Showing papers on "Renewal theory published in 2011"

The Fractional Poisson Process and the Inverse Stable Subordinator

Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

Dynamic scheduling of a two-class queue with setups

Abstract: We analyze two scheduling problems for a queueing system with a single server and two customer classes. Each class has its own renewal arrival process, general service time distribution, and holding cost rate. In the first problem, a setup cost is incurred when the server switches from one class to the other, and the objective is to minimize the long-run expected average cost of holding customers and incurring setups. The setup cost is replaced by a setup time in the second problem, where the objective is to minimize the average holding cost. By assuming that a recently derived heavy traffic principle holds not only for the exhaustive policy but for nonexhaustive policies, we approximate (under standard heavy traffic conditions) the dynamic scheduling problems by diffusion control problems. The diffusion control problem for the setup cost problem is solved exactly, and asymptotics are used to analyze the corresponding setup time problem. Computational results show that the proposed scheduling policies are within several percent of optimal over a broad range of problem parameters.

New rates for exponential approximation and the theorems of Rényi and Yaglom

Abstract: We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Renyi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classical theorem of Yaglom on the exponential asymptotic behavior of a critical Galton–Watson process conditioned on nonextinction. The primary tools are an adaptation of Stein’s method, Stein couplings, as well as the equilibrium distributional transformation from renewal theory.

Stability in a system subject to noise with regulated periodicity.

Abstract: The stability of a simple dynamical system subject to multiplicative one-side pulse noise with hidden periodicity is investigated both analytically and numerically. The stability analysis is based on the exact result for the characteristic functional of the renewal pulse process. The influence of the memory effects on the stability condition is analyzed for two cases: (i) the dead-time-distorted poissonian process, and (ii) the renewal process with Pareto distribution. We show that, for fixed noise intensity, the system can be stable when the noise is characterized by high periodicity and unstable at low periodicity.

Asymptotics for the ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims

Abstract: In this paper, we consider two dependent classes of insurance business with heavy-tailed claims. The dependence comes from the assumption that claim arrivals of the two classes are governed by a common renewal counting process. We study two types of ruin in the two-dimensional framework. For each type of ruin, we establish an asymptotic formula for the finite-time ruin probability. These formulae possess a certain uniformity feature in the time horizon. Copyright © 2010 John Wiley & Sons, Ltd.

Modeling Nonsaturated IEEE 802.11 DCF Networks Utilizing an Arbitrary Buffer Size

Abstract: We propose an approximate model for a nonsaturated IEEE 802.11 DCF network. This model captures the significant influence of an arbitrary node transmit buffer size on the network performance. We find that increasing the buffer size can improve the throughput slightly but can lead to a dramatic increase in the packet delay without necessarily a corresponding reduction in the packet loss rate. This result suggests that there may be little benefit in provisioning very large buffers, even for loss-sensitive applications. Our model outperforms prior models in terms of simplicity, computation speed, and accuracy. The simplicity stems from using a renewal theory approach for the collision probability instead of the usual multidimensional Markov chain, and it makes our model easier to understand, manipulate and extend; for instance, we are able to use our model to investigate the important problem of convergence of the collision probability calculation. The remarkable improvement in the computation speed is due to the use of an efficient numerical transform inversion algorithm to invert generating functions of key parameters of the model. The accuracy is due to a carefully constructed model for the service time distribution. We verify our model using ns-2 simulation and show that our analytical results based on an M/G/1/K queuing model are able to accurately predict a wide range of performance metrics, including the packet loss rate and the waiting time distribution. In contradiction to claims by other authors, we show that 1) a nonsaturated DCF model like ours that makes use of decoupling assumptions for the collision probability and queuing dynamics can produce accurate predictions of metrics other than just the throughput, and 2) the actual service time and waiting time distributions for DCF networks have truncated heavy-tailed shapes (i.e., appear initially straight on a log-log plot) rather than exponential shapes. Our work will help developers select appropriate buffer sizes for 802.11 devices, and will help system administrators predict the performance of applications.

The Fractional Poisson Process and the Inverse Stable Subordinator

Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

Optimal dividend-payout in random discrete time

Abstract: Assume that the surplus process of an insurance company is described by a general Levy process and that possible dividend pay-outs to shareholders are restricted to random discrete times which are determined by an independent renewal process. Under this setting we show that the optimal dividend pay-out policy is a band-policy. If the renewal process is a Poisson process, it is further shown that for Cramer–Lundberg risk processes with exponential claim sizes and its diffusion limit the optimal policy collapses to a barrier-policy. Finally, a numerical example is given for which the optimal bands can be calculated explicitly. The random observation procedure studied in this paper also allows for an interpretation in terms of a random walk model with a certain type of random discounting.

