Interarrival Distributions Research Articles

Bounds for the distribution and moments of the forward and backward recurrence times in a renewal process

We obtain a variety of lower and upper bounds for the bivariate tail of the joint distribution between the (forward and backward) recurrence times in a renewal process. Some of these bounds are valid generally, some others in the case where the interarrival distribution F exhibits monotone aging. We also give various bounds for the moments of the recurrence times. Our results improve upon existing bounds for these quantities in the literature. We also discuss the case where the aging of F is non-monotonic. Finally, we present numerical examples to illustrate the performance of the new bounds.

Bounds for the Ruin Probability in the Sparre–Andersen Model

We obtain the upper and lower bounds for the ruin probability in the Sparre–Andersen model. These bounds are established under various conditions: when the adjustment coefficient exists, when it does not exist, and when the interarrival distribution belongs to certain aging classes. Additionally, we improve the Lundberg upper bound for the ruin probability.

Diagnosing the coexistence of Poissonity and self-similarity in memory workloads

Accurately modeling the inter-arrival distribution is critical for synthesizing memory traces to evaluate performance. The intensity and complexity of memory accesses have increased greatly in modern applications. This motivates us to re-examine two fundamental issues: (1) Can the arrival process of memory accesses in modern workloads still be described as a Poisson process? (2) Do modern memory workloads present self-similarity? Via analyzing SPEC CPU2017, this paper demonstrates the coexistence of both Poissonity and self-similarity in modern memory workloads, which seems to reconcile the contradiction between Poisson and self-similar characteristics. Specifically, memory access intervals at a small time scale, such as milliseconds, approximately follow an exponential distribution and memory accesses are independent of each other. At such time scales, memory access series can be approximated by a Poisson process. For the aggregated process of memory access series at a large time scale, such as several minutes, the non-degenerate structures of the auto-correlation function show the presence of self-similarity in memory workloads. All Hurst parameters estimated are greater than 0.5, confirming the presence of self-similarity.

The Zeros of the Partition Function of the Pinning Model

We aim at understanding for which (complex) values of the potential the pinning partition function vanishes. The pinning model is a Gibbs measure based on discrete renewal processes with power law inter-arrival distributions. We obtain some results for rather general inter-arrival laws, but we achieve a substantially more complete understanding for a specific one parameter family of inter-arrivals. We show, for such a specific family, that the zeros asymptotically lie on (and densely fill) a closed curve that, unsurprisingly, touches the real axis only in one point (the critical point of the model). We also perform a sharper analysis of the zeros close to the critical point and we exploit this analysis to approach the challenging problem of Griffiths singularities for the disordered pinning model. The techniques we exploit are both probabilistic and analytical. Regarding the first, a central role is played by limit theorems for heavy tail random variables. As for the second, potential theory and singularity analysis of generating functions, along with their interplay, will be at the heart of several of our arguments.

Synchronized abandonment in discrete-time renewal input queues with vacations

<p style='text-indent:20px;'>We consider a discrete-time infinite buffer renewal input queue with multiple vacations and synchronized abandonment. Waiting customers get impatient during the server's vacation and decide whether to take service or abandon simultaneously at the vacation completion instants. Using the supplementary variable technique and difference operator method, we obtain an explicit expression to find the steady-state system-length distributions at pre-arrival, random, and outside observer's observation epochs. We provide the stochastic decomposition structure for the number of customers and discuss the various performance measures. With the help of numerical experiments, we show that the method formulated in this work is analytically elegant and computationally tractable. The results are appropriate for light-tailed inter-arrival distributions and can also be leveraged to find heavy-tailed inter-arrival distributions.</p>

Covariance between the forward recurrence time and the number of renewals

Recurrence times and the number of renewals in $(0,t]$ are fundamental quantities in renewal theory. Firstly, it is proved that the upper orthant order for the pair of the forward and backward recurrence times may result in NWUC (NBUC) interarrivals. It is also demonstrated that, under DFR interarrival times, the backward recurrence time is smaller than the forward recurrence time in the hazard rate order. Lastly, the sign of the covariance between the forward recurrence time and the number of renewals in $(0,t]$ at a fixed time point t and when $t\to \infty $ is studied assuming that the interarrival distribution belongs to certain ageing classes.

On the local time of a recurrent random walk on ℤ²

We prove a functional limit theorem for the number of visits by a planar random walk on Z 2 \mathbb {Z}^2 with zero mean and finite second moment to the points of a fixed finite set P ⊂ Z 2 P\subset \mathbb {Z}^2 . The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.

Probabilistic Analysis of a Buffer Overflow Duration in Data Transmission in Wireless Sensor Networks

One of the most important problems of data transmission in packet networks, in particular in wireless sensor networks, are periodic overflows of buffers accumulating packets directed to a given node. In the case of a buffer overflow, all new incoming packets are lost until the overflow condition terminates. From the point of view of network optimization, it is very important to know the probabilistic nature of this phenomenon, including the probability distribution of the duration of the buffer overflow period. In this article, a mathematical model of the node of a wireless sensor network with discrete time parameter is proposed. The model is governed by a finite-buffer discrete-time queueing system with geometrically distributed interarrival times and general distribution of processing times. A system of equations for the tail cumulative distribution function of the first buffer overflow period duration conditioned by the initial state of the accumulating buffer is derived. The solution of the corresponding system written for probability generating functions is found using the analytical approach based on the idea of embedded Markov chain and linear algebra. Corresponding result for next buffer overflow periods is obtained as well. Numerical study illustrating theoretical results is attached.

