bluuebunny/crossref_metadata_2025 huggingface dataset:
Taylor's formula and preservation of generalized convexity for positive linear operators
Authors: Adell José A. , Lekuona Alberto
Published: July 2016

In this paper, we consider positive linear operators L representable in terms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.
Poisson approximations for conditional r-scan lengths of multiple renewal processes and application to marker arrays in biomolecular sequences
Authors: Chen Chingfer , Karlin Samuel
Published: July 2016

This study is motivated by problems of molecular sequence comparison for multiple marker arrays with correlated distributions. In this paper, the model assumes two (or more) kinds of markers, say Markers A and B, distributed along the DNA sequence. The two primary conditions of interest are (i) many of Marker B (say ≥ m) occur, and (ii) few of Marker B (say ≤ l) occur. We title these the conditional r-scan models, and inquire on the extent to which Marker A clusters or is over-dispersed in regions satisfying condition (i) or (ii). Limiting distributions for the extremal r-scan statistics from the A array satisfying conditions (i) and (ii) are derived by extending the Chen-Stein Poisson approximation method.
Assessing an intuitive condition for stability under a range of traffic conditions via a generalised Lu-Kumar network
Authors: Niño-Mora José , Glazebrook Kevin D.
Published: July 2016

We argue the importance both of developing simple sufficient conditions for the stability of general multiclass queueing networks and also of assessing such conditions under a range of assumptions on the weight of the traffic flowing between service stations. To achieve the former, we review a peak-rate stability condition and extend its range of application and for the latter, we introduce a generalisation of the Lu–Kumar network on which the stability condition may be tested for a range of traffic configurations. The peak-rate condition is close to exact when the between-station traffic is light, but degrades as this traffic increases.
Smoothing effect of the superposition of homogeneous sources in tandem networks
Authors: Hordijk Arie , Liu Zhen , Towsley Don
Published: July 2016

We analyze the smoothing effect of superposing homogeneous sources in a network. We consider a tandem queueing network representing the nodes that customers generated by these sources pass through. The servers in the tandem queues have different time varying service rates. In between the tandem queues there are propagation delays. We show that for arbitrary arrival and service processes which are mutually independent, the sum of unfinished works in the tandem queues is monotone in the number of homogeneous sources in the increasing convex order sense, provided the total intensity of the foreground traffic is constant. The results hold for both fluid and discrete traffic models.
The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier
Authors: Martin-Löf Anders
Published: July 2016

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n
1/3, m ≈ bn
1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n
2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t
2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.
The expected time until absorption when absorption is not certain
Authors: Walker D. M.
Published: July 2016

This paper considers continuous-time Markov chains whose state space consists of an irreducible class, 𝒞, and an absorbing state which is accessible from 𝒞. The purpose is to provide a way to determine the expected time to absorption conditional on such time being finite, in the case where absorption occurs with probability less than 1. The results are illustrated by applications to the general birth and death process and the linear birth, death and catastrophe process.
The output process of an MMPP/M/1 queue
Authors: Bean Nigel , Green David , Taylor Peter
Published: July 2016

Olivier and Walrand (1994) claimed that the departure process of an MMPP/M/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. They also conjectured that the departure process of an MAP/PH/1 queue is not an MAP unless the queue is a stationary M/M/1 queue. We show that their proof of the first result has an algebraic error, which leaves open the above question of whether the departure process of an MMPP/M/1 can be an MAP.
Extension of Wald's first lemma to Markov processes
Authors: Moustakides George V.
Published: July 2016

Let ξ0,ξ1,ξ2,… be a homogeneous Markov process and let S

n
denote the partial sum S

n
= θ(ξ1) + … + θ(ξ
n
), where θ(ξ) is a scalar nonlinearity. If N is a stopping time with 𝔼N < ∞ and the Markov process satisfies certain ergodicity properties, we then show that 𝔼S

N
= [lim
n→∞𝔼θ(ξ
n
)]𝔼N + 𝔼ω(ξ0) − 𝔼ω(ξ
N
). The function ω(ξ) is a well defined scalar nonlinearity directly related to θ(ξ) through a Poisson integral equation, with the characteristic that ω(ξ) becomes zero in the i.i.d. case. Consequently our result constitutes an extension to Wald's first lemma for the case of Markov processes. We also show that, when 𝔼N → ∞, the correction term is negligible as compared to 𝔼N in the sense that 𝔼ω(ξ0) − 𝔼ω(ξ
N
) = o(𝔼N).
Extinction of population-size-dependent branching processes in random environments
Authors: Wang Han-xing
Published: July 2016

