Interarrival Time Distributions Research Articles

Approximation of single-class queueing networks with downtime-induced traffic variability

Queuing networks have been used with partial success for analytical modelling of manufacturing systems. In this paper, we consider a tandem system with high traffic variability caused by downtime events in the first queue. We propose improved approximation for departure variability in order to predict the waiting duration at the bottleneck queue located last in the line. We demonstrate that existing methods do not properly approximate such systems and provide some reasons and insights. Thus, a new decomposition method which employs the variability function principles is proposed. We differentiate between two components of the departure variability in multi-class systems: the ‘within-class effect’ – the variability caused by the class’ own inter-arrival and service time distributions – and the ‘between-class effect’ – the variability caused by interactions with other classes. Our analysis shows that the first effect can be approximated by existing multi-class decomposition methods, while the second effect requires a new development. Our proposed approximation for between-class effect is based on simulating a proper sub-system. The method enables modelling different policies of downtimes (e.g. FCFS, Priority). Numerical experiments show relative errors much smaller vs. existing procedures.

Power series expansions for the distribution and mean value function of a geometric process with Weibull interarrival times

AbstractThe geometric process is considered when the distribution of the first interarrival time is assumed to be Weibull. Its one‐dimensional probability distribution is derived as a power series expansion of the convolution of the Weibull distributions. Further, the mean value function is expanded into a power series using an integral equation. © 2014 Wiley Periodicals, Inc. Naval Research Logistics, 61: 599–603, 2014

An empirical study on TCP flow interarrival time distribution for normal and anomalous traffic

SummaryIn this paper, we study the effects of anomalies on the distribution of TCP flow interarrival time process. We show empirically that despite the variety of data networks in size, number of users, applications, and load, the interarrival times of normal flows comply with the Weibull distribution, whereas specific irregularities (anomalies) causes deviations from the distribution. We first estimate the scale and shape parameters and then check the discrepancy of the data from a Weibull distribution with the estimated parameters. We also utilize the Weibull counting model to recheck the conformance of small flow interarrival times with the distribution. We perform our experiments on a diverse variety of traffic data sets from backbone connections to endpoints of academic and commercial networks. Moreover, we propose a window‐based anomaly detection method as a possible application of our findings in which we first estimate the Weibull parameters of interarrival times in each window and then check the discrepancy of the data with a Weibull distribution with the estimated parameters and set an alarm whenever the difference is significant. We apply this method on one of our data sets and present the results to clarify the idea and show its capability in detecting volume anomalies. Copyright © 2014 John Wiley & Sons, Ltd.

G-RAND: A phase-type approximation for the nonstationary [formula omitted] queue

We present a Markov model to analyze the queueing behavior of the nonstationary G(t)/G(t)/s(t)+G(t) queue. We assume an exhaustive service discipline (where servers complete their current service before leaving) and use acyclic phase-type distributions to approximate the general interarrival, service, and abandonment time distributions. The time-varying performance measures of interest are: (1) the expected number of customers in queue, (2) the variance of the number of customers in queue, (3) the expected number of abandonments, and (4) the virtual waiting time distribution of a customer arriving at an arbitrary moment in time. We refer to our model as G-RAND since it analyzes a general queue using the randomization method. A computational experiment shows that our model allows the accurate analysis of small- to medium-sized problem instances.

Modeling and evaluating the AVS/RS with tier-to-tier vehicles using a semi-open queueing network

The Autonomous Vehicle Storage and Retrieval System (AVS/RS) with tier-to-tier vehicles is modeled as a semi-open queueing network (SOQN). Different storage/retrieval requests in the AVS/RS are modeled as different classes of customers in the SOQN model. Analyzing multiple configurations of an SOQN via computer simulation is time-consuming. Therefore, this article uses some analytical methods to evaluate the performances of a multi-class, multi-stage SOQN with general service time and interarrival time distributions. Two synchronization policies are also compared and the results show that an AVS/RS with virtual synchronization has a better performance than that with physical synchronization.

Interarrival Distribution of a Long-Range Dependent Workload Process

We derive the interarrival distribution of a workload input process which is a variation of the infinite source Poisson process for packet traffic. It accounts for long-range dependence and self-similarity exhibited by real traces in the Internet. The packet generation process is compound Poisson over each session which has a heavy tailed distribution. Considering the dependence induced by the workload, we derive the conditional distribution of the next interarrival time given that a packet has just arrived. This allows the use of the workload as general arrivals to a queueing system for further performance analysis

Fractional Skellam processes with applications to finance

Abstract The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.

