Interarrival Time Distributions Research Articles

The Stability Analysis of a Double-X Queuing Network Occurring in the Banking Sector

We model a common teller–customer interaction occurring in the Ghanaian banking sector via a Double-X queuing network consisting of three single servers with infinite-capacity buffers. The servers are assumed to face independent general renewal of customers and independent identically distributed general service times, the inter-arrival and service time distributions being different for each server. Servers, when free, help serve customers waiting in the queues of other servers. By using the fluid limit approach, we find a sufficient stability condition for the system, which involves the arrival and service rates in the form of a set of inequalities. Finally, the model is validated using an illustrative example from a Ghanaian bank.

Performance Analysis of Packet Aggregation Mechanisms and Their Applications in Access (e.g., IoT, 4G/5G), Core, and Data Centre Networks.

The transmission of massive amounts of small packets generated by access networks through high-speed Internet core networks to other access networks or cloud computing data centres has introduced several challenges such as poor throughput, underutilisation of network resources, and higher energy consumption. Therefore, it is essential to develop strategies to deal with these challenges. One of them is to aggregate smaller packets into a larger payload packet, and these groups of aggregated packets will share the same header, hence increasing throughput, improved resource utilisation, and reduction in energy consumption. This paper presents a review of packet aggregation applications in access networks (e.g., IoT and 4G/5G mobile networks), optical core networks, and cloud computing data centre networks. Then we propose new analytical models based on diffusion approximation for the evaluation of the performance of packet aggregation mechanisms. We demonstrate the use of measured traffic from real networks to evaluate the performance of packet aggregation mechanisms analytically. The use of diffusion approximation allows us to consider time-dependent queueing models with general interarrival and service time distributions. Therefore these models are more general than others presented till now.

Application-Oriented Traffic Modeling of WiFi-Based Internet of Things Gateways

Many Internet of Things (IoT) devices generate relatively small-sized data and have limited energy supply. These two factors limit their ability to connect directly to cloud servers through a wireless backbone network without imposing a burden on this network in providing efficient data transfer. In this article, we consider an IoT network architecture where a number of different IoT devices send their data wirelessly to an IoT gateway (or a fog node) via a WiFi network. We focus on characterizing incoming traffic patterns to the gateway for three typical IoT applications with real-time and nonreal-time data transfer requirements, such as video surveillance, smart city, and e-healthcare. Our study is based on generating real IoT traffic traces in a lab environment from various sensors and devices for the aforementioned applications and employing these traces to emulate a network of IoT nodes connected to a gateway via WiFi. In the conducted experiments, different homogenous and nonhomogeneous traffic patterns of the selected applications are examined for synchronized and unsynchronized data sources. Based on our empirical data, the experimental results reveal that the packet interarrival time distribution at the gateway is close to generalized Pareto distribution for homogeneous eHealth and smart city traffic, whereas the Weibull distribution is the nearest to model the empirical packet interarrival time for the rest of the examined traffic patterns. Moreover, we show that employing the experimental findings to analyze the delay performance of connecting the gateway to the cloud, given certain backbone network resources, leads to accurate results.

M/PH/1 queueing model with re-servicing

In this paper, we analyze an M/PH/1 queuing system with optional re-service. In this queueing system the interarrival time distribution is exponential with mean service time and re-service time distribution follows continuous time phase-type distribution and each customer will be re-serviced with probability ‘p’. We have obtained the expressions of various system performance measures and maximum likelihood method using E-M algorithm and Bayesian procedure based on MCMC method were proposed to estimate system parameters. Numerical illustration of various special cases of the queueing system was carried out using simulated data sets.

