Bulk Service Queue Research Articles

Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues

We consider decay properties including the decay parameter, invariant measures and quasi-stationary distributions for a Markovian bulk-arrival and bulk-service queue which stops when the waiting line is empty. Investigating such a model is crucial for understanding the busy period and other related properties of the Markovian bulk-arrival and bulk-service queuing processes. The exact value of the decay parameter λ C is first obtained. We show that the decay parameter can be easily expressed explicitly. The invariant measures and quasi-distributions are then revealed. We show that there exists a family of invariant measures indexed by λ∈[0,λ C ]. We then show that under some mild conditions there exists a family of quasi-stationary distributions also indexed by λ∈[0,λ C ]. The generating functions of these invariant measures and quasi-stationary distributions are presented. We further show that this stopped Markovian bulk-arrival and bulk-service queueing model is always λ C -transient. Some deep properties regarding λ C -transience are examined and revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

Markovian bulk-arrival and bulk-service queues with state-dependent control

We study a modified Markovian bulk-arrival and bulk-service queue incorporating state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated and the relationship w...

DESIGN OF A QUEUEING MODEL FOR ANALYSISING THE BEHAVIOUR OF POWER TRANSMISSION SYSTEM

This paper deals with the Difference Queueing model for the power transmission system. The device called “Thyristor” which consists of three terminals modeled with queues. Gohain et al. (1979) have explained the difference equation technique which is applied for bulk service queues. Later on, Pochee(1998) has discussed about the performance of computer mirroring system. In this paper a comparison is made between the two queueing models M/M/1 and M/DTQ/1. Under high load conditions it is proved that M/DTQ/1 model, approximated the behaviour of the transmission system. Generation of pulses and utilization can be made moderate by using this queueing model. Based on the numerical calculations and graphical representations M/DTQ/1 can be considered as an appropriate model for transmission system.

A Simple and Complete Solution to the Stationary Queue-Length Probabilities of a Bulk-Arrival Bulk-Service Queue

A complete solution for the stationary queue-length distribution of a bulk-arrival, bulk-service (GIX/MY/1) queue is presented. Beginning with a known expression for the probability generating function of the stationary pre-arrival-epoch queue-length distribution, the roots method is used to invert it and determine all probabilities. Next, using level crossing arguments, theoretical relationships between pre-arrival and arbitrary-epoch probabilities are developed. These relationships are then used to directly determine a complete set of probabilities for the arbitrary- epoch queue-length distribution. Finally, selected examples are presented. These demonstrate how, given arbitrary arrival time, arrival group size and service batch size probability distributions, a complete solution for the stationary queue-length probabilities can be readily determined.

A quorum queueing system with an unreliable server

We analyze a bulk service queueing system with an unreliable server, Poisson input, and general service and repair times. A stability condition and steady state system size distributions are obtained. The optimal management policy is discussed and illustrative examples are provided.

Multiserver bulk service discrete-time queue with finite buffer and renewal input

This paper analyzes a discrete-time finite-buffer multi-server bulk-service queueing system in which the interarrival- and service-times are, respectively, arbitrarily and geometrically distributed. Using the supplementary variable and the imbedded Markov-chain techniques, the queue is analyzed for the early arrival system. We obtain state probabilities at prearrival, arbitrary and outside observer’s observation epochs. Some performance measures, waiting-time distribution in the queue along with some numerical results, and special cases of the model have also been discussed. Finally, it is shown that in the limiting case the results obtained in this paper tend to the continuous-time counterpart.

Optimal management policy for a single and bulk service queue under Bernoulli vacation schedules

This paper generalises the single server queue with single and bulk service characterised by the bilevel service discipline introduced by Baburaj (1999). In this system if the size (n) is not more than r, then the server takes a single customer for service according to FCFS rule and if the size is more than r then he serves min{n,R} (R > r) units in a batch according to FCFS rule. While Baburaj assumes exponential service times we consider general service time distributions. We also consider that the server implements a Bernoulli vacation schedule, so that a vacation is taken with some positive probability when a service is completed. The first objective is to derive the probability generating function of the number of customers in the system at a service completion epoch and at an arbitrary instant of time, as well as the performance characteristics of the system. The second objective is to develop an optimal management policy through a quick search algorithm.

