Interdeparture Time Research Articles

Characterization of the departure process in a closed fork–join synchronization network

In this article we consider a closed fork–join synchronization Markovian network, where a queueing model consisting of two finite input buffers, B 1 and B 2, fed by arrivals from two finite populations of sizes K 1 and K 2 is investigated. The first population feeds the first buffer and the second population feeds the second buffer. As soon as there is one part in each buffer, two parts one from each buffer are joined and exit immediately. We provide model analysis and characterization of the departure process; in particular we provide the marginal distribution of inter-departure times.

Two-moment approximations for analysing a series of automated workstations

In this research we develop two-moment approximation equations for the analysis of a series of automated workstations with infinite buffer capacity. An automated workstation is a server with deterministic processing times that experiences random operating times between failures, and then subsequent random repair times. The distribution of job inter-departure times from a workstation receiving input from an automated workstation is derived. From this distribution, expressions for the mean and variance of job inter-departure times from a workstation are obtained. These are then used as a basis for ‘linking’ workstations in a two-workstation aggregation approach. This method is compared to modelling such a system as a series of G/G/1 queues. The analysis method is easily implemented in a spreadsheet. Extensive computational experiments demonstrate the effectiveness of the overall approach.

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.

Models of the departure process of a BMAP/MAP/1 queue

We propose a family of finite approximations for the departure process of a BMAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to M/G/1-type Markov processes. The proposed approximations are indexed by a parameter n ( n < 1), which determines the size of the output model as n + 1 block levels of the M/G/1-type process. This output approximation preserves exactly the marginal distribution of the true departure process and the lag correlations of the inter-departure times up to lag n -2. Experimental results support the applicability of the proposed approximation in traffic-based decomposition of queueing networks.

Characterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation

We propose a family of finite approximations for the departure process of a BMAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to M/G/1-type Markov processes. The proposed approximations are indexed by a parameter n(n > 1), which determines the size of the output model as n + 1 block levels of the M/G/1-type process. This output approximation preserves exactly the marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n − 2. Experimental results support the applicability of the proposed approximation in traffic-based decomposition of queueing networks.

Kernel Density Estimation for Interdeparture Time of GI/G/1 Queues

The departure process of a single queue has been studied since the 1960s. Due to its inherent complexity, closed form solutions for the distribution of the departure process are nearly intractable. In this study, kernel type estimators of the density of interdeparture time in a GI/G/1 queue are studied. Uniform strong consistency of the estimators in a GI/G/1 queue and their rates of convergence are obtained. The stochastic processes are shown to satisfy the strong mixing condition with random instants of sampling. With the analysis presented, we provide a novel analytic tool for studying the departure process in a general queueing model.

Performance evaluation of an open distributed platform for realistic traffic generation

Network researchers have dedicated a notable part of their efforts to the area of modeling traffic and to the implementation of efficient traffic generators. We feel that there is a strong demand for traffic generators capable to reproduce realistic traffic patterns according to theoretical models and at the same time with high performance. This work presents an open distributed platform for traffic generation that we called distributed internet traffic generator (D-ITG), capable of producing traffic (network, transport and application layer) at packet level and of accurately replicating appropriate stochastic processes for both inter departure time (IDT) and packet size (PS) random variables. We implemented two different versions of our distributed generator. In the first one, a log server is in charge of recording the information transmitted by senders and receivers and these communications are based either on TCP or UDP. In the other one, senders and receivers make use of the MPI library. In this work a complete performance comparison among the centralized version and the two distributed versions of D-ITG is presented.

Characterizing the idle time of a nonexponential server system

Understanding the behavior of an idle time of a limited resource is the key to increase productivity in service operations. When the system consists of nonexponential properties of time distributions it becomes difficult to provide results for the general case. We derive the MacLaurin series for the moments of the idle time with respect to the parameters in the service time and interarrival time distributions for a GI/G/1 queue. The light traffic derivatives are obtained to investigate the quality of a well-known MacLaurin series. The expected error bound under this approach is identified. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The result for the idle period is easily taken as input to the calculation of other performance measures of the system, e.g., cycle time or interdeparture time distributions. Numerical examples are given to illustrate these results.

