This paper introduces a new form of local balance and the corresponding product-form results. It is shown that these new product-form results allow capacity constraints at the stations of a queueing network without reversibility assumptions and without special blocking protocols. In particular, exact product-form results for heavily loaded queueing networks are obtained.

This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: ‘saturate' the queues which are fed by the external arrival stream; look at the ‘intensity' μ of the departure stream in this saturated system; then stability holds whenever the intensity of the arrival process, say λ satisfies the condition λ < μ, whereas the network is unstable if λ > μ. Whenever the stability condition is satisfied, it is also shown that certain state variables associated with the network admit a finite stationary regime which is constructed pathwise using a Loynes-type backward argument. This framework involves two main pathwise properties, external monotonicity and separability, which are satisfied by several classical queueing networks. The main tool for the proof of this rule is subadditive ergodic theory. It is shown that, for various problems, the proposed method provides an alternative to the methods based on Harris recurrence and regeneration; this is particularly true in the Markov case, where we show that the distributional assumptions commonly made on service or arrival times so as to ensure Harris recurrence can in fact be relaxed.

Hunt and Kurtz [9] consider a loss network as the number of circuits and the offered traffics become large. They prove a functional law of large numbers for such a network and illustrate their results with some simple examples. In this paper we apply their results to slightly more complicated examples to illustrate other, and sometimes surprising, behaviour of the loss networks in heavy traffic. The networks we consider operate under somewhat unusual routing rules but this is to enable us to produce the behaviour in networks with only a few links. In larger, real-world networks it is likely that much more ‘natural' and intuitively appealing routing rules could produce similar undesirable behaviour.

The DNA of higher animals replicates by an interesting mechanism. Enzymes recognise specific sites randomly scattered on the molecule and establish a bidirectional process of unwinding and replication from these sites. We investigate the limiting distribution of the completion time for this process by considering related coverage problems investigated by Janson (1983) and Hall (1988).

A variety of performance measures of a GI/G/1 queue are explicitly related to the idle-period distribution of the queue, suggesting that the system analysis can be accomplished by the analysis of the idle period. However, the ‘stand-alone' relationship for the idle-period distribution of the GI/G/1 queue (i.e. the counterpart of Lindley's equation) has not been found in the literature. In this paper we develop a non-linear integral equation for the idle period distribution of the GI/G/1 queue. We also show that this non-linear system defines a unique solution. This development makes possible the analysis of the GI/G/1 queue in a different perspective.

In this paper, an optimal maintenance model for standby systems is studied. An inspection–repair–replacement policy is employed. Assume that the state of the system can only be determined through an inspection which may incorrectly identify the system state. After each inspection, if the system is identified as in the down state, a repair action will be taken. It will be replaced some time later by a new and identical one. The problem is to determine an optimal policy so that the availability of the system is high enough at any time and the long-run expected cost per unit time is minimized. An explicit expression for the long-run expected cost per unit time is derived. For a geometric model, a simple algorithm for the determination of an optimal solution is suggested.

Models for epidemic spread of infections are formulated by defining intensities for relevant counting processes. It is assumed that an infected individual passes through k stages of infectivity. The times spent in the different stages are random. Many well-known models for the spread of infections can be described in this way. The models can also be applied to describe other processes of epidemic character (such as models for rumour spreading). Asymptotic results are derived both for the size and for the duration of the epidemic.

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

Consider a homogeneous Poisson process in with density ρ, and add the origin as an extra point. Now connect any two points x and y of the process with probability g(x − y), independently of the point process and all other pairs, where g is a function which depends only on the Euclidean distance between x and y, and which is nonincreasing in the distance. We distinguish two critical densities in this model. The first is the infimum of all densities for which the cluster of the origin is infinite with positive probability, and the second is the infimum of all densities for which the expected size of the cluster of the origin is infinite. It is known that if , then the two critical densities are non-trivial, i.e. bounded away from 0 and ∞. It is also known that if g is of the form , for some r > 0, then the two critical densities coincide. In this paper we generalize this result and show that under the integrability condition mentioned above the two critical densities are always equal.

