Interarrival Time Distributions Research Articles

Polling models with renewal arrivals: A new method to derive heavy-traffic asymptotics

We consider asymmetric cyclic polling systems with an arbitrary number of queues, general service-time distributions, zero switch-over times, gated service at each queue, and with general renewal arrival processes at each of the queues. For this classical model, we propose a new method to derive closed-form expressions for the expected delay at each of the queues when the load tends to 1, under proper heavy-traffic (HT) scalings. In the literature on polling models, rigorous proofs of HT limits have only been obtained for polling models with Poisson-type arrival processes, whereas for renewal arrivals HT limits are based on conjectures [E.G. Coffman, A.A. Puhalskii, M.I. Reiman, Polling systems with zero switch-over times: A heavy-traffic principle, Ann. Appl. Probab. 5 (1995) 681–719; E.G. Coffman, A.A. Puhalskii, M.I. Reiman, Polling systems in heavy-traffic: A Bessel process limit, Math. Oper. Res. 23 (1998) 257–304; T.L. Olsen, R.D. van der Mei, Periodic polling systems in heavy-traffic: Renewal arrivals, OR Lett. 33 (2005) 17–25]. Therefore, the main contribution of this paper lies in the fact that we propose a new method to rigorously prove HT limits for a class of non-Poisson-type arrivals. The results are remarkably simple and provide new fundamental insight and reveal explicitly how the expected delay at each of the queues depends on the system parameters, and in particular on the interarrival-time distributions at each of the queues. The results also suggest simple approximations for the expected delay in stable polling systems. Numerical results show that the approximations are highly accurate when the system load is roughly 90% or more.

Lattice Path Approach for Busy Period Density ofGIb/G/1 Queues UsingC2Coxian Distributions

In this paper busy period analysis of non-Markovian queuing system GIb/G/1, starting initially with i0 batches of customers, is carried out via lattice path approach. Both interarrival and service time distributions are approximated by 2-phase Cox distributions, C2, that have Markovian property, amenable to the application of lattice paths combinatorial analysis. Arrivals occur in batches of size b. Distributions having rational Laplace-Stieltjes transform and square coefficient of variation lying in [1/2, ∞] form a very wide class of distributions. As any distribution of this class can be approximated by a C2, the use of C2, therefore, has led us to achieve results applicable to almost any real life queuing system GIb/G/1 occurring in computer systems, communication systems, manufacturing systems, etc. Numerical computations have been performed for different sets of values of the parameters involved using software Mathematica and presented graphically.

Bayesian prediction of the transient behaviour and busy period in short- and long-tailed [formula omitted] queueing systems

Bayesian inference for the transient behaviour and duration of a busy period in a single server queueing system with general, unknown distributions for the interarrival and service times is investigated. Both the interarrival and service time distributions are approximated using the dense family of Coxian distributions. A suitable reparameterization allows the definition of a non-informative prior, and Bayesian inference is then undertaken using reversible jump Markov chain Monte Carlo methods. An advantage of the proposed procedure is that heavy-tailed interarrival and service time distributions such as the Pareto can be well approximated. The proposed procedure for estimating the system measures is based on recent theoretical results for the Coxian/Coxian/1 system. A numerical technique is developed for every MCMC iteration so that the transient queue length and waiting time distributions and the duration of a busy period can be estimated. The approach is illustrated with both simulated and real data.

Estimation of the mean waiting time of a customer subject to balking: a simulation study

To a customer, the waiting time for order processing for a product or service is important information for order placement. If the time foreseen for order fulfillment is long, the order might be lost to a competitor. In particular, modern principles of supply chain management highly suggest information sharing between entities in the chain and information technology has enabled customers to conveniently consider the waiting time for a potential balking decision. To help determine the design and operation of a manufacturing or service system in which a customer may balk based on the foreseen waiting time, this paper develops procedures to estimate the average waiting time of an order. Either the procedures allow the maximum waiting time for a balking decision to be random or do not require knowledge of the arrival process of customers before balking if the balking limit is known. For generality of the model, this paper considers general inter-arrival and service time distributions, and uses the simulation and regression approach.

