A discrete-time queueing model with abandonments

The paper studies a queuing model with Poisson arrival process and bulk service. The server serves the customers in batches of fixed size b, and the service time is assumed to be exponentially distribution. The model is analyzed to find the steady-state distribution of the number of customers stranded following each service. The approach adopted is based on discrete-time Markov chains, instead of Laplace transforms that is usually used in literature. A simulation study is carried out to estimate the expected number of stranded customers at any point of time, its variance and the downside risk for given values of the system parameters.

Performance Evaluation, 2015

This paper studies a discrete-time queueing system where each customer has a maximum allowed sojourn time in the system, referred to as the "deadline" of the customer. More specifically, we model the deadlines of the consecutive customers as independent and geometrically distributed random variables. Customers enter the system according to a general independent arrival process, i.e., the numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. Service times of the customers are deterministically equal to one slot each. For this queueing model, we are able to obtain exact formulas for such quantities as the generating function and the expected value of the system content, the mean customer delay and the deadline-expiration ratio. These formulas, however, contain infinite sums and infinite products, which implies that truncations are required to actually compute numerical values. Therefore, we also derive some easy-to-evaluate approximate results for the main performance measures, based on a polynomial approximation technique. We believe this technique, in its own right, is also one of the major (methodological) contributions of the paper. Possible applications of this type of queueing model are numerous: the (variable) deadlines could model, for instance, the fact that customers may become impatient and leave the queue unserved if they have to wait too long in line, but they could also reflect the fact that the service of a customer is not useful anymore if it cannot be delivered soon enough, etc.

Neural, Parallel, & Scientific Computations, 2020

In this paper, we presents two discrete queueing inventory models with positive service time and lead time where customers arrive according to a Bernoulli process and service time follows a geometric distribution. In model 1, we assume that an arriving customer joins the system only if the number in the queue is less than the number of items in the inventory at that epoch. In model 2, it is assumed that if the inventory level is greater than reorder level, s at the time of arrival of a customer, then he necessarily joins. However, if it is less than or equal to s (but larger than zero) then he joins only if the number of customers present is less than the on hand inventory. We analyse this queueing system using the matrix geometric method and we derive an explicit expression for the stability condition of the model-2. We obtain the steady-state behaviour of these systems and several system performance measures. An average system cost function is constructed for the models and are investigated numerically. The influence of various parameters on the system performances are also discussed through numercal example.

IEEE Journal on Selected Areas in Communications, 1991

This paper is concerned with an M / G / I FCFS queue with twd types of customers, viz. (1) ordinary customers who arrive according to a Poisson process, and (2) permanent customers, who immediately return to the end of the queue after having received a service. The influence of the permanent customers on queue length and sojourn times of the Poisson customers is studied, using results from queueing theory and from the theory of branching processes. In particular it is shown, for the case that the service time distributions of the Poisson customers and all K permanent customers are negative exponential with identical means, that the queue length and sojourn time distributions of the Poisson customers are the ( K + 1 )-fold convolution of those for the case without permanent customers.

This paper deals with the steady-state solution of the queueing system: M X /H k /1/N with reneging in which (i) units arrive in batches of random size with the interarrival times of batches following negative exponential distribution, (ii) the batches are served in order of their arrival; and (iii) the service time distribution is hyperexponential with k branches. Recurrence relations connecting the various probabilities introduced are found. Some measures of effectiveness are deduced and some special cases are also obtained. Keywords: queueing system: M X /H k /1/N, hyperexponential DESCRIPTION OF THE SYSTEM Morse [4] discussed the steady-state queueing system in which the service channel consists of two branches, the units arrive singly and the capacity of the waiting space is infinite. Gupta and Goyal [1] studied a similar system by using the generating functions with k branches in the service channel, the units arrive singly and the capacity of the waiting space is finite. Habib [3] and Gupta and Goyal [2] treated the system M X /Hk/1. White et al. [6] solved the system: M/H2 /2/2 numerically. All the previous studies are without balking and reneging. In the present system, it is assumed that the units arrive at the system in batches of random size X, i. e., at each moment of arrival, there is a probability Cj = Pr (X= j) that j units arrive simultaneously, and the interarrival times of batches follow a negative exponential distribution with time independent parameter . Let  Cj t, (j = 1, 2,..., N), be the first order probability that a batches of j units comes in time t. The service channel is busy if a unit is present in any one of the k branches and in this case the arrival units form a queue and the capacity of the system is N. The unit at the head of the queue requires service in the r th branch with probability *. The service time distribution in the r th branch is * The variation of the subscripts i,j,r,s is from 1 to k, unless otherwise explicitly mentioned.

PLOS ONE, 2016

In a queueing system with the dropping function the arriving customer can be denied service (dropped) with the probability that is a function of the queue length at the time of arrival of this customer. The potential applicability of such mechanism is very wide due to the fact that by choosing the shape of this function one can easily manipulate several performance characteristics of the queueing system. In this paper we carry out analysis of the queueing system with the dropping function and a very general model of arrival process-the model which includes batch arrivals and the interarrival time autocorrelation, and allows for fitting the actual shape of the interarrival time distribution and its moments. For such a system we obtain formulas for the distribution of the queue length and the overall customer loss ratio. The analytical results are accompanied with numerical examples computed for several dropping functions.

Mathematical Problems in Engineering, 2013

A multiserver queueing system with infinite and finite buffers, two types of customers, and two types of servers as a model of a call center with a call-back for lost customers is investigated. Type 1 customers arrive to the system according to a Markovian arrival process. All rejected type 1 customers become type 2 customers. Typer,r=1,2, servers serve typercustomers if there are any in the system and serve typer′,r′=1,2, r′≠r,customers if there are no typercustomers in the system. The service times of different types of customers have an exponential distribution with different parameters. The steady-state distribution of the system is analyzed. Some key performance measures are calculated. The Laplace-Stieltjes transform of the sojourn time distribution of type 2 customers is derived. The problem of optimal choice of the number of each type servers is solved numerically.

2016

Queueing Theory is one of the most commonly used mathematical tool for the performance evaluation of systems. The aim of the book is to present the basic methods, approaches in a Markovian level for the analysis of not too complicated systems. The main purpose is to understand how models could be constructed and how to analyze them. It is intended not only for students of computer science, engineering, operation research, mathematics but also those who study at business, management and planning departments, too. It covers more than one semester and has been tested by graduate students at Debrecen University over the years. It gives a very detailed analysis of the involved queueing systems by giving density function, distribution function, generating function, Laplace-transform, respectively. Furthermore, Java-applets are provided to calculate the main performance measures immediately by using the pdf version of the book in a WWW environment. I have attempted to provide examples for ...

2011

In the queueing theory, it is assumed that customer arrivals correspond to a Poisson process and service time has the exponential distribution. Using these assumptions, the behaviour of the queueing system can be described by means of Markov chains and it is possible to derive the characteristics of the system. In the paper, these theoretical approaches are presented on several types of systems and it is also shown how to compute the characteristics in a situation when these assumptions are not satisfied Keywords—Queueing theory, Poisson process, Markov chains.