi(e). At the instant of failure of every element the renewal work begins and the renewal time has distribution function Gi(t). Let E_ be the set of failed system states. The goal of this study is the asymptotic estimation of the distribution of the time until the first system failure, .

We obtain the equilibrium solutions of an R-out-of-N system subject to random breakdown. There are M spares and a single repairman, who instals good spares into the system when breakdowns occur and also repairs the failed items. Installation has pre-emptive repeat priority over repairs. Arbitrary distributions of installation and repair times are allowed. Equilibrium availability and downtime distributions are obtained from the equilibrium state distribution.

Consider a slotted communication channel which carries packetized voice and which can transmit exactly one packet every timeslot. Assume that every conversation routed over this channel generates exactly one packet every N timeslots. We study, for the case of an infinite buffer and an N-periodic strongly interchangeable input process, buffer behavior and packet delays as functions of the number of calls routed over the channel, as long as that number is less than or equal to N. Among our results are a simple algorithm which computes the marginal distribution of the buffer content and an algorithm with complexity of order N4 which computes the distribution of the maximal buffer content (over a period of N timeslots).

In this paper, we study a repair replacement model for a stochastically deteriorating system. For the expected discounted reward case, we show that the optimal replacement policy is of the form ‘replace at the time of the Nth failure'.

Non-exponential queueing networks with blocking are studied under two types of blocking protocols: the ‘stop' and the ‘repeat' protocol under which services are to be either stopped or repeated respectively upon blocking. The two protocols are shown to be stationary equivalent when the system satisfies a partial balance condition. This condition is related to product-form results. The results are illustrated by closed and open network applications with reversible and nonreversible routing.

For , we obtain a K′- capacity queue from a K- capacity queue through a random time change and a truncation, provided arrivals are Poisson or service is exponential. In the case of an M/G/1/K queue, the time change erases service intervals that begin with more than K′ customers in the systems. This construction yields a straightforward sample path proof of Keilson's result on the proportionality of the ergodic queue length probabilities in M/G/1/K queues. The same approach proves a strengthened result for ‘detailed' state probabilities. It also reproduces a proportionality result for a vacation model, due to Keilson and Servi. A ‘dual' construction yields a different kind of proportionality for the G/M/1/K queue.

Product-form equilibrium distributions in networks of queues in which customers move singly have been known since 1957, when Jackson derived some surprising independence results. A product-form equilibrium distribution has also recently been shown to be valid for certain queueing networks with batch arrivals, batch services and even correlated routing.

This paper derives the joint equilibrium distribution of states immediately before and after a batch of customers is released into the network. The results are valid for either discrete- or continuous-time queueing networks: previously obtained results can be seen as marginal distributions within a more general framework. A generalisation of the classical ‘arrival theorem' for continuous-time networks is given, which compares the equilibrium distribution as seen by arrivals to the time-averaged equilibrium distribution.

This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.

We consider the asymmetric simple exclusion process which starts from a product measure such that all the sites to the left of zero (including zero) are occupied and the right of 0 (excluding 0) are empty. We label the particle initially at 0 as the leading particle. We study the long-term behaviour of this process near large sites when the leading particle's holding time is different from that of the other particles. In particular, we assume that the leading particle moves at a slower rate than the other particles. We call this modified asymmetric simple exclusion process the road-hog process. Coupling and stochastic ordering techniques are used to derive the density profile of this process. Road-hog processes are useful in modelling series of exponential queues with Poisson and non-Poisson input process. The density profiles dramatically illustrate the flow of customers through the queues.

We introduce a new class of queueing networks in which customers are either ‘negative' or ‘positive'. A negative customer arriving to a queue reduces the total customer count in that queue by 1 if the queue length is positive; it has no effect at all if the queue length is empty. Negative customers do not receive service. Customers leaving a queue for another one can either become negative or remain positive. Positive customers behave as ordinary queueing network customers and receive service. We show that this model with exponential service times, Poisson external arrivals, with the usual independence assumptions for service times, and Markovian customer movements between queues, has product form. It is quasi-reversible in the usual sense, but not in a broader sense which includes all destructions of customers in the set of departures. The existence and uniqueness of the solutions to the (nonlinear) customer flow equations, and hence of the product form solution, is discussed.

We study sojourn times in M/G/1 multiple vacation systems and multiqueue cyclic-service (polling) systems with instantaneous Bernoulli feedback. Three service disciplines, exhaustive, gated, and 1-limited, are considered for both M/G/1 vacation and polling systems. The Laplace-Stieltjes transforms of the sojourn time distributions in the three vacation systems are derived. For polling systems, we provide explicit expressions for the mean sojourn times in symmetric cases. Furthermore a pseudo-conservation law with respect to the mean sojourn times is derived for a polling system with a mixture of the three service disciplines.