QoS Scheduler/Shaper for Optical Coarse Packet Switching IP-Over-WDM Networks

For IP-over-WDM networks, optical coarse packet switching (OCPS) has been proposed to circumvent optical packet switching limitations by using in-band-controlled per-burst switching and advocating traffic control enforcement to achieve high bandwidth utilization and quality-of-service (QoS). In this paper, we first introduce the OCPS paradigm. Significantly, we present a QoS-enhanced traffic control scheme exerted during packet aggregation at ingress nodes, aiming at providing delay and loss class differentiations for OCPS networks. Serving a dual purpose, the scheme is called (/spl psi/,/spl tau/)-Scheduler/Shaper, where /spl psi/ and /spl tau/ are the maximum burst size and burst assembly time, respectively. To provide delay class differentiation, for IP packet flows designated with delay-associated weights, (/spl psi/,/spl tau/)-Scheduler performs packet scheduling and assembly into bursts based on their weights and a virtual window of size /spl psi/. The guaranteed delay bound for each delay class is quantified via the formal specification of a stepwise service curve. To provide loss class differentiation, (/spl psi/,/spl tau/)-Shaper facilitates traffic shaping with larger burst sizes assigned to higher loss priority classes. To examine the shaping effect on loss performance, we analytically derive the departure process of (/spl psi/,/spl tau/)-Shaper. The aggregate packet arrivals are modeled as a two-state Markov modulated Bernoulli process (MMBP) with batch arrivals. Analytical results delineate that (/spl psi/,/spl tau/)-Shaper yields substantial reduction, proportional to the burst size, in the coefficient of variation of the burst interdeparture time. Furthermore, we conduct extensive simulations on a 24-node ARPANET network to draw packet loss comparisons between OCPS and just-enough-time (JET)-based OBS. Simulation results demonstrate that, through burst size adjustment, (/spl psi/,/spl tau/)-Shaper effectively achieves differentiation of loss classes. Essentially, compared to JET-based OBS using out-of-band control and offset-time-based QoS strategy, OCPS is shown to achieve invariably superior packet loss probability for a high-priority class, facilitating better differentiation of loss traffic classes.

An Analytical Approach to Obtain the Interdeparture Time Characteristics in a Multistage VoIP Network

In this paper, we investigate the evolution of the interarrival and interdeparture times between voice packets when they are proceeding through a number of network nodes. We model the arrival process in a node as the superposition of a single tagged stream and an independent background process that aggregates the remaining traffic sources. Because we assume that the load of a single voice stream is very low compared to the load of the aggregate traffic, we can represent the tagged voice packets as markers (packets with size zero). We will establish an expression for the probability generating function (pgf) of the interdeparture time of the voice packets after one stage and use this interdeparture-time pgf as the pgf of the interarrival time of the voice packets in the next stage, in order to assess the evolution of the interarrival-time characteristics throughout the network.

Inter-Departure Times in Base-Stock Inventory-Queues

An inventory-queue is an inventory system controlled by a processing station with queueing. They are natural building blocks for supply chain models. An important and largely open issue for inventory queues is the characterization of their departure processes. In an inventory-queue, departures are triggered either by a job arrival when the output buffer is not empty or otherwise by a service completion. Such departures are more difficult to analyze than departures from a standard queue. The main results in this study are expressions for the probability distributions and squared coefficient of variations of inter-departure times for base-stock inventory-queues with birth–death production processes.

Filtering recursions for calculating likelihoods for queues based on inter-departure time data

We consider inference for queues based on inter-departure time data. Calculating the likelihood for such models is difficult, as the likelihood involves summing up over the (exponentially-large) space of realisations of the arrival process. We demonstrate how a likelihood recursion can be used to calculate this likelihood efficiently for the specific cases of M/G/1 and Er/G/1 queues. We compare the sampling properties of the mles to the sampling properties of estimators, based on indirect inference, which have previously been suggested for this problem.

Characterization of the departure process from anME/ME/1queue

In this paper we propose a family of finite approximations for the departure process of an ME/ME/1 queue indexed by a parameter k defined as the system size of the finite approximation. The approximations capture the interdeparture times from an ME/ME/1 queue exactly and preserve the lag correlations of inter-event times of the departures from an ME/ME/1 queue up to lag ( k - 1).

