Inqueueing theory, a discipline within the mathematicaltheory of probability, abulk queue^[1] (sometimesbatch queue^[2]) is a generalqueueing 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 aproduct form stationary distribution under certain conditions.^[9] Under heavy traffic conditions a bulk queue is known to behave like areflected Brownian motion.^[10][11]

InKendall's notation 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 anM/M/1 queue where the arrivals are in batches determined by the random variableX and the services in bulk determined by the random variableY. In a similar way, theGI/G/1 queue is extended to GI^X/G^Y/1.^[1]

Customers arrive at random instants according to aPoisson 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 aMarkov decision process.^[13]

Optimal service-provision procedures to minimize long run expected cost have been published.^[4]

The waiting time distribution of bulk Poisson arrival is presented in.^[14]

• Single queueing nodes