We generalize a population-size-dependent branching process to a more general branching model called the population-size-dependent branching process in random environments. For the model where {Z

n
}
n≥0 is associated with the stationary environment ξ− = {ξ
n
}
n≥0, let B = {ω : Z

n
(ω) = for some n}, and q(ξ−) = P(B | ξ−, Z
0 = 1). The result is that P(q(̅ξ) = 1) is either 1 or 0, and sufficient conditions for certain extinction (i.e. P(q(ξ−) = 1) = 1) and for non-certain extinction (i.e. P(q(ξ−) < 1) = 1) are obtained for the model.
Avoiding the Braess paradox in non-cooperative networks
Authors: Korilis Yannis A. , Lazar Aurel A. , Orda Ariel
Published: July 2016

The exponential growth of computer networking demands massive upgrades in the capacity of existing networks. Traditional capacity design methodologies, developed with the single-class networking paradigm in mind, overlook the non-cooperative structure of modern networks. Consequently, such design approaches entail the danger of degraded performance when resources are added to a network, a phenomenon known as the Braess paradox.

The present paper proposes methods for adding resources efficiently to a non-cooperative network of general topology. It is shown that the paradox is avoided when resources are added across the network, rather than on a local scale, and when upgrades are focused on direct connections between the sources and destinations. The relevance of these results for modern networks is demonstrated.
Rendezvous search on a graph
Authors: Alpern Steve , Baston V. J. , Essegaier Skander
Published: July 2016

Two agents are placed randomly on nodes of a known graph. They are aware of their own position, up to certain symmetries of the graph, but not that of the other agent. At each step, each agent may stay where he is or move to an adjacent node. Their common aim is to minimize the expected number of steps required to meet (occupy the same node). We consider two cases determined by whether or not the players are constrained to use identical strategies. This work extends that of Anderson and Weber on ‘discrete locations’ (complete graph) and is related to continuous (time and space) rendezvous as formulated by Alpern. Probabilistic notions arise in the random initial placement, in the random symmetries determining spatial uncertainty of agents, and through the use of mixed strategies.
Linear stochastic fluid networks
Authors: Kella Offer , Whitt Ward
Published: July 2016

We introduce open stochastic fluid networks that can be regarded as continuous analogues or fluid limits of open networks of infinite-server queues. Random exogenous input may come to any of the queues. At each queue, a c.d.f.-valued stochastic process governs the proportion of the input processed by a given time after arrival. The routeing may be deterministic (a specified sequence of successive queue visits) or proportional, i.e. a stochastic transition matrix may govern the proportion of the output routed from one queue to another. This stochastic fluid network with deterministic c.d.f.s governing processing at the queues arises as the limit of normalized networks of infinite-server queues with batch arrival processes where the batch sizes grow. In this limit, one can think of each particle having an evolution through the network, depending on its time and place of arrival, but otherwise independent of all other particles. A key property associated with this independence is the linearity: the workload associated with a superposition of inputs, each possibly having its own pattern of flow through the network, is simply the sum of the component workloads. As with infinite-server queueing models, the tractability makes the linear stochastic fluid network a natural candidate for approximations.
Quasistationary distributions for continuous time Markov chains when absorption is not certain
Authors: Pollett P. K.
Published: July 2016

Recently, Elmes et al. (see [2]) proposed a definition of a quasistationary distribution to accommodate absorbing Markov chains for which absorption occurs with probability less than 1. We will show that the probabilistic interpretation pertaining to cases where absorption is certain (see [13]) does not hold in the present context. We prove that the state probabilities at time t conditional on absorption taking place after t, generally depend on t. Conditions are derived under which there is no initial distribution such that the conditional state probabilities are stationary.
Elementary new proofs of classical limit theorems for Galton–Watson processes
Authors: Geiger Jochen
Published: July 2016

Classical results describe the asymptotic behaviour of a Galton–Watson branching process conditioned on non-extinction. We give new proofs of limit theorems in critical and subcritical cases. The proofs are based on the representation of conditioned Galton–Watson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned Galton–Watson family tree along the line of descent of the left-most particle.
The distribution of general final state random variables for stochastic epidemic models
Authors: Ball Frank , O'Neill Philip
Published: July 2016

In this paper we introduce the notion of general final state random variables for generalized epidemic models. These random variables are defined as sums over all ultimately infected individuals of random quantities of interest associated with an individual; examples include final severity. By exploiting a construction originally due to Sellke (1983), exact results concerning the final size and general final state random variables are obtained in terms of Gontcharoff polynomials. In particular, our approach highlights the way in which these polynomials arise via simple probabilistic arguments. For ease of exposition we focus initially upon the single-population case before extending our arguments to multi-population epidemics and other variants of our basic model.
A brokered market with heterogeneous suppliers and consumers
Authors: Righter Rhonda
Published: July 2016

