Steady-state System Length Distributions Research Articles

Synchronized abandonment in discrete-time renewal input queues with vacations

<p style='text-indent:20px;'>We consider a discrete-time infinite buffer renewal input queue with multiple vacations and synchronized abandonment. Waiting customers get impatient during the server's vacation and decide whether to take service or abandon simultaneously at the vacation completion instants. Using the supplementary variable technique and difference operator method, we obtain an explicit expression to find the steady-state system-length distributions at pre-arrival, random, and outside observer's observation epochs. We provide the stochastic decomposition structure for the number of customers and discuss the various performance measures. With the help of numerical experiments, we show that the method formulated in this work is analytically elegant and computationally tractable. The results are appropriate for light-tailed inter-arrival distributions and can also be leveraged to find heavy-tailed inter-arrival distributions.</p>

Performance analysis of renewal input queues with multiple vacations and synchronized abandonment

ABSTRACT This paper studies a single server renewal input queueing model with multiple vacations in the presence of impatient customers. The server follows an exhaustive vacation policy whenever the system is empty. During vacation, waiting customers become impatient and decide independently whether to abandon the system simultaneously or avail service at the vacation completion instant. Using the supplementary variable and difference equation techniques, we obtain the steady-state system-length distributions at pre-arrival and arbitrary epochs. We present the stochastic decomposition structure for the number of customers in the system and derive several performance measures including the sojourn time. Moreover, we examine the impact of various parameters on the system performance measures. A potential application of the model in the inventory management of an e-commerce company with fresh agricultural products is illustrated. Numerical results discuss the effects of the system parameters and provide insight into the performance of such systems.

Modelling and analysis of a discrete-time GI^X/Geom/1/K queue with N threshold policy

This paper presents modelling and analysis of a discrete-time GIX/Geom/1/K queueing system (where K is the capacity of the system) with N threshold policy for the early arrival system (EAS). The server is turned off when the system is vacant and checks the queue length every time for an arrival of a batch. As soon as the queue length reaches a pre-specified value N(1 ≤ N ≤ K), the server turns on and serves continuously until the system becomes vacant. We obtain the steady state system length distributions at pre-arrival, arbitrary and outside observer's epochs using the combination of the supplementary and the imbedded markov chain techniques. Various performance characteristics like average number of users in the queue/system, blocking probabilities of users (first-, an arbitrary- and last-user of an arriving batch) and average waiting time are obtained analytically with numerical analysis. The numerical analysis data are presented in graphical format for blocking probabilities under different buffer size values.

Ant colony optimisation for impatient customer queue under N-policy and Bernoulli schedule vacation interruption

This paper analyses a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption under N-policy. Arriving customer decide either to join the system or to balk. After joining the queue, the customer may renege. The server leaves for working vacation after serving the queue exhaustively. At a service completion instant during working vacation, the server continue the vacation with probability q or if there are at least N customers in the queue the server interrupts the vacation and resumes regular service with probability 1 − q. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed and are mutually independent. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimisation are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

Ant colony optimisation for impatient customer queue under N-policy and Bernoulli schedule vacation interruption

This paper analyses a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption under N-policy. Arriving customer decide either to join the system or to balk. After joining the queue, the customer may renege. The server leaves for working vacation after serving the queue exhaustively. At a service completion instant during working vacation, the server continue the vacation with probability q or if there are at least N customers in the queue the server interrupts the vacation and resumes regular service with probability 1 − q. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed and are mutually independent. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimisation are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging

This paper analyzes a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption. Arriving customers decide either to join the system or to balk. After joining the queue the customers may renege based on their desire for service or their unwillingness for waiting. At a service completion instant during working vacations, the server can decide either to continue the vacation with probability $q$ or interrupt the vacation and resume regular service period with probability $1-q$. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimization are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

On renewal input state dependent working vacations queue with impatient customers

This paper studies the effect of balking and reneging in a finite buffer renewal input queue with multiple working vacations. Arriving customers balk with a probability and renege according to exponential distribution. As soon as the system becomes empty, the server leaves for a working vacation. During working vacations the server works at a different rate rather than completely stopping service. Service times during a service period, during a working vacation period and vacation times are assumed to be exponentially distributed with state dependent rates. Employing the supplementary variable and recursive techniques, the steady-state system length distributions at pre-arrival and arbitrary epochs have been derived. Some performance measures of the model and cost analysis through simulated annealing have been discussed. Finally, sensitivity analysis is carried out through some numerical experiments.

Analysis of B M A P/M S P/1 Queue

The analysis for the B M A P/M S P/1 queueing system is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution at random epoch. We obtain the steady-state system-length distributions at various epochs as well as of the actual sojourn-time distribution of an arbitrary customer in an arriving batch.

Analytic and computational analysis of the discrete-time GI/D-MSP/1 queue using roots

This paper presents a simple closed-form analysis for evaluating system-length distributions at various epochs of the discrete-time GI/D-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution at prearrival epochs. We provide the steady-state system-length distribution at random epoch by using the classical argument based on Markov renewal theory. The queueing-time distribution has also been investigated. Numerical aspects have been tested for a variety of interarrival- and service-time distributions and a sample of numerical outputs is presented.

Finite Buffer GI/M(n)/1 Queue with Bernoulli-Schedule Vacation Interruption under N-Policy.

We study a finite buffer N-policy GI/M(n)/1 queue with Bernoulli-schedule vacation interruption. The server works with a slower rate during vacation period. At a service completion epoch during working vacation, if there are at least N customers present in the queue, the server interrupts vacation and otherwise continues the vacation. Using the supplementary variable technique and recursive method, we obtain the steady state system length distributions at prearrival and arbitrary epochs. Some special cases of the model, various performance measures, and cost analysis are discussed. Finally, parameter effect on the performance measures of the model is presented through numerical computations.

