Queueing systems with distinguished arrivals are described on the basis of Markov arrival processes with marked transitions. Customers are distinguished by their types of arrival. Usually, the queues observed by customers of different types are different, especially for queueing systems with bursty arrival processes. We study queueing systems from the points of view of customers of different types. A detailed analysis of the fundamental period, queue lengths and waiting times at the epochs of arrivals is given. The results obtained are the generalizations of the results of the MAP/G/1 queue.

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.

Let N be a stationary Markov-modulated marked point process on ℝ with intensity β ∗ and consider a real-valued functional ψ(N). In this paper we study expansions of the form Eψ(N) = a 0 + β ∗ a 1 + ·· ·+ (β∗ )nan + o((β ∗)n ) for β ∗→ 0. Formulas for the coefficients ai are derived in terms of factorial moment measures of N. We compute a 1 and a 2 for the probability of ruin φ u with initial capital u for the risk process in the Markov-modulated environment; a 0 = 0. Moreover, we give a sufficient condition for ϕu to be an analytic function of β ∗. We allow the premium rate function p(x) to depend on the actual risk reserve.

We consider a heavy traffic regime in queueing systems with identical service. These systems belong to the class of multi-phase systems with dependent service times in different service nodes. We study the limit behaviour of the waiting time vector in heavy traffic. Both transient behaviour and the stationary regime are considered. Our analysis is based on the conception of ‘approximated functionals', which appeared to be fruitful in weak convergence theory of stochastic processes.

An exponential inequality is generalized to one involving the tail of a decreasing failure rate distribution. The results are then applied in various situations, notably when the exponential inequality does not apply.

Using sample path analysis we show that under the same load the mean delay in queue in the M/G/2 system is smaller than that in the corresponding M/G/1 system, when the service time has either the DMRL or NBU property and the service discipline is FCFS. The proof technique uses a new device that equalizes the work in a two server system with that in a single sterver system. Other interesting quantities such as the average difference in work between the two servers in the GI/G/2 system and an exact alternate derivation of the mean delay in the M/M/2 system from sample path analysis are presented. For the same load, we also show that the mean delay in the M/G/C system with general service time distribution is smaller than that in the M/G/1 system when the traffic intensity is less than 1/c.

The comparison of optimal strategies in a simple stochastic replacement model for two types of machines with identical cost characteristics when one of them is more reliable than the other is conducted. It is proven that the scheduled replacement period for the more reliable type is always less than for the less reliable one. An example is presented when even the expected period of use of a more reliable machine is less than the expected period for the less reliable one. Some related problems are briefly discussed.

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

This paper presents an analysis of the single-server discrete-time finite-buffer queue with general interarrival and geometric service time, GI/Geom/1/N. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variations of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both cases, steady-state distributions for outside observers as well as at random and prearrival epochs have been obtained. The waiting time analysis has also been carried out. Results for the Geom/G/1/N queue with LAS-DA have been obtained from the GI/Geom/1/N queue with EAS. We also give various performance measures. An algorithm for computing state probabilities is given in an appendix.

We obtain a necessary and sufficient condition for the interaction between a service system and an external environment under which the stationary joint distribution of the set of busy channels and the state of the external environment is given by a product-form formula.

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ···,n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.

Conditions that have been widely used to establish stochastic ordering of random vectors include those given by Kamae et al. (1977), Barlow and Proschan (1975) and Langberg (1988). In this note we offer conditions that are weaker than those above, thus providing potential application to a wider class of problems.

This paper considers fluid queuing models of Markov-modulated traffic that, due to large differences in the time-scales of events, possess structural characteristics that yield a nearly completely decomposable (NCD) state-space. Extension of domain decomposition and aggregation techniques that apply to approximating the eigensystem of Markov chains permits the approximate subdivision of the full system to a number of small, independent subsystems (decomposition phase), plus an ‘aggregative' system featuring a state-space that distinguishes only one index per subsystem (aggregation phase). Perturbation analysis reveals that the error incurred by the approximation is of an order of magnitude equal to the weak coupling of the NCD Markov chain.

The study in this paper is then extended to the structure of NCD fluid models describing source superposition (multiplexing). It is shown that efficient spectral factorization techniques that arise from the Kronecker sum form of the global matrices can be applied through and combined with the decomposition and aggregation procedures. All structural characteristics and system parameters are expressible in terms of the individual sources multiplexed together, rendering the construction of the global system unnecessary.

Finally, besides providing efficient computational algorithms, the work in this paper can be recast as a conceptual framework for the better understanding of queueing systems under the presence of events happening in widely differing time-scales.

Conditions for a correlated cumulative shock model under which the system failure time is HNBUE are given. It is shown that the proof of a theorem given by Sumita and Shanthikumar (1985) relative to this property is not correct and a correct proof of the theorem is given.

The cumulative distribution of the finite sum of the binary sequence of order k is studied and some of its applications discussed. Certain properties of this sequence are investigated and uniformly superior bounds for the cumulative distribution under minimal information on the ‘success' probabilities are derived.

First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.