Showing papers on "Renewal theory published in 2001"

Event-driven power management

Abstract: Energy consumption of electronic devices has become a serious concern in recent years. Power management (PM) algorithms aim at reducing energy consumption at the system-level by selectively placing components into low-power states. Formerly, two classes of heuristic algorithms have been proposed for PM: timeout and predictive. Later, a category of algorithms based on stochastic control was proposed for PM. These algorithms guarantee optimal results as long as the system that is power managed can be modeled well with exponential distributions. We show that there is a large mismatch between measurements and simulation results if the exponential distribution is used to model all user request arrivals. We develop two new approaches that better model system behavior for general user request distributions. Our approaches are event-driven and give optimal results verified by measurements. The first approach we present is based on renewal theory. This model assumes that the decision to transition to low-power state can be made in only one state. Another method we developed is based on the time-indexed semi-Markov decision process (TISMDP) model. This model has wider applicability because it assumes that a decision to transition into a lower-power state can be made upon each event occurrence from any number of states. This model allows for transitions into low-power states from any state, but it is also more complex than our other approach. It is important to note that the results obtained by renewal model are guaranteed to match results obtained by TISMDP model, as both approaches give globally optimal solutions. We implemented our PM algorithms on two different classes of devices: two different hard disks and client-server wireless local area network systems such as the SmartBadge or a laptop. The measurement results show power savings ranging from a factor of 1.7 up to 5.0 with insignificant variation in performance.

Nonparametric Estimation With Recurrent Event Data

Abstract: The problem of nonparametric estimation for the distribution function governing the time to occurrence of a recurrent event in the presence of censoring is considered. We derive Nelson–Aalen and Kaplan–Meier-type estimators for the distribution function, and establish their respective finite-sample and asymptotic properties. We allow for random observation periods for each subject under study and explicitly account for the informative sum-quota nature of the data accrual scheme. These allowances complicate technical matters considerably and, in particular, invalidate the direct use of martingale methods. Consistency and weak convergence of our estimators are obtained by extending an approach due to Sellke, who considered a single renewal process (i.e., recurrent events on a single subject) observed over an infinite time period. A useful feature of the present analysis is that strong parallels are drawn to the usual “single-event” setting, providing a natural route toward developing extensions that involve...

Shocks, runs and random sums

Abstract: In this paper we study random variables related to a shock reliability model. Our models can be used to study systems that fail when k consecutive shocks with critical magnitude (e.g. above or below a certain critical level) occur. We obtain properties of the distribution function of the random variables involved and we obtain their limit behaviour when k tends to infinity or when the probability of entering a critical set tends to zero. This model generalises the Poisson shock model.

Metastability in stochastic dynamics of disordered mean-field models

Abstract: We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of “admissible transitions”. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor. The distributions of the rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.

On random motions with velocities alternating at Erlang-distributed random times

Abstract: We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

Moments of compound renewal sums with discounted claims

Abstract: Delbaen and Haezendonck [Ins. Math. Econ. 6 (1987) 85] and Willmot [Scand. Actuarial J. 1 (1989) 1] give an analytical expression for the net premium density of a compound Poisson present value risk (CPPVR) process. Their calculation is based, essentially, on the independence of the increments of the CPPVR process. In this paper, under regularity conditions, we derive the first two moments of a compound renewal present value risk (CRPVR) process using renewal theory arguments. Some examples, extensions and limiting results are also given.

Recursive Moments of Compound Renewal Sums with Discounted Claims

Abstract: Under regularity conditions, Le´veille´& Garrido [6] gives a derivation of the first two moments (resp. asymptotic) of a Compound Renewal Present Value Risk (CRPVR) process using renewal theory arguments. In this paper, with the same procedure and assuming that all the moments of the claim severity and the claims number process exist, we get recursive formulas for all the moments (resp. asymptotic) of the CRPVR process.