On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations

This paper considers a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations, where upon arrival, a negative customer removes one positive (ordinary) customer in service if any is present and disappears immediately; otherwise, it has no effect on the system if the system is empty. As soon as the system becomes empty, the server immediately takes a working vacation. If there are no customers in the system at the end of the working vacation, the server takes another working vacation with probability $p$ or remains dormant in the system with probability $1-p$. Otherwise, the server starts to serve the customers with the normal service rate immediately if there are some customers at the end of a working vacation. This pattern does not terminate until the server has taken $J$ successive working vacations. Steady-state system length distributions at various epochs such as, pre-arrival, arbitrary and outside observer's observation epochs have been obtained. Based on the various system length distributions, we also give some important performance measures including blocking probabilities, mean queue length, probability mass function of waiting time and other performance measures along with some numerical examples. Then, we use the parabolic method to search the optimum value of the normal service rate under a established cost function.

Discrete-time renewal input queue with balking and multiple working vacations

This paper analyzes a discrete-time finite buffer GI/Geo/1 queue with multiple working vacations wherein the customers may balk due to impatience. The service times during working vacation and vacation times are assumed to be geometrically distributed. Embedded Markov chain and supplementary variable techniques have been adopted to obtain the steady-state system length distributions at pre-arrival and arbitrary epochs, respectively. Various performance measures of the model and waiting-time distribution are also presented. A variety of numerical results showing the effect of model parameters on key performance measures are demonstrated.

Discrete-time renewal input state dependent queue with working vacations and change over time

This paper considers a discrete-time renewal input state dependent queue with multiple working vacations and change over time. Using the supplementary variable and recursive techniques, the steady state system length distributions at different time epochs have been derived. Based on the system length distributions, some performance measures of the model have been discussed. The optimum values of service rates have been found through quadratic fit search method under a given cost structure.

DISCRETE-TIME GIX/GEO/1/N QUEUE WITH WORKING VACATIONS AND VACATION INTERRUPTION

In this paper, we study a discrete-time finite buffer batch arrival queue with multiple geometric working vacations and vacation interruption: the server serves the customers at the lower rate rather than completely stopping during the vacation period and can come back to the normal working level once there are customers after a service completion during the vacation period, i.e., a vacation interruption happens. The service times during a service period, service times during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer's observation epochs. We also present probability generation function (p.g.f.) of actual waiting-time distribution in the system and some performance measures.

A Non-Markovian Multistage Batch Arrival Queue with Breakdown and Reneging

The present investigation deals with analysis of non-Markovian queueing model with multistage of services. When the server is unavailable during the system breakdown (or) vacation periods, we consider reneging to prevail. Supplementary variable techniques have been adopted to obtain steady state system length distributions. The numerical illustrations are provided to validate the tractability of performance measures as far as computational aspect is concerned. Numerical results in the form of graphical representation are also presented. Practical large scale industry applications are described to justify our model.

On The Renewal Input Batch-arrival Queue Under Single And Multiple Working Vacation Policy With Application To EPON

This paper analyzes renewal input batch arrival queues with multiple and single working vacation policy. We obtain the steady state system-length distributions at pre-arrival and arbitrary epochs. The Laplace-Stieltjes transforms (LST) of the sojourn time distribution for the first customer and an arbitrary customer in an accepted batch have been derived. Also the probability density function of sojourn time distribution for the first customer in an accepted batch has been obtained. The proposed analysis is based on the roots of the characteristics equations formed using the probability generating functions (p.g.f.) of embedded pre-arrival epoch probabilities. Several performance measures such as the mean system-length and mean sojourn times of first and an arbitrary customer in an accepted batch are presented. The suggested queues have potential application in ethernet passive optical network (EPON).

A Roots Method in GI/PH/1 Queueing Model and Its Application

In this paper, we introduce a roots method that uses the roots inside the unit circle of the associated characteristics equation to evaluate the steady-state system-length distribution at three epochs (pre-arrival, arbitrary, and post-departure) and sojourn-time distribution in GI/PH/1 queueing model. It is very important for an air base to inspect airplane oil because low-quality oil leads to drop or breakdown of an airplane. Since airplane oil inspection is composed of several inspection steps, it sometimes causes train congestion and delay of inventory replenishments. We analyzed interarrival time and inspection (service) time of oil supply from the actual data which is given from one of the ROKAF's (Republic of Korea Air Force) bases. We found that interarrival time of oil follows a normal distribution with a small deviation, and the service time follows phase-type distribution, which was first introduced by Neuts to deal with the shortfalls of exponential distributions. Finally, we applied the GI/PH/1 queueing model to the oil train congestion problem and analyzed the distributions of the number of customers (oil trains) in the queue and their mean sojourn-time using the roots method suggested by Chaudhry for the model GI/C-MSP/1.

Discrete-time [formula omitted] queue with negative customers and multiple working vacations

Using the supplementary variable and the embedded Markov chain method, we consider a discrete-time batch arrival finite capacity queue with negative customers and working vacations, where the RCH killing policy and partial batch rejection policy are adopted. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. Furthermore, we consider the influence of system parameters on several performance measures to demonstrate the correctness of the theoretical analysis.

ANALYTICALLY EXPLICIT RESULTS FOR THE GI/C-MSP/1/∞ QUEUEING SYSTEM USING ROOTS

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

A short note on the [formula omitted] queueing system

In this paper, we present the steady-state system-length distributions at intermediate, outside observer’s, post-departure and pre-service completion epochs for the G I / G e o / 1 queue. It is also shown that in the limiting case, the results given in this paper tend to the continuous-time counterpart.