Batch Service Queue Research Articles

A batch service queueing system with multiple vacations, setup time and server's choice of admitting reservice

In this paper, a bulk queueing system with server’s choice of admitting reservice, multiple vacations and setup time is considered. At a service completion, the leaving batch may request for reservice with probability ʌ and it is not mandatory to accept it; the server admits this request with a probability α. After the reservice or service completion without request for reservice, if the queue length is less than a, the server leaves for a vacation of random length. After this vacation, if the queue length is still less than ‘a’, the server leaves for another vacation, etc., until he finally finds at least ‘a’ customers waiting for service. At a vacation completion epoch, if the server finds at least ‘a’ customers waiting for service, he requires a setup time to start the service. After a setup time or on service completion or on reservice completion, if the server finds at least ‘a’ customers waiting for service say, [ he serves a batch of min (ξ, b) customers, where t ba. The probability generating func...

Discrete-time renewal input multiple vacation queue with accessible and non-accessible batches

This paper analyzes a discrete-time finite buffer single server accessible and non-accessible batch service queue with multiple vacations. The inter-arrival times of customers are arbitrarily distributed. The service times and vacation times are both geometrically distributed. The server starts service in batches of maximum size b with a minimum threshold value a, otherwise it goes for vacation. There is a provision to include the late arriving customers in the ongoing service batch if the size of the batch being served is less than a threshold value d (a ≤ d ≤ b) and the service times are not affected by such inclusions. Using the supplementary variable and the embedded Markov chain techniques, we obtain the steady state probabilities at pre-arrival and arbitrary epochs. Some performance measures, waiting time distribution in the queue, and special cases of the model have also been discussed. Finally, sensitivity analysis is carried out in the form of table and graphs.

Monotonicity properties of user equilibrium policies for parallel batch systems

We study a simple network with two parallel batch-service queues, where service at a queue commences when the batch is full and each queue is served by infinitely many servers. A stream of general arrivals observe the current state of the system on arrival and choose which queue to join to minimize their own expected transit time. We show that for each set of parameter values there exists a unique user equilibrium policy and that it possesses various monotonicity properties. User equilibrium policies for probabilistic routing are also discussed and compared with the state-dependent setting.

Analysis of Finite-Buffer Discrete-Time Batch-Service Queue with Multiple Vacations

This paper investigates a discrete-time single-server finite-buffer batch-service queue with multiple vacations wherein arrivals occur according to a discrete-time renewal process. Service and vacation times are mutually independent and geometrically distributed. The service is performed in batches and the server takes vacations when the system does not have any waiting customers at a service completion epoch or a vacation completion epoch. The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system. Using the supplementary variable and the embedded Markov chain techniques, we obtain the queue-length distributions at pre-arrival, arbitrary and outside observer's observation epochs. Some performance measures, waiting-time distribution in the system and cost analysis have also been discussed. Finally, a variety of numerical results as tables and graphs showing the effect of model parameters on key performance measures are presented.

Analysis of discrete-time batch service renewal input queue with multiple working vacations

This paper investigates a discrete-time single server batch service queue with multiple working vacations wherein arrivals occur according to a discrete-time renewal process. The server works with a different service rate rather than completely stopping during the vacation period. The service is performed in batches and the server takes a vacation when the system does not have any waiting customers at a service completion epoch or a vacation completion epoch. We present a recursive method, using the supplementary variable technique to obtain the steady-state queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The displacement operator method is used to solve simultaneous non-homogeneous difference equations. Some performance measures and waiting-time distribution in the system have also been discussed. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

Analysis of threshold-based batch-service queueing systems with batch arrivals and general service times

Most research concerning batch-service queueing systems has focussed on some specific aspect of the buffer content. Further, the customer delay has only been examined in the case of single arrivals. In this paper, we examine three facets of a threshold-based batch-service system with batch arrivals and general service times. First, we compute a fundamental formula from which an entire gamut of known as well as new results regarding the buffer content of batch-service queues can be extracted. Second, we produce accurate light- and heavy-traffic approximations for the buffer content. Third, we calculate various quantities with regard to the customer delay. This paper thus provides a whole spectrum of tools to evaluate the performance of batch-service systems.