Bayesian control of the number of servers in a [formula omitted] queueing system

In this paper we consider the problem of designing a GI / M / c queueing system. Given arrival and service data, our objective is to choose the optimal number of servers so as to minimize an expected cost function which depends on quantities, such as the number of customers in the queue. A semiparametric approach based on Erlang mixture distributions is used to model the general interarrival time distribution. Given the sample data, Bayesian Markov Chain Monte Carlo methods are used to estimate the system parameters and the predictive distributions of the usual performance measures. We can then use these estimates to minimize the steady-state expected total cost rate as a function of the control parameter c. We provide a numerical example based on real data obtained from a bank in Madrid.

A recursive method for the F-policy G/M/1/K queueing system with an exponential startup time

This paper deals with the optimal control of a finite capacity G/M/1 queueing system combined the F -policy and an exponential startup time before start allowing customers in the system. The F -policy queueing problem investigates the most common issue of controlling arrival to a queueing system. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distribution of the number of customers in the system. We illustrate a recursive method by presenting three simple examples for exponential, 3-stage Erlang, and deterministic interarrival time distributions, respectively. A cost model is developed to determine the optimal management F -policy at minimum cost. We use an efficient Maple computer program to determine the optimal operating F -policy and some system performance measures. Sensitivity analysis is also studied.

Analysis and modeling of job arrivals in a production grid

In this paper we present an initial analysis of job arrivals in a production data-intensive Grid and investigate several traffic models for the interarrival time processes. Our analysis focuses on the heavy-tail behavior and autocorrelations, and the modeling is carried out at three different levels: Grid, Virtual Organization (VO) , and region . A set of m-state Markov modulated Poisson processes (MMPP) is investigated, while Poisson processes and hyperexponential renewal processes are evaluated for comparison studies. We apply the transportation distance metric from dynamical systems theory to further characterize the differences between the data trace and the simulated time series, and estimate errors by bootstrapping . The experimental results show that MMPPs with a certain number of states are successful to a certain extent in simulating the job traffic at different levels, fitting both the interarrival time distribution and the autocorrelation function. However, MMPPs are not able to match the autocorrelations for certain VOs, in which strong deterministic semi-periodic patterns are observed. These patterns are further characterized using different representations. Future work is needed to model both deterministic and stochastic components in order to better capture the correlation structure in the series.

Estimation of the mean waiting time of a customer subject to balking: a simulation study

To a customer, the waiting time for order processing for a product or service is important information for order placement. If the time foreseen for order fulfillment is long, the order might be lost to a competitor. In particular, modern principles of supply chain management highly suggest information sharing between entities in the chain and information technology has enabled customers to conveniently consider the waiting time for a potential balking decision. To help determine the design and operation of a manufacturing or service system in which a customer may balk based on the foreseen waiting time, this paper develops procedures to estimate the average waiting time of an order. Either the procedures allow the maximum waiting time for a balking decision to be random or do not require knowledge of the arrival process of customers before balking if the balking limit is known. For generality of the model, this paper considers general inter-arrival and service time distributions, and uses the simulation and regression approach.

Value of capacity pooling in supply chains with heterogeneous customers

We investigate the value of pooling capacity in supply chains that serve product demands of different variabilities. We build and analyze models that integrate production queuing models with base stock inventory systems serving demands with different inter-arrival time distributions. The first model combines hyperexponential and exponential demand inter-arrival time distributions. Exact analysis of the model allows us to develop insights into the impact of the difference in demand variabilities on the value of pooling capacity. Simulation experiments allow us to validate these insights for more general settings. We then find one special case that combines exponential and deterministic demand arrivals with deterministic service, where pooling capacity results in increasing the total cost.

Approximated Headway Distributions of Free-Flowing Traffic on Ohio Freeways for Work Zone Traffic Simulations

Headway or interarrival time (IAT) data of successive vehicles in free-flowing traffic ahead of work zones for each lane on six different Ohio freeways (two with two lanes in one direction, three with three lanes, and one with four lanes) for 3 days were measured and analyzed. Approximated IAT distributions were generated as a function of the traffic volume for each lane, and relationships between traffic volumes and approximate cumulative IAT distributions were established; this allowed a direct conversion from hourly traffic counts to corresponding cumulative IAT distributions. On the basis of the validation of these IAT distributions with IAT data collected in an independent study, this conversion method produces fairly accurate cumulative IAT distributions for selected hourly traffic volumes. A set of cumulative IAT distribution spreadsheets may be downloaded at webce.ent.ohiou.edu/orite/cumulativeIATdistributions.html. It was also found that the same approximated cumulative IAT distribution can be used to model and simulate free-flowing traffic at other freeway locations in Ohio. It is not known whether these universal approximated IAT distributions would apply for freeways in other states.