Switching diffusion approximations for optimal power management in parallel processing systems

In this paper, we investigate optimal power management in parallel processing systems composed of one queue and several identical processing stations. Power consumption is controlled by setting some of the stations into an inactive state, where they consume less power but are unable to provide service. This way, we are faced with the conflicting objective of minimizing power consumption while maintaining a desired quality of service. A distinguishing feature here, regarding most previous literature on this subject, is that we consider systems operating a policy that may turn the reserve machines on or off with setup times and under general inter-arrival or service time distributions, subject to some conditions. When these condition fail, we also provide a model with general inter-arrival times and exponentially distributed service times. To some extent, a controlled switching diffusion obtained in this paper via heavy traffic analysis and stochastic optimal control theory are the technical underpinning of the paper. We also propose a numerical approach to the solutions of the optimal control problems based on the Markov chain approximation method. Finally, we consider some numerical experiments that illustrate the efficiency of the proposed approach.

First-detection time of a quantum state under random probing

We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the $n\mathrm{th}$ attempt. We calculate as well the mean and mean square, both of the number of the first successful detection attempt and the time until first detection. We present explicit results for a particle initially localized at a site on a ring of size $L$, probed at some arbitrary given site, in the case when the detection intervals are distributed exponentially. We prove that, for all interval distributions and finite-dimensional Hamiltonians, the mean detection time is equal to the mean attempt number times the mean time interval between attempts. We further prove that for the return problem when the initial and target state are identical, the total detection probability is unity and the mean attempts until detection is an integer, which is the size of the Hilbert space (symmetrized about the target state). We study an interpolation between the fixed time interval case to an exponential distribution of time intervals via the Gamma distribution with constant mean and varying width. The mean arrival time as a function of the mean interval changes qualitatively as we tune the interarrival time distribution from very narrow ($\ensuremath{\delta}$ peaked) to exponential, as resonances are wiped out by the randomness of the sampling.

Heavy traffic analysis for single-server SRPT and LRPT queues via EDF diffusion limits

Extending the results of Kruk (Queueing theory and network applications. QTNA 2019. Lecture notes in computer science, vol 11688. Springer, Cham, pp 263–275, 2019), we derive heavy traffic limit theorems for a single server, single customer class queue in which the server uses the Shortest Remaining Processing Time (SRPT) policy from heavy traffic limits for the corresponding Earliest Deadline First queueing systems. Our analysis allows for correlated customer inter-arrival and service times and heavy-tailed inter-arrival and service time distributions, as long as the corresponding stochastic primitive processes converge weakly to continuous limits under heavy traffic scaling. Our approach yields simple, concise justifications and new insights for SRPT heavy traffic limit theorems of Gromoll et al. (Stoch Syst 1(1):1–16, 2011). Corresponding results for the longest remaining processing time policy are also provided.

Analysis and Simulation of Intervention Strategies against Bus Bunching by means of an Empirical Agent-Based Model

In this paper, we propose an empirically based Monte Carlo bus-network (EMB) model as a test bed to simulate intervention strategies to overcome the inefficiencies of bus bunching. The EMB model is an agent-based model which utilizes the positional and temporal data of the buses obtained from the Global Positioning System (GPS) to constitute (1) a set of empirical velocity distributions of the buses and (2) a set of exponential distributions of interarrival time of passengers at the bus stops. Monte Carlo sampling is then performed on these two derived probability distributions to yield the stochastic dynamics of both the buses’ motion and passengers’ arrival. Our EMB model is generic and can be applied to any real-world bus network system. In particular, we have validated the model against the Nanyang Technological University’s Shuttle Bus System by demonstrating its accuracy in capturing the bunching dynamics of the shuttle buses. Furthermore, we have analyzed the efficacy of three intervention strategies: holding, no-boarding, and centralized-pulsing, against bus bunching by incorporating the rule set of these strategies into the model. Under the scenario where the buses have the same velocity, we found that all three strategies improve both the waiting and travelling times of the commuters. However, when the buses have different velocities, only the centralized-pulsing scheme consistently outperforms the control scenario where the buses periodically bunch together.

