In queueing theory, a discipline within the mathematical theory of probability, a bulk queue (sometimes batch queue) is a general queueing model where jobs arrive in and/or are served in groups of random size.:vii 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 under certain conditions. Under heavy traffic conditions a bulk queue is known to behave like a reflected Brownian motion.
In Kendall's notation for single queueing nodes, the random variable denoting bulk arrivals or service is denoted with a superscript, for example MX/MY/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 is extended to GIX/GY/1.
Customers arrive at random instants according to a Poisson process 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.
Optimal service-provision procedures to minimize long run expected cost have been published.
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