Stationary analysis of a BMAP/R/1 queue with R-type multiple working vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple working vacation policy. The service time during a working vacation is generally distributed random variable which is independent of the service times during a normal busy period as well as the arrival process. Duration of service times during a normal busy period and duration of working vacation times follow the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The service time during a normal busy period, working vacation time, and the service time during a working vacation are independent of each other as well as of the arrival process. If a working vacation terminates while service is going on for a customer at head of the queue in vacation mode then, the server switches to normal mode and the customer at head of the queue is entitled to receive a full service time in the normal busy period irrespective of the amount of service received by the customer at head of the queue during the previous working vacation period. We obtain system-length distributions at various epoch, such as post-departure, pre-arrival, arbitrary, and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain system-length distribution at post-departure epoch. Later, we use supplementary variable technique and simple algebraic manipulations to obtain system-length distribution at arbitrary epoch using the system-length distribution at post-departure epoch. Some important performance measures, such as mean system lengths and mean waiting time have been obtained. Finally, some numerical results have been presented in the form of tables and graphs to show the applicability of the results obtained in this article. The model has potential application in areas of computer and communication networks, such as ethernet passive optical network (EPON).     

Busy period in batch queues

in a recent paper Takacs [1] generalized the classical ballot problem and used the results of the classical ballot problem to analyze the busy period in queues involving arrivals in batches or departures in batches. In this paper the generalized ballot problem is used to analyze the busy period in queues having both batch arrivals and batch departures.     

THE EFFECT OF POSTPONABLE INTERRUPTIONS ON A M/G/1 SYSTEM WITH IMPATIENT CUSTOMERS

This paper considers a waiting line system where units become impatient after having waited for certain time and leave the queue (renege) without being serviced. The servicing of the units is subject to interruption by the arrival of an "interruption " possessing a priority for service over the ordinary units, head-of-the-line priority discipline being prevalent. The busy period process is investigated first, making use of the supplementary variable method. Later, the general process is studied in terms of the busy period process and renewal distributions. Lastly, the ergodic properties of the general process are examined by appealing to some results of renewal theory.     

Structural properties and estimation in m|EK| 1 queue

In this paper we study the distribution of the number of customers served in a busy period in the framework of modified power series distribution introduced by Gupta (197U) and obtain the moments and probability generating function of this distribution. We also study the maximum likelihood estimation of the parameter θand the variance and the asymptotic bias of the MLE are also obtained. The minimum variance unbiased estimate of θ^ris investigated and an estimate of the probabilities is given.     

An M/G/1 clearing queueing system with setup time and multiple vacations for an unreliable server

This paper studies an M/G/1 clearing queueing system with setup time and multiple vacations, in which all present customers in the system are served simultaneously and breakdowns may occur in busy or setup period. We investigate the stationary distribution of system size and the Laplace–Stieltjes transform of sojourn time. In addition, various performance measures are discussed, such as the mean system size at arbitrary time and the mean length of a vacation circle. Moreover, a cost analysis is carried out for this queueing system. Numerical results are presented to study the sensitivity of the system parameters on the expected cost function and system performances.     

Analysis of a batch arrival queue with vacation policy and exceptional service

ABSTRACT

We consider the distributions of operating characteristics of an M^[x]/G/1 queue under vacation policies, where the first customer of each busy period receives an exceptional service. When all the customers are served in the system exhaustively, the server deactivates and operates one of two vacation policies: (1) multiple vacation policy and (2) single vacation policy. We develop the performance measures for both systems. Finally, some numerical illustrations are also given. These two vacation models have potential applications in day-to-day life, such as post offices, banks, hospitals, etc.     

A Bitcoin-inspired infinite-server model with a random fluid limit

The synchronization process inherent to the Bitcoin network gives rise to an infinite-server model with the unusual feature that customers interact. Among the closed-form characteristics that we derive for this model is the busy period distribution which, counterintuitively, does not depend on the arrival rate. We explain this by exploiting the equivalence between two specific service disciplines, which is also used to derive the model’s stationary distribution. Next to these closed-form results, the second major contribution concerns an asymptotic result: a fluid limit in the presence of service delays. Since fluid limits arise under scalings of the law-of-large-numbers type, they are usually deterministic, but in the setting of the model discussed in this paper the fluid limit is random (more specifically, of growth-collapse type).     

Some applications of lagrangian distributions in queueing theory and epidemiology

In this note we demonstrate that the Lagrangian distributions have applications in queueing theory and theory of epidemics. The Lagrangian distribution appears as the distribution describing the number of customers served in a busy period under certain conditions. Also, the Lagrangian distribution describes the distribution of number of persons infected by a certain infectious disease.     

Bayesian estimation for the M/G/1 queue using a phase-type approximation

This article deals with Bayesian inference and prediction for M/G/1 queueing systems. The general service time density is approximated with a class of Erlang mixtures which are phase-type distributions. Given this phase-type approximation, an explicit evaluation of measures such as the stationary queue size, waiting time and busy period distributions can be obtained. Given arrival and service data, a Bayesian procedure based on reversible jump Markov Chain Monte Carlo methods is proposed to estimate system parameters and predictive distributions.     

Multivariate lagrange distributions and a subfamiliy

Second order moments about its means, i.e. the variances and covari-ances for multivariate Lagrange distributions are derived in a matrix form. A subfamily of multivariate Lagrange distributions which can be characterized as the distributions of customers served in a busy period in queues with some conditions are considered. Theorems about their probability functions, one of which is a multivariate generalization of a formula by Takà cs(1989). are given and the means and second order moments about its means are considered. As an example, a multivariate Borel-Tanner distribution is derived.     