Statistical inference for doubly geometric process with Weibull interarrival times

In recent years, the doubly geometric process is started to use as a model to fit the data from a series of events since it provides a more flexible model for wider application than the geometric process. In the applications of the doubly geometric process, the estimation problem associated with the process can arise. In this study, we deal with this problem for the stochastic process model determined by doubly geometric process when the distribution of the first interarrival time is assumed to be Weibull. The model parameters of the doubly geometric process and the parameters of Weibull distribution are estimated by using the maximum likelihood method. The joint asymptotic distribution of the estimators is obtained. We investigate some statistical properties of the estimators such as asymptotic unbiasedness and consistency. The proposed estimators are evaluated through a computer simulation study with different parameter values and sample sizes. Two goodness-of-fitness tests are applied to show that the Weibull distribution may be used for the two real world data sets which can be modeled by doubly geometric process. Finally, we present the estimators of the model and distribution parameters for these data sets.

Power series expansions for the probability distribution, mean value and variance functions of a geometric process with gamma interarrival times

The geometric process is used widely as a stochastic monotone model in many areas since its introduction. In this study, this process is considered when the distribution of the first interarrival time follows a gamma distribution. One dimensional probability distribution of the process is obtained by expanding the convolution of gamma distributions into a power series. Further, its mean value, second moment and variance functions are derived as a power series expansion with the help of the integral equations given for mean value and second moment functions. The illustrative examples are also given. Finally, a real-world data set is considered to see the applicability of the results.

Multi-Hazard Portfolio Loss Estimation for Time-Dependent Shaking and Tsunami Hazards

Megathrust subduction earthquakes generate intense ground shaking and massive tsunami waves, posing major threat to coastal communities. The occurrence of such devastating seismic events is uncertain and depends on their recurrence characteristics (e.g. inter-arrival time distribution and parameters) as well as elapsed time since the last major event. Current standard probabilistic loss models for earthquakes and tsunamis are based on a time-independent Poisson process and uniform earthquake slip distribution. Thereby, considerations of more realistic time-dependent earthquake occurrence and heterogeneous earthquake slip distribution are necessary. This study presents an innovative computational framework for conducting a time-dependent multi-hazard loss estimation of a building portfolio subjected to megathrust subduction earthquakes and tsunamis. The earthquake occurrence is represented by a set of multiple renewal models, which are implemented using a logic-tree approach, whereas earthquake rupture characterization is based on stochastic source models with variable fault geometry and heterogeneous slip distribution. By integrating these hazard components with seismic and tsunami fragility functions, multi-hazard loss potential for a coastal community can be evaluated quantitatively by considering different possibilities of earthquake recurrence and rupture characteristics. To demonstrate the implementation of the developed time-dependent multi-hazard loss model, the Tohoku region of Japan is considered.

Approximations in Performance Analysis of a Controllable Queueing System with Heterogeneous Servers

The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of customers in the system reaches its minimal value. The Markov decision model and the policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, is of threshold type defined as a sequence of threshold levels which specifies the queue lengths for the usage of any slower server. To avoid time-consuming calculations for systems with a large number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this solution with the real values. We develop also the simple lower and upper bound methods based on approximation by an equivalent heterogeneous queueing system with a preemption to measure the mean number of customers in the system operating under the optimal policy. Finally, the simulation technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of inter-arrival and service time distributions.

Queue-Length, Waiting-Time and Service Batch Size Analysis for the Discrete-Time GI/D-MSP$^{\text {(a,b)}}/1/\infty $ Queueing System

This paper analyzes an infinite-buffer single-server bulk-service queueing system in which customers arrive according to a discrete-time renewal process. The customers are served under the discrete-time Markovian service process according to the general bulk-service rule. The matrix-geometric method is used to obtain the queue-length distribution at prearrival epoch. The queue-length distributions at other various time epochs are also derived based on prearrival epoch probabilities. A simple approach has been developed to compute the waiting-time distribution of an arriving customer. We also carried out closed-form analytical expression for the service batch size distribution of an arriving customer. Some numerical results are provided in the form of tables for a variety of interarrival-time distributions and model parameters to understand the system behaviour.

