# Bulk queue

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]

## References

1. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
2. Özden, Eda. Discrete Time Analysis of Consolidated Transport Processes. KIT Scientific Publishing. p. 14. ISBN 3866448015.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
3. Chaudhry, M. L.; Templeton, James G. C. (1983). A first course in bulk queues. Wiley. ISBN 0471862606.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
4. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
5. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
6. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
7. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
8. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
9. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
10. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
11. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
12. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).
13. Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).