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In this paper, we introduce a family of processes with values on the nonnegative integers that describes the dynamics of populations where individuals are allowed to have different types of interactions. The types of interactions that we consider include pairwise interactions, such as competition, annihilation, and cooperation; and interactions among several individuals that can be viewed as catastrophes. We call such families of processes branching processes with interactions. Our aim is to study their long-term behaviour under a specific regime of the pairwise interaction parameters that we introduce as the subcritical cooperative regime. Under such a regime, we prove that a process in this class comes down from infinity and has a moment dual which turns out to be a jump-diffusion that can be thought as the evolution of the frequency of a trait or phenotype, and whose parameters have a classical interpretation in terms of population genetics. The moment dual is an important tool for characterizing the stationary distribution of branching processes with interactions whenever such a distribution exists; it is also an interesting object in its own right.

This paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

Using a result of Blanchet and Wallwater (2015: Exact sampling of stationary and time-reversed queues. ACM TOMACS, 25, 26) for exactly simulating the maximum of a negative drift random walk queue endowed with independent and identically distributed (iid) increments, we extend it to a multi-dimensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a first-in-first-out (FIFO) multi-server queue in which the arrival process is a general renewal process and the service times are iid; the FIFO GI/GI/c queue with 2 \le c < 1. Our method utilizes dominated coupling from the past (DCFP) as well as the Random Assignment (RA) discipline, and complements the earlier work in which Poisson arrivals were assumed, such as the recent work of Connor and Kendall (2015: Perfect simulation of M/G/c queues. Advances in Applied Probability, 47, 4). We also consider the models in continuous-time and show that with mild further assumptions, the exact simulation of those stationary distributions can also be achieved. We also give, using our FIFO algorithm, a new exact simulation algorithm for the stationary distribution of the infinite server case, the GI/GI/\infty model. Finally, we even show how to handle Fork-Join queues, in which each arriving customer brings c jobs, one for each server., 32 pages, 4 figures

Let (Mn,Sn)n≥0 be a Markov random walk with positive recurrent driving chain (Mn)n≥0 on the countable state space 𝒮 with stationary distribution π. Suppose also that lim supn→∞Sn=∞ almost surely, so that the walk has almost-sure finite strictly ascending ladder epochs σn>. Recurrence properties of the ladder chain (Mσn>)n≥0 and a closely related excursion chain are studied. We give a necessary and sufficient condition for the recurrence of (Mσn>)n≥0 and further show that this chain is positive recurrent with stationary distribution π> and 𝔼π>σ1>n,𝑆̂n)n≥0, obtained by time reversal and called the dual of (Mn,Sn)n≥0, is positive divergent, i.e. 𝑆̂n→∞ almost surely. Simple expressions for π> are also provided. Our arguments make use of coupling, Palm duality theory, and Wiener‒Hopf factorization for Markov random walks with discrete driving chain.

We are interested in a large queue in a $GI/G/k$ queue with heterogeneous servers. For this, we consider tail asymptotics and weak limit approximations for the stationary distribution of its queue length process in continuous time under a stability condition. Here, two weak limit approximations are considered. One is when the variances of the inter-arrival and/or service times are bounded, and the other is when they get large. Both require a heavy traffic condition. Tail asymptotics and heavy traffic approximations have been separately studied in the literature. We develop a unified approach based on a martingale produced by a good test function for a Markov process to answer both problems., This is a supplemented version for the paper to appear in Advanced Applied Probability in 2017

In this paper we generalize a bounded Markov process, described by Stoyanov and Pacheco-González for a class of transition probability functions. A recursive integral equation for the probability density of these bounded Markov processes is derived and the stationary probability density is obtained by solving an equivalent differential equation. Examples of stationary densities for different transition probability functions are given and an application for designing a robotic coverage algorithm with specific emphasis on particular regions is discussed.

We examine the long-time behavior of forward rates in the framework of Heath-Jarrow-Morton-Musiela models with infinite-dimensional Lévy noise. We give an explicit condition under which the rates have a mean reversion property. In a special case we show that this condition is fulfilled for any Lévy process with variance smaller than a given constant, depending only on the state space and the volatility.

