In queueing theory, a discipline within the mathematical theory of probability, a bulk queue[1] (sometimes batch queue[2]) is a general queueing model where jobs arrive in and/or are served in groups of random size.[3]: vii Batch arrivals have been used to describe large deliveries[4] and batch services to model a hospital out-patient department holding a clinic once a week,[5] a transport link with fixed capacity[6][7] and an elevator.[8]
Networks of such queues are known to have a product form stationary distribution under certain conditions.[9] Under heavy traffic conditions a bulk queue is known to behave like a reflected Brownian motion.[10][11]
Kendall's notation
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.[1]
Bulk service
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[12]) are served at a rate with independent distribution.[5] The equilibrium distribution, mean and variance of queue length are known for this model.[5]
The optimal maximum size of batch, subject to operating cost constraints, can be modelled as a Markov decision process.[13]
Bulk arrival
Optimal service-provision procedures to minimize long run expected cost have been published.[4]
Waiting Time Distribution
The waiting time distribution of bulk Poisson arrival is presented in.[14]