Interdeparture Time Distribution Research Articles

Study of a two types of general heterogeneous service queueing system in a single server with optional repeated service and feedback queue

This paper addresses a model on a single server queue and two service representatives. After a customer is served, he/she has the three options: opting for receive the same service again (re-service), joining as a new customer for another regular service (feedback), or leaving the service system altogether. To ensure the queueing system is Markovian, we introduce an additional variable (supplementary variable) and using this approach, we derive the explicit distribution of queue size at random and departure epochs. Additionally, we determine the distribution of response time, inter-departure time, and busy period. By using the embedded Markov chain technique we have also derived the queue size distribution at departure epoch. We have also presented the cost analysis of the model with some numerical examples. The numerical illustration validates our findings and provides valuable insights into the queuing system.

Marginal and joint distribution of inter-departure times for a PH/PH/c queue

In this paper, we introduce the Laplace–Stieltjes transform (LST) of the inter-departure time distribution in a PH/PH/c queue, and the two-dimensional joint LST of two consecutive inter-departure times to construct their correlation structure. We exploit the properties of phase-type (PH) random variables, as well as the steady-state distribution of the underlying continuous-time Markov chain in a PH/PH/c queue to construct these LSTs. We demonstrate our approach through numerical examples, while validating the results. Later, we analyze the correlation between two consecutive inter-departure times for various PH/PH/c queues. We observe that, if the fundamental elements of the queue have high (low) variability, then the correlation is positive (negative).

Reducing the variability of inter-departure times of a single-server queueing system – The effects of skewness

A critical performance measure in serial production lines is the variability of inter-departure times of a single-server queue. Increasing upstream inter-departure time variability generates greater downstream variability, diminishing overall line performance. Theory suggests that the variability of inter-departure times of a single-server queue is reduced by decreasing the variance of inter-arrival and service times. This study investigates the effects of the skewness of inter-arrival and service time distributions on the variability of inter-departure times. Contrary to previous results suggesting that mean waiting times of a GI/G/1 queue can be reduced by increasing inter-arrival time skewness, this experimental study of a GI/G/1 queue with triangular inter-arrival and service times shows that the inter-departure time coefficient of variation is reduced through a combination of negative inter-arrival time skewness and positive service time skewness. These results also suggest that the absolute value of the negative autocorrelation between consecutive departures is reduced by the same combination of negative inter-arrival time skewness and positive service time skewness for low values of server’s utilization, while positive skewness for both inter-arrival and service times reduces this value for high values of server’s utilization. Finally, it was found that queue capacity constraints increase the coefficient of variation of inter-departure times, as has been previously suggested, as well as the skewness and the absolute correlation values of the inter-departure time distribution.

On the Inter-Departure Times in <inline-formula> <tex-math notation="LaTeX">$M/\widetilde{D}/1/{B}_{on}$ </tex-math> </inline-formula> Queue With Queue-Length Dependent Service and Deterministic/Exponential Vacations

We derive the distribution of inter-departure times of a finite-buffer single-server queue with Poisson arrival process and queue-length-dependent service times, where the server goes to vacation if either the queue is emptied or a limited number $(R_{1})$ of packets are served, whichever occurs first, in the current busy period. We consider two types of vacation distributions: 1) deterministic and 2) exponential. Queue-length distribution at embedded points is derived first, then, the distribution and variance of inter-departure times are derived, for both types of vacations. The simulation results are in good agreement with the derived analytical results. The above framework would be useful at the receiver in modeling and analyzing the jitter and the waiting time of time-division multiplexing (TDM) emulated packets in TDM over packet-switched network (TDM over PSN) technology as a function of a buffer size.