Covariance of discounted compound renewal sums with a stochastic interest rate

Abstract: Formulas have been obtained for the moments of the discounted aggregate claims process, for a constant instantaneous interest rate, and for a claims number process that is an ordinary or a delayed renewal process. In this paper, we present explicit formulas on the first two moments and the joint moment of this risk process, for a non-trivial extension to a stochastic instantaneous interest rate. Examples are given for Erlang claims number processes, and for the Ho–Lee–Merton and the Vasicek interest rate models.

Collision local time of transient random walks and intermediate phases in interacting stochastic systems

Abstract: In a companion paper (M. Birkner, A. Greven, F. den Hollander, Quenched LDP for words in a letter sequence, Probab. Theory Relat. Fields 148 , no. 3/4 (2010), 403-456), a quenched large deviation principle (LDP) has been established for the empirical process of words obtained by cutting an i.i.d. sequence of letters into words according to a renewal process. We apply this LDP to prove that the radius of convergence of the generating function of the collision local time of two independent copies of a symmetric and strongly transient random walk on $\mathbb{Z}^d$, $d\geq1$ , both starting from the origin, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. We conjecture that the same holds when the random walk is transient but not strongly transient. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments.

Introduction to Point Processes

Abstract: Point process models are useful for describing phenomena occurring at random locations and/or times. Following a review of basic concepts, some important models are surveyed including Poisson processes, renewal processes, Hawkes processes, and Markovian point processes. Techniques for estimation, simulation, and residual analysis for point processes are also briefly discussed. Keywords: Poisson process; marked point process; Hawkes process; renewal process; conditional intensity; clustering density; maximum likelihood; residual analysis

Cost analysis and minimization of movement-based location management schemes in wireless communication networks: a renewal process approach

Abstract: The paper makes new contributions to cost analysis and minimization of movement-based location management schemes in wireless communication networks. The main contributions of the paper are three-fold. First, we consider two different call handling models, that is, the call plus location update (CPLU) model and the call without location update (CWLU) model. We point out that all existing analysis of location update cost of a movement-based location management scheme (MBLMS) do not accurately capture the essence of the two models. Second, we analyze the exact location update cost of an MBLMS under both CPLU and CWLU models using a renewal process approach which has rarely been used before. We find that the location update cost of an MBLMS under the CWLU model is much easier to analyze than that of an MBLMS under the CPLU model. Furthermore, an MBLMS operated under the CWLU model has lower location update cost than an MBLMS operated under the CPLU model. Third, we are able to derive a closed form solution to the movement threshold that minimizes the total cost of location management in an MBLMS for the CPLU model when the inter-call time has an arbitrary distribution and the cell residence time has an Erlang distribution, and for the CWLU model when both inter-call time and cell residence time have arbitrary distributions. Such closed form solutions have not been available in the existing literature.

Reliability analysis of a one-unit system with finite vacations

Abstract: This paper introduces finite vacations from a reliability theory viewpoint and deals with a one-unit repairable system with a repairman who takes finite vacations. A total probability decomposition method for analyzing the reliability problems of the system is presented with the help of the renewal process theory. With the decomposition method, two key reliability characteristics of the system i.e. the availability and mean failure number of the system are discussed under some assumptions. It is important that the steady-state availability and steady-state failure frequency of the system are obtained. Furthermore, three special cases of the model have showed that the vacation number taken by the repairman affects the performance of the system and the results presented in this paper are more general than the existing results in some literatures.

Refinement of the spectral asymptotics of generalized Krein Feller operators

Abstract: Abstract Spectral asymptotics of operators of the form are investigated. In the case of self-similar measures μ and ν it turns out that the eigenvalue counting function N(x) under both Dirichlet and Neumann conditions behaves like xγ as x → ∞, where the spectral exponent γ is given in terms of the scaling numbers of the measures. More precisely, it holds that In the present paper, we give a refinement of this spectral result, i.e. we give a sufficient condition under which the term N(x)x –γ converges. We show, using renewal theory, that the behaviour of N(x)x –γ depends essentially on whether the set of logarithms of the scaling numbers of the measures is arithmetic.