We consider a brokered market with heterogeneous suppliers and consumers of identical items or services. Each supplier determines a minimum acceptable price for the items it supplies, based on its costs and target rate of return. The broker begins with a fixed set of items from the suppliers. Customers arrive periodically to the market and offer a bid for an item. If a customer wants more than one item, a set of bids is made. If the broker accepts a bid it must be assigned an item with minimum accceptable price less than or equal to the bid, and the price paid is the bid price. The broker has the option of rejecting a bid. The broker makes acceptance and assignment decisions to maximize the total expected discounted revenue. We show the optimal policy has a simple threshold structure where thresholds can be easily determined by finding the individually optimal policies subject to priorities.
On the convergence to stationarity of the many-server Poisson queue
Authors: Stadje Wolfgang , Parthasarathy P. R.
Published: July 2016

We consider the many-server Poisson queue M/M/c with arrival intensity λ, mean service time 1 and λ/c < 1. Let X(t) be the number of customers in the system at time t and assume that the system is initially empty. Then p

n
(t) = P(X(t) = n) converges to the stationary probability π
n
= P(X = n). The integrals ∫0
∞[E(X)-E(X(t))]dt and ∫0
∞[P(X≤n) − P(X(t)≤n)]dt are a measure of the speed of convergence towards stationarity. We express these integrals in terms of λ and c.
A BMAP/SM/1 queueing system with Markovian arrival input of disasters
Authors: Dudin Alexander , Nishimura Shoichi
Published: July 2016

Disaster arrival in a queuing system with negative arrivals causes all customers to leave the system instantaneously. Here we obtain a queue-length and virtual waiting (sojourn) time distribution for the more complicated system BMAP/SM/1 with MAP input of disasters.
On the transient delays of M/G/1 queues
Authors: Wang Chia-Li
Published: July 2016

A recent study on the GI/G/1 queue derives the Maclaurin series for the moments of the waiting time and the delay to respect to some parameters. By the same approach, we obtain an identity on the moments of the transient delay of the M/G/1 queue. This identity allows us to understand the transient behavior of the process better. We apply the identity with other established results to study convergence rate and stochastic concavity of the transient delay process, and to derive bounds and approximations of the moments. Our approximation and bound both have simple closed forms and are asymptotically exact as either the traffic intensity goes to zero or the process approaches stationarity. Performance of the approximation of several M/G/1 queues is illustrated by numerical experiments. It is interesting to note that our results can also help to gain variance reduction in simulation.
Generalized adaptive exponential smoothing of observations from an ergodic hidden Markov model
Authors: Herkenrath Ulrich
Published: July 2016

We consider a sequence of observations which is generated by a so-called hidden Markov model. An exponential smoothing procedure applied to such an observation sequence generates an inhomogeneous Markov process as a sequence of smoothed values. If the state sequence of the underlying hidden Markov model is moreover ergodic, then for two classes of smoothing functions the strong ergodicity of the sequence of smoothed values is proved. As a consequence a central limit theorem and a law of large numbers hold true for the smoothed values. The proof uses general results for so-called convergent inhomogeneous Markov processes. The procedure proposed by the author can be applied to some time series discussed in the literature.
Asymptotic distribution of sum and maximum for Gaussian processes
Authors: Ho Hwai-Chung , McCormick William P.
Published: July 2016

Let {X

n
, n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = E
X
0
X

n
. Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.
A unified approach to the study of tail probabilities of compound distributions
Authors: Cai Jun , Garrido José
Published: July 2016

We consider the tail probabilities of a class of compound distributions. First, the relations between reliability distribution classes and heavy-tailed distributions are discussed. These relations reveal that many previous results on estimating the tail probabilities are not applicable to heavy-tailed distributions.

Then, a generalized Wald's identity and identities for compound geometric distributions are presented in terms of renewal processes. Using these identities, lower and upper bounds for the tail probabilities are derived in a unified way for the class of compound distributions, both under the conditions of NBU and NWU tails, which include exponential tails, as well as under the condition of heavy-tailed distributions.

Finally, simplified bounds are derived by the technique of stochastic ordering. This method removes some unnecessary technical assumptions and corrects errors in the proof of some previous results.
Local risk minimization and numéraire
Authors: Biagini F. , Pratelli M.
Published: July 2016

The ‘change of numéraire’ technique has been introduced by Geman, El Karoui and Rochet for pricing and hedging contingent claims in the case of complete markets. In this paper we study the ‘change of numéraire’ using the ‘locally risk-minimizing approach’, when the market is not complete. We prove that, if the stochastic process which represents the prices is continuous, the l.r.m. strategy is invariant by a change of numéraire (this result is false in the right-continuous case, as is shown by some counterexamples).