Ageing of a data-intensive legacy system: symptoms and remedies

Abstract: This study generalizes some of the symptoms of ageing of a legacy system. Each symptom is specified by metrics and the results of the measurements made suggest what operations should be undertaken to renew the software. The study is based on retrospective analysis of data collected during the execution of a large renewal process of a very old legacy system. It therefore provides evidence of the expected efficacy of such renewal processes and can be used to decide how best to plan them and manage them in order to increase their efficacy. It can also be used to define the reengineering requirements to ensure long life to the system despite successive evolutions of the application and the operation domain. The metrics can provide a basis for monitoring a software system to ensure that its quality does not degrade to such an extent that the most costly and risky renewal processes then have to be performed to improve it. Finally, the paper points out the problems with renewal processes that still remain open. Copyright © 2001 John Wiley & Sons, Ltd.

A note on stochastic comparisons of excess lifetimes of renewal processes

Abstract: In this paper we provide new results about stochastic comparisons of the excess lifetime at different times of a renewal process when the interarrival times belong to several ageing classes. We also provide a preservation result for the new better than used in the Laplace transform order ageing class for series systems.

Some new approximations for the renewal function

Abstract: In this paper, an approximation for computing the (renewal function) solution of renewal-type integral equations is presented. This approximation is a modified rational function near the origin and switches to an asymptotic linear function for larger values of the time variable. This approach is similar to that of Garg and Kalagnanam (IEEE Trans. Reliability 1998, 47 (1), 66–72) but differs near the origin and has a different switch-over-point in general. Examples are presented for the truncated normal and Weibull distributions. Some alternative models are also discussed.

On the semi-Markovian random walk with two reflecting barriers

Abstract: In this paper, the semi-Markovian random walk with two reflecting barriers is constructed mathematically and non-stationary distribution functions of it are expressed by means of the probability characteristics of renewal process {T n } and random walk {Y n } without barriers. In particular, when the time between two jump instants has exponential or Erlang distribution, explicit formulae are obtained for non-stationary distribution functions of the process. Moreover, explicit expressions are given for expected value, variance and moment generating function of the first reflection moment, an important boundary functional, of the process from lower reflecting barrier.

Geometric renewal convergence rates from hazard rates

Abstract: This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

Monotone Markov processes with respect to the reversed hazard rate ordering: an application to reliability

Abstract: We consider a repairable system with a finite state space which evolves in time according to a Markov process as long as it is working. We assume that this system is getting worse and worse while running: if the up-states are ranked according to their degree of increasing degradation, this is expressed by the fact that the Markov process is assumed to be monotone with respect to the reversed hazard rate and to have an upper triangular generator. We study this kind of process and apply the results to derive some properties of the stationary availability of the system. Namely, we show that, if the duration of the repair is independent of its completeness degree, then the more complete the repair, the higher the stationary availability, where the completeness degree of the repair is measured with the reversed hazard rate ordering.

Markov Renewal Processes

Abstract: The study of the semi-Markov process is closely related to the theory of Markov renewal processes (MRP) which can be considered as an extension of the classical renewal theory (see, e.g., Feller [32], Cox [23]).

Time dependent analysis of multivariate marked renewal processes

Abstract: The paper examines multivariate delayed marked renewal processes, of which one component is formed by a delayed compound Poisson process observed at epochs of some point process. In addition, the values of these observations (and other components) are watched when crossing their respective thresholds and the value of the original Poisson process at any moment of time, past the first passage time, is the objective of this investigation. The results (which are imperative for classes of semiregenerative processes) are given in closed analytical forms and illustrated on various stochastic models.

Two control policies for stochastic eoq-type models

Abstract: We present the production version of two EOQ-type models in heavy traffic. The output is interpreted by random demand and the input by a deterministic production plus random returns. In the ON part of the cycle, the inventory content is a reflected Brownian motion, and in the OFF part, it is a Brownian motion with a negative drift. The ON/OFF periods generate an alternative renewal process and the content-level process is a regenerative process. Two control policies are considered. In one policy that is natural under conditions of continuous review, production is stopped when the content level in the ON period reaches a predetermined level q. In the other policy, which resembles periodic review, production is stopped when the ON time reaches a predetermined time t0.