We consider a nearly unstable, or near unit root, AR(1) process with regularly varying innovations. Two different approximations for the stationary distribution of such processes exist: a Gaussian approximation arising from the nearly unstable nature of the process and a heavy-tail approximation related to the tail asymptotics of the innovations. We combine these two approximations to obtain a new uniform approximation that is valid on the entire real line. As a corollary, we obtain a precise description of the regions where each of the Gaussian and heavy-tail approximations should be used.

In this paper we consider reflected diffusions with positive and negative jumps, constrained to lie in the nonnegative orthant of ℝ n . We allow for the drift and diffusion coefficients, as well as for the directions of reflection, to be random fields over time and space. We provide a boundary behavior characterization, generalizing known results in the nonrandom coefficients and constant directions of the reflection case. In particular, the regulator processes are related to semimartingale local times at the boundaries, and they are shown not to charge the times the process expends at the intersection of boundary faces. Using the boundary results, we extend the conditions for product-form distributions in the stationary regime to the case when the drift and diffusion coefficients, as well as the directions of reflection, are random fields over space.

We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.

Autoregressive arrival models are described by a few parameters and provide a simple means to obtain analytical models for matching the first- and second-order statistics of measured data. We consider a discrete-time queueing system where the service time of a customer occupies one slot and the arrival process is governed by a discrete autoregressive process of order 1 (a DAR(1) process) which is characterized by an arbitrary stationary batch size distribution and a correlation coefficient. The tail behaviors of the queue length and the waiting time distributions are examined. In particular, it is shown that, unlike in the classical queueing models with Markovian arrival processes, the correlation in the DAR(1) model results in nongeometric tail behavior of the queue length (and the waiting time) if the stationary distribution of the DAR(1) process has infinite support. A complete characterization of the geometric tail behavior of the queue length (and the waiting time) is presented, showing the impact of the stationary distribution and the correlation coefficient when the stationary distribution of the DAR(1) process has finite support. It is also shown that the stationary distribution of the queue length (and the waiting time) has a tail of regular variation with index -β − 1, by finding an explicit expression for the tail asymptotics when the stationary distribution of the DAR(1) process has a tail of regular variation with index -β.

The ‘square root formula’ in the Internet transmission control protocol (TCP) states that if the probability p of packet loss becomes small and there is independence between packets, then the stationary distribution of the congestion window W is such that the distribution of W√p is almost independent of p and is completely characterizable. This paper gives an elementary proof of the convergence of the stationary distributions for a much wider class of processes that includes classical TCP as well as T. Kelly's ‘scalable TCP’. This paper also gives stochastic dominance results that translate to a rate of convergence.

We present recursions for the total number,Sn, of mutations in a sample ofnindividuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rater>0. The coalescent is allowed to have simultaneous multiple collisions of ancestral lineages, which corresponds to the existence of large families in the underlying population model. For the subclass of Λ-coalescent processes allowing for multiple collisions, such that the measure Λ(dx)/xis finite, we prove thatSn/(nr) converges in distribution to a limiting variable,S, characterized via an exponential integral of a certain subordinator. When the measure Λ(dx)/x2is finite, the distribution ofScoincides with the stationary distribution of an autoregressive process of order 1 and is uniquely determined via a stochastic fixed-point equation of the formwith specific independent random coefficientsAandB. Examples are presented in which explicit representations for (the density of)Sare available. We conjecture thatSn/E(Sn)→1 in probability if the measure Λ(dx)/xis infinite.

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate function r(x). We deal with the stationary distribution of such a GCP, (Xt)t≥0, and the distributions of the hitting times Ta = inf{t ≥ 0 : Xt = a}, a > 0. After presenting the general theory of these GCPs, several important special cases are studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[Ta, σ] in this case (where σ is the time of the first jump), and apply it to determine the long-run average cost of running a certain Markov-modulated disaster-ridden system.