We also give an extension of Merton's formula to the case of stochastic volatility.
A general shock model for a reliability system
Authors: Skoulakis Georgios
Published: July 2016

We study a reliability system subject to shocks generated by a renewal point process. When a shock occurs, components fail independently of each other with equal probabilities that are random numbers drawn from a distribution that may differ from shock to shock. We first consider the case of a parallel system and derive closed expressions for the Laplace-Stieltjes transform and the expectation of the time to system failure and for its density in the case that the distribution function of the renewal process possesses a density. We then treat a more general system structure, which has some very important special cases, such as k-out-of-n:F systems, and derive analogous formulae.
Power-law correlations, related models for long-range dependence and their simulation
Authors: Gneiting Tilmann
Published: July 2016

Martin and Walker ((1997) J. Appl. Prob.
34, 657–670) proposed the power-law ρ(v) = c|v|-β, |v| ≥ 1, as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of c is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family ρ(v) = (1 + |v/c|α)-β/α allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.
Non-product form of two-dimensional fluid networks with dependent Lévy inputs
Authors: Kella Offer
Published: July 2016

We show that the stationary distribution of a two-dimensional stochastic fluid network with (possibly dependent) Lévy inputs does not have product form other than in truly obvious cases. This is in contrast to queueing networks, where product form exists for non-obvious situations in which the inputs are independent, and for Brownian networks, where it typically exists for cases where the driving processes are actually dependent.
A new representation for the characteristic function of strictly geo-stable vectors
Authors: Klebanov Lev B. , Mittnik Stefan , Rachev Svetlozar T. , Volkovich Vladimir E.
Published: July 2016

A new representation for the characteristic function of the multivariate strictly geo-stable distribution is presented. The representation is appealing from a parametric viewpoint: its parameters have an intuitive probabilistic interpretation; and it is particularly useful for estimating the parameters of the geo-stable distribution.
The travel time in carousel systems under the nearest item heuristic
Authors: Litvak Nelly , Adan Ivo
Published: July 2016

A carousel is an automated warehousing system consisting of a large number of drawers rotating in a closed loop. In this paper, we study the travel time needed to pick a list of items when the carousel operates under the nearest item heuristic. We find a closed form expression for the distribution and all moments of the travel time. We also analyse the asymptotic behaviour of the travel time when the number of items tends to infinity. All results follow from probabilistic arguments based on properties of uniform order statistics.
Bounds for the American perpetual put on a stock index
Authors: Paulsen V.
Published: July 2016

Let us consider n stocks with dependent price processes each following a geometric Brownian motion. We want to investigate the American perpetual put on an index of those stocks. We will provide inner and outer boundaries for its early exercise region by using a decomposition technique for optimal stopping.
Stochastic processes involving random deletion
Authors: Rothmann Mark D. , El Barmi Hammou
Published: July 2016

We consider a system where units having magnitudes arrive according to a nonhomogeneous Poisson process, remain there for a random period and then depart. Eventually, at any point in time only a portion of those units which have entered the system remain. Of interest are the finite time properties and limiting behaviors of the distribution of magnitudes among the units present in the system and among those which have departed from the system. We will derive limiting results for the empirical distribution of magnitudes among the active (departed) units. These results are also shown to extend to systems having stages or steps through which units must proceed. Examples are given to illustrate these results.
Some asymptotic results for transient random walks with applications to insurance risk
Authors: Baltrūnas Aleksandras
Published: July 2016

We consider a real-valued random walk which drifts to -∞ and is such that the step distribution is heavy tailed, say, subexponential. We investigate the asymptotic tail behaviour of the distribution of the upwards first passage times. As an application, we obtain the exact rate of convergence for the ruin probability in finite time. Our result supplements similar theorems in risk theory.
A new discrete distribution arising in a model of DNA replication
Authors: Cowan Richard
Published: July 2016

During DNA replication, small fragments of DNA are formed. These have been observed experimentally and the mechanism of their formation modelled mathematically. Using the stochastic model of Cowan and Chiu (1992), (1994), we find the probability distribution of the number of fragments. A new discrete distribution arises. The work has interest as an application of the recent theory on quasirenewal equations in Piau (2000).
A reconsideration of Lotka's extinction probability using a bisexual branching process
Authors: Hull David M.
Published: July 2016

