Pre-arrival Epochs Research Articles

A Review of Birth-Death and Other Markovian Discrete-Time Queues

In this review article, we consider discrete-time birth-death processes and their applications to discrete-time queues. To make the analysis simpler to follow, we focus on transform-free methods and consider instances of non-birth-death Markovian discrete-time systems. We present a number of results within one discrete-time framework that parallels the treatment of continuous time models. This approach has two advantages; first, it unifies the treatment of several discrete-time models in one framework, and second, it parallels to the extent possible the treatment of continuous time models. This allows us to draw parallels and contrasts between the discrete and continuous time queues. Specifically, we focus on birth-death applications to the single server discrete-time model with Bernoulli arrivals and geometric service times and provide the reader with a simple rigorous detailed analysis that covers all five scheduling rules considered in the literature, with attention to stationary distributions at slot edges, slot centers, and prearrival epochs. We also cover the waiting time distributions. Moreover, we cover three Markovian models that fit the global balance equations. Our approach provides interesting insights into the behavior of discrete-time queues. The article is intended for those who are familiar with queueing theory basics and would like a simple, yet rigorous introductory treatment to discrete-time queues.

A population model with Markovian arrival process and binomial correlated catastrophes

Stochastic population models with mild catastrophes have gained much attention in recent years due to their wide application in a variety of areas including computer-communications systems. This article considers a population model in which both the arrival process of individuals and catastrophes occur as per the Markovian arrival process (MAP) and are independent of each other. The killing mechanism takes place according to the binomial distribution. The steady-state analysis of the model is carried out using the vector generating function approach and the population size distributions at an arbitrary, post-catastrophe, and pre-arrival epochs are presented in terms of the roots of the characteristic equation. Several performance measures of the system are studied in detail, and the impact of critical parameters is duly investigated.

Analytic and Computational Analysis of GI/Ma,b/c Queueing System

Bulk-service queueing systems have been widely applied in many areas in real life. While single-server queueing systems work in some cases, multi-servers can efficiently handle most complex applications. Bulk-service, multi-server queueing systems (compared to well-developed single-server queueing systems) are more complex and harder to deal with, especially when the inter-arrival time distributions are arbitrary. This paper deals with analytic and computational analyses of queue-length distributions for a complex bulk-service, multi-server queueing system GI/Ma,b/c, wherein inter-arrival times follow an arbitrary distribution, a is the quorum, and b is the capacity of each server; service times follow exponential distributions. The introduction of quorum a further increases the complexity of the model. In view of this, a two-dimensional Markov chain has to be involved. Currently, it appears that this system has not been addressed so far. An elegant analytic closed-form solution and an efficient algorithm to obtain the queue-length distributions at three different epochs, i.e., pre-arrival epoch (p.a.e.), random epoch (r.e.), and post-departure epoch (p.d.e.) are presented, when the servers are in busy and idle states, respectively.

Analysis of BMAP/MSP/1 queue with MAP generated negative customers and disasters

This paper studies an infinite-buffer single server queueing model with negative customers and disasters. The ordinary (positive) customers arrive in the system according to the batch Markovian arrival process and join the queue for service. The negative customer and disaster are characterized by two independent Markovian arrival processes. A negative customer removes the positive customer undergoing service, if any, and a disaster makes the system empty by simultaneously removing all the positive customers present in the system. We obtain the steady-state vector generating function (VGF) of the distribution of the number of positive customers in the system at arbitrary epoch. Further, the inversion of the VGF is carried out using the roots method to obtain the distribution. Moreover, the probability vectors at post-departure and pre-arrival epochs are also obtained. The results obtained throughout the analysis are computationally tractable, as illustrated by few numerical examples. Finally, we discuss the impact of the correlation of the arrival process of negative customers and disasters on the performance of the system through some graphical representations.

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.

Computational and Numerical Investigation of the Batch Markovian Arrival Process Subject to Renewal Generated Geometric Catastrophes.

In this paper, we consider a stochastic model in which the population grows according to the batch Markovian arrival process and is subjected to renewal generated geometric catastrophes. Our analytical work starts from the vector generating function (VGF) of the population size at post-catastrophe epoch. We develop a methodology for extracting the population size distribution at post-catastrophe epoch from the VGF, which is based on the inversion of VGF using the roots method. The method is analytically quite simple and easy to implement. Further, we obtain the population size distribution at arbitrary, pre-catastrophe and pre-arrival epochs along with their factorial moments. To show the applicability and correctness of the proposed methodology, we match our results with the available ones in special cases and present several numerical examples for different inter-catastrophe time distributions. Moreover, we investigate the effect of key parameters on the system performance and display the results in the form of graphs along with a detailed description.