Packet Inter-arrival Time Distribution in Academic Computer Network

The article presents the statistical analysis results of network packet inter-arrival time distribution in academic computer network. Most popular transport protocols TCP and UDP are addressed in the research. Data was gathered using NetFlow protocol. Network traffic was divided into sections according its direction and usage trends, then packet inter-arrival time distributions were found. Kolmogorov-Smirnov test was used to evaluate goodness-of-fit of packet inter-arrival time distributions and it was determined, that Pareto Second Kind distribution fits the majority of the experimental distributions. DOI: http://dx.doi.org/10.5755/j01.eee.20.3.6683

Measuring the variance of customer waiting time in service operations

Purpose – Waiting lines and delays have become commonplace in service operations. As a result, customer waiting time guarantee is a widely used competition strategy in service industries. To implement waiting time guarantee strategy, managers need to not only know the average of waiting time, but also the variance around average waiting time. This paper aims to discuss these issues. Design/methodology/approach – This research provides a mathematically exact expression for the coefficient of variation of waiting time for Markov queues. It then applies the concept of isomorphism to approximate the variance of customer waiting time in a general queue. Simulation experiments are conducted to verify the accurate approximations. Findings – A significant feature of the approximation method is that it is mathematically tractable and can be implemented in a spreadsheet format. It provides a practical way to estimate the variance of customer waiting time in practice. The results demonstrate the usefulness of the queuing models in providing guidance on implementing appointment scheduling and waiting time guarantee strategy. Also, the spreadsheet can be used to conduct what-if analysis by inputting different parameters. Originality/value – This paper develops a simple, easy-to-use spreadsheet model to estimate the standard deviation of waiting time. The approximation requires only the mean and standard deviation or the coefficient of variation of the inter-arrival and service time distributions, and the number of servers. A spreadsheet model is specifically designed to analyze the variance of waiting time.

Product-Form Solutions for Integrated Services Packet Networks and Cloud Computing Systems

We iteratively derive the product-form solutions of stationary distributions for a type of preemptive priority multiclass queueing networks with multiserver stations. This type of queueing systems can typically be used to model the stochastic dynamics of some large scale backbone networks with multiprocessor shared-memory switches or local (edge) cloud computing centers with parallel-server pools. The queueing networks are Markovian with exponential interarrival and service time distributions. The obtained iterative solutions can be used to conduct performance analysis or as comparison criteria for approximation and simulation studies. Numerical comparisons with existing Brownian approximating model (BAM) related to general interarrival and service times are provided to show the effectiveness of our current designed algorithm and our previous derived BAM. Furthermore, based on the iterative solutions, we can also give some analysis concerning network stability for some cases of these queueing systems, which provides some insight for more general study.

Delay Moment Bounds for Multiserver Queues with Infinite Variance Service Times

In this paper we consider First-In-First-Out (FIFO) queues. For such queues, upper bounds for stationary mean delay expressed in terms of the moments of the interarrival and service time distributions are particularly useful. For many years, most upper bounds for mean delay, , required service times to have finite second moment; it was not until Scheller-Wolf and Sigman (1997b) developed bounds using a new delay recursion that delay bounds were obtained when service times had infinite second moment. However, these bounds required that the system load, , where is the generic service time and is the generic interarrival time, satisfy . Since that paper first appeared, new recursions for delay have appeared in the literature which imply that improved bounds can be obtained. This paper updates and extends the moment bounds of Scheller-Wolf and Sigman (1997b) in response to these developments, to include all cases when . We show that, in general, bounds can be improved when there is a gap between the order of service time moments necessary to ensure finite mean delay (see Scheller-Wolf (2003)) and the actual service time moment that is finite in the system.

Human Mobility Patterns in Cellular Networks

This letter investigates inter-arrival time, dwell time distributions and other mobility patterns in mobile cellular networks. It has been generally assumed and widely accepted that both inter-arrival time and dwell time distributions can be well approximated by exponential distribution. However, based on real cellular data measurements, we evaluate the fitness of various typical statistical distributions such as power-law, exponential, Weibull, lognormal and Rayleigh distributions, and find that a power-law distribution fits both inter-arrival time and dwell time more precisely. Besides, mobility patterns in daytime, night, rural and urban areas provide further demonstrations of the power-law model. Moreover, new models on the distributions of inter-departure time and the number of arrived subscribers are also proposed to characterize other mobility patterns, and the corresponding simulation results are well consistent with the empirical ones.

On the exact distribution and mean value function of a geometric process with exponential interarrival times

The geometric process is considered when the distribution of the first interarrival time is assumed to be exponential. An analytical expression for the one dimensional probability distribution of this process is obtained as a solution to a system of recursive differential equations. A power series expansion is derived for the geometric renewal function by using an integral equation and evaluated in a computational perspective. Further, an extension is provided for the power series expansion of the geometric renewal function in the case of the Weibull distribution.