It is generally recognized that Alfred Lotka made the first application of
standard Galton-Watson branching process theory to calculate an extinction probability in a specific population (using asexual reproduction). This note applies bisexual Galton-Watson branching process theory to the calculation of an extinction probability from Lotka's data, yielding a somewhat higher value.
An asymptotic expansion for the expectation of an age-dependent branching process with a submultiplicative estimate of the remainder
Authors: Sgibnev M. S.
Published: July 2016

An asymptotic expansion for the expected number,μ(t), of particles of an age-dependent branching process is obtained with a general submultiplicative estimate for the remainder term. The influence of the roots of the characteristic equation on the asymptotic behaviour ofμ(t) is taken into account.
Differentiability and monotonicity of expected passage time in Euclidean first-passage percolation
Authors: Howard C. Douglas
Published: July 2016

In first-passage percolation (FPP) models, the passage timeTℓfrom the origin to the pointℓeℓsatisfiesf(ℓ) := ETℓ= μℓ+o(ℓ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously thatf(ℓ) is eventually monotonically increasing. Here, for the Poisson-based Euclidean FPP of Howard and Newman (Prob. Theory Relat. Fields108(1997), 153–170), we prove an explicit formula forf′(ℓ). In all dimensions, for certain values of the model's only parameter we havef′(ℓ) ≥C> 0 for largeℓ.
Transient analysis of state-dependent queueing networks via cumulant functions
Authors: Matis Timothy I. , Feldman Richard M.
Published: July 2016

A new procedure that generates the transient solution of the first moment of the state of a Markovian queueing network with state-dependent arrivals, services, and routeing is developed. The procedure involves defining a partial differential equation that relates an approximate multivariate cumulant generating function to the intensity functions of the network. The partial differential equation then yields a set of ordinary differential equations which are numerically solved to obtain the first moment.
Availability analysis of periodically inspected systems with random walk model
Authors: Cui Lirong , Xie M.
Published: July 2016

In this paper, the instantaneous availability of a system maintained under periodic inspection is investigated using random walk models. Two cases are considered. In the first model, the system is repaired or modified and it is assumed to be as good as new upon periodic inspection and maintenance. In the second model, the system is not modified after the inspection if the system is still working, and the condition of the system is assumed to be the same as that before the inspection. For both models the failures only can be found through the inspection. Perfect repair or replacement of a failed system is assumed to be carried out, but the time it takes can be constant or of a random length. The relationship between this problem and the random walk model in a two-dimensional plane is described. Several new results are also shown.
Convergence of the number of failed components in a Markov system with nonidentical components
Authors: Bon Jean-Louis , Păltănea Eugen
Published: July 2016

For most repairable systems, the number N(t) of failed components at time t appears to be a good quality parameter, so it is critical to study this random function. Here the components are assumed to be independent and both their lifetime and their repair time are exponentially distributed. Moreover, the system is considered new at time 0. Our aim is to compare the random variable N(t) with N(∞), especially in terms of total variation distance. This analysis is used to prove a cut-off phenomenon in the same way as Ycart (1999) but without the assumption of identical components.
Convergence of simulated annealing using Foster-Lyapunov criteria
Authors: Andrieu Christophe , Breyer Laird A. , Doucet Arnaud
Published: July 2016

Simulated annealing is a popular and much studied method for maximizing functions on finite or compact spaces. For noncompact state spaces, the method is still sound, but convergence results are scarce. We show here how to prove convergence in such cases, for Markov chains satisfying suitable drift and minorization conditions.
A probabilistic method for Navier-Stokes vortices
Authors: He Xinyu
Published: July 2016

Consider a Navier-Stokes incompressible turbulent fluid in R
2. Let x(t) denote the position coordinate of a moving vortex with initial circulation Γ0 > 0 in the fluid, subject to a force F. Define x(t) as a stochastic process with continuous sample paths described by a stochastic differential equation. Assuming a suitable notion of weak rotationality, it is shown that the stochastic equation is equivalent to a linear partial differential equation for the complex function ψ, i∂ψ/∂t = [-Γ0Δ + F] ψ, where |ψ|2 = ρ(x,t), ρ being the probability density function of finding the vortex centre in position x at time t.
Optimal estimation of diffusion processes hidden by general obstacles
Authors: Kim Hyung Geun , Nam Dougu
Published: July 2016

Let X

t
be an n-dimensional diffusion process and S(t) be a set-valued function. Suppose X

t
is invisible when it is hidden by S(t), but we can see the process exactly otherwise. In this paper, we derive the optimal estimator E[f(X
1) | X

s

1

X

s
∉S(s), 0 ≤ s ≤ 1] for a bounded Borel function f. We illustrate some computations for Gauss-Markov processes.
Accurate bounds for the asymptotic constant in a statistical multiplexer with homogeneous generalized binary Markov sources
Authors: Xue J. , Alfa Attahiru Sule
Published: July 2016