Reliability assessment for discrete time shock models via phase‐type distributions

In this paper, particular shock models are studied for the case when the times between successive shocks and the magnitudes of shocks have discrete phase‐type distributions. The well‐known shock models such as delta shock model, extreme shock model, and the mixed shock model which is obtained by combining delta and extreme shock models are considered. The probability generating function and recursive equation for the distribution of the system's lifetime are obtained for the cases when the interarrival times between shocks and the magnitudes of shocks are independent and when they are dependent. System reliability is computed for particular interarrival distributions such as geometric, negative Binomial and generalized geometric distributions.

Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process

The best known result about the joint distribution of the backward and forward recurrence times in a renewal process concerns the asymptotic behavior for the tail of that bivariate distribution. In the present paper we study the joint behavior of the recurrence times at a fixed time point t, and we obtain both exact results and bounds for their joint tail behavior. We also obtain results about the joint moments of these two random variables and we show in particular that the expectation of the product between the two recurrence times increases with time when the interarrival distribution has a decreasing failure rate. The results are illustrated by some numerical examples.

The Limit of Stationary Distributions of Many-Server Queues in the Halfin–Whitt Regime

We consider the so-called GI/GI/N queue, in which a stream of jobs with independent and identically distributed service times arrive as a renewal process to a common queue that is served by [Formula: see text] identical parallel servers in a first-come, first-served manner. We introduce a new representation for the state of the system and, under suitable conditions on the service and interarrival distributions, establish convergence of the corresponding sequence of centered and scaled stationary distributions in the so-called Halfin–Whitt asymptotic regime. In particular, this resolves an open question posed by Halfin and Whitt in 1981. We also characterize the limit as the stationary distribution of an infinite-dimensional, two-component Markov process that is the unique solution to a certain stochastic partial differential equation. Previous results were essentially restricted to exponential service distributions or service distributions with finite support, for which the corresponding limit process admits a reduced finite-dimensional Markovian representation. We develop a different approach to deal with the general case when the Markovian representation of the limit is truly infinite dimensional. This approach is more broadly applicable to a larger class of networks.

Stochastic calculus in a risk model with stochastic return on investments

ABSTRACT In this paper, we consider a perturbed risk model with stochastic return on investments. After discretizing of the renewal risk process, using the elementary properties of the classical conditional expectation, the Markov property and considerating the ordinary differential equation of a common density function of the inter-arrival times, we introduce a general integro-differential equation for the Laplace transform of the time of ruin with a positive initial surplus. The special cases, for different inter-arrival times distributions, are given in some details in the discrete renewal risk process.

Contact process under renewals II

We continue the study of renewal contact processes initiated in a companion paper, where we showed that if the tail of the interarrival distribution μ is heavier than t−α for some α<1 (plus auxiliary regularity conditions) then the critical value vanishes. In this paper we show that if μ has decreasing hazard rate and tail bounded by t−α with α>1, then the critical value is positive in the one-dimensional case. A more robust and much simpler argument shows that the critical value is positive in any dimension whenever the interarrival distribution has a finite second moment.

Discrete-time queue with batch renewal input and random serving capacity rule: $$GI^X/ Geo^Y/1$$ G I X / G e o Y / 1

In this paper, we provide a complete analysis of a discrete-time infinite buffer queue in which customers arrive in batches of random size such that the inter-arrival times are arbitrarily distributed. The customers are served in batches by a single server according to the random serving capacity rule, and the service times are geometrically distributed. We model the system via the supplementary variable technique and further use the displacement operator method to solve the non-homogeneous difference equation. The analysis done using these methods results in an explicit expression for the steady-state queue-length distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the underlying characteristic equation. Our approach enables one to estimate the asymptotic distribution at a pre-arrival epoch by a unique largest root of the characteristic equation lying inside the unit circle. With the help of few numerical results, we demonstrate that the methodology developed throughout the work is computationally tractable and is suitable for light-tailed inter-arrival distributions and can also be extended to heavy-tailed inter-arrival distributions. The model considered in this paper generalizes the previous work done in the literature in many ways.

Pólya urns with immigration at random times

We study the number of white balls in a classical Pólya urn model with the additional feature that, at random times, a black ball is added to the urn. The number of draws between these random times are i.i.d. and, under certain moment conditions on the inter-arrival distribution, we characterize the limiting distribution of the (properly scaled) number of white balls as the number of draws goes to infinity. The possible limiting distributions obtained in this way vary considerably depending on the inter-arrival distribution and are difficult to describe explicitly. However, we show that the limits are fixed points of certain probabilistic distributional transformations, and this fact provides a proof of convergence and leads to properties of the limits. The model can alternatively be viewed as a preferential attachment random graph model where added vertices initially have a random number of edges, and from this perspective, our results describe the limit of the degree of a fixed vertex.