Analysis of a renewal batch arrival queue with a fault-tolerant server using shift operator method

Motivated by the fault tolerance for manufacturing, we investigate a renewal input bulk arrival queue with a fault-tolerant server, in which the server can keep working with a low service rate even if the partial failure occurs. Only when there are no customers in the system, the partial failure can be removed. To explore the performance measures of the queue, a more generic and simpler algorithm based on the right shift operator method for solving difference equations is employed to obtain the queue-length distributions at different time epochs. The significant feature of this algorithm lies in that it does not require the derivation of the transition probability matrix for the corresponding embedded Markov chain. Furthermore, we can resort to the queue-length distribution at the pre-arrival epoch to quickly get the expected sojourn time for an arbitrary customer. Finally, with the help of Padacute{mathrm{e}} approximation, several representative numerical examples are illustrated in tables and graphs, under which we show how to verify the correctness of our theoretical results through Little’s law.

Alternative approach based on roots for computing the stationary queue-length distributions in GIX/M(1,b)/1 single working vacation queue

The purpose of this paper is to present an alternative algorithm for computing the stationary queue-length and system-length distributions of a single working vacation queue with renewal input batch arrival and exponential holding times. Here we assume that a group of customers arrives into the system, and they are served in batches not exceeding a specific number b. Because of batch arrival, the transition probability matrix of the corresponding embedded Markov chain for the working vacation queue has no skip-free-to-the-right property. Without considering whether the transition probability matrix has a special block structure, through the calculation of roots of the associated characteristic equation of the generating function of queue-length distribution immediately before batch arrival, we suggest a procedure to obtain the steady-state distributions of the number of customers in the queue at different epochs. Furthermore, we present the analytic results for the sojourn time of an arbitrary customer in a batch by utilizing the queue-length distribution at the pre-arrival epoch. Finally, various examples are provided to show the applicability of the numerical algorithm.

Modelling and analysis of D-BMAP/D-MSP/1 queue using RG-factorization

ABSTRACT The current investigation of our paper is concentrated on an infinite-buffer single-server queueing model with discrete-time batch Markovian arrival process and Markovian service process. The steady-state behaviour of this discrete-time queueing system at various significant time epochs are presented. We first analyze the system-length distribution at outside observer’s epoch using the -factorization technique combined with the spectral expansion method to compute the stochastic matrix . For the purpose of comparative study, a comprehensive analysis of the system-length distribution at outside observer’s epoch is also carried out using the roots method. Next, we obtain the explicit closed-form expressions of the system-length distributions at pre-arrival, intermediate, post-departure and random epochs by developing the relations among them in equilibrium state. The analysis of waiting-time distribution in the queue measured in slots of an arbitrary customer in an arriving batch is also discussed. Computational experiences with adequate numerical results are conferred in the structure of tables and graphs. Finally, we suggest that our queueing model seems to be fitted for outpatient service of any specialized hospital in a smart city.

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.

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.

A discrete-time batch transmission channel with random serving capacity under batch-size-dependent service:

Discrete-time queues have increasingly diverse spectrum of applications in the modern packet-basedcommunication systems. Due to the wide range of applicability of such queues, we analyze a discrete-time queue with group-arrival and batch-service, where transmission time depends on the batch-size. The arrival occurs according tothe batch Bernoulli process and service is provided according to the random serving capacity rule. First, we obtain the bivariate probability generating function of the joint distribution of queue and server content at post transmission epoch. After the determination of unknown probabilities, the complete joint distribution has been extracted. We also acquire the probability distribution at random and pre-arrival epochs. An approximation of the tail distribution is also discussed so that it will be useful to improve the cell loss ratio. Some assorted numerical examples are incorporated to validate the analytic procedure and results.

Analysis of stationary queue-length distributions of the BMAP/R(a,b)/1 queue

This paper investigates an infinite waiting space single-server queueing system in which customers arrive according to a batch Markovian arrival process. The server serves the customers in batches of maximum size ‘b’ with a minimum threshold value ‘a’. The service time of each batch follows R-type distribution, where R-type distribution represents a class of distributions whose Laplace-Stieltjes transform is rational or approximated rational function, which is independent of service batch size and the arrival process. We determine the queue-length distribution at departure epoch in terms of roots of the associated characteristic equation of the vector probability generating function. To show the strength and advantage of roots method, we also provide a comprehensive analysis of the queue-length distribution at departure epoch using the matrix-analytic method (MAM). To determine the queue-length distribution at random epoch, we obtain a relation between the queue-length distributions at departure and random epochs using the supplementary variable technique with remaining service time of a batch in service as the supplementary variable. We also derive the queue-length distributions at pre-arrival epoch of an arrived batch and post-arrival epoch of a random customer of an arrived batch. Some numerical results are demonstrated for different service-time distributions to show the key performance measures of the system.