Discrete-time bulk-service queue with two heterogeneous servers

This paper analyzes a discrete-time bulk-service queueing system with two heterogeneous servers, i.e., two batch servers working with different service rates. The interarrival times of customers and service times of batches are assumed to be independent and geometrically distributed. Applications can be found in a wide variety of real systems including servers formed with different types of processors as a consequence of system updates, nodes in telecommunications network with links of different capacities, nodes in wireless systems serving different mobile users. We obtain closed-form expressions for the steady-state probabilities at arbitrary epoch with the help of displacement operator method and derive the outside observer’s observation epoch probabilities and waiting time distribution measured in slots. Computational experiences with a variety of numerical results in the form of tables and graphs are discussed. Moreover, some queueing models discussed in the literature are derived as special cases of our model. Finally, it is shown that in the limiting case the results obtained in this paper tend to the continuous-time counterpart.

Steady state analysis and computation of the GI[ x] / Mb/1/ L queue with multiple working vacations and partial batch rejection

This paper analyzes a finite-buffer bulk-arrival bulk-service queueing system with multiple working vacations and partial batch rejection in which the inter-arrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the embedded Markov chain techniques, we obtain the waiting queue-length distributions at pre-arrival and arbitrary epochs. We also present Laplace–Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.

A Hysteretic Bulk Quorum Queue with a Choice of Service and Optional Re-Service

We derive in this paper the optimal control policy for a two-phase bulk service queueing system where the group of customers has the option to choose the service type in either phase of service. The group has also the option not to take the second phase. The server takes multiple vacations during the idle period which is terminated according to the N-policy. Also, a setup time is required following an idle period, and the server follows the Bernoulli single vacation schedule discipline. The arrival process is compound Poisson.

Analyzing a multiserver bulk-service finite-buffer queue

This paper analyzes a finite-buffer multiserver bulk-service queueing system in which the interarrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the queue-length distributions at prearrival and arbitrary epochs. We also present Laplace–Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.

A bulk service queue with a choice of service and re-service under Bernoulli schedule

This paper investigates the queueing process of a bulk service queueing system under Bernoulli schedule. It also generalizes some known scenarios concerning the choice of service and re-service types. The queueing process is studied both in discrete time and in continuous time. Performance measures are derived and used to implement an optimal management policy of the system.

Bounds and Approximations for the Fixed-Cycle Traffic-Light Queue

This paper deals with the fixed-cycle traffic-light (FCTL) queue, where vehicles arrive at an intersection controlled by a traffic light and form a queue. The traffic light alternates between green and red periods, and delayed vehicles are assumed to depart during the green period at equal time intervals. The key performance characteristic in the FCTL queue is the so-called mean overflow, defined as the mean queue length at the end of a green period. An exact solution for the mean overflow is available, but it has been considered to be of little practical value because it requires some numerical procedures. Therefore, most of the literature on the FCTL queue is about deriving approximations for the mean overflow. In deriving these approximations, most authors first approximate the FCTL queue by a bulk-service queue, approximate the mean overflow in the bulk-service queue, and use this as an approximation for the mean overflow in the FCTL queue. So far no quantitative comparison of both models has been given. We compare both models and assess the quality of the approximation for various settings of the parameter values. In this comparison and throughout the paper we do not restrict ourselves to Poisson arrivals, but consider a more general arrival process instead. We discuss the numerical issues that need to be resolved to calculate the exact expression for the mean overflow in both queues and show that clear computational schemes are available. Next, we present several bounds and approximations of the mean overflow that do not require numerical procedures. In particular, we derive a new approximation based on the heavy traffic limit and a scaling argument. We compare the new bounds and approximation with the existing ones. We elaborate on the impact of several parameters, like the length of the green and red period and the variance of the arrival distribution. Each of these parameters turns out to be crucial.

