A discrete-time on–off source queueing system with negative customers

This paper studies a discrete-time single-server infinite-capacity queueing system with correlated arrivals, geometrically distributed service times and negative customers. Positive customers are generated by a Bernoulli bursty source, with geometrically distributed lengths of the on-periods and off-periods. Negative customers arrive to the system according to a geometrical arrival process which is independent of the positive arrival process. A negative customer removes a positive customer in service if any, but has no effect on the system if it finds the system empty. We analyze the Markov chain underlying the queueing system and evaluate the performance of the system based on generating functions technique. Closed-form expressions of some performance measures of the system are obtained, such as stationary probability generating functions of queue length, unfinished work, sojourn time distribution and so on. Finally, the effect of several parameters on the system is shown numerically.     

The model of queueing network with parallel routes

A queueing network that has several parallel routes of data delivery from the source node to the destination node is studied. In the presence of several free channels, information is transmitted from the source node simultaneously on all routes beginning from these channels. Using matrix-analytical methods, stationary characteristics of network operation are obtained.     

Markov queueing system with finite buffer and negative customers affecting the queue end

A multiserver queueing system with finite buffer, Markov input flow, and Markov (general) service process of all customers on servers with the number of process states and intensities of inter-phase transitions depending on the number of customers in the system is considered. A Markov flow of negative customers arrives to the system; one negative customer “kills” one positive customer at the end of the queue. A recurrent algorithm for computing stationary probabilities of system states is obtained; and a method for calculating stationary distribution of waiting time before starting service of a positive customer is proposed.     

Exponential queuing system with negative customers and bunker for ousted customers

Consideration was given to the queuing system with Poisson flows of incoming positive and negative customers. For the positive customers, there is an infinite-capacity buffer. The arriving negative customer knocks out a positive customer queued in the buffer and moves it to an infinite-capacity buffer of ousted customers (bunker). If the buffer is empty, then the negative customer discharges the system without affecting it. After servicing the current customer, the server receives a customer from the buffer or, if the buffer is empty, the bunker. The customers arriving from both the buffer and bunker are distributed exponentially with the same parameter. Relations for calculation of the stationary distributions of the queues in the buffer and bunker were obtained.     

A discrete-time retrial queueing system with recurrent customers

We consider a discrete-time single-server retrial queue with general service times and two classes of customers: transit and a fixed number of recurrent customers. After service completion, recurrent customers always return to the orbit and transit customers leave the systems forever. In this work we study the influence of recurrent customers. This structure appears in many applications on computer and communication networks, but also has theoretical interest. The explicit expressions of generating functions of the stationary distribution of the Markov chain are given. We also provide the main reliability indexes and numerical examples with the use of discrete Fourier transform inversion.     

A queueing system with batch arrival of customers in sessions

This paper describes and analyzes a single-server queueing model with a finite buffer and session arrivals. Generation of the sessions is described by the Markov Arrival Process (MAP). Arrival of the groups of the requests within any admitted session is described by the Terminating Batch Markov Arrival Process (TBMAP). Service time of the request has Phase (PH) type distribution. The number of the sessions that can be simultaneously admitted to the system is under control.     

Analysis of the multi-server Markov queuing system with unlimited buffer and negative customers

Consideration was given to the multi-server queuing system with unlimited buffer, Markov input flow, and Markov (general) process of servicing all customers on servers with the number of process states and intensities of the inter-phase passage depending on the number of customers in the system. Additionally, a Markov flow of negative customers arrives to the system, the arriving negative customer killing the last queued positive customer. A recurrent algorithm to calculate the stationary probabilities of system states was obtained, and a method of calculation of the stationary distribution of the waiting time before starting servicing of a positive customer was proposed.     

A Multiserver Retrial Queueing System with Batch Markov Arrival Process

A sufficient condition for a BMAP/M/N system to have a stationary state probability distribution and an algorithm for computing this distribution are investigated.     

A new solution for a queueing model of a manufacturing cell with negative customers under a rotation rule

In this paper, we consider a queueing model extension for a manufacturing cell composed of a machining center and several parallel downstream production stations under a rotation rule. A queueing model is extended with the arrival processes of negative customers to capture failures of production stations, reorganization of works and disasters in the manufacturing cell. We present an exact solution for the steady-state probabilities of the proposed queueing model. The solution does not require the approximation of the infinite sum. In addition, it provides an alternative way to compute the rate matrix for the matrix-geometric method as well.     