Subexponential Distributions — Large Deviations with Applications to Insurance and Queueing Models

This paper presents a fine large‐deviations theory for heavy‐tailed distributions whose tails are heavier than exp(?√t and have finite second moment. Asymptotics for first passage times are derived. The results are applied to estimate the finite time ruin probabilities in insurance as well as the busy period in a GI/G/1 queueing model.     

Performance analysis of machine repair system with single working vacation

Abstract

This paper studies a machine repair problem with repairman’s single working vacation in which repairman works with a lower repair rate rather than completely terminating repair during vacation period. Employing Markov process theory and matrix analytical method, various system performance measures are obtained in transient and stationary regimes. Moreover, we deduce the system reliability, the mean time to failure, the repairman’s busy period and the waiting time of failed machine by using the probabilistic properties of phase type distribution. Further, some numerical examples are provided. Finally, a cost model is developed to determine the optimum value of operating machines.     

On some probability distributions associated with random walks

The probability distribution of the total number of games to ruin in a gambler's ruin random walk with initial position n, the probability distribution of the total size of an epidemic starting with n cases and the probability distribution of the number of customers served during a busy period M/M/1 when the service starts with n waiting customers are identical. All these can be easily obtained by using Lagrangian expansions instead of long combinatorial methods. The binomial, trinomial, quadrinomial and polynomial random walks of a particle have been considered with an absorbing barrier at 0 when the particle starts its walks from a point n, and the pgfs. and the probability distributions of the total number of jumps (trials) before absorption at 0 have been obtained. The values for the mean and variance of such walks have also been given.     

A two phases queueing system with Bernoulli vacation schedule under multiple vacation policy

This paper deals with a single server Poisson arrival queue with two phases of heterogeneous service along with a Bernoulli schedule vacation model, where after two successive phases service the server either goes for a vacation with probability p (0≤p≤1) or may continue to serve the next unit, if any, with probability q(=1−p). Further the concept of multiple vacation policy is also introduced here. We obtained the queue size distributions at a departure epoch and at a random epoch, Laplace Stieltjes Transform of the waiting time distribution and busy period distribution along with some mean performance measures. Finally we discuss some statistical inference related issues.     

Testing for autocorrelation among common stock returns

We consider the null distribution of autocorrelation coefficients for stock returns when the variance of the returns is infinite. We show that the empirical autocorrelations then tend to zero faster than in the standard case and that they tend, after suitable normalisation, in distribution to a rather complicated nonnormal law. An empirical application to the 14 most busy German stocks reveals that the significance of observed correlations is thereby in general reduced.     

Analysis of an unreliable MX/G1G2/1/repeated service queue with delayed repair under randomized vacation policy

Abstract

In this article we consider an unreliable M^X/G/1 queue with two types of general heterogeneous service and optional repeated service subject to server’s break down and delayed repair under randomized vacation policy. We assume that customer arrive to the system according to a compound Poisson process. The server provides two types of general heterogeneous service and a customer can choose either type of service before its service start. After the completion of either type of service, the customer has the further option to repeat the same type of service once again. While the server is working with any types of service or repeated service, it may breakdown at any instant. Further the concept of randomized vacation is also introduced. For this model, we first derive the joint distribution of state of the server and queue size by considering both elapsed and remaining time, which is one of the objective of this article. Next, we derive Laplace Stieltjes transform of busy period distribution. Finally, we obtain some important performance measure and reliability indices of this model.     

Explicit formulae and convergence rate for the system M α /G/1/N as N→∞

The paper deals with the system M ^α /G/1/N with a finite number of waiting places in which arrivals can occur in a group. The number of customers in the line and the virtual waiting time are studied both in the transient and in the stationary regime. Special attention is paid to the stationary distributions of these functionals as N→∞. The number of customers lost during a busy period is considered as well.     

Lattice paths combinatorics applied to transient queue length distribution of C2/M/1 queues and busy period analysis of bulk queues C2/M/1

This paper aims at deriving explicit transient queue length distribution for GI/M/1 system and busy period analysis of bulk queue GI^b/M/1 through lattice paths (LPs) combinatorics. The general interarrival time distribution is approximated by two-phase Cox distribution, C₂, that has Markovian property, enabling us to represent the processes by two-dimensional LPs. As distributions C₂ cover a wide class of distributions that have rational Laplace–Stieltjes transforms (LSTs) with square coefficient of variation lying in , the results obtained are applicable to a large class of real life situations. Some numerical results for the C₂^b/M/1 model are also given.     

Performance analysis of the retrial queues with finite number of sources and service interruptions

This paper aims at presenting an analytic approach for investigating a single-server retrial queue with finite population of customers where the server is subject to interruptions. A free source may generate a primary call to request service. If the server is free upon arrival, the call starts to be served and the service times are independent, generally distributed random variables. During the service time the source cannot generate a new primary call. After service the source moves into the free state and can generate a new primary call. There is no waiting space in front of the server, and a call who finds the server unavailable upon arrival joins an orbit of unsatisfied customers. The server is subject to interruptions during the service processes. When the server is interrupted, the call being served just before server interruption goes to the retrial orbit and will retry its luck after a random amount of time until it finds the server available. The recovery times of the interrupted server are assumed to be generally distributed. Our analysis extends previous work on this topic and includes the analysis of the arriving customer’s distribution, the busy period, and the waiting time process.