Batch arrival

The purpose of this tutorial is to survey queueing networks, a class of stochastic models extensively applied to represent and analyze resource sharing systems such as communication and computer systems. Queueing networks (QNs) have been proved to be a powerful and versatile tool for system performance evaluation and prediction. First we briefly survey QNs that consist of a single service center, i.e., the basic queueing systems defined under various hypotheses, and we discuss their analysis to evaluate a set of performance indices, such as resource utilization and throughput and customer response time. Their solution is based on the introduction of an underlying stochastic Markov process. Then, we introduce QNs that consist of a set of service centers representing the system resources that provide service to a collection of customers that represent the users. Various types of customers define the customers classes in the network that are gathered in chains. We consider various analytical methods to analyze QNs with single-class and multiple-class. We mostly focus on product-form QNs that have a simple closed form expression of the stationary state distribution that allows to define efficient algorithms to evaluate average performance measures. We review the basic results, stating from the BCMP theorem that defines a large class of product-form QNs, and we present the main solution algorithms for single-class e multiple-class QNs. We discuss some interesting properties of QNs including the arrival theorem, exact aggregation and insensitivity. Finally, we discuss some particular models of product-form QNs that allow to represent special system features such as state-dependent routing, negative customers, customers batch arrivals and departures and finite capacity queues. The class of QN models is illustrated through some application examples of to analyze computer and communication systems.

This paper deals with the optimal control policy of
two-phase service  M
X
/ Ek /1  queues with vacation, N-policy,
server break downs and balking. Generating functions method is
used to derive the system characteristics.  The total expected cost
function is developed to determine the optimal threshold of N at
a minimum cost. Numerical experiment is performed to validate
the analytical results. The sensitivity analysis has been carried
out to examine the effect of different parameters in the system

We analyze a Processor-Sharing queue with Batch arrivals. Our analysis is based on the integral equation derived by Kleinrock, Muntz and Rodemich. Using the contraction mapping principle, we demonstrate the existence and uniqueness of a solution to the integral equation. Then we provide asymptotical analysis as well as tight bounds for the expected response time conditioned on the service time. In particular, the asymptotics for large service times depends only on the first moment of the service time distribution and on the first two moments of the batch size distribution. That is, similarly to the Processor-Sharing with single arrivals, in the Processor-Sharing queue with batch arrivals the expected conditional response time is finite even when the service time distribution has infinite second moment. Finally, we show how the present results can be applied to the Multi-Level Processor-Sharing scheduling.

Inspired by the need for performability models for HSDPA user equipment, a Markovian queue with varying number of servers is conceived. The arrival and the service processes, the number of allocated or active servers of the queue are inherently, and independently (or jointly) Markov modulated. Batch arrivals, batch services, autocorrelation of inter-arrival times, and autocorrelation of batch sizes can be

Perfect simulation, or coupling from the past, is an ecient technique for sampling the steady state of monotone discrete time Markov chains. Indeed, one only needs to consider two trajectories corresponding to minimal and maximal state in the system. We show here that even for non-monotone sys- tems one only needs to compute two trajectories: an in- mum and supremum

In this paper we study a system with two stations in tandem.There is an infinite buffer in front of the first station and no buffer between the two stations. The customers come in batches; the batches contain random numbers of customers and arrive according to a Poisson process. Assuming arbitrary service time distributions at the two stations, we derive an

We consider a queueing network with concurrent batch arrivals and batch services. In addition to the batch movement assumptions made in the regular queueing literature, in this paper we assume that the batch movement can also be triggered by arrivals and/or departures. Specifically, an arriving or departing batch may induce another event to occur before they are routed. This triggered event may either be the addition of a batch of customers to the network, or the removal of a batch of customers from the network. This special feature makes it useful in the study of discrete event dynamic systems with concurrent or simultaneous sequential activities. By assuming that there is an additional arrival process to the network, we show that its stationary distribution has a product form, which depends on a set of non-standard nonlinear traffic equations. Furthermore, this simple product form solution is a stochastic upper bound for the network without the additionally introduced arrivals. Be...

The authors present and solve a discrete-time G/sup (X)//D/1-S queuing system with a finite queue size and batch arrivals with a general batch size distribution. The motivation for this model arises from performance modeling of a statistical multiplexer with synchronous transmission of fixed-size data units in synchronous time slots. The arrival process to the multiplexer, for example, may originate from a number of independent sources with packets of variable lengths. Hence, a packet arrival corresponds to an arrival of a batch of data units. Different performance measures such as percentage of packet loss and data-unit loss are considered under two different admission policies of packets into the queue.>