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by the R-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. The RG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector.

In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density in R + n that satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.

Consider a continuous-time Markov chain in random environment. We study certain forms of interaction between the process of interest and the environmental process, under which the stationary joint distribution is tractable. Moreover, we obtain necessary and sufficient conditions for a product-form stationary distribution. A number of examples that illustrate the applicability of our results in queueing and population growth models are also included. © Applied Probability Trust 2005.

In this paper, we study a fluid model with partial message discarding and early message discarding, in which a finite buffer receives data (or information) from N independent on/off sources. All data generated by a source during one of its on periods is considered as a complete message. Our discarding scheme consists of two parts: (i) whenever some data belonging to a message has been lost due to overflow of the buffer, the remaining portion of this message will be discarded, and (ii) as long as the buffer content surpasses a certain threshold value at the instant an on period starts, all information generated during this on period will be discarded. By applying level-crossing techniques, we derive the equations for determining the system's stationary distribution. Further, two important performance measures, the probability of messages being transmitted successfully and the goodput of the system, are obtained. Numerical results are provided to demonstrate the effect of control parameters on the performance of the system.

We consider spectrally positive Lévy processes with regularly varying Lévy measure and study conditional limit theorems that describe the way that various rare events occur. Specifically, we are interested in the asymptotic behaviour of the distribution of the path of the Lévy process (appropriately scaled) up to some fixed time, conditionally on the event that the process exceeds a (large) positive value at that time. Another rare event we study is the occurrence of a large maximum value up to a fixed time, and the corresponding asymptotic behaviour of the (scaled) Lévy process path. We study these distributional limit theorems both for a centred Lévy process and for one with negative drift. In the latter case, we also look at the reflected process, which is of importance in applications. Our techniques are based on the explicit representation of the Lévy process in terms of a two-dimensional Poisson random measure and merely use the Poissonian properties and regular variation estimates. We also provide a proof for the asymptotic behaviour of the tail of the stationary distribution for the reflected process. The work is motivated by earlier results for discrete-time random walks (e.g. Durrett (1980) and Asmussen (1996)) and also by their applications in risk and queueing theory.

We study a Markovian model for a perishable inventory system with random input and an external source of obsolescence: at Poisson random times the whole current content of the system is spoilt and must be scrapped. The system can be described by its virtual death time process. We derive its stationary distribution in closed form and find an explicit formula for the Laplace transform of the cycle length, defined as the time between two consecutive item arrivals in an empty system. The results are used to compute several cost functionals. We also derive these functionals under the corresponding heavy traffic approximation, which is modeled using a Brownian motion in [0,1] reflected at 0 and 1 and restarted at 1 at the Poisson disaster times.

We consider a device that is subject to three types of failures: repairable, non-repairable and failures due to wear-out. This last type is also non-repairable. The times when the system is operative or being repaired follow phase type distributions. When a repairable failure occurs, the operating time of the device decreases, in that the lifetimes between failures are stochastically decreasing according to a geometric process. Following a non-repairable failure or after a previously fixed number of repairs occurs, the device is replaced by a new one. Under these conditions, the functioning of the device can be modelled by a Markov process. We consider two different models depending on whether or not the phase of the operational system at the instants of failure is remembered or not. For both models we derive the stationary distribution of the Markov process, the availability of the device, the rate of occurrence of the different types of failures, and certain quantities of interest.

We consider a fluid model similar to that of Kella and Whitt [32], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established by Kella and Whitt are extended to the finite buffer case: it is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N On-Off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue.

A new Markov process is introduced, describing growth or spread in two dimensions, via the aggregation of particles or the filling of cells. States of the process are configurations of part of the boundary of the growing aggregate, and transitions are captures or escapes of single particles. For suitably chosen transition rates, the process is dynamically reversible, leading to an explicit stationary distribution and a statistical description of the boundary. The growth rate is calculated and growth behaviour described. Different asymptotic relations between transition rates lead to different growth patterns or regimes. Besides the regimes familiar in polymer crystal growth, several new ones are described. The aggregate can have a porous structure resembling thin solid films deposited from vapour. Two measures of porosity, one for the boundary and one for the bulk, are calculated. The process is relevant to growing colonies of bacteria or the like, to the spread of epidemics and grass or forest fires, and to voter models.

