Stieltjes Transform Research Articles

Fractional Integral and Generalized Stieltjes Transforms for Hypergeometric Functions as Transmutation Operators

For each of the eight $n$-th derivative parameter changing formulas for Gauss hypergeometric functions a corresponding fractional integration formula is given. For both types of formulas the differential or integral operator is intertwining between two actions of the hypergeometric differential operator (for two sets of parameters): a so-called transmutation property. This leads to eight fractional integration formulas and four generalized Stieltjes transform formulas for each of the six different explicit solutions of the hypergeometric differential equation, by letting the transforms act on the solutions. By specialization two Euler type integral representations for each of the six solutions are obtained.

A new method to build spatio-temporal covariance functions: analysis of ozone data

Statistical analysis of natural phenomena with spatial and temporal correlations requires the specification of the correlation structure via a covariance function. A separable spatio-temporal covariance function is usually used for the ease of application. Nonetheless, the separability of the spatio-temporal covariance function can be unrealistic in many settings, where it is required to use a non-separable spatio-temporal covariance function. In this paper, the role of Stieltjes transformation in the construction of non-separable spatio-temporal covariance function is investigated. Then, structural copula function is applied to construct a family of non-separable spatio-temporal covariance function. Afterwards, it is proved that this family of covariance functions does not possess any dimple which exists in some Gneiting’s models. Finally, a modified genetic algorithm is applied to explore the spatio-temporal correlation structure of Ozone data in Tehran, Iran.

Exact reconstruction using Beurling minimal extrapolation

We show that measures with finite support on the real line are the unique solution to an algorithm, named generalized minimal extrapolation, involving only a finite number of generalized moments (which encompass the standard moments, the Laplace transform, the Stieltjes transformation, etc.).Generalized minimal extrapolation shares related geometric properties with the basis pursuit approach of Chen et al. (1998) [5]. Indeed we also extend some standard results of compressed sensing (the dual polynomial, the nullspace property) to the signed measure framework.We express exact reconstruction in terms of a simple interpolation problem. We prove that every nonnegative measure, supported by a set containing s points, can be exactly recovered from only 2s+1 generalized moments. This result leads to a new construction of deterministic sensing matrices for compressed sensing.

Orchestrating parallel TCP connections: Cyclic and probabilistic polling policies

The standard Transmission Control Protocol (TCP) is based on an additive rate increase in the absence of congestion, and on multiplicative decrease triggered by congestion signals. However, it does not scale well as the distances, or as the speed of the network, increase. Thus, we study some of the solutions that have been proposed to encounter this problem. These solutions include (i) splitting the transmission from a source to its destination into several parallel connections, and (ii) using Scalable TCP, which is a more aggressive version of TCP. The connection whose rate decreases when a signal arrives is chosen either at random or according to a round robin policy. Our analysis concentrates on a centrally controlled TCP system having N connections. We consider both Additive Increase Multiplicative Decrease (AIMD) and Multiplicative Increase Multiplicative Decrease (MIMD) control mechanisms. The Laplace–Stieltjes Transforms (LST) of the transmission rate of each connection at a polling instant, as well as at an arbitrary moment, are derived. Explicit results are obtained for the mean transmission rate and (in contrast to most polling models) for its second moment. For the AIMD procedure under the cyclic visit policy we show that, for both dynamic (Hamiltonian-type) and static visit orders in each cycle, the connections should be visited following a simple index rule in order to achieve maximum throughput. For the probabilistic visit policy we obtain the set of optimal probabilities that maximizes mean throughput. The analysis of the probabilistic MIMD models uses transformations yielding a system’s law of motion equivalent to that of an M/G/1 queue with batch service. The MIMD control mechanism with probabilistic strategy is further analyzed for the case where the transmission rate is bounded above.

Computation of the spectral density of two-point functions: Complex masses, cut rules, and beyond

We present a steepest descent calculation of the Kallen-Lehmann spectral density of two-point functions involving complex conjugate masses in Euclidean space. This problem occurs in studies of (gauge) theories with Gribov-like propagators. As the presence of complex masses and the use of Euclidean space brings the theory outside of the strict validity of the Cutkosky cut rules, we discuss an alternative method based on the Widder inversion operator of the Stieltjes transformation. It turns out that the results coincide with those obtained by naively applying the cut rules. We also point out the potential usefulness of the Stieltjes (inversion) formalism when non-standard propagators are used, in which case cut rules are not available at all.