SET-VALUED PERFORMANCE APPROXIMATIONS FOR THE QUEUE GIVEN PARTIAL INFORMATION

In order to understand queueing performance given only partial information about the model, we propose determining intervals of likely values of performance measures given that limited information. We illustrate this approach for the mean steady-state waiting time in the $GI/GI/K$ queue. We start by specifying the first two moments of the interarrival-time and service-time distributions, and then consider additional information about these underlying distributions, in particular, a third moment and a Laplace transform value. As a theoretical basis, we apply extremal models yielding tight upper and lower bounds on the asymptotic decay rate of the steady-state waiting-time tail probability. We illustrate by constructing the theoretically justified intervals of values for the decay rate and the associated heuristically determined interval of values for the mean waiting times. Without extra information, the extremal models involve two-point distributions, which yield a wide range for the mean. Adding constraints on the third moment and a transform value produces three-point extremal distributions, which significantly reduce the range, producing practical levels of accuracy.

Impact of dependence on single-server queueing systems

This study uses advanced simulation input modeling – the Vector-Auto-Regressi-ve-to-Anything (VARTA) method – to study the impact of bivariate and temporal dependencies among interarrival and service times on the performance of single-server queues. Our initial experiments, with the M/M/1 queue, show that there is nonmonotonic behavior of average waiting time with respect to negative autocorrelation in interarrival and/or service times at high utilization levels; such nonmonotonic behavior with negative autocorrelation in service times is well-known in literature, we are first to show its existence for interarrival times. Our use of VARTA allows us to extend our simulation approach to study dependence among interarrival and service times in nonexponential distributions, enabling us to compare their effects to the M/M/1. We find that the impact of dependence on the performance under nonexponential distributions of interarrival and service times is primarily determined by the second moment of the distribution. Greater (lower) variance of the nonexponential distribution increases (decreases) the average waiting time.

Monitoring Regularly Maintained Systems Based on the Renewal Process with Generalized Exponential Distribution of Time between Events

Abstract In the modern industrial age, regular system maintenance is an integral process because systems, both engineering and nonengineering, deteriorate over time. Statistical process monitoring techniques, especially control charts, are very helpful in monitoring the performance of such systems and consequently help in decision making on whether maintenance is advantageous or the system has reached a phase where no further maintenance is useful. The aim of this article is to introduce an approach that would integrate maintenance decisions with statistical quality monitoring techniques by assuming an exponentiated class of distributions of interarrival times of the renewal process. This article includes generalized exponential, generalized Rayleigh, and exponentiated Pareto distributions as special cases. An extensive simulation study to compute average run length and average length of inspection (or average time to signal) is conducted, and a comparison is given with approximated exponential control charts. The effect of cost for the regularly maintained generalized exponential systems is also discussed in this study. We focus on the generalized exponential distribution because of its practical relevance in reliability and closeness to Weibull distribution. Finally, an illustrative example is also a part of this article.

Extended analysis and computationally efficient results for the queueing system

We consider an infinite waiting space queueing system in which customers arrive according to a renewal process. The server serves the customers in batches under general bulk-service rule. The successive service times of batches are mutually independent and have a common exponential distribution. To obtain the queue-length distributions at prearrival and random epochs, we have used the displacement operator which assists to solve simultaneous non-homogeneous difference equations. An analytically simple and computationally efficient procedures have developed to compute the waiting-time distribution of an arrival customer. Our formula to determine the waiting-time distribution is also useful even when multiple poles occur in the Laplace-Stieltjes transform of the interarrival time distribution. We have determined closed-form analytical expression for the distribution of size of a service batch of an arrived customer. We also have derived the results of some particular well-known queueing models from our model. Some numerical results are provided in the form of tables for a variety of interarrival-time distributions to demonstrate the variation of performance measures of the system.