The simple harmonic urn

Abstract: We study a generalized Polya urn model with two types of ball. If the drawn ball is red, it is replaced together with a black ball, but if the drawn ball is black it is replaced and a red ball is thrown out of the urn. When only black balls remain, the roles of the colors are swapped and the process restarts. We prove that the resulting Markov chain is transient but that if we throw out a ball every time the colors swap, the process is recurrent. We show that the embedded process obtained by observing the number of balls in the urn at the swapping times has a scaling limit that is essentially the square of a Bessel diffusion. We consider an oriented percolation model naturally associated with the urn process, and obtain detailed information about its structure, showing that the open subgraph is an infinite tree with a single end. We also study a natural continuous-time embedding of the urn process that demonstrates the relation to the simple harmonic oscillator; in this setting, our transience result addresses an open problem in the recurrence theory of two-dimensional linear birth and death processes due to Kesten and Hutton. We obtain results on the area swept out by the process. We make use of connections between the urn process and birth–death processes, a uniform renewal process, the Eulerian numbers, and Lamperti’s problem on processes with asymptotically small drifts; we prove some new results on some of these classical objects that may be of independent interest. For instance, we give sharp new asymptotics for the first two moments of the counting function of the uniform renewal process. Finally, we discuss some related models of independent interest, including a “Poisson earthquakes” Markov chain on the homeomorphisms of the plane.

The Geometric Markov Renewal Processes with Application to Finance

Abstract: We introduce the geometric Markov renewal processes as a model for a security market and study this processes in a series scheme. We consider its approximations in the form of averaged, merged and double averaged geometric Markov renewal processes. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes are presented. Martingale properties, infinitesimal operators of geometric Markov renewal processes are presented and a Markov renewal equation for expectation is derived. As an application, we consider the case of two ergodic classes. Moreover, we consider a generalized binomial model for a security market induced by a position dependent random map as a special case of a geometric Markov renewal process.

The on---off network traffic model under intermediate scaling

Abstract: The result provided in this paper helps complete a unified picture of the scaling behavior in heavy-tailed stochastic models for transmission of packet traffic on high-speed communication links. Popular models include infinite source Poisson models, models based on aggregated renewal sequences, and models built from aggregated on---off sources. The versions of these models with finite variance transmission rate share the following pattern: if the sources connect at a fast rate over time the cumulative statistical fluctuations are fractional Brownian motion, if the connection rate is slow the traffic fluctuations are described by a stable Levy motion, while the limiting fluctuations for the intermediate scaling regime are given by fractional Poisson motion. In this paper, we prove an invariance principle for the normalized cumulative workload of a network with m on---off sources and time rescaled by a factor a. When both the number of sources m and the time scale a tend to infinity with a relative growth given by the so-called 'intermediate connection rate' condition, the limit process is the fractional Poisson motion. The proof is based on a coupling between the on---off model and the renewal type model.

Modeling Cross Correlation in Three-Moment Four-Parameter Decomposition Approximation of Queueing Networks

Abstract: In two-moment decomposition approximations of queueing networks, the arrival process is modeled as a renewal process, and each station is approximated as a GI/G/1 queue whose mean waiting time is approximated based on the first two moments of the interarrival times and the service times The departure process is also approximated as a renewal process even though the autocorrelation of this process may significantly affect the performance of the subsequent queue depending on the traffic intensity When the departure process is split into substreams by Markovian random routing, the split processes typically are modeled as independent renewal processes even though they are correlated with each other This cross correlation might also have a serious impact on the queueing performance In this paper, we propose an approach for modeling both the cross correlation and the autocorrelation by a three-moment four-parameter decomposition approximation of queueing networks The arrival process is modeled as a nonrenewal process by a two-state Markov-modulated Poisson process, viz, MMPP(2) The cross correlation between randomly split streams is accounted for in the second and third moments of the merged process by the innovations method The main contribution of the present research is that both the cross correlation and the autocorrelation can be modeled in parametric decomposition approximations of queueing networks by integrating the MMPP(2) approximation of the arrival/departure process and the innovations method We also present numerical results that strongly support our refinements

On the Asymptotic Internal Path Length and the Asymptotic Wiener Index of Random Split Trees

Abstract: The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.

Hazard rate properties of a general counting process stopped at an independent random time

Abstract: In this work we provide sufficient conditions under which a general counting process stopped at a random time independent from the process belongs to the reliability decreasing reversed hazard rate (DRHR) or increasing failure rate (IFR) class. We also give some applications of these results in generalized renewal and trend renewal processes stopped at a random time.

Random point sets and their diffraction

Abstract: The diffraction of various random subsets of the integer lattice ℤ d , such as the coin tossing and related systems, are well understood. Here, we go one important step beyond and consider random point sets in ℝ d . We present several systems with an effective stochastic interaction that still allow for explicit calculations of the autocorrelation and the diffraction measure. We concentrate on one-dimensional examples for illustrative purposes, and briefly indicate possible generalisations to higher dimensions. In particular, we discuss the stationary Poisson process in ℝ d and the renewal process on the line. The latter permits a unified approach to a rather large class of one-dimensional structures, including random tilings. Moreover, we present some stationary point processes that are derived from the classical random matrix ensembles as introduced in the pioneering work of Dyson and Ginibre. Their reconsideration from the diffraction point of view improves the intuition on systems with randomness and ...