Stationary Analysis of an Infinite-Buffer D-MAP/D-MSP/1 Queueing System

This paper deals with the analysis of an infinite-buffer single-server D-MAP/D-MSP/1 queueing system. This queueing system can be analyzed by representing level-independent quasi-birth-and-death process in tridiagonal structure. The proposed analysis is based on the use of matrix-geometric method in conjunction with the spectral method to obtain the system-length distribution at outside observer’s epoch. We derive the stationary system-length distributions at random, prearrival, intermediate and post-departure epochs using the system-length distribution at outside observer’s epoch. The waiting-time distribution in the queue measured in slots of an arriving customer is also carried out. Computational procedures along with numerical results are provided to confirm the correctness of our analytical results.

Analysis of a geometric catastrophe model with discrete-time batch renewal arrival process

Discrete-time stochastic models have been extensively studied since the past few decades due to its huge application in areas of computer-communication networks and telecommunication systems. However, the growing use of the internet often makes these systems vulnerable to catastrophe/ virus attack leading to the removal of some or all the elements from the system. Taking note of this, we consider a discrete-time model where the population (in the form of packets, data, etc.) is assumed to grow in batches according to renewal process and is likely to be affected by catastrophes which occur according to Bernoulli process. The catastrophes have a sequential impact on the population and it destroys each individual at a time with probabilityp. This destruction process stops as soon as an individual survives or when the entire population becomes extinct. We analyze both late and early arrival systems independently and using supplementary variable and shift operator methods obtain explicit expressions of steady-state population size distribution at pre-arrival and arbitrary epochs. We deduce some important performance measures and further show that for both the systems the tail probabilities at pre-arrival epoch can be well approximated using a single root of the characteristic equation. In order to illustrate the computational procedure, we present some numerical results and also investigate the change in the behavior of the model with the change in parameter values.

Analytical and computational studies of the queue

This article studies a single server queueing system in which customers arrive according to a batch Markovian arrival process and customers are served in batches of random capacity ‘Y’ with a minimum threshold value ‘a’. The service time of each batch is arbitrarily distributed and independent of service batch size. We first construct the vector difference equations by observing two consecutive departure epochs and then convert them into the vector probability generating function. The queue-length distribution at departure epoch is extracted in terms of roots of the associated characteristic equation. A comparative study is also carried out to compare the roots method with that of the matrix-analytic method. To obtain the queue-length distribution at random epoch, we establish a relationship between the queue-length distributions at departure and random epochs with the help of Markov renewal theory. We obtain the queue-length distribution at prearrival epoch of an arriving batch and the queue-length distribution at post-arrival epoch of a random customer of an arrived batch. Further, we derive the results of some particular well-known queueing models from our model. The results are illustrated by numerical examples for different service-time distributions to show the variation of the key performance measures.

Analysis of a population model with batch Markovian arrivals influenced by Markov arrival geometric catastrophes

We consider a population model in which individuals arrive as per the batch Markovian arrival process. The population is influenced by catastrophes which occur according to Markovian arrival process and it eliminates each individual of the population in a sequential order with probability p until the one individual survives or the entire population is annihilated. We first obtain the steady-state vector generating function of the population size distribution at arbitrary epoch and then the distribution is extracted in term of roots of the associated characteristic equation. Further, we obtain the population size distribution at post-catastrophe and pre-arrival epochs. Finally, a few numerical results are given.

Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization

<p style='text-indent:20px;'>This paper analyzes an infinite-buffer single-server queueing system wherein customers arrive in batches of random size according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process. Based on the censoring technique, the UL-type <inline-formula><tex-math id="M1">\begin{document}$ RG $\end{document}</tex-math></inline-formula>-factorization for the Toeplitz type block-structured Markov chain is used to obtain the prearrival epoch probabilities. The random epoch probabilities are obtained with the help of classical principle based on Markov renewal theory. The system-length distributions at outside observer's, intermediate and post-departure epochs are obtained by making relations among various time epochs. The analysis of waiting-time distribution measured in slots of an arbitrary customer in an arrival batch has also been investigated. In order to unify the results of both discrete-time and its continuous-time counterpart, we give a brief demonstration to get the continuous-time results from those of the discrete-time ones. A variety of numerical results are provided to illustrate the effect of model parameters on the performance measures.

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.

Performance analysis of the GI/D-MSP/1 queue with N-policy and its optimal control

ABSTRACT This paper investigates a discrete-time GI/D-MSP/1/\\infty queueing system under N-policy with renewal input. The service process is correlated and its structure is constructed through discrete-time Markovian service process. The idle server resumes to serve the customers as soon as the number of waiting customers in the queue reaches a predefined threshold value N and serves the customers exhaustively until the system becomes empty. We use the matrix-geometric method to derive the system-length distribution at prearrival epoch. Employing the Markov renewal theory, we obtain the system-length distribution at random epoch. We also carried out the system-length distributions at outside observer’s, intermediate and post-departure epochs. Further, we find the waiting-time distribution in the queue measured in slots of an arrival customer. An expected linear cost function per unit time is considered to determine the optimal value of N, which minimises the expected cost function. Some numerical results are demonstrated to measure the effects of N, interarrival-time distributions and other model parameters.