# 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 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 is extended to GI^{X}/G^{Y}/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

- ↑
^{1.0}^{1.1}Chiamsiri, Singha; Leonard, Michael S. (1981). "A Diffusion Approximation for Bulk Queues".*Management Science*.**27**(10): 1188–1199. JSTOR 2631086. - ↑ Özden, Eda.
*Discrete Time Analysis of Consolidated Transport Processes*. KIT Scientific Publishing. p. 14. ISBN 3866448015. - ↑ Chaudhry, M. L.; Templeton, James G. C. (1983).
*A first course in bulk queues*. Wiley. ISBN 0471862606. - ↑
^{4.0}^{4.1}Berg, Menachem; van der Duyn Schouten, Frank; Jansen, Jorg (1998). "Optimal Batch Provisioning to Customers Subject to a Delay-Limit".*Management Science*.**44**(5): 684–697. JSTOR 2634473. - ↑
^{5.0}^{5.1}^{5.2}Bailey, Norman T. J. (1954). "On Queueing Processes with Bulk Service".*Journal of the Royal Statistical Society, Series B*.**61**(1): 80–87. JSTOR 2984011. - ↑ Deb, Rajat K. (1978). "Optimal Dispatching of a Finite Capacity Shuttle".
*Management Science*.**24**(13): 1362–1372. JSTOR 2630642. - ↑ Glazer, A.; Hassin, R. (1987). "Equilibrium Arrivals in Queues with Bulk Service at Scheduled Times".
*Transportation Science*.**21**(4): 273–278. JSTOR 25768286. doi:10.1287/trsc.21.4.273. - ↑ Marcel F. Neuts (1967). "A General Class of Bulk Queues with Poisson Input" (PDF).
*The Annals of Mathematical Statistics*.**38**(3): 759–770. JSTOR 2238992. doi:10.1214/aoms/1177698869. - ↑ Henderson, W.; Taylor, P. G. (1990). "Product form in networks of queues with batch arrivals and batch services".
*Queueing Systems*.**6**: 71. doi:10.1007/BF02411466. - ↑ Iglehart, Donald L.; Ward, Whitt (1970). "Multiple Channel Queues in Heavy Traffic. II: Sequences, Networks, and Batches" (PDF).
*Advances in Applied Probability*. Applied Probability Trust.**2**(2): 355–369. JSTOR 1426324. Retrieved 30 Nov 2012. - ↑ Harrison, P. G.; Hayden, R. A.; Knottenbelt, W. (2013). "Product-forms in batch networks: Approximation and asymptotics" (PDF).
*Performance Evaluation*.**70**(10): 822. doi:10.1016/j.peva.2013.08.011. - ↑ Downton, F. (1955). "Waiting Time in Bulk Service Queues".
*Journal of the Royal Statistical Society, Series B*. Royal Statistical Society.**17**(2): 256–261. JSTOR 2983959. - ↑ Deb, Rajat K.; Serfozo, Richard F. (1973). "Optimal Control of Batch Service Queues".
*Advances in Applied Probability*.**5**(2): 340–361. JSTOR 1426040.