We introduce a batch service discipline, called assemble-transfer batch service, for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers there; the node is emptied otherwise. We assume a Markovian setting for the arrival process, service times and routing, where batch sizes are generally distributed. Under the assumption that extra batches arrive while nodes are empty, and under a stability condition, it is shown that the stationary distribution of the queue length has a geometric product form over the nodes if and only if certain conditions are satisfied for the extra arrivals. This gives a new class of queueing networks which have tractable stationary distributions, and simultaneously shows that the product form provides a stochastic upper bound for the stationary distribution of the corresponding queueing network without the extra arrivals.

The quasi-stationary distribution of the closed stochastic SIS model changes drastically as the basic reproduction ratio R 0 passes the deterministic threshold value 1. Approximations are derived that describe these changes. The quasi-stationary distribution is approximated by a geometric distribution (discrete!) for R 0 distinctly below 1 and by a normal distribution (continuous!) for R 0 distinctly above 1. Uniformity of the approximation with respect to R 0 allows one to study the transition between these two extreme distributions. We also study the time to extinction and the invasion and persistence thresholds of the model.

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.

This paper deals with a service system in which the processor must serve two types of impatient units. In the case of blocking, the first type units leave the system whereas the second type units enter a pool and wait to be processed later.We develop an exhaustive analysis of the system including embedded Markov chain, fundamental period and various classical stationary probability distributions. More specific performance measures, such as the number of lost customers and other quantities, are also considered. The mathematical analysis of the model is based on the theory of Markov renewal processes, in Markov chains of M/G/l type and in expressions of ‘Takács' equation' type.

A stochastic process, called reallocatable GSMP (RGSMP for short), is introduced in order to study insensitivity of its stationary distribution. RGSMP extends GSMP with interruptions, and is applicable to a wide range of queues, from the standard models such as BCMP and Kelly's network queues to new ones such as their modifications with interruptions and Serfozo's (1989) non-product form network queues, and can be used to study their insensitivity in a unified way. We prove that RGSMP supplemented by the remaining lifetimes is product-form decomposable, i.e. its stationary distribution splits into independent components if and only if a version of the local balance equations hold, which implies insensitivity of the RGSMP scheme in a certain extended sense. Various examples of insensitive queues are given, which include new results. Our proofs are based on the characterization of a stationary distribution for SCJP (self-clocking jump process) of Miyazawa (1991).

Let X1, X2, · ··, Xn be a sequence of n random variables taking values in the ξ -letter alphabet . We consider the number N = N(n, k) of non-overlapping occurrences of a fixed k-letter word under (a) i.i.d. and (b) stationary Markovian hypotheses on the sequence , and use the Stein–Chen method to obtain Poisson approximations for the same. In each case, results and couplings from Barbour et al. (1992) are used to show that the total variation distance between the distribution of N and that of an appropriate Poisson random variable is of order (roughly) O(kS(k)), where S(k) denotes the stationary probability of the word in question. These results vastly improve on the approximations obtained in Godbole (1991). In the Markov case, we also make use of recently obtained eigenvalue bounds on convergence to stationarity due to Diaconis and Stroock (1991) and Fill (1991).

This paper studies computer simulation methods for estimating the sensitivities (gradient, Hessian etc.) of the expected steady-state performance of a queueing model with respect to the vector of parameters of the underlying distribution (an example is the gradient of the expected steady-state waiting time of a customer at a particular node in a queueing network with respect to its service rate). It is shown that such a sensitivity can be represented as the covariance between two processes, the standard output process (say the waiting time process) and what we call the score function process which is based on the score function. Simulation procedures based upon such representations are discussed, and in particular a control variate method is presented. The estimators and the score function process are then studied under heavy traffic conditions. The score function process, when properly normalized, is shown to have a heavy traffic limit involving a certain variant of two-dimensional Brownian motion for which we describe the stationary distribution. From this, heavy traffic (diffusion) approximations for the variance constants in the large sample theory can be computed and are used as a basis for comparing different simulation estimators. Finally, the theory is supported by numerical results.