Disorder and critical phenomena: the $$\alpha =0$$ α = 0 copolymer model

The copolymer model is a disordered system built on a discrete renewal process with inter-arrival distribution that decays in a regularly varying fashion with exponent $$1+ \alpha \;\geqslant \;1$$ . It exhibits a localization transition which can be characterized in terms of the free energy of the model: the free energy is zero in the delocalized phase and it is positive in the localized phase. This transition, which is observed when tuning the mean h of the disorder variable, has been tackled in the physics literature notably via a renormalization group procedure that goes under the name of strong disorder renormalization. We focus on the case $$\alpha =0$$ —the critical value $$h_c(\beta )$$ of the parameter h is exactly known (for every strength $$\beta $$ of the disorder) in this case—and we provide precise estimates on the critical behavior. Our results confirm the strong disorder renormalization group prediction that the transition is of infinite order, namely that when $$h\searrow h_c(\beta )$$ the free energy vanishes faster than any power of $$h-h_c(\beta )$$ . But we show that the free energy vanishes much faster than the physicists’ prediction.

A General Approximation to the Distribution of Count Data with Applications to Inventory Modeling

We derive a general approximation to the distribution of count data based on the first two moments of the underlying interarrival distribution. The result is a variant of the Birnbaum-Saunders (BISA) distribution. This distribution behaves like the lognormal in several respects; however, we show that the BISA can fit both simulated and empirical data better than the lognormal and that the BISA possesses additive properties that the lognormal does not. This results in computational advantages for operational models that involve summing random variables. Moreover, although the BISA can be fit to count data (as we demonstrate empirically), it can also be fit directly to transaction-level interarrival data. This provides a simple, practical way to sidestep distributional fitting problems that arise from count data that is censored by inventory stockouts. In numerical experiments involving dynamic inventory models, we compare the BISA distribution to other commonly used distributions and show how it leads to better managerial decisions.

Method of Obtaining Inter-Arrival Distribution between Two Consecutive Heart Attacks Using Palm Probability

An attempt has been made to obtain the inter-arrival times of two consecutive heart attacks by using the technique of Palm probability. The arrival rate of heart attacks is taken to be Poisson, which is assumed to vary individual to individual following gamma distribution leading to a dependent process. In this communication, a methodology has been used to estimate the consecutive failure time of heart attack with compound Poisson inputs weighted by gamma distribution based on hospital discharge data relating to heart failures.

Inter-arrival time distribution of passengers at service facilities in underground subway stations: A case study of the metropolitan city of Chengdu in China

Inter-arrival time distribution of passengers plays an important role in the capacity design of service facilities such as, fare gate, ticket vending machine, and passageways, in an underground subway station. An inaccurate inter-arrival time distribution likely causes traffic congestion or resource wastage at service facilities. In this study, to obtain accurate inter-arrival time distribution, we collected an inter-arrival time of passengers at existing service facilities in three underground subway stations of a metropolitan city, Chengdu, China. We fitted eight types of distributions, including Hyper-Erlang distribution (HErD), which is firstly introduced in the capacity design of service facilities in an underground subway station, to the observed data set based on maximum likelihood estimation. Results showed that the HErD works the best in terms of fitting quality and flexibility. We also fitted eight types of distribution to the observed inter-arrival time data at service facilities of seventy-seven underground subway stations of another metropolitan city—Shenzhen, China—to confirm our findings. Results also showed the HErD still performs the best. Simulation is also conducted to examine the effect of inter-arrival distributions on the performance of service facilities. To estimate future inter-arrival time based on HErD for the capacity design of service facilities to be constructed in the planning period, we developed a basic parameter estimation model according to two given design parameters namely, long-term peak-hour volume and peak-hour factor. However, the proposed model did not work well because the HErD has many free parameters to be estimated. Thus, we derived a method to reduce the number of free parameters, and then we proposed an improved parameter estimation model of HErD to describe future inter-arrival time distribution based on given long-term peak-hour volume and peak-hour factor.

Generalized Pareto processes and fund liquidity risk

Motivated by the modelling of liquidity risk in fund management in a dynamic setting, we propose and investigate a class of time series models with generalized Pareto marginals: the autoregressive generalized Pareto process (ARGP), a modified ARGP and a thresholded ARGP. These models are able to capture key data features apparent in fund liquidity data and reflect the underlying phenomena via easily interpreted, low-dimensional model parameters. We establish stationarity and ergodicity, provide a link to the class of shot-noise processes, and determine the associated interarrival distributions for exceedances. Moreover, we provide estimators for all relevant model parameters and establish consistency and asymptotic normality for all estimators (except the threshold parameter, which is to be estimated in advance). Finally, we illustrate our approach using real-world fund redemption data, and we discuss the goodness-of-fit of the estimated models.