Markovian queueing systems with retrials and heterogeneous servers

The asymptotic performances of a random access and an ordered entry G/M/K/Oqueueing system with a stationary counting arrival process, K heterogeneous parallel servers, no waiting room and retrials are approximated based on a two-parameter method. In a random access system, units upon arrival are randomly assigned to one of the servers. In an ordered entry system, servers are indexed from 1 to K, and units first arrive at server i and if the server is found to be busy, those units arrive at server (i + 1), for i = 1 to K − 1. In both queueing systems, if units are not processed by one of the servers, those units are not lost, instead they retry to receive service by merging with the incoming arrival units.

To approximate the asymptotic performance of the above queueing systems, a recursive algorithm is suggested, and appropriate performance measures are presented to be used as comparison criteria at the design stage. Furthermore, numerical results are provided and approximation outcomes are compared against those from a simulation study.     

A closed-form solution on a level-dependent Markovian arrival process with queueing application

Abstract This paper reports a closed-form solution of the arrival events for a particular level-dependent Markovian arrival process (MAP). We apply the Baker–Hausdorff Lemma to the matrix expression of the number of arrival events in (0, t]. The successful derivation depends on the fact that the matrices representing the MAP have a specific structure. We report the results of numerical experiments indicating that the closed-form solution is less time-consuming than the uniformization technique for large values of t. As an application, we consider a finite-capacity, multi-server queueing model with impatient customers for possible use in automatic call distribution (ACD) systems. Our primary interest lies in performance measures related to customer waiting time, and we demonstrate how the closed-form solution is applicable to performance analysis.     

A queueing system with negative claims and a bunker for superseded claims in discrete time

We consider a discrete time single-line queueing system with independent geometric streams of regular and negative claims, infinite buffer, and geometric service. A negative claim pushes a regular claim out of the buffer queue and moves it to a bunker of infinite capacity. If the buffer is empty, a negative claim leaves the system without any effect. After servicing a claim, the system receives the next claim from the buffer, if it is not empty, or from the bunker. We obtain relations that allow computing stationary distributions for queues in the buffer and the bunker.     

On temporal characteristics in an exponential queueing system with negative claims and a bunker for ousted claims

We consider a queueing system with Poisson input streams of positive and negative claims, an infinite collector, and exponential service. A negative claim ousts a positive claim out of the collector queue and moves it to a bunker of unbounded capacity. If the collector is empty then a negative claim leaves the system with no influence on it. After a claim is serviced, the device receives the next claim from the collector or, if the collector is empty, from the bunker. For different combinations of FIFO and LIFO orders of choosing a claim for service from the collector’s queue, choosing a claim for service from the bunker’s queue, and ousting claims from the collector to the bunker, we obtain formulas for computing the stationary waiting time distribution for a claim to begin service and other temporal characteristics.     

One multiserver mixed queueing system with the limited waiting time

In this paper, a M/G/n/c multiserver queueing system with basic and standby servers is studied. Customers servicing is disturbed by failures of servers that make up a simplest flow. After the failure, the server needs a random time for renewal. It is also assumed that customers have limited, exponentially distributed waiting time in the system. The system is studied in both stationary and nonstationary modes.     

The stationary distribution invariance of states in a closed queueing network with temporarily non-active customers

A stationary functioning of a closed queueing network with temporarily non-active customers is analyzed. Non-active customers are located at network nodes in queues, being not serviced. For a customer, the feasibility of passing from its ordinary state to the temporarily non-active state (and backwards) is provided. Service times of customers at different nodes possess arbitrary distributions. Finally, the stationary distribution invariance of network states is established with respect to the functional form of customer service time distributions under fixed first-order moments.     

A repairable queueing model with two-phase service,start-up times and retrial customers

A repairable queueing model with a two-phase service in succession, provided by a single server, is investigated. Customers arrive in a single ordinary queue and after the completion of the first phase service, either proceed to the second phase or join a retrial box from where they retry, after a random amount of time and independently of the other customers in orbit, to find a position for service in the second phase. Moreover, the server is subject to breakdowns and repairs in both phases, while a start-up time is needed in order to start serving a retrial customer. When the server, upon a service or a repair completion finds no customers waiting to be served, he departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service and repair times are arbitrarily distributed. For such a system the stability conditions and steady state analysis are investigated. Numerical results are finally obtained and used to investigate system performance.