Stationary distributions of the R[X]/R/1 cross-correlated queue

ABSTRACTWe consider an infinite buffer single server queue wherein batch interarrival and service times are correlated having a bivariate mixture of rational (R) distributions, where R denotes the class of distributions with rational Laplace–Stieltjes transform (LST), i.e., ratio of a polynomial of degree at most n to a polynomial of degree n. The LST of actual waiting time distribution has been obtained using Wiener–Hopf factorization of the characteristic equation. The virtual waiting time, idle period (actual and virtual) distributions, as well as inter-departure time distribution between two successive customers have been presented. We derive an approximate stationary queue-length distribution at different time epochs using the Markovian assumption of the service time distribution. We also derive the exact steady-state queue-length distribution at an arbitrary epoch using distributional form of Little’s law. Finally, some numerical results have been presented in the form of tables and figures.

On the rate-jitter performance of jitter-buffer in TDMoPSN: study using queueing models with a state-dependent service

Time-division multiplexing over packet-switched network (TDMoPSN) is an intermediate phase of transition from current synchronous TDM to future all-optical converged network. TDMoPSN is exclusively used to transport interactive voice traffic transparently over a PSN (e.g., IP, MPLS or Ethernet). The goal of this paper is to reduce rate-jitter that is introduced into a stream of packets carrying TDM payload. We have proposed two online algorithms, algorithm-A and algorithm-B, to reduce the rate-jitter and shown analytically that the rate-jitter achieved by algorithm-A is strictly less than the rate-jitter of online algorithm proposed by Mansour et al. [1]. We have used three stochastic processes, namely Poisson, Markov-modulated Poisson process (MMPP) and an arrival process with Pareto-distributed inter-arrival times (see [2---4]) for modeling the arrival of TDM packets (say, IP packets with single or many TDM frames as payloads) at the destination. We undertook statistical analysis of the proposed algorithms by modeling the jitter-buffer as $$M/\widetilde{D}/1/\, B_{on}$$M/D~/1/Bon and $$MMPP/\widetilde{D}/1/B_{on}$$MMPP/D~/1/Bon queues, to derive steady-state queue-length distribution, mean waiting time and distribution of inter-departure times. We also simulate the most realistic queueing model $$Pareto/\widetilde{D}/1/B_{on}$$Pareto/D~/1/Bon of our study and evaluated its performance with respect to the metrics: rate-jitter, mean waiting time, packet loss probability and steady-state queue-length distribution. Simulation results show that our proposed algorithms far outperforms the scheme proposed in [1]. We also present simulation results to verify the correctness of analytical queueing models. The algorithms proposed here are more general (for TDMoPSNs) and can be used to study TDMoIP, pseudowire, CES over metro Ethernet network (MEN), etc.

Extracting Message Inter-Departure Time Distributions from the Human Electroencephalogram

The complex connectivity of the cerebral cortex is a topic of much study, yet the link between structure and function is still unclear. The processing capacity and throughput of information at individual brain regions remains an open question and one that could potentially bridge these two aspects of neural organization. The rate at which information is emitted from different nodes in the network and how this output process changes under different external conditions are general questions that are not unique to neuroscience, but are of interest in multiple classes of telecommunication networks. In the present study we show how some of these questions may be addressed using tools from telecommunications research. An important system statistic for modeling and performance evaluation of distributed communication systems is the time between successive departures of units of information at each node in the network. We describe a method to extract and fully characterize the distribution of such inter-departure times from the resting-state electroencephalogram (EEG). We show that inter-departure times are well fitted by the two-parameter Gamma distribution. Moreover, they are not spatially or neurophysiologically trivial and instead are regionally specific and sensitive to the presence of sensory input. In both the eyes-closed and eyes-open conditions, inter-departure time distributions were more dispersed over posterior parietal channels, close to regions which are known to have the most dense structural connectivity. The biggest differences between the two conditions were observed at occipital sites, where inter-departure times were significantly more variable in the eyes-open condition. Together, these results suggest that message departure times are indicative of network traffic and capture a novel facet of neural activity.

