queueing theory Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...

, a discipline within the mathematical

theory of probability Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, a bulk queue (sometimes batch queue) is a general

queueing model Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...

where jobs arrive in and/or are served in groups of random size. Batch arrivals have been used to describe large deliveries and batch services to model a hospital out-patient department holding a clinic once a week, a transport link with fixed capacity and an elevator. Networks of such queues are known to have a

product form stationary distribution In probability theory, a product-form solution is a particularly efficient form of solution for determining some metric of a system with distinct sub-components, where the metric for the collection of components can be written as a product of the ...

under certain conditions. Under heavy traffic conditions a bulk queue is known to behave like a

reflected Brownian motion In probability theory, reflected Brownian motion (or regulated Brownian motion, both with the acronym RBM) is a Wiener process in a space with reflecting boundaries. In the physical literature, this process describes diffusion in a confined spac ...

Kendall's notation

Kendall's notation In queueing theory, a discipline within the mathematical theory of probability, Kendall's notation (or sometimes Kendall notation) is the standard system used to describe and classify a queueing node. D. G. Kendall proposed describing queueing mod ...

for single queueing nodes, the random variable denoting bulk arrivals or service is denoted with a superscript, for example M^''X''/M^''Y''/1 denotes an M/M/1 queue where the arrivals are in batches determined by the random variable ''X'' and the services in bulk determined by the random variable ''Y''. In a similar way, the

GI/G/1 queue In queueing theory, a discipline within the mathematical theory of probability, the G/G/1 queue represents the queue length in a system with a single server where interarrival times have a general (meaning arbitrary) distribution and service times h ...

is extended to GI^''X''/G^''Y''/1.

Bulk service

Customers arrive at random instants according to a

Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...

and form a single queue, from the front of which batches of customers (typically with a fixed maximum size) are served at a rate with independent distribution. The equilibrium distribution, mean and variance of queue length are known for this model. The optimal maximum size of batch, subject to operating cost constraints, can be modelled as a Markov decision process.

Bulk arrival

Optimal service-provision procedures to minimize long run expected cost have been published.

Waiting Time Distribution

The waiting time distribution of bulk Poisson arrival is presented in.

References

{{Stochastic processes Single queueing nodes