Analysis of a Fork/Join Synchronization Station with Inputs from Coxian Servers in a Closed Queuing Network

Fork/join stations are commonly used to model the synchronization constraints in queuing models of computer networks, fabrication/assembly systems and material control strategies for manufacturing systems. This paper presents an exact analysis of a fork/join station in a closed queuing network with inputs from servers with two-phase Coxian service distributions, which models a wide range of variability in the input processes. The underlying queue length and departure processes are analyzed to determine performance measures such as throughput, distributions of the queue length and inter-departure times from the fork/join station. The results show that, for certain parameter settings, variability in the arrival processes has a significant impact on system performance. The model is also used to study the sensitivity of performance measures such as throughput, mean queue lengths, and variability of inter-departure times for a wide range of input parameters and network populations.

Distributed Neural Networks for Quality of Service Estimation in Communication Networks

We study an original scheme based on distributed feedforward neural networks, aimed at modelling several queueing systems in cascade fed with bursty traffic. For each queueing system, a neural network is trained to anticipate the average number of waiting packets, the packet loss rate and the coefficient of variation of the packet inter-departure time, given the mean rate, the peak rate and the coefficient of variation of the packet inter-arrival time. The latter serves for the calculation of the coefficient of variation of the cell inter-arrival time of the aggregated traffic which is fed as input to the next neural network along the path. The potential of this method is successfully illustrated on several single server FIFO (First In, First Out) queues and on small queueing networks made up from a combination of queues in tandem and in parallel fed by a superposition of ideal sources. Our long-term goal is the design of preventive control strategy in a multiservice communication network.

The departure process of a quorum queueing system

We study in this paper the departure process of a bulk service queueing system. The server operates under a minimum batch size strategy. We characterize the departure process through the distribution of the interdeparture times of batches and of customers, the distribution of the number of customers in a batch, and the coefficient of correlation between the interdeparture time of a batch and the number of customers in the batch. A numerical illustration is presented.

Gradient Estimation for a Class of Systems with Bulk Services: A Problem in Public Transportation

This paper deals with a system where batch arrivals wait in a station until a server (a train) is available, at which moment it services all customers in waiting. This is an example of a bulk server, which has many applications in public transportation, telecommunications, computer resource allocation, and multiple access telecommuncation networks, among others. We consider a subway model and focus on a metro line serving a particular metro station. Denote the planned inter-departure time of this line by x. The metro station is served by several other lines and passengers change trains at the station. Traveling times of trains are assumed to be given by fixed times and an additional stochastic noise. We perform a sensitivity analysis of the total delay of passengers waiting for the x line with respect to x. We establish a smoothed perturbation analysis (SPA), a measure-valued differentiation (MVD), and a score function (SF) estimator. Numerical experiments are performed to compare the ensuing estimators. It turns out that the SPA and MVD estimators are intrinsically different and the model presented in this paper may serve as a counter-example to the widespread belief that SPA and MVD yield similar estimators.

Renewal approximations for the departure processes of batch systems

In this paper we consider three standard batch arrival/service systems and develop renewal process approximations for the inter-departure time squared coefficient of variations. The systems considered are: (i) batch arrival with individual service; (ii) batching for setup reduction for a single server system; and (iii) batch processing. The random version of the batch arrival with individual service case naturally results from random branching of individuals following a batch service process. This case is also developed with the intent of improving the modeling results for downstream workstations following a batch service process.

Two-Class Priority Queueing System with State-Dependent Arrivals

In this paper, we present a performance analysis of a 2-dimensional preemptive priority queueing system with state-dependent arrivals. Using a Markovian formulation we first compute the steady state distribution for the queue length of both classes. Then, waiting times and busy periods are characterized through (i) first and second moments and (ii) the approximation of their cumulative distribution functions (cdf) and Laplace–Stieltjes transforms (LST). We derive these approximations connecting bounds in the Laplace domain with bounds on the original time domain. We also, study the behavior of the inter-departure time for each class. Finally, we conclude the paper with a set of computational experiments testing our results.

Analysis of assembly systems for interdeparture time variability and throughput

This paper studies the effect of the number of component stations (parallelism), work transfer, processing time distributions, buffers and buffer allocation schemes on throughput and interdeparture time variability of assembly systems. As an alternative to work transfer, variability transfer is introduced and its effectiveness is assessed. Previous research has indicated that the optimal throughput displays an anomaly at certain processing time distributions and, this phenomenon is now thoroughly analyzed and the underlying details are uncovered. This study also yields several new findings that convey important practical implications.