Statistics of the occupation time for a class of Gaussian Markov processes

Abstract: We revisit the work of Dhar and Majumdar (1999 Phys. Rev. E 59 6413) on the limiting distribution of the temporal mean Mt = t-1∫0tdu sign yu, for a Gaussian Markovian process yt depending on a parameter α, which can be interpreted as Brownian motion in the time scale t' = t2α. This quantity, the mean `magnetization', is simply related to the occupation time of the process, that is the length of time spent on one side of the origin up to time t. Using the fact that the intervals between sign changes of the process form a renewal process on the time scale t', we determine recursively the moments of the mean magnetization. We also find an integral equation for the distribution of Mt. This allows a local analysis of this distribution in the persistence region (Mt→±1), as well as its asymptotic analysis in the regime where α is large. Finally, we put the results thus found in perspective with those obtained by Dhar and Majumdar by another method, based on a formalism due to Kac.

Stationarity of delayed subordinators

Abstract: It is shown that, analogous to partial-sum processes in renewal theory, nondecreasing Levy processes (subordinators) can be delayed such as to show a certain stationarity

Graphical methods for reliability of repairable equipment and maintenance planning

Abstract: Graphical methods using total time on test (TTT) plots introduced by Barlow and Campo and latter advocated by Bergman and Klefso is handy to analyze maintenance/overhaul periods of repairable equipment. Simple statistical and graphical techniques are used to analyze failure data which exhibit independently and identically distributed (IID) pattern. Weibull or other types of renewal process models are used to model failure times. TTT plots can be used to monitor health of equipment in terms of constant failure rate/increasing or decreasing failure rate. It is presumed that these equipment are as good as new after each failure-repair process. The TTT plots can be drawn using Kaplan-Meier or piecewise exponential or maximum likelihood estimators. However, data, which does not satisfy IID assumption and where trend is observed. Nonhomogenous poisson process models, can be used, which represent repairable equipment with minimal repair. Maintenance planning with optimal overhaul periods based on minimum cost per unit time can be obtained graphically for repairable equipments experiencing increasing failure rate (IFR). Further, repairable equipments, whose performance is effected by concurrent variables, can be modelled by proportional hazard models. Graphical methods can be used to arrive at maintenance intervals. This paper discusses some of the graphical tools mentioned above to analyze failure data and plan maintenance intervals for material handling equipments operating in the mining industry.

A renewal cluster model for the inter-arrival times of rainfall events

Abstract: A statistical model, based on a renewal cluster point process, is proposed and used to infer the distributional properties of dry periods in a continuous-time record. The model incorporates a mixed probability distribution in which inter-arrival times are classified into two distinct types, representing cyclonic and anticyclonic weather. This results in rainfall events being clustered in time, and enables objective probabilistic statements to be made about storm properties, e.g. the expected number of events in a storm cluster. The model is fitted to data taken from a gauge near Wellington, New Zealand, by maximizing the likelihood function with respect to the parameters. The Akaike Information Criteria is used to select the best fitting distributions from a range of candidates. The Log-Normal distribution is found to provide the best fit to the times between successive storm clusters, whilst the Weibull distribution is found to provide the best fit to the times between successive events in the same storm cluster. Harmonic curves are used to provide a parsimonious parameterization, allowing for the seasonal variation in precipitation. Under the fitted model, the interval series is transformed into a residual series, which is assessed to determine overall goodness-of-fit. Copyright © 2001 Royal Meteorological Society

Approximating the finite-time ruin probability under interest force

Abstract: We present an algorithm to determine both a lower and an upper bound for the finite-time probability of ruin for a risk process with constant interest force. We split the time horizon into smaller intervals of equal length and consider the probability of ruin in case premium income for a time interval is received at the beginning (resp. end) of that interval, which yields a lower (resp. upper) bound. For both bounds we present a renewal equation which depends on the distribution of the present value of the aggregate claim amount in a time interval. This distribution is determined through a generalization of Panjer’s [ASTIN Bulletin 12 (1981) 22] recursive method.

