In queueing theory, a discipline within the mathematical theory of probability, a bulk queue (sometimes batch queue) is a general queueing model where jobs arrive in and/or are served in groups of random size.:vii Batch arrivals have been used to describe large deliveries and batch services to model a hospital out-patient department holding a clinic once a week, a transport link with fixed capacity and an elevator.
Networks of such queues are known to have a product form stationary distribution under certain conditions. Under heavy traffic conditions a bulk queue is known to behave like a reflected Brownian motion.
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.
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) are served at a rate with independent distribution. 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.
- 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.
- 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.
- 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. 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. 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.