Control charts for the queue length in a G/G/S system

Statistical-process-control-based monitoring of performance measures of queuing systems has so far eluded mainstream practices within the quality engineering discipline. There are two reasons for this. First, most measures have highly skewed distributions, and, secondly, no general theoretical formulae exist that may provide a platform to calculate control limits for the general G/G/S queue. In this paper, Shewhart-like general control charts for attributes, recently introduced, are combined with existent highly accurate approximations for the steady-state probabilities of the G/G/S queue, to develop control charts for the queue size in a G/G/S system. The effectiveness of this approach is discussed and demonstrated by applying the proposed control charts to a sample of queues with various inter-arrival time and service time distributions.

A preliminary performance comparison of five machine learning algorithms for practical IP traffic flow classification

The identification of network applications through observation of associated packet traffic flows is vital to the areas of network management and surveillance. Currently popular methods such as port number and payload-based identification exhibit a number of shortfalls. An alternative is to use machine learning (ML) techniques and identify network applications based on per-flow statistics, derived from payload-independent features such as packet length and inter-arrival time distributions. The performance impact of feature set reduction, using Consistency-based and Correlation-based feature selection, is demonstrated on Naïve Bayes, C4.5, Bayesian Network and Naïve Bayes Tree algorithms. We then show that it is useful to differentiate algorithms based on computational performance rather than classification accuracy alone, as although classification accuracy between the algorithms is similar, computational performance can differ significantly.

Control Policies for Inventory with Service Time

This article introduces an additional control policy—the N-policy–into (s, S) inventory system with positive service time. Under specified interarrival and service time distributions, which are independent of each other, we obtain the necessary and sufficient condition for the system to be stable. We also obtain the optimal values of the control variables s, S, and N; it is seen that the cost function attains the minimum value at s = 0. It is also shown that the cost function is separately convex in the variables S and N. Numerical illustrations are provided. Several measures of performance of the system are evaluated.

Nonparametric Estimation of the Renewal Function by Empirical Data

We consider an estimate of the renewal function (rf) H(t) using a limited number of independent observations of the interarrival times for an unknown interarrival-time distribution (itd). The nonparametric estimate is derived from the rf-representation as a series of distribution functions (dfs) of consecutive arrival times using a finite summation and approximations of the latter by empirical dfs. Due to the limited number of observed interarrival times, the estimate is accurate just for closed time intervals [0, t]. An important aspect is given by the selection of an optimal number of terms k of the finite sum. Here two methods are proposed: (1) an a priori choice of k as function of the sample size l which provides almost surely (a.s.) the uniform convergence of the estimate to the rf for light- and heavy-tailed itds if the time interval is not too large, and (2) a data-dependent selection of k by a bootstrap method. To evaluate both the efficiency of the estimate and the selection methods of k, a Monte Carlo study is performed.

Estimating the Distribution of a Renewal Process from Times at which Events from an Independent Process Are Detected

The analysis of length-biased data has been mostly limited to the interarrival interval of a renewal process covering a specific time point. Motivated by a surveillance problem, we consider a more general situation where this time point is random and related to a specific event, for example, status change or onset of a disease. We also consider the problem when additional information is available on whether the event intervals (interarrival intervals covering the random event) end within or after a random time period (which we call a window period) following the random event. Under the assumptions that the occurrence rate of the random event is low and the renewal process is independent of the random event, we provide formulae for the estimation of the distribution of interarrival times based on the observed event intervals. Procedures for testing the required assumptions are also furnished. We apply our results to human immunodeficiency virus (HIV) test data from public test sites in Seattle, Washington, where the random event is HIV infection and the window period is from the onset of HIV infection to the time at which a less sensitive HIV test becomes positive. Results show that the estimator of the intertest interval length distribution from event intervals ending within the window period is less biased than the estimator from all event intervals; the latter estimator is affected by right truncation. Finally, we discuss possible applications to estimating HIV incidence and analyzing length-biased samples with right or left truncated data.

