Connecting Renewal Age Processes with M/D/1 and M/D/∞ Queues Through Stick Breaking

We consider a Markovian growth collapse process on the state space E = [0, ∞) which evolves as follows. Between random downward jumps the process increases with slope one. Both the jump intensity and the jump sizes depend on the current state of the process. We are interested in the behavior of the first hitting time τ y = inf{t ≥ 0|X t = y} as y becomes large and the growth of the maximum process M t = sup{X s |0 ≤ s ≤ t} as t → ∞. We consider the recursive sequence of equations Am n = m n−1 , m 0 ≡ 1, where A is the extended generator of the MGCP, and show that the solution sequence (which is essentially unique and can be given in integral form) is related to the moments of τ y . The Laplace transform of τ y can be expressed in closed form (in terms of an integral involving a certain kernel) in a similar way. We derive asymptotic results for the running maximum: (i) if m 1 (y) is of rapid variation, we have M t /m −1 (t)

show abstract

“…If b − a ≥ −1 then, by Theorem 9 in Appendix A, The case where λ(x) ≡ λ is constant has been discussed in [26].…”

Section: Separable Jump Measuresmentioning

confidence: 99%

“…In this case we write f ∈ R a . The function f is called rapidly varying if (26) holds with a = ∞ (letting λ ∞ = 0 for λ < 1 and λ ∞ = ∞ for λ > 1) and slowly varying if (26) holds with a = 0. The convergence in (26) is uniform for…”

Section: Appendix a Regular And Rapid Variationmentioning

confidence: 99%

Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

Löpker

Stadje

2011

Journal of Applied Probability

View full text Add to dashboard Cite

show abstract

“…In other words, the information transferred by runoff cannot spread beyond a given wet zone, and the runoff process regenerates itself at an occurrence independently of the past. Powerful asymptotic results are obtained thanks to this property [Stidham, 1972;Van Leeuwaarden et al, 2010], which will be further discussed in section 3.5.…”

Section: Fðyþgðx2yþdymentioning

confidence: 99%

“…This class of stochastic processes is used to model the independent, identically distributed occurrences at which the process restarts itself. Each wet pixel located downslope of a dry pixel represents a regeneration point that is governed by a Poisson process [Van Leeuwaarden et al, 2010]. Therefore, the succession of wet zones along the domain is the aggregation of independent, identically distributed blocks of random length [Asmussen, 2003].…”

Section: Fðyþgðx2yþdymentioning

confidence: 99%

1-D steady state runoff production in light of queuing theory: Heterogeneity, connectivity, and scale

Harel

Mouche

2013

Water Resour. Res.

View full text Add to dashboard Cite

[1] We used the frameworks of queuing theory and connectivity to study the runoff generated under constant rainfall on a one-dimensional slope with randomly distributed infiltrability. The equivalence between the stationary runoff-runon equation and the customers waiting time in a single server queue provides a theoretical link between the statistical description of infiltrability and that of runoff flow rate. Five distributions of infiltrability, representing soil heterogeneities at different scales, are considered: four uncorrelated (exponential, bimodal, lognormal, uniform) and one autocorrelated (lognormal, with or without a nugget). The existing theoretical results are adapted to the hydrological framework for the exponential case, and new theoretical developments are proposed for the bimodal law. Numerical simulations validate these results and improve our understanding of runoff-runon for all of the distributions. The quantities describing runoff generation (runoff one-point statistics) and its organization into patterns (patterns statistics and connectivity) are studied as functions of rainfall rate. The variables describing the wet areas are also compared to those describing the rainfall excess areas, i.e., the areas where rainfall exceeds infiltrability. Preliminary results concerning the structural and functional connectivity functions are provided, as well as a discussion about the origin of scale effects in such a system. We suggest that the upslope no-flow boundary condition may be responsible for the dependence of the runoff coefficient on the scale of observation. Queuing theory appears to be a promising framework for runoff-runon modeling and hydrological connectivity problems.Citation: Harel, M.-A., and E. Mouche (2013), 1-D steady state runoff production in light of queuing theory: Heterogeneity, connectivity, and scale, Water Resour. Res., 49,[7973][7974][7975][7976][7977][7978][7979][7980][7981][7982][7983][7984][7985][7986][7987][7988][7989][7990][7991]

show abstract

“…We may note that this probability is simply the probability that the largest interevent time of a Poisson(µ) process M on the interval [0, x − t] is at most t (counting 0 and x − t as epochs). An explicit expression for this is known (see [12], and also [11] and [18]), given the number, m = M(x − t), of Poisson epochs, but is an alternating series with order m terms, so using this formula would not reduce the complexity from O(x) and could potentially be numerically unstable. We have therefore not pursued this approach, but suggest a different solution in the next section.…”

Section: Computational Effortmentioning

confidence: 99%

Importance Sampling for Failure Probabilities in Computing and Data Transmission

Asmussen

2009

J. Appl. Probab.

View full text Add to dashboard Cite

In this paper we study efficient simulation algorithms for estimating P(X > x), where X is the total time of a job with ideal time T that needs to be restarted after a failure. The main tool is importance sampling, where a good importance distribution is identified via an asymptotic description of the conditional distribution of T given X > x. If T ≡ t is constant, the problem reduces to the efficient simulation of geometric sums, and a standard algorithm involving a Cramér-type root, γ (t), is available. However, we also discuss an algorithm that avoids finding the root. If T is random, particular attention is given to T having either a gamma-like tail or a regularly varying tail, and to failures at Poisson times. Different types of conditional limit occur, in particular exponentially tilted Gumbel distributions and Pareto distributions. The algorithms based upon importance distributions for T using these asymptotic descriptions have bounded relative error as x → ∞ when combined with the ideas used for a fixed t. Nevertheless, we give examples of algorithms carefully designed to enjoy bounded relative error that may provide little or no asymptotic improvement over crude Monte Carlo simulation when the computational effort is taken into account. To resolve this problem, an alternative algorithm using two-sided Lundberg bounds is suggested.

show abstract

Connecting Renewal Age Processes with M/D/1 and M/D/∞ Queues Through Stick Breaking

Cited by 5 publications

References 16 publications

Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

1-D steady state runoff production in light of queuing theory: Heterogeneity, connectivity, and scale

Importance Sampling for Failure Probabilities in Computing and Data Transmission

Contact Info

Product

Resources

About