Regenerative processes in the infinite mean cycle case

Abstract: A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (1/2, 1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.

Finite-size corrections to Poisson approximations of rare events in renewal processes

Abstract: Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.

On some distributional properties of a first-order nonnegative bilinear time series model

Abstract: We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

The impact of long-range dependent traffic in a CDMA system supporting real-time services

Abstract: This paper examines the impact of long-range dependent real-time variable bit rate (rt-VBR) traffic in a multi-service CDMA system that has been dimensioned using a short-range dependent traffic model. A simulation of the system is performed using both a self-similar Pareto renewal process and a Poisson process as traffic models. The importance of traffic model choice when dimensioning the system is established by comparing the QoS delivered under each traffic model. Simulation results show that the factor of performance difference between the two models is heavily dependent upon the level of self-similarity in the traffic, traffic load and propagation conditions. This difference is most significant when: the Hurst parameter exceeds approximately 0.75; there is a medium traffic load; and the probability of propagation loss is low. The scheduling policy is shown to skew the relative importance of the traffic model to each service.

Rekorde in der Erneuerungstheorie

Abstract: The first chapter of the thesis deals with the von Neumann parallel addition, an algorithm that computes the sum of two binary integers. We show that for two independent random vectors which are uniformly distributed on {0, 1}, n ∈ N, the number An of steps required by the algorithm, which is the main focus here, is connected with the possibly uncompleted record lifetime of an alternating renewal process up to time n. We prove that the distribution of Amk − blog2 mkc + 1 with mk ∈ N for all k ∈ N and limk→∞ mk 2k+2 = β ∈ (1, 2) converges with k → ∞ in total variation distance to the distribution of dZ + log2 βe where Z is a Gumbel(log 2)distributed random variable. Also, the expected value of Amk − log2 mk converges to φ(β) + γ log 2 − 1. Here γ is the Euler-Mascheroni-constant and φ is a function in β ∈ (1, 2). Further we show that φ(β) is not constant in β ∈ (1, 2); in particular E(An)− log2 n does not converge with n→∞. In the second chapter we consider a renewal process with lifetimes X (N) 1 , X (N) 2 , . . . that are uniformly distributed on {1, . . . , N}, N ∈ N, and analyse the asymptotic behaviour of the number Y (N) of resulting record lifetimes and of the total amount Z of time spent in these records as N → ∞. It follows that Y (N) as well as Z are each concentrated about their expected value, i.e. we have Y /E(Y ) P → 1 and Z/E(Z) P → 1 with N → ∞. Furthermore, we even get (Y (N) − E(Y ))/ √ Var(Y (N)) D → Z and (Z − E(Z))/ √ Var(Z(N)) D → Z with N → ∞ and Z ∼ N(0, 1). To obtain the asymptotic behaviour of Z we define a random variable Z̃ which describes the sum of those lifetimes in the sequence (N −X k )k∈N that are smallest at the time of their appearance. We show that (Z̃ − N)/N converges with N → ∞ to −1 + ∑∞ i=1 ∏i j=1 Uj where (Un)n∈N is a sequence of independent and on (0, 1) uniformly distributed random variables. Among other things, it follows that Z̃, contrary to Y (N) and Z, is neither asymptotically concentrated about its mean nor asymptotically normally distributed. In the last chapter the main focus is on the Poisson process with rate λ > 0. We prove that the total amount of time spent in record lifetimes up to time t behaves as 1 2λ (log t) with t→∞. To obtain this result we analyse the behaviour of the sum Zn of the record lifetimes within the first n lifetimes and also the sum Tn of the first n record lifetimes. While for Tn we can show concentration about the mean as well as a central limit theorem, for Zn we only have the concentration about its mean and the stochastic boundedness of (Zn − (logn) 2 2λ )/ 1 λ √ (log n)3.