The age of the arrival process in the G/M/1 and M/G/1 queues

This paper shows that in the G/M/1 queueing model, conditioning on a busy server, the age of the inter-arrival time and the number of customers in the queue are independent. The same is the case when the age is replaced by the residual inter-arrival time or by its total value. Explicit expressions for the conditional density functions, as well as some stochastic orders, in all three cases are given. Moreover, we show that this independence property, which we prove by elementary arguments, also leads to an alternative proof for the fact that given a busy server, the number of customers in the queue follows a geometric distribution. We conclude with a derivation for the Laplace Stieltjes Transform (LST) of the age of the inter-arrival time in the M/G/1 queue.

Analysis of a finite buffer model with two servers and two nonpreemptive priority classes

In this paper, we analyze a finite buffer queueing model with two servers and two nonpreemptive priority service classes. The arrival streams are independent Poisson processes, and the service times of the two classes are exponentially distributed with different means. One of the two servers is reserved exclusively for one class with high priority and the other server serves the two classes according to a nonpreemptive priority service schedule. For the model, we describe its dynamic behavior by a four-dimensional continuous-time Markov process. Applying recursive approaches we present the explicit representation for the steady-state distribution of this Markov process. Then, we calculate the Laplace–Stieltjes Transform and the steady-state distribution of the actual waiting times of two classes of customers. We also give some numerical comparison results with other queueing models.

General decrementing service M/G/1 queue with multiple adaptive vacations

In this paper, we introduce the multiple adaptive vacation policy and the general decrementing service rule based on the classical M/G/1 queueing systems, and obtain the P.G.F. (Probability Generating Function) of stationary queue length by using the embedded Markov chain method and regeneration cycle approach. Then, the LST (Laplace Stieltjes Transform) of stationary waiting time is also derived according to the independence between the waiting time and arrival process. At last some special cases are given to show the general properties of the new model, and some numerical results are shown to compare the mean queue length and waiting time of special cases.

Algorithms for the Laplace–Stieltjes Transforms of First Return Times for Stochastic Fluid Flows

We derive several algorithms, including quadratically convergent algorithms, which can be used to calculate the Laplace–Stieltjes transforms of the time taken to return to the initial level in the Markovian stochastic fluid flow model. We give physical interpretations of the algorithms and consider their numerical analysis. The numerical performance of the algorithms, which depends on the physical properties of the process, is discussed and illustrated with simple examples. Besides the powerful algorithms, this paper contributes interesting theoretical results. In particular, the methodology for constructing these algorithms is a valuable contribution to the theory of fluid flow models. Moreover, useful physical interpretations of the algorithms, and related expressions, given in terms of the fluid flow model, can assist in further analysis and help in a better understanding of the model.

Sur une mesure rendant orthogonaux les polynômes secondaires

This Note presents a method for defining, from a probability density function ρ on an interval, a new measure turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system. The construction uses the associated Stieltjes transformations of the two measures and we can express the solution thanks to the inverse formula of Stieltjes–Perron. Through this study, we see interesting isometric relations using the operator T ρ creating the secondary polynomials. To cite this article: R. Groux, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The distribution of the first [formula omitted] point in the classical risk model with interest

In this paper we investigate the distribution function and the Laplace–Stieltjes Transform(L-S-T) of the first β point s 1 β of the surplus process { U ( t ) , t ⩾ 0 } using its strong Markov property and the theory of renewal measure. We find the distribution function of s 1 β in some cases.

Analysis of BMAP/G/1 Queue with Reservation of Service

In this article, we study BMAP/G/1 queue with service time distribution depending on number of processed items. This type of queue models the systems with the possibility of preliminary service. For the considered system, an efficient algorithm for calculating the stationary queue length distribution is proposed, and Laplace–Stieltjes transform of the sojourn time is derived. Little's law is proved. An optimization problem is considered.