A highly symmetric loss network is considered, the symmetric star network previously considered by Whitt and Ziedins and Kelly. As the number of links, K, becomes large, the state space for this process also grows, so we consider a functional of the network, one which contains all information relevant to blocking probabilities within the network but which is easier to analyse. We show that this reduced process obeys a functional law of large numbers and a functional central limit theorem, the limit in this latter case being an Ornstein-Uhlenbeck diffusion process. Finally, by considering the network in equilibrium, we are able to prove that the well-known Erlang fixed point approximation for blocking probabilities is correct to within o(K -1/2) as K → ∞.

The Laplace transform of the probability distribution of the end-to-end delay in tandem networks is obtained where the first and/or second queue are G-queues, i.e. they have negative arrivals. For the most general case the method is based on the solution of a boundary value problem on a closed contour in the complex plane, which itself reduces to the solution of a Fredholm integral equation of the second kind. We also consider the dependence or independence of the sojourn times at each queue in the two special cases where only one of the queues is a G-queue, the other having no negative arrivals.

There are some generalised semi-Markov processes (GSMP) which are insensitive, that is the value of some performance measures for the system depend only on the mean value of lifetimes and not on their actual distribution. In most cases this is not true and a performance measure can take on a number of values depending on the lifetime distributions. In this paper we present a method for finding tight bounds on the sensitivity of performance measures for the class of GSMPs with a single generally distributed lifetime. Using this method we can find upper and lower bounds for the value of a function of the stationary distribution as the distribution of the general lifetime ranges over a set of distributions with fixed mean. The method is applied to find bounds on the average queue length of the Engset queue and the time congestion in the GI/M/n/n queueing system.

Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group , and relates integrations concerning them to each other. For , we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently obtained as a generalization for Little's formula. The generalized Mecke formula, which is called GMF, can be also viewed as a generalization of the stationary version of H = λG. Under the stationary and ergodic assumptions, we apply it to derive many sample path formulas which have been known as extensions of H = λG. This will make clear what kinds of probabilistic conditions are sufficient to get them. We also mention a further generalization of Mecke's formula.

We study certain stochastic processes arising in probabilistic modelling. We discuss the limit behavior of these processes and estimate the rate of convergence to the limit.

A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fixed insensitive type, if the lifetime of this clock is changed to infinity, then the background process is stationary under a certain time change. This implies that the expected time required for the tagged clock to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity results are known for two special RGSMPs: ordinary GSMP and Kelly's symmetric queue. Our results not only extend them to a general RGSMP but also give more detailed formulas, which allow us to calculate for instance the expected attained sojourn time while the background process is in a given state. Furthermore, we remark that analogous results hold for GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).

In this paper we consider the analysis of call blocking at a single resource with differing capacity requirements as well as differing arrival rates and holding times. We include in our analysis trunk reservation parameters which provide an important mechanism for tuning the relative call blockings to desired levels. We base our work on an asymptotic regime where the resource is in heavy traffic. We further derive, from our asymptotic analysis. methods for the analysis of finite systems. Empirical results suggest that these methods perform well for a wide class of examples.

Monotonicity and correlation results for queueing network processes, generalized birth–death processes and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.

In this paper, we discuss a variety of methods for computing the Wiener-Hopf factorization of a finite Markov chain associated to a fluctuating additive functional. The importance of this is that the equilibrium law of a fluid model can be expressed in terms of these Wiener–Hopf factors. The diagonalization methods considered are actually quite efficient, and provide an effective solution to the problem.

In this paper we study the departure process of M/G/1 queueing models with a single server vacation and multiple server vacations. The arguments employed are direct probability decomposition, renewal theory and the Laplace–Stieltjes transform. We discuss the distribution of the interdeparture time and the expected number of departures occurring in the time interval (0, t] from the beginning of the state i (i = 0, 1, 2, ···), and provide a new method for analysis of the departure process of the single-server queue.

This paper studies the absorption time of an integer-valued Markov chain with a lower-triangular transition matrix. The main results concern the asymptotic behavior of the absorption time when the starting point tends to infinity (asymptotics of moments and central limit theorem). They are obtained using stochastic comparison for Markov chains and the classical theorems of renewal theory. Applications to the description of large random chains of partitions and large random ordered partitions are given.