This paper considers the asymptotic tail distribution of the number of cells queued in a statistical multiplexer fed with homogeneous generalized binary Markov sources. As the asymptotic decay rate is easy to obtain, we focus our effort on bounding the asymptotic constant, which is dependent on the initial phase combination of the sources and is hard to compute even for a moderate number of sources. We derive upper and lower bounds for the asymptotic constant, taking the initial phase combination into account. Numerical experiments show the accuracy of these bounds. They also show that, while the asymptotic decay rates are the same, the variation of initial phase combination of the sources may significantly affect the asymptotic constants.
On stability of queueing networks with job deadlines
Authors: Ward Amy R. , Bambos Nicholas
Published: July 2016

In this paper, we consider a single-server queue with stationary input, where each job joining the queue has an associated deadline. The deadline is a time constraint on job sojourn time and may be finite or infinite. If the job does not complete service before its deadline expires, it abandons the queue and the partial service it may have received up to that point is wasted. When the queue operates under a first-come-first served discipline, we establish conditions under which the actual workload process—that is, the work the server eventually processes—is unstable, weakly stable, and strongly stable. An interesting phenomenon observed is that in a nontrivial portion of the parameter space, the queue is weakly stable, but not strongly stable. We also indicate how our results apply to other nonidling service disciplines. We finally extend the results for a single node to acyclic (feed-forward) networks of queues with either per-queue or network-wide deadlines.
The coupon-collector's problem revisited
Authors: Adler Ilan , Oren Shmuel , Ross Sheldon M.
Published: July 2016

Consider the classical coupon-collector's problem in which items of m distinct types arrive in sequence. An arriving item is installed in system i ≥ 1 if i is the smallest index such that system i does not contain an item of the arrival's type. We study the expected number of items in system j at the moment when system 1 first contains an item of each type
Biased movement at a boundary and conditional occupancy times for diffusion processes
Authors: Ovaskainen Otso , Cornell Stephen J.
Published: July 2016

Motivated by edge behaviour reported for biological organisms, we show that random walks with a bias at a boundary lead to a discontinuous probability density across the boundary. We continue by studying more general diffusion processes with such a discontinuity across an interior boundary. We show how hitting probabilities, occupancy times and conditional occupancy times may be solved from problems that are adjoint to the original diffusion problem. We highlight our results with a biologically motivated example, where we analyze the movement behaviour of an individual in a network of habitat patches surrounded by dispersal habitat.
On the structure of value functions for threshold policies in queueing models
Authors: Bhulai Sandjai , Koole Ger
Published: July 2016

We study the multiserver queue with Poisson arrivals and identical independent servers with exponentially distributed service times. Customers arriving at the system are admitted or rejected according to a fixed threshold policy. Moreover, the system is subject to holding, waiting, and rejection costs. We give a closed-form expression for the average costs and the value function for this multiserver queue. The result will then be used in a single step of policy iteration in the model where a controller has to route to several finite-buffer queues with multiple servers. We numerically show that the improved policy has a close to optimal value.
Mixed distributions in Sattolo's algorithm for cyclic permutations via randomization and derandomization
Authors: Mahmoud Hosam M.
Published: July 2016

We investigate the limit distributions associated with cost measures in Sattolo's algorithm for generating random cyclic permutations. The number of moves made by an element turns out to be a mixture of 1 and 1 plus a geometric distribution with parameter ½, where the mixing probability is the limiting ratio of the rank of the element being moved to the size of the permutation. On the other hand, the raw distance traveled by an element to its final destination does not converge in distribution without norming. Linearly scaled, the distance converges to a mixture of a uniform and a shifted product of a pair of independent uniforms. The results are obtained via randomization as a transform, followed by derandomization as an inverse transform. The work extends analysis by Prodinger.
On the moment problem for random sums
Authors: Gut Allan
Published: July 2016

A recent paper by Lin and Stoyanov is devoted to the moment problem for geometrically compounded sums. The aim of this note is to provide affirmative answers to their conjectures.
Signatures of indirect majority systems
Authors: Boland Philip J.
Published: July 2016