Departure process in a mixed fork–join synchronization network

In this article we consider a mixed fork–join synchronization Markovian queueing network consisting of two input buffers, B 1 and B 2 , fed by arrivals from two populations. The first is a finite population of size K and second population is infinite. The first population feeds the first buffer and the second population feeds the second finite 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, performance measures, and characterization of the departure process; in particular we provide the marginal distribution of inter-departure times.

Marked Markovian Arrivals in a Tandem G-Network with Blocking

Queueing networks with blocking have broad applications in computer modelling and manufacturing. The present paper focusses on the MMAP[2]/M/1/ ∞ →·/M/1/K + 1 G-queue with blocking. This network consists of a sequence of two single-server stations with an infinite queue allowed before the first server and an intermediate queue of finite capacity K ≥ 0 allowed between servers. This restriction results in the blocking of the first server whenever a unit having completed its service in Station 1 cannot enter into Station 2 due to K + 1 units are accommodated into Station 2. There are two types of arrivals, called units and signals, which are modelled by a single Markovian arrival process with marked transitions. Each unit is served at Stations 1 and 2 in that order, and then it exits of the network. A signal induces the last unit in queue or in service, if there is one, to leave the network instantly, and it has no effect otherwise. Our purpose is to study the influence of the dependence between units and signals on the performance evaluation of the continuous-time Markov chain describing the state of the network at arbitrary times, which constitutes a quasi-birth-and-death process. We present tractable formulas for a variety of probabilistic descriptors, with special emphasis on the distribution of inter-departure times.

Modeling and Synthesis of Traffic in Optical Burst-Switched Networks

We analyze burst assembly process as the main building block of the optical burst switching (OBS) paradigm. The analysis is performed for time-based, volume-based, as well as hybrid burst assemblers. Under the assumption that the process of packet arrival to the assembly buffer is Poisson, exact analytical expressions are derived for length and interdeparture time of bursts that are generated by these three classes of assembly algorithms. Furthermore, we consider the issue of generating burst trace, which arises during the performance evaluation of OBS networks through discrete-event simulation. In such a simulation study, a significant part of the simulation time, particularly in the case of a network with a large number of ingress nodes, is used by the implementation of the burst assembly algorithms. This is due to the fact that each data burst is a result of aggregating several short-length packets, which-in a straightforward approach-have to be individually generated and, afterward, ldquomeltedrdquo into the burst. We present a novel approach to fast generation of bursts, which is based on the analytical models that are developed for burst length and burst interdeparture time distributions as well as an efficient generation technique (composition) supporting the generation of these distributions. The analysis is completed by numerical results that validate the accuracy of developed models and demonstrate the speedup gain of using proposed burst generation algorithms.

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.

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.

PERFORMANCE ANALYSIS OF MULTI-STAGE INTERCONNECTION NETWORKS WITH DETERMINISTIC SERVICE TIMES

In this paper we present a performance model for a constant service time, globally synchronized multistage interconnection network that is an improvement on earlier work in this field. The events at the network input are assumed to have a geometric distribution in time. As the events are combined within the network, it can happen that two events arrive simultaneously at one of the network server elements, called a double arrival rate stream (DAR) server. Analytical methods are described to derive the response time and inter-departure time distribution of the double arrival rate constant service time server, which has a queuing property (DAR/D/1). Due to the queuing effect of the servers in the first stage of the network, their outputs show a "distorted" geometric distribution, which is known to introduce analytical difficulties in the performance evaluation of the subsequent stages. A method is presented to derive the response time of servers in the second stage, when the network has a uniform workload distribution. It is shown that the knowledge of the response times of the first and second stage of the network is sufficient to predict the total response time of a large network with an accuracy better than 3%, even under a heavy workload.

