Bulk queue explained

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] 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, a transport link with fixed capacity^{[5] [6]} and an elevator.^[7]

Networks of such queues are known to have a product form stationary distribution under certain conditions.^[8] Under heavy traffic conditions a bulk queue is known to behave like a reflected Brownian motion.^{[9] [10]}

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.

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^[11]) are served at a rate with independent distribution.^[12] The equilibrium distribution, mean and variance of queue length are known for this model.

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.

Waiting Time Distribution

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

Notes and References

1. Singha . Chiamsiri . Michael S. . Leonard . 1981 . A Diffusion Approximation for Bulk Queues . . 27 . 10 . 1188–1199 . 2631086 . 10.1287/mnsc.27.10.1188.
2. Book: Özden, Eda . Discrete Time Analysis of Consolidated Transport Processes . 14. KIT Scientific Publishing . 978-3866448018. 2012 .
3. Book: A first course in bulk queues . M. L. . Chaudhry . James G. C. . Templeton . Wiley . 1983 . 978-0471862604.
4. Menachem . Berg . Frank . van der Duyn Schouten . Jorg . Jansen . 1998 . Optimal Batch Provisioning to Customers Subject to a Delay-Limit . . 44 . 5 . 684–697 . 2634473 . 10.1287/mnsc.44.5.684.
5. Rajat K. . Deb . 1978 . Optimal Dispatching of a Finite Capacity Shuttle . . 24 . 13 . 1362–1372 . 2630642 . 10.1287/mnsc.24.13.1362.
6. A. . Glazer . R. . Hassin . 1987 . Equilibrium Arrivals in Queues with Bulk Service at Scheduled Times . Transportation Science . 21 . 4 . 273–278 . 25768286 . 10.1287/trsc.21.4.273.
7. Marcel F. Neuts . 1967 . A General Class of Bulk Queues with Poisson Input . The Annals of Mathematical Statistics . 38 . 3 . 759–770 . 2238992 . 10.1214/aoms/1177698869 . free .
8. Henderson . W. . Taylor . P. G. . 10.1007/BF02411466 . Product form in networks of queues with batch arrivals and batch services . Queueing Systems. 6 . 71–87 . 1990 .
9. Iglehart . Donald L. . Ward . Whitt . Ward Whitt . 1970 . Multiple Channel Queues in Heavy Traffic. II: Sequences, Networks, and Batches . Advances in Applied Probability . 2 . 2 . 355–369 . 1426324 . 30 Nov 2012 . 10.1017/s0001867800037435.
10. Harrison . P. G. . Peter G. Harrison . Hayden . R. A. . Knottenbelt . W. . 10.1016/j.peva.2013.08.011 . Product-forms in batch networks: Approximation and asymptotics . . 70 . 10 . 822 . 2013 . 10.1.1.352.5769 . 2015-09-04 . https://web.archive.org/web/20160303205349/http://pubs.doc.ic.ac.uk/batches-heavy-traffic-journal/batches-heavy-traffic-journal.pdf . 2016-03-03 . dead .
11. Downton . F. . 1955 . Waiting Time in Bulk Service Queues . . 17 . 2 . 256–261 . . 2983959 .
12. On Queueing Processes with Bulk Service . Norman T. J. . Bailey . . 80–87 . 2984011 . 61 . 1 . 1954.
13. Rajat K. . Deb . Richard F. . Serfozo . 1973 . Optimal Control of Batch Service Queues . Advances in Applied Probability . 5 . 2 . 340–361 . 1426040. 10.2307/1426040 .
14. Jyotiprasad . Medhi . 1975 . Waiting Time Distribution in a Poisson Queue with a General Bulk Service Rule . Management Science . 21 . 7 . 777–782 . 2629773 . 10.1287/mnsc.21.7.777.