If τ is the lifetime of a coherent system, then the signature of the system is the vector of probabilities that the lifetime coincides with the ith order statistic of the component lifetimes. The signature can be useful in comparing different systems. In this treatment we give a characterization of the signature of a system with independent identically distributed components in terms of the number of path sets in the system as well as in terms of the number of what we call ordered cut sets. We consider, in particular, the signatures of indirect majority systems and compare them with the signatures of simple majority systems of the same size. We note that the signature of an indirect majority system of size r × s = n is symmetric around , and use this to show that the expected lifetime of an r × s = n indirect majority system exceeds that of a simple (direct) majority system of size n when the components are exponentially distributed with the same parameter.
Convergence to the coalescent with simultaneous multiple mergers
Authors: Sagitov Serik
Published: July 2016

The general coalescent process with simultaneous multiple mergers of ancestral lines was initially characterized by Möhle and Sagitov (2001) in terms of a sequence of measures defined on the finite-dimensional simplices. A more compact characterization of the general coalescent requiring a single probability measure Ξ defined on the infinite simplex Δ was suggested by Schweinsberg (2000). This paper presents a simple criterion of weak convergence to the Ξ-coalescent. In contrast to the earlier criterion of Möhle and Sagitov (2001) based on the moment conditions, the key condition here is expressed in terms of the joint distribution of the ranked offspring sizes. This criterion interprets a vector in Δ as the ranked fractions of the total population size assigned to sibling groups constituting a (rare) generation, where a merger might occur. An example of the general coalescent is developed on the basis of the Poisson–Dirichlet distribution. It suggests a simple algorithm of simulating the Kingman coalescent with occasional (simultaneous) multiple mergers.
Convergence of lower records and infinite divisibility
Authors: Bose Arup , Gangopadhyay Sreela , Sarkar Anish , Sengupta Arindam
Published: July 2016

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.
On the geometric growth in controlled branching processes with random control function
Authors: González M. , Molina M. , del Puerto I.
Published: July 2016

The limit behaviour of a controlled branching process with random control function is investigated. A necessary condition and a sufficient condition for the geometric growth of such a process are established by considering the L
1-convergence. Finally, taking into account the classical X log+
X criterion in branching processes, a necessary and sufficient condition is provided.
On the entropy for semi-Markov processes
Authors: Girardin Valerie , Limnios Nikolaos
Published: July 2016

The aim of this paper is to define the entropy of a finite semi-Markov process. We define the entropy of the finite distributions of the process, and obtain explicitly its entropy rate by extending the Shannon–McMillan–Breiman theorem to this class of nonstationary continuous-time processes. The particular cases of pure jump Markov processes and renewal processes are considered. The relative entropy rate between two semi-Markov processes is also defined.
On moments and tail behaviors of storage processes
Authors: Kohatsu-Higa Arturo , Yamazato Makoto
Published: July 2016

We study the existence of moments and the tail behavior of the densities of storage processes. We give sufficient conditions for existence and nonexistence of moments using the integrability conditions of submultiplicative functions with respect to Lévy measures. We then study the asymptotical behavior of the tails of these processes using the concave or convex envelope of the release rate function.
Stochastic order for invariant measures of the contact process
Authors: Andjel E. D.
Published: July 2016

We show that the second lowest invariant measure of the contact process, introduced by Salzano and Schonmann (1997), is stochastically lower that any invariant measure which puts no mass on the empty configuration.
Ruin probabilities for competing claim processes
Authors: Huzak Miljenko , Perman Mihael , Šikić Hrvoje , Vondraček Zoran
Published: July 2016

Let C
1, C
2,…,C

m
be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x + ct - C(t). The ruin probability is given by the well-known Pollaczek–Khinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators C

i
. Formulae for the probability that ruin is caused by C

i
are derived. These formulae can be extended to perturbed risk processes of the type X(t) = x + ct - C(t) + Z(t), where Z is a Lévy process with mean 0 and no positive jumps.
Markov-renewal fluid queues
Authors: Latouche Guy , Takine Tetsuya
Published: July 2016

We consider a fluid queue controlled by a semi-Markov process and we apply the Markov-renewal approach developed earlier in the context of quasi-birth-and-death processes and of Markovian fluid queues. We analyze two subfamilies of semi-Markov processes. In the first family, we assume that the intervals during which the input rate is negative have an exponential distribution. In the second family, we take the complementary case and assume that the intervals during which the input rate is positive have an exponential distribution. We thoroughly characterize the structure of the stationary distribution in both cases.
Joint exceedances of the ARCH process
Authors: Gomes M. Ivette , de Haan Laurens , Pestana Dinis
Published: July 2016

We examine the joint finite structure of extremes of the ARCH process and find an unexpected phenomenon: when assessing probabilities of failure during some finite time interval in the future, the extremal index seems not to be the object to look at. Two possible ramifications of this phenomenon are put forward.
Preservation of positive and negative orthant dependence concepts under mixtures and applications
Authors: Belzunce Félix , Semeraro Patrizia
Published: July 2016

