Systems with Queueing and their Simulation

Thailand Statistician, 2017

The paper investigates a M / M ( b , b ) /1 queuing model with bulk service. The server serves the customers in batches of fixed size b , and the service time is assumed to be exponentially distribution. Customers arrive to the system as a Poisson process and may renege after waiting in the queue for an exponentially distributed time. The reneging of a customer depends on the state of the system. The model is analyzed to find the different measures of effectiveness of the model. The approach adopted is based on embedded Markov chains.

Methodology and Computing in Applied Probability, 2019

We consider a single-server queueing system with server vacations as the important component of the polling queueing model of a real-world system. Period of continuous operation of the server (the maximum server attendance time) is restricted, but the service of a customer cannot be interrupted when this period expires. Such features are inherent for many realworld systems with resource sharing. We assume that the customers arrival is described by the Markovian Arrival Process and service, vacation and maximum server attendance times have a phase-type distribution. We derive the stationary distributions of the system states and waiting time. Taking in mind the necessity of further application of the results to modeling the polling queueing systems, the distribution of the server visiting time is derived. Extensive numerical results are presented. They highlight that an account of the coefficient of variation of vacation and maximum attendance time is very important for exact evaluation of the key performance measures of the system, while only the results for the coefficient of variation equal to zero or one are known in the literature. Impact of the possible customers impatience, which is intuitively important because the time-limited service is considered, is confirmed by the results of the numerical experiments. Optimization problem of matching the durations of vacation and maximum attendance time is considered.

2016

The paper investigates a M/M/1 queuing model with bulk service customers in batches of fixed size b, and the service time is assumed to be exponentially distribution. Customers arrive to the system as a Poisson process and may renege after waiting in the queue for an exponentially distributed time the system. The model is analyzed to find the different measures of effectiveness of the model approach adopted is based on embedded Markov chains ______________________________

Journal of the Institute of Engineering, 2016

This paper deals with the study of Erlangian queueing system with time dependent framework. Under our study we find (i) the expected number of customers in the queue (ii) the expected waiting time before being served (iii) the expected time spent in the system (iv) the expected number of customers in the system. Customers arrive in the system in Poisson fashion with rate and served in arbitrary service time distribution with rate µ.The probability generating function technique and Laplace transform method have been used. The numerical computation has also been obtained for applicability of the model.

Springer eBooks, 2010

Queueing systems with batch service have been investigated extensively during the past decades. However, nearly all the studied models share the common feature that an uncorrelated arrival process is considered, which is unrealistic in several real-life situations. In this paper, we study a discrete-time queueing model, with a server that only initiates service when the amount of customers in system (system content) reaches or exceeds a threshold. Correlation is taken into account by assuming a discrete batch Markovian arrival process (D-BMAP), i.e. the distribution of the number of customer arrivals per slot depends on a background state which is determined by a first-order Markov chain. We deduce the probability generating function of the system content at random slot marks and we examine the influence of correlation in the arrival process on the behavior of the system. We show that correlation merely has a small impact on the threshold that minimizes the mean system content. In addition, we demonstrate that correlation might have a significant influence on the system content and therefore has to be included in the model.

Electronics and Communications in Japan (Part I: Communications), 1990

ABSTRACT This paper presents an excellent two-moment approximation via an N-environment diffusion model of a queueing system with Markov modulated Poisson process (MMPP) arrivals and general service time distribution. By choosing residual workload as a performance criterion, an approximation error of less than 3 percent can be achieved.We give a closed form solution for all moments of the waiting time distribution with two-state MMPP arrivals. We also propose an algorithm for the approximate solution of single server queueing systems with n-state MMPP arrivals. The proposed sufficiency condition for the existence of a stationary solution has not been proved for the n-state case with n ≥ 3.

Industrial Engineering and Management Systems, 2016

We consider a multi-server queueing system without buffer and with two types of customers as a model of operation of a mobile network cell. Customers arrive at the system in the marked Markovian arrival flow. The service times of customers are exponentially distributed with parameters depending on the type of customer. A part of the available servers is reserved exclusively for service of first type customers. Customers who do not receive service upon arrival, can make repeated attempts. The system operation is influenced by random factors, leading to a change of the system parameters, including the total number of servers and the number of reserved servers. The behavior of the system is described by the multi-dimensional Markov chain. The generator of this Markov chain is constructed and the ergodicity condition is derived. Formulas for computation of the main performance measures of the system based on the stationary distribution of the Markov chain are derived. Numerical examples are presented.