Rare event simulation of non-Markovian queueing networks using RESTART method

RESTART is an accelerated simulation technique that allows the probabilities of rare events to be evaluated. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event of interest is higher. These regions are defined by means of a function of the system state called the importance function. An appropriate choice of the importance function is crucial for the effective application of RESTART because, although the rare event estimator is unbiased for any importance function, the acceleration achieved is closely dependent on the selected function. Formulas for obtaining suitable importance functions to estimate overflow probabilities, previously provided for Jackson networks, are extended here to non-Markovian queueing networks. This extension is made by introducing an innovative concept, the effective load of a node, defined as the actual load of a node of a Jackson network which has a similar queue length distribution. The formulas are tested in four network topologies, ranging from a two-node network with strong feedback to a 15-node network with multiple feedbacks, with different interarrival and service time distributions. The paper shows how probabilities of rare events are accurately estimated in all the tested cases with short computational time. The large variety of cases simulated suggests that the proposed importance function may be suitable for many other queueing networks.

A Markov Model for Measuring Service Levels in Nonstationary G(t)/G(t)/s(t)+G(t) Queues

We present a Markov model to approximate the queueing behavior at the G(t)=G(t)=s(t) + G(t) queue with exhaustive discipline and abandonments. The performance measures of interest are: (1) the average number of customers in queue, (2) the variance of the number of customers in queue, (3) the average number of abandonments and (4) the virtual waiting time distribution of a customer when arriving at an arbitrary moment in time. We use acyclic phase-type distributions to approximate the general interarrival, service and abandonment time distributions. An ecient, iterative algorithm allows the accurate analysis of small- to medium-sized problem instances. The validity and accuracy of the model are assessed using a simulation study.

Beta/M/1 Model for Machine Type Communication

In context of Machine Type Communication (MTC) applications, network has to face increased load as well as possible surges of MTC traffic due to massive concurrent data and signaling transmission, in which case the interarrival time distribution of MTC devices may follow Beta distribution. In this paper, we model the network with MTC applications as a Beta/M/1 queue model. By deducing the moment generation function's analytical expression of Beta distributions with integral shape parameters, the full performance analysis of Beta/M/1 model is presented. The conclusions drawn from numerical analysis can be used as a preliminary guide to appropriately choose the model for different MTC application scenarios and promptly judge the effectiveness of the chosen model.

A tandem [formula omitted] queue with blocking

Tandem queues are widely used in mathematical modeling of random processes describing the operation of manufacturing systems, supply chains, computer and telecommunication networks. Although there exists a lot of publications on tandem queueing systems, analytical research on tandem queues with non-Markovian input is very limited. In this paper, the results of analytical investigation of two-node tandem queue with arbitrary distribution of inter-arrival times are presented. The first station of the tandem is represented by a single-server queue with infinite waiting room. After service at the first station, a customer proceeds to the second station that is described by a single-server queue without a buffer. Service times of a customer at the first and the second server have PH (Phase-type) distributions. A customer, who completes service at the first server and meets a busy second server, is forced to wait at the first server until the second server becomes available. During the waiting period, the first server becomes blocked, i.e., not available for service of customers. We calculate the joint stationary distribution of the system states at the embedded epochs and at arbitrary time. The Laplace–Stieltjes transform of the sojourn time distribution is derived. Key performance measures are calculated and numerical results presented.

Characterization theorem on losses in [formula omitted] queues

In this paper, we prove a characterization theorem on the number of losses during a busy period in GIX/GIY/1/n queueing systems, in which the interarrival time distribution belongs to the class NWUE.

Computing the Distribution Function of the Number of Renewals

The method of Laplace transforms is used to find the distribution function, mean, and variance of the number of renewals of a renewal process whose inter-arrival time distribution has a rational Laplace transform. Where the Laplace transform is not rational, we use the Pade approximation method. We apply our method to certain examples and the results are compared to those reported by other researchers.

On time to buffer saturation in a GI/M/1/N-type queue

A GI/M/1/N-type queueing system with independent and generally distributed interarrival times and exponential service times is investigated. A system of equations for conditional distributions of the time to the first buffer saturation is built. The solution is written using a special-type sequence defined by ``input'' distributions of the system. The formula of total probability is used to derive a representation for the distribution of the time to the $k$th buffer saturation for k>=2. Moreover, special cases of the Poisson arrival process and the system with one-place buffer are discussed. Sample numerical results for the 3-Erlang and deterministic distributions of interarrival times are attached as well.