Probabilistic Space- and Time-Interaction Modeling of Mainshock Earthquake Rupture Occurrence

ABSTRACTThis article presents a probabilistic formulation for modeling earthquake rupture processes of mainshocks. A correlated multivariate Bernoulli distribution is used to model rupture occurrence. The model captures time interaction through the use of Brownian passage-time distributions to assess rupture interarrival in multiple sections of the fault, and it also considers spatial interaction through the use of spatial correlograms. The correlograms represents the effect of rupture nucleation and propagation. This model is proposed as an attractive alternative to existing probabilistic models because it (1) incorporates time and space interactions of mainshocks, (2) preserves the marginal distributions of interarrival times after including spatial rupture interactions, that is, model consistency, and (3) has an implicit physical interpretation aligned with rupture behavior observations. The proposed model is applied to assess the occurrence of large interface earthquakes in the subduction fault along the coast of Lima, Peru. The model matches well both the annual magnitude exceedance rates and the average seismic moment release in the tectonic region. The Akaike information criterion (AIC) test confirms that our model performs statistically better than models that do not capture earthquake space interactions. AIC also shows that the spherical correlogram outperforms the exponential correlogram at reproducing earthquake data. Finally, time-dependent seismic hazard in the region is calculated, and the results demonstrate that by accounting for recent earthquake occurrences, the inclusion of time-dependent effects can reduce the 30 yr seismic hazard by a factor of 4.

Real Smart Home Data-Assisted Statistical Traffic Modeling for the Internet of Things

The majority of practical studies and analyses in the context of the Internet of Things (IoT) have been carried out assuming that data packet generation follows theoretical models (typically a Poisson process with exponentially distributed packet interarrival times) without previous experimental validation and supporting evidence. In contrast to this approach, this article proposes a novel experimental and mathematical framework to determine statistical models for IoT data traffic. Based on empirical data generated by common smart home devices (e.g., ambient temperature, luminous intensity, atmospheric pressure, and motion sensors) recorded over a full year using an experimental IoT subsystem, this article first shows that real IoT traffic does not follow the Poisson process model conventionally assumed in the literature, but rather depends on the type of application. Consequently, we estimate the empirical statistical distribution of the interarrival between data packets for several smart home applications. The empirical distribution of the packet interarrival times is fitted with some well-established classical statistical distributions using the method of moments as well as maximum-likelihood estimation techniques, and the goodness of fit is quantified using the Kolmogorov–Smirnov (KS) test. Moreover, we also carry out a regression analysis to provide mathematical relations between the distribution parameters and the considered physical input parameters (ambient temperature, luminous intensity, and atmospheric pressure), which is particularly useful in practical scenarios. Furthermore, an exhaustive analysis of the variation of parameters over different time scales and the autocorrelation characteristics of the data packet generation are included as well. In summary, this article provides accurate traffic models suitable for real-life IoT scenarios that can be used for an adequate design and optimization of future communication networks to efficiently support IoT services.

Variation across Scales: Measurement Fidelity under Twitter Data Sampling

A comprehensive understanding of data quality is the cornerstone of measurement studies in social media research. This paper presents in-depth measurements on the effects of Twitter data sampling across different timescales and different subjects (entities, networks, and cascades). By constructing complete tweet streams, we show that Twitter rate limit message is an accurate indicator for the volume of missing tweets. Sampling also differs significantly across timescales. While the hourly sampling rate is influenced by the diurnal rhythm in different time zones, the millisecond level sampling is heavily affected by the implementation choices. For Twitter entities such as users, we find the Bernoulli process with a uniform rate approximates the empirical distributions well. It also allows us to estimate the true ranking with the observed sample data. For networks on Twitter, their structures are altered significantly and some components are more likely to be preserved. For retweet cascades, we observe changes in distributions of tweet inter-arrival time and user influence, which will affect models that rely on these features. This work calls attention to noises and potential biases in social data, and provides a few tools to measure Twitter sampling effects.