Discrete-time bulk-service queues with accessible and non-accessible batches

This paper analyzes a discrete-time single-server infinite- (finite-) buffer bulk-service queues. The inter-arrival times of successive arrivals are assumed to be independent and geometrically distributed. The customers are served by a single-server in accessible or non-accessible batches. The service times of batches are also assumed to be independent and geometrically distributed. The steady-state probabilities with computational experiences have been presented.

Cascades of queues

This paper presents results from a bulk service queueing system, where the main service takes place at fixed time intervals, but where back-up (or relief) services may be introduced in times of heavy demand on the system. On occasions the back-up service itself may receive further back-up services, thus leading to the concept of a cascade of queues flowing from the original main service. Analytic expressions are derived for the mean queue lengths at the points of delivery of service, and the computational requirements are discussed in detail. Sample numerical results and a simple cost analysis are provided.

Models for data transmission delay in cable networks

Cable networks have been upgraded in recent years to support bidirectional traffic rather than just unidirectional television signals. Thus, it has become possible to use cable networks for internet browsing and telephony. In this paper, we propose delay models to determine how this bidirectional traffic is best handled and to evaluate the resulting performance of the network, notably its delay properties. It is shown that these models, variations on the repairman problem and the bulk service queue, lead to actual improvements in the data transmission schedules for cable networks. Moreover, these models can be combined in order to arrive at approximations for the average packet delay. Our theoretical calculations are backed up by simulations.

A bulk quorum queueing system with a random setup time under N-policy and with Bernoulli vacation schedule

This paper is concerned with the optimal management of a batch arrival, bulk service queueing system with random set-up time under Bernoulli vacation schedule and N-policy. If the number of customers in the system at a service completion is larger than some integer r, then the server starts processing a group of r customers. If, on the other hand, it is smaller than r, then the server idles and waits for the line to grow up to some other integer N, (N ≥ r). Using the embedded Markov chain and semi-regenerative techniques, we obtain all the system characteristics required to build a linear cost structure. Then, we implement a simple search procedure to determine the optimal thresholds levels r and N. An illustrative example is presented.

A Quorum Queueing System with a Random Setup Time under N-policy and with Bernoulli Vacation Schedule

We aim in this paper at designing an optimal management policy for a bulk service queueing system with random set-up time under Bernoulli vacation schedule and N-policy. We first study the discrete time parameter and continuous time parameter stochastic processes and derive all the quantities required to build a linear cost structure. Then an algorithm is suggested to determine the optimal management policy. An illustrative example is presented to show how to implement this policy and a sensitivity analysis is conducted to determine the effect of the system parameters.

Factorization and Stochastic Decomposition Properties in Bulk Queues with Generalized Vacations

This paper considers a class of stationary batch-arrival, bulk-service queues with generalized vacations. The system consists of a single server and a waiting room of infinite capacity. Arrivals of customers follow a batch Markovian arrival process. The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in groups of fixed size B. For this class of queues, we show that the vector probability generating function of the stationary queue length distribution is factored into two terms, one of which is the vector probability generating function of the conditional queue length distribution given that the server is on vacation. The special case of batch Poisson arrivals is carefully examined, and a new stochastic decomposition formula is derived for the stationary queue length distribution.

Analytic Computation Schemes for the Discrete-Time Bulk Service Queue

In commonly used root-finding approaches for the discrete-time bulk service queue, the stationary queue length distribution follows from the roots inside or outside the unit circle of a characteristic equation. We present analytic representations of these roots in the form of sample values of periodic functions with analytically given Fourier series coefficients, making these approaches more transparent and explicit. The resulting computational scheme is easy to implement and numerically stable. We also discuss a method to determine the roots by applying successive substitutions to a fixed point equation. We outline under which conditions this method works, and compare these conditions with those needed for the Fourier series representation. Finally, we present a solution for the stationary queue length distribution that does not depend on roots. This solution is explicit and well-suited for determining tail probabilities up to a high accuracy, as demonstrated by some numerical examples.