# 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. Chiamsiri, Singha; Leonard, Michael S. (1981). "A Diffusion Approximation for Bulk Queues". Management Science. 27 (10): 1188–1199. JSTOR 2631086.
2. Özden, Eda. Discrete Time Analysis of Consolidated Transport Processes. KIT Scientific Publishing. p. 14. ISBN 3866448015.
3. Chaudhry, M. L.; Templeton, James G. C. (1983). A first course in bulk queues. Wiley. ISBN 0471862606.
4. 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. 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.
6. Deb, Rajat K. (1978). "Optimal Dispatching of a Finite Capacity Shuttle". Management Science. 24 (13): 1362–1372. JSTOR 2630642.
7. Glazer, A.; Hassin, R. (1987). "Equilibrium Arrivals in Queues with Bulk Service at Scheduled Times". Transportation Science. 21 (4): 273–278. doi:10.1287/trsc.21.4.273. JSTOR 25768286.
8. Marcel F. Neuts (1967). "A General Class of Bulk Queues with Poisson Input" (PDF). The Annals of Mathematical Statistics. 38 (3): 759–770. doi:10.1214/aoms/1177698869. JSTOR 2238992.
9. 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.
10. 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.
11. 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.
12. 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.
13. Deb, Rajat K.; Serfozo, Richard F. (1973). "Optimal Control of Batch Service Queues". Advances in Applied Probability. 5 (2): 340–361. JSTOR 1426040.