Solving Ck/Cm/1/N queues by using characteristic roots in matrix analytic methods

In this paper, we study a C k / C m /1/ N open queueing system with finite capacity. We investigate the property which shows that a product of the Laplace Stieltjes Transforms of interarrival and service times distributions satisfies an equation of a simple form. According to this equation, we present that the stationary probabilities on the unboundary states can be written as a linear combination of vector product-forms. Each component of these products is expressed in terms of roots of an associated characteristic polynomial. As a result, we carry out an algorithm for solving stationary probabilities in C k / C m /1/ N systems, which is independent of N , hence greatly reducing the computational complexity.

Queues with Dependency Between Interarrival and Service Times Using Mixtures of Bivariates

We analyze queueing models where the joint density of the interarrival time and the service time is described by a mixture of joint densities. These models occur naturally in multiclass populations serviced by a single server through a single queue. Other motivations for this model are to model the dependency between the interarrival and service times and consider queue control models. Performance models with component heavy tailed distributions that arise in communication networks are difficult to analyze. However, long tailed distributions can be approximated using a finite mixture of exponentials. Thus, the models analyzed here provide a tool for the study of performance models with heavy tailed distributions. The joint density of A and X, the interarrival and service times respectively, f(a,x), will be of the form where p i > 0 and . We derive the Laplace Stieltjes Transform of the waiting time distribution. We also present and discuss some numerical examples to describe the effect of the various parameters of the model.

Tauberian-type theorems with application to the Stieltjes transformation

In the first part, we define the spaceL'(r)and the modified Stieltjes transformation introduced by Lavoine and Misra (1979) and Marichev (1983), respectively. In the second part of the paper, we extend Tauberian-type theorems for the distributional Stieltjes transformations to the distributional modified Stieltjes transformations.

A two phases queueing system with Bernoulli vacation schedule under multiple vacation policy

This paper deals with a single server Poisson arrival queue with two phases of heterogeneous service along with a Bernoulli schedule vacation model, where after two successive phases service the server either goes for a vacation with probability p ( 0 ≤ p ≤ 1 ) or may continue to serve the next unit, if any, with probability q ( = 1 − p ) . Further the concept of multiple vacation policy is also introduced here. We obtained the queue size distributions at a departure epoch and at a random epoch, Laplace Stieltjes Transform of the waiting time distribution and busy period distribution along with some mean performance measures. Finally we discuss some statistical inference related issues.

A finite‐source queuing model for the IEEE 802.11 DCF

AbstractThe most mature medium access control protocol for wireless local area networks is the IEEE 802.11 standard. The primary access mode of this protocol is based on the mechanism of carrier sense multiple access with binary exponential backoff. We develop a finite‐source feedback queueing model for this access mode. In this model, we derive expressions for the throughput and the first moment of the delay and a set of equations to compute the Laplace Stieltjes Transform and higher moments of the delay. The accuracy of the model is verified by simulation. Copyright © 2005 AEIT.

Handoff counting in hierarchical cellular system with overflow scheme

In mobile computing scenario, the number of handoff (called as handoff counting) is very important for the signalling network design and traffic load analysis. In this paper, we study the handoff counting characteristics in terms of probability mass function (pmf) and statistical moments in the hierarchical cellular system applying overflow scheme under the general cell residence time and general call holding time distribution. The analytical result and the algorithm for the evaluation of handoff counting are, respectively, presented for the cases whether or not the Laplace–Stieltjes Transform of call holding time exists. The numerical results reveal that as the call holding time is heavy-tailed, the handoff counting follows the heavy-tailed tendency. In addition, the handoff counting pmf as well as the average value shows a significant different pattern with the utilization of different call holding time distribution.

Performance of correlated queues: the impact of correlated service and inter-arrival times

In this paper, we consider correlated queues where the service time of a packet is strongly correlated with its inter-arrival time due to the finite transmission capacity of input and output links. We present an analytical method to derive the LST (Laplace Stieltjes Transform) of the (actual) waiting time distribution and the system time distribution. To investigate the impact of such correlation between service and inter-arrival times on the system performance, we consider a counterpart GI/G/1 queue where the service time and inter-arrival time distributions are the same as in our correlated system, but they are independent. Some numerical examples are provided to show that such correlation gives significant impact on the system performance.

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.