Bulk queue

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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]

Waiting Time Distribution

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

References

^ ^a ^b Chiamsiri, Singha; Leonard, Michael S. (1981). "A Diffusion Approximation for Bulk Queues". Management Science. 27 (10): 1188–1199. doi:10.1287/mnsc.27.10.1188. JSTOR 2631086.
^ Özden, Eda (2012). Discrete Time Analysis of Consolidated Transport Processes. KIT Scientific Publishing. p. 14. ISBN 978-3866448018.
^ Chaudhry, M. L.; Templeton, James G. C. (1983). A first course in bulk queues. Wiley. ISBN 978-0471862604.
^ ^a ^b 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. doi:10.1287/mnsc.44.5.684. JSTOR 2634473.
^ ^a ^b ^c 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. doi:10.1287/mnsc.24.13.1362. JSTOR 2630642.
^ 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.
^ 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.
^ Henderson, W.; Taylor, P. G. (1990). "Product form in networks of queues with batch arrivals and batch services". Queueing Systems. 6: 71–87. 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. 2 (2): 355–369. doi:10.1017/s0001867800037435. 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. CiteSeerX 10.1.1.352.5769. doi:10.1016/j.peva.2013.08.011. Archived from the original (PDF) on 2016-03-03. Retrieved 2015-09-04.
^ Downton, F. (1955). "Waiting Time in Bulk Service Queues". Journal of the Royal Statistical Society, Series B. 17 (2). Royal Statistical Society: 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. doi:10.2307/1426040. JSTOR 1426040.
^ Medhi, Jyotiprasad (1975). "Waiting Time Distribution in a Poisson Queue with a General Bulk Service Rule". Management Science. 21 (7): 777–782. doi:10.1287/mnsc.21.7.777. JSTOR 2629773.

[chiamsiri-1] Chiamsiri, Singha; Leonard, Michael S. (1981). "A Diffusion Approximation for Bulk Queues". Management Science. 27 (10): 1188–1199. doi:10.1287/mnsc.27.10.1188. JSTOR 2631086.

[2] Özden, Eda (2012). Discrete Time Analysis of Consolidated Transport Processes. KIT Scientific Publishing. p. 14. ISBN 978-3866448018.

[chaudhry-3] Chaudhry, M. L.; Templeton, James G. C. (1983). A first course in bulk queues. Wiley. ISBN 978-0471862604.

[berg-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. doi:10.1287/mnsc.44.5.684. JSTOR 2634473.

[bailey-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. doi:10.1287/mnsc.24.13.1362. 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: 71–87. 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. 2 (2): 355–369. doi:10.1017/s0001867800037435. 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. CiteSeerX 10.1.1.352.5769. doi:10.1016/j.peva.2013.08.011. Archived from the original (PDF) on 2016-03-03. Retrieved 2015-09-04.

[12] Downton, F. (1955). "Waiting Time in Bulk Service Queues". Journal of the Royal Statistical Society, Series B. 17 (2). Royal Statistical Society: 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. doi:10.2307/1426040. JSTOR 1426040.

[14] Medhi, Jyotiprasad (1975). "Waiting Time Distribution in a Poisson Queue with a General Bulk Service Rule". Management Science. 21 (7): 777–782. doi:10.1287/mnsc.21.7.777. JSTOR 2629773.

[1]

v t e Queueing theory
Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue
Arrival processes	Poisson point process Markovian arrival process Rational arrival process
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