In this paper we calculate posterior distributions associated with a version of the Poisson–Dirichlet distribution called the GEM. The GEM has been shown (by several authors) to be the limiting stationary distribution for allele frequencies listed in age order associated with the neutral infinite alleles model. In view of this result, we use our posterior distributions to calculate Bayes estimators for the frequency of the oldest allele given a sample.

A general framework of continuous-time queueing networks is studied with simultaneous state dependent service completions such as due to concurrent servicing or discrete-time slotting and with state dependent batch routings such as most typically modelling blocking. By using a key notion of group-local-balance, necessary and sufficient conditions are given for the stationary distribution to be of product form. These conditions and a constructive computation of the product form are based upon merely local solutions of the group-local-balance equations which can usually be solved explicitly for concrete networks. Moreover, a decomposition theorem is presented to separate service and routing conditions. General batch service and batch routing examples yielding a product form are hereby concluded. As illustrated by various examples, known results on both discrete- and continuoustime queueing networks are unified and extended. DISCRETE-TIME QUEUEING NETWORK; BATCH SERVICE; BATCH ROUTING; BLOCKING; GROUP-LOCAL-BALANCE

In this paper, we study a synchronous computer network model which may be described as follows. There are n service stations around a circle and there are k people to be serviced with at most one person per station. Time is slotted and in each time slot b adjacent people with an empty station ahead can move synchronously and simultaneously to the next counterclockwise station with probability pb. We show that the stationary distribution is uniform on all possible states, and using the Descartes' rule of signs we find the optimal value of k (fixed p and n) that yields maximal ‘throughput', i.e., the expected number of people served per time slot in equilibrium. We also briefly study an asynchronous model (introduced by D. Sarkar) where in each time slot a person moves with probability p to the next counterclockwise station if it is empty. We find, among other results, the new stationary distribution (Ramakrishnan et al. (1989) also find independently the stationary distribution and they also find the optimal value of k (fixed p and n) that yields maximal throughput in this model). We give tables comparing the synchronous and asynchronous models. Some applications which motivate this study are briefly presented.

The neutral two-locus model in population genetics is reformulated as a measure-valued diffusion process and is shown under certain conditions to have a unique stationary distribution and be weakly ergodic. The limits of the process and its stationary distribution as the recombination parameter tends to infinity are found. Genealogies are incorporated into the model, and it is shown that a random sample of sizenfrom the population at stationarity has a common ancestor.

It has recently been shown that for the G/G/1 queue, virtual delay and attained waiting time have the same stationary distribution. We present a sample-path derivation of this result.

A construction is given of a process in which X(t) represents the content at time t of a dam whose cumulative input process is a Lévy process with measure v and whose release rate at time t is r(X(t)). It is assumed only that r(0) = 0 and that r is strictly positive and left-continuous with strictly positive finite right limits on (0,∞). The sample-paths of X are shown to satisfy the storage equation The process X is analyzed using renewal theory and stochastic comparison techniques, and necessary and sufficient conditions are found in terms of v and r for X to have a stationary distributionπ. These generalize previous results which were obtained under the assumption that v is finite. Conditions for Πto have an atom at 0 are considered in some detail, and related results on the positivity of the expected occupation time of level 0 are given.Necessary and sufficient conditions for the existence of Πare expressed in terms of the existence of non-negative integrable solutions of certain integral equations and conditions are given under which such solutions are necessarily stationary densities for X. A simple sufficient condition for X to have a stationary distribution is found in terms of and in the case when r is non-decreasing the condition is shown to be also necessary. Finally some examples are considered; these show that the results described above unify various known conditions in special cases, and confirm several conjectures in the related literature.