A stochastic model of raindrop release: Application to the simulation of point rain observations

A rainfall simulator has been developed based on simple assumptions on the release of raindrops from cloud bases and on a previously found model of raindrop interarrival time and size distributions at ground level. The simulator is designed so that the main characteristics of rain observed on the ground are preserved: size distribution, interarrival time distribution and autocovariance of drop sizes. Results show that derived characteristics of rain, such as rain rate, are in good agreement with observations.

Understanding interarrival and interdeparture time statistics from interactions in queuing systems

We discuss the statistics of long queues, in which the interdeparture time statistics is dominated by spatial interactions among the elements in a queue rather than the arrival or exit processes. Based on a Fokker–Planck approach, it is possible to calculate the stationary distance distribution among the elements in a queue as a function of their interaction potential. The results relate to the ones known from Random Matrix Theory. Together with the velocity distribution, one can determine the time-gap distribution as well. This yields an analytical approach to the interdeparture and interarrival time distributions of queuing systems with spatially interacting elements. While these distributions are usually determined from empirical data or from theoretical assumptions about the arrival or exit process, we offer here an alternative interpretation of interdeparture time distributions as an effect of interactions in a queue. This is relevant for the understanding of traffic and production systems and for the optimization of the statistical behavior of some queuing systems.

The Investigation of Self-similar Traffic for OBS Assembly

This paper investigates the traffic properties before and after assembly in edge node of Ethernet over optical burst switching (OBS) network for the first time. The features of the edge access system are that (i) Forty Ethernet sources aggregate into optical burst edge node. Moreover they all generate self-similar traffic and have the same parameters. (ii) The packet size of sources follows Pareto distribution with the range from 64 to 1518 bytes. (iii) Packets are assembled into bursts according to four classes and fifteen output ports. Burst and interarrival time distributions are simulated under time-based and length-based assemble schemes. Self-similar traffic Hurst parameter is compared through R/S and V/T plot. Finally three self-similar traffic generating methods are given. Simulation results demonstrate that, multi-source traffic increases self-similar degree, however after assembly, time-assembly scheme can decrease self-similar degree, and aggregated burst size is close to Gaussian distribution. Length-assembly method has no effect on the self-similarity of input traffic. RMD is fit for study of burst network with large self-similarity.

Analysis of GI/M/s/c queues using uniformisation

We combine uniformisation, a powerful numerical technique for the analysis of continuous time Markov chains, with the Markov chain embedding technique to analyze GI/M/s/c queues. The main steps of the proposed approach are the computation of (1) the mixed-Poisson probabilities associated to the number of arrival epochs in the uniformising Poisson process between consecutive customer arrivals to the system; and (2) the conditional embedded uniformised transition probabilities of the number of customers in the queueing system immediately before customer arrivals to the system. To show the performance of the approach, we analyze queues with Pareto interarrival times using a stable recursion for the associated mixed-Poisson probabilities whose computation time is linear in the number of computed coefficients. The results for queues with Pareto interarrival times are compared with those obtained for queues with other interarrival time distributions, including exponential, Erlang, uniform and deterministic interarrival times. The obtained results show that much higher loss probabilities and mean waiting times in queue may be obtained for queues with Pareto interarrival times than for queues with the other mentioned interarrival time distributions, specially for small traffic intensities.

Time-Domain Mathematical Model of Impulsive Em Noises Emitted from Discharges

Impulsive electromagnetic (EM) noises emitted from discharges have been measured and characterized in ultra-wideband VHF channel. Statistical characteristics of EM pulses emitted from single discharge source are investigated as fundamental study. Firstly, the reproducibility of impulsive EM noises is confirmed by calculating their auto-correlations. The frequency spectrums of such pulses are found to be unique for each source. Next, statistical analysis is performed on the amplitude and inter-arrival time of received EM pulses. It is found that the probability distributions of amplitude and inter-arrival time of EM pulses follow normal and exponential distributions, respectively. Taking into account these ultra-wideband characteristics of impulsive EM noises from discharges, a mathematical model of EM pulses emitted from discharges is developed in the time domain. This is represented as a simple analytic formula with three parameters, which are derived from measurements. This model has been validated via measurement where multiple discharge sources exist.