Interdeparture time analysis of CBR traffic in AAL multiplexer with bursty background traffic

The authors present a discrete time model of a queue to represent an AAL multiplexer with two types of traffic and to analyse cell delay variation of constant bit rate (CBR) traffic in ATM networks. It is assumed that a single periodically deterministic CBR traffic stream and several variable size bursty background traffic streams are multiplexed in the AAL multiplexer and one ATM cell payload stream goes out. The AAL multiplexer is modelled as a BX + D/D/1 queue with all infinite buffer and the departure process of CBR cell payloads is analysed by using the queueing model with the superposed input traffic of CBR and bursty background streams. Both the packet and cell delay time distributons of the CBR traffic and the interdeparture time distribution of the CBR cells are obtained. The authors also give some numerical results for the performance measures.

The interdeparture time distribution for each class in the ∑ Mi[ X] / Gi/1 queue with set-up times and repeated server vacations

This paper studies the interdeparture time distribution of one class of customers who arrive at a single server queue where customers of several classes are served and where the server takes a vacation whenever the system becomes empty or is empty when the server returns from a vacation. Furthermore, the first customer in the busy period is allowed to have an exceptional service time (set-up time), depending on the class to which this customer belongs. Batches of customers of each class arrive according to independent Poisson processes and compete with each other on a FIFO basis. All customers who belong to the same class are served according to a common generally distributed service time. Service times, batch sizes and the arrival process are all assumed to be mutually independent. Successive vacation times of the server form independent and identically distributed sequences with a general distribution. For this queueing model we obtain the Laplace transform of the interdeparture time distribution for each class of customers whose batch size is geometrically distributed. No explicit assumptions of the batch size distributions of the other classes of customers are necessary to obtain the results. The paper ends by showing how the mathematical results can be used to evaluate a protocol that controls access to a shared medium of an ATM passive optical network. The numerical results presented in the last section of this paper show that the bundle spacing principle that is used by the permit distribution algorithm of this protocol introduces high delays and in many cases also more variable interdeparture times for the ATM cells of individual connections. An alternative algorithm is proposed that does not cope with these performance short comings and at the same time conserves the good properties of the protocol.

Derivation of the N-step interdeparture time distribution in GI/ G/1 queueing systems

The departure process of a queueing system 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 paper, we derive a closed form expression for the distribution of interdeparture time in a GI/ G/1 queueing model. Without loss of generality, we consider an embedded Markov chain in a general K M / G/1 queueing system, in which the interarrival time distribution is Coxian and service time distribution is general. Closed form solutions of the equilibrium distribution are derived for this model and the Laplace–Stieltjes transform (LST) of the distribution of interdeparture times is presented. An algorithmic computing procedure is given and numerical examples are provided to illustrate the results. With the analysis presented, we provide a novel analytic tool for studying the departure process in a general queueing model.

Waiting and Interdeparture Times in Priority Queues with Poisson- and General-Arrival Streams

This paper presents new results for the interdeparture time distributions in single-server priority queues with an arbitrary number of high-priority Poisson-arrival classes and one lowest-priority general-arrival class. Little-known results for the waiting time distributions in the same queue are presented, including a new proof for the powerful result that the waiting time distribution in the non-preemptive priority queue is insensitive to the interarrival time distributions of lower-priority classes, beyond their mean.

Traffic profiles in ATM networks

A lot of studies have been made to characterize and model sources of ATM traffic (voice, data, video) and to evaluate the performance of a multiplexer whose input is a superposition of these sources, using different methods and techniques (fluid flow, matrix-analytic, etc.). However, in order to better understand the end-to-end performance of ATM connections, characterizations and models of ATM traffic inside the network (i.e. after passage through one or more network elements) are needed. In this paper we intend to study the following problems: (i) Traffic profile of an ATM connection after being policed, in particular worst case traffic, and evaluation of the performance of the related statistical multiplexer. (ii) Traffic profile of the output of a multiplexer (characterized by means of the interdeparture time distribution and the index of dispersion for counts and the index of dispersion for interarrival times). (iii) Traffic profile of a single connection after passing a multiplexer. The aim is to obtain useful characterizations and models of ATM traffic in order to evaluate the performance and the efficiency of ATM network elements and of traffic control functions.