A renewal input single and batch service queues with accessibility to batches

This paper presents a renewal input single and batch service infinite (finite) buffer queueing system with accessibility to the batches. The inter-arrival times of successive arrivals are assumed to be independent and arbitrarily distributed. The customers are served by a single-server either one at a time or in batches with accessibility to the batches depending upon the number of customers present in the system. The service times are assumed to be independent and exponentially distributed. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining inter-arrival time, to obtain the steady-state probability distribution of the number of customers in the system/queue. A cost model is developed to determine the optimum service rates which minimizes the total expected cost for assumed numerical values of system parameters. Various performance measures are provided based on this analysis.

Renewal input infinite buffer batch service queue with single exponential working vacation and accessibility to batches

This paper considers an infinite buffer single server batch service queue with single exponential working vacation policy. The inter-arrival times are generally independent and identically distributed random variables and the service times are exponential. The server accesses new arrivals even after service has started on any batch of initial number a. This operation continues till the service time of the ongoing batch is completed or the maximum accessible limit d of the batch being served is attained whichever occurs first. The supplementary variable technique and the recursive method are used to develop the steady-state queue length distributions at pre-arrival and arbitrary epochs. Some performance measures and numerical results are discussed.

A queueing model for general group screening policies and dynamic item arrivals

Classification of items as good or bad can often be achieved more economically by examining the items in groups rather than individually. If the result of a group test is good, all items within it can be classified as good, whereas one or more items are bad in the opposite case. Whether it is necessary to identify the bad items or not, and if so, how, is described by the screening policy. In the course of time, a spectrum of group screening models has been studied, each including some policy. However, the majority ignores that items may arrive at random time epochs at the testing center in real life situations. This dynamic aspect leads to two decision variables: the minimum and maximum group size. In this paper, we analyze a discrete-time batch-service queueing model with a general dependency between the service time of a batch and the number of items within it. We deduce several important quantities, by which the decision variables can be optimized. In addition, we highlight that every possible screening policy can, in principle, be studied, by defining the dependency between the service time of a batch and the number of items within it appropriately.

User equilibria for a parallel queueing system with state dependent routing

Consider a system of two queues in parallel, one of which is a ?|M|1 single-server infinite capacity queue, and the other a ?|G (N)|? batch service queue. A stream of general arrivals choose which queue to join, after observing the current state of the system, and so as to minimize their own expected delay. We show that a unique user equilibrium (user optimal policy) exists and that it possesses various monotonicity properties, using sample path and coupling arguments. This is a very simplified model of a transportation network with a choice of private and public modes of transport. Under probabilistic routing (which is equivalent to the assumption that users have knowledge only of the mean delays on routes), the network may exhibit the Downs---Thomson paradox observed in transportation networks with expected delay increasing as the capacity of the ?|M|1 queue (private transport) is increased. We give examples where state-dependent routing mitigates the Downs---Thomson effect observed under probabilistic routing, and providing additional information on the state of the system to users reduces delay considerably.

Finite buffer renewal input multiple exponential vacations queue with accessible and non-accessible batches

In this paper, we study a finite buffer single server accessible and non-accessible batch service queue with multiple exponential vacations. The inter-arrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable technique (SVT) and the embedded Markov chain technique (EMCT), we obtain the steady state system (queue) length distributions at pre-arrival and arbitrary epochs. Some numerical results are presented in the form of table and graphs.

Complete characterisation of the customer delay in a queueing system with batch arrivals and batch service

Whereas the buffer content of batch-service queueing systems has been studied extensively, the customer delay has only occasionally been studied. The few papers concerning the customer delay share the common feature that only the moments are calculated explicitly. In addition, none of these surveys consider models including the combination of batch arrivals and a server operating under the full-batch service policy (the server waits to initiate service until he can serve at full capacity). In this paper, we aim for a complete characterisation—i.e., moments and tail probabilities - of the customer delay in a discrete-time queueing system with batch arrivals and a batch server adopting the full-batch service policy. In addition, we demonstrate that the distribution of the number of customer arrivals in an arbitrary slot has a significant impact on the moments and the tail probabilities of the customer delay.