Journal of Management and Science, 2016

Queueing models are stochastic models that represent the probability that a queueing system will be found in a particular configuration or state. Several interesting stationary queueing systems have been solved analytically; on the other hand, non-stationary queueing systems are relatively unexplored. The present study analyses the waiting times of a non-stationary M/M/1 queueing system using simulation methods.

Queueing Systems, 1988

This paper gives an overview of those aspects of simulation methodology that are (to some extent) peculiar to the simulation of queueing systems. A generalized semi-Markov process framework for describing queueing systems is used through much of the paper. The main topics covered are: output analysis for simulation of transient and steady-state quantities, variance reduction methods that exploit queueing structure, and gradient estimation methods for performance parameters associated with queueing networks.

The comparative analysis of the results of simulation models of limited and unlimited queuing service systems is reviewed. Here, when the restriction put on the time of presence of requests is violated, the loss of requests occurs. The analysis of possible situations leading to the loss in such systems is one of the important issues. These situations occur practically in service processes of different technical systems. An experiment was conducted; results were obtained and comparatively analyzed for both cases.

RUDN JOURNAL OF MATHEMATICS, INFORMATION SCIENCES AND PHYSICS

We investigate the queueing system in which the losses of incoming orders due to the introduction of a special renovation mechanism are possible. The introduced queueing system consists of server with a general distribution of service time and a buffer of unlimited capacity. The incoming flow of tasks is a Poisson one. The renovation mechanism is that at the end of its service the task on the server may with some probability empty the buffer and leave the system, or with an additional probability may just leave the system. In order to study the characteristics of the system the Markov chain embedded upon the end of service times is introduced. Under the assumption of the existence of a stationary regime for the embedded Markov chain the formula for the probability generation function is obtained. With the help of the probability generation function such system characteristics as the probability of the system being empty, the average number of customers in the system, the probability of a task not to be dropped, the distribution of the service waiting time for non-dropped tasks, the average service waiting time for non-dropped requests are derived.

Annals of Operations Research, 2018

In this paper, we study the M n /M n /c/K + M n queueing system where customers arrive according to a Poisson process with state-dependent rates. Moreover, the rates of the exponential service times and times to abandonment of the queued customers can also change whenever the system size changes. This implies that a customer may experience different service rates throughout the time she is being served. Similarly, a queued customer can change her patience time limits while waiting in the queue. Thus, we refer to the analyzed system as the "sensitive" Markovian queue. We conduct an exact analysis of this system and obtain its steady-state performance measures. The steady-state system size distribution yields itself via a birth-death process. The times spent in the queue by an arbitrary or an eventually served customer are represented as the times until absorption in two continuous-time Markov chains and follow Phase-type distributions with which the queueing time distributions and moments are obtained. Then, we demonstrate how the M n /M n /c/K + M n queue can be employed to approximately yet accurately estimate the performance measures of the M n /GI/c/K + GI type call center.

Mathematics

We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the nodes is exponentially distributed. After service completion at a node, the serviced customer can transit to another node or leave the network forever. The main features of the model are the mutual dependence of processes of customer arrivals and retrials and the impatience and non-persistence of customers. Dynamics of the network are described by a multidimensional Markov chain with infinite state space, state inhomogeneous behavior and special structure of the infinitesimal generator. The explicit form of the generator is derived. An effective algorithm for computing the stationary distribution of this chain is recommended. The expressions for computation of the key performance measures of the...

Mathematics

In this paper, we discuss the waiting-time distribution for a finite-space, single-server queueing system, in which customers arrive singly following a Poisson process and the server operates under (a,b)-bulk service rule. The queueing system has a finite-buffer capacity ‘N’ excluding the batch in service. The service-time distribution of batches follows a general distribution, which is independent of the arrival process. We first develop an alternative approach of obtaining the probability distribution for the queue length at a post-departure epoch of a batch and, subsequently, the probability distribution for the queue length at a random epoch using an embedded Markov chain, Markov renewal theory and the semi-Markov process. The waiting-time distribution (in the queue) of a random customer is derived using the functional relation between the probability generating function (pgf) for the queue-length distribution and the Laplace–Stieltjes transform (LST) of the queueing-time distri...

2000

This paper focuses on the study of finite capacity queues with compound Poisson arrivals and two GE-type service completions using dual embedded Markov chains. New closed form expressions for queue length distribution and blocking probability are obtained. Informa- tion theoretic interpretations are given based on the principles of Maximum Entropy. Moreover, numerical results are included to illustrate performance dierentiations when