In this paper we consider some dependence properties and orders among multivariate distributions, and we study their preservation under mixtures. Applications of these results in reliability, risk theory, and mixtures of discrete distributions are provided.
Path reversal, islands, and the gapped alignment of random sequences
Authors: Spouge John L.
Published: July 2016

In bioinformatics, the notion of an ‘island’ enhances the efficient simulation of gapped local alignment statistics. This paper generalizes several results relevant to gapless local alignment statistics from one to higher dimensions, with a particular eye to applications in gapped alignment statistics. For example, reversal of paths (rather than of discrete time) generalizes a distributional equality, from queueing theory, between the Lindley (local sum) and maximum processes. Systematic investigation of an ‘ownership’ relationship among vertices in ℤ2formalizes the notion of an island as a set of vertices having a common owner. Predictably, islands possess some stochastic ordering and spatial averaging properties. Moreover, however, the average number of vertices in a subcritical stationary island is 1, generalizing a theorem of Kac about stationary point processes. The generalization leads to alternative ways of simulating some island statistics.
Occupation time processes of super-Brownian motion with cut-off branching
Authors: Dong Zhao , Feng Shui
Published: July 2016

In this article we investigate a class of superprocess with cut-off branching, studying the long-time behavior of the occupation time process. Persistence of the process holds in all dimensions. Central-limit-type theorems are obtained, and the scales are dimension dependent. The Gaussian limit holds only when d ≤ 4. In dimension one, a full large deviation principle is established and the rate function is identified explicitly. Our result shows that the super-Brownian motion with cut-off branching in dimension one has many features that are similar to super-Brownian motion in dimension three.
Random walks on a complete graph: a model for infection
Authors: Datta Nilanjana , Dorlas Tony C.
Published: July 2016

We introduce a new model for the infection of one or more subjects by a single agent, and calculate the probability of infection after a fixed length of time. We model the agent and subjects as random walkers on a complete graph of N sites, jumping with equal rates from site to site. When one of the walkers is at the same site as the agent for a length of time τ, we assume that the infection probability is given by an exponential law with parameter γ, i.e. q(τ) = 1 - e-γ
τ
. We introduce the boundary condition that all walkers return to their initial site (‘home’) at the end of a fixed period T. We also assume that the incubation period is longer than T, so that there is no immediate propagation of the infection. In this model, we find that for short periods T, i.e. such that γ
T ≪ 1 and T ≪ 1, the infection probability is remarkably small and behaves like T
3. On the other hand, for large T, the probability tends to 1 (as might be expected) exponentially. However, the dominant exponential rate is given approximately by 2γ/[(2+γ)N] and is therefore small for large N.
Pair formation in a Markovian arrival process with two event labels
Authors: Neuts Marcel F. , Alfa Attahiru Sule
Published: July 2016

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.
A Lindley-type equation arising from a carousel problem
Authors: Vlasiou M. , Adan I. J. B. F. , Wessels J.
Published: July 2016

In this paper we consider a system with two carousels operated by one picker. The items to be picked are randomly located on the carousels and the pick times follow a phase-type distribution. The picker alternates between the two carousels, picking one item at a time. Important performance characteristics are the waiting time of the picker and the throughput of the two carousels. The waiting time of the picker satisfies an equation very similar to Lindley's equation for the waiting time in the PH/U/1 queue. Although the latter equation has no simple solution, we show that the one for the waiting time of the picker can be solved explicitly. Furthermore, it is well known that the mean waiting time in the PH/U/1 queue depends on the complete interarrival time distribution, but numerical results show that, for the carousel system, the mean waiting time and throughput are rather insensitive to the pick-time distribution.
Extinction times for a general birth, death and catastrophe process
Authors: Cairns Ben , Pollett P. K.
Published: July 2016

The birth, death and catastrophe process is an extension of the birth–death process that incorporates the possibility of reductions in population of arbitrary size. We will consider a general form of this model in which the transition rates are allowed to depend on the current population size in an arbitrary manner. The linear case, where the transition rates are proportional to current population size, has been studied extensively. In particular, extinction probabilities, the expected time to extinction, and the distribution of the population size conditional on nonextinction (the quasi-stationary distribution) have all been evaluated explicitly. However, whilst these characteristics are of interest in the modelling and management of populations, processes with linear rate coefficients represent only a very limited class of models. We address this limitation by allowing for a wider range of catastrophic events. Despite this generalisation, explicit expressions can still be found for the expected extinction times.