Delay analysis of two batch-service queueing models with batch arrivals: Geo X /Geo c /1

In this paper, we compute the probability generating functions (PGF’s) of the customer delay for two batch-service queueing models with batch arrivals. In the first model, the available server starts a new service whenever the system is not empty (without waiting to fill the capacity), while the server waits until he can serve at full capacity in the second model. Moments can then be obtained from these PGF’s, through which we study and compare both systems. We pay special attention to the influence of the distribution of the arrival batch sizes. The main observation is that the difference between the two policies depends highly on this distribution. Another conclusion is that the results are considerably different as compared to Bernoulli (single) arrivals, which are frequently considered in the literature. This demonstrates the necessity of modeling the arrivals as batches.

On the batch arrival batch service queue with finite buffer under server’s vacation: MX/GY/1/N queue

This paper considers a finite-buffer batch arrival and batch service queue with single and multiple vacations. The steady-state distributions of the number of customers in the queue at service completion, vacation termination, departure, arbitrary and pre-arrival epochs have been obtained. Finally, various performance measures such as average queue length, average waiting time, probability that the server is busy, blocking probabilities, etc. are discussed along with some numerical results. The effect of certain model parameters on the key performance measures have also been investigated. The model has potential application in several areas including manufacturing, internet web-server and telecommunication systems.

Analysis of the MAP/G(a,b)/1/N queue with multiple vacations

This paper deals with a batch service queue and multiple vacations. The system consists of a single server and a waiting room of finite capacity. Arrival of customers follows a Markovian arrival process ( MAP ). The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in batches of maximum size ‘ b ’ with a minimum threshold value ‘ a ’. We obtain the queue length distributions at various epochs along with some key performance measures. Finally, some numerical results have been presented.

Equivalences of Batch-Service Queues and Multi-Server Queues and Their Complete Simple Solutions in Terms of Roots

In this article, we first present a unified discussion of several equivalence relationships among (as well as between) batch-service queues and multi-server queues, in terms of the stationary queue-length and waiting-time distributions. Then, we present a complete and simple solution for the queue-length and waiting-time distributions of the discrete-time multi-server deterministic-service Geo/D/b queue, in terms of roots of the so-called characteristic equation. This solution also represents the solutions for the other equivalent queues, as a result of the equivalence relationships. To aid in the applications of these results, sample numerical results are presented at the end.

The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations

We consider a finite buffer batch service queueing system with multiple vacations wherein the input process is Markovian arrival process (MAP). The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service- and vacation- times are arbitrarily distributed. We obtain the queue length distributions at service completion, vacation termination, departure, arbitrary and pre-arrival epochs. Finally, some performance measures such as loss probability, average queue lengths are discussed. Computational procedure has been given when the service- and vacation- time distributions are of phase type (PH-distribution).

ANALYSIS OF A GI/G Y /1 SYSTEM IN DISCRETE-TIME

ABSTRACT We consider a class of single server queueing systems in which customers arrive singly and service is provided in batches, depending on the number of customers waiting when the server becomes free. Service is independent of the batch size. This system could also be considered as a batch service queue in which a server visits the queue at arbitrary times and collects a batch of waiting customers for service, or waits for a customer to arrive if there are no waiting customers. A waiting server immediately collects and processes the first arriving customer. The system is considered in discrete time. The interarrival times of customers and the inter-visit times of the server, which we call the service time, have general distributions and are represented as remaining time Markov chains. We analyze this system using the matrix-geometric method and show that the resulting R matrix can be determined explicitly in some special cases and the stationary distributions are known semi-explicitly in some other special cases.

Analytic and numerical aspects of batch service queues with single vacation

This paper deals with an M/ G/1 batch service queue where customers are served in batches of maximum size b with a minimum threshold value a. The server takes a single vacation when he finds less than a customers after the service completion. The vacation time of the server is arbitrarily distributed. Using the supplementary variable method we obtain the probability generating functions of the queue length distributions at various epochs. We also obtain relations among queue length distributions at arbitrary, service (vacation) termination epochs. Further their evaluation is also discussed. Finally, some numerical results and graphs are presented.

On the stochastic domination for batch-arrival, batch-service and assemble-transfer queueing networks

Stochastic monotonicity properties for various classes of queueing networks have been established in the literature mainly with the use of coupling constructions. Miyazawa and Taylor (1997) introduced a class of batch-arrival, batch-service and assemble-transfer queueing networks which can be thought of as generalized Jackson networks with batch movements. We study conditions for stochastic domination within this class of networks. The proofs are based on a certain characterization of the stochastic order for continuous-time Markov chains, written in terms of their associated intensity matrices.