bulk queue
In queueing theory, a discipline within the mathematical theory of probability, a bulk queue{{cite journal | first1 = Singha | last1 = Chiamsiri | first2 = Michael S. | last2= Leonard | year = 1981 | title = A Diffusion Approximation for Bulk Queues | journal = Management Science | volume = 27 | issue = 10 | pages = 1188–1199 | jstor = 2631086 | doi=10.1287/mnsc.27.10.1188}} (sometimes batch queue{{cite book | title = Discrete Time Analysis of Consolidated Transport Processes | last = Özden |first = Eda | page = 14| publisher = KIT Scientific Publishing | isbn = 978-3866448018| year = 2012 }}) is a general queueing model where jobs arrive in and/or are served in groups of random size.{{cite book | title = A first course in bulk queues | first1 = M. L. | last1 = Chaudhry | first2= James G. C. |last2= Templeton | publisher = Wiley | year = 1983 | isbn=978-0471862604}}{{rp|vii}} Batch arrivals have been used to describe large deliveries{{cite journal | first1 = Menachem | last1 = Berg | first2 = Frank | last2 = van der Duyn Schouten | first3 = Jorg | last3 = Jansen | year = 1998 | title = Optimal Batch Provisioning to Customers Subject to a Delay-Limit | journal = Management Science | volume = 44 | issue = 5 | pages = 684–697 | jstor = 2634473 | doi=10.1287/mnsc.44.5.684}} and batch services to model a hospital out-patient department holding a clinic once a week, a transport link with fixed capacity{{cite journal | first = Rajat K. | last = Deb | year = 1978 | title = Optimal Dispatching of a Finite Capacity Shuttle | journal = Management Science | volume = 24 | issue = 13 | pages = 1362–1372 | jstor = 2630642 | doi=10.1287/mnsc.24.13.1362}}{{cite journal | first1= A. |last1=Glazer | first2= R. |last2= Hassin | year = 1987 | title = Equilibrium Arrivals in Queues with Bulk Service at Scheduled Times | journal = Transportation Science | volume = 21 | issue = 4 | pages = 273–278 | jstor = 25768286 | doi = 10.1287/trsc.21.4.273}} and an elevator.{{cite journal | author = Marcel F. Neuts | year = 1967 | title = A General Class of Bulk Queues with Poisson Input | journal = The Annals of Mathematical Statistics | volume = 38 | issue = 3 | pages = 759–770 | jstor = 2238992 | doi = 10.1214/aoms/1177698869 | url = http://www.stat.purdue.edu/research/technical_reports/pdfs/1-527/tr-046.pdf| doi-access = free }}
Networks of such queues are known to have a product form stationary distribution under certain conditions.{{Cite journal | last1 = Henderson | first1 = W. | last2 = Taylor | first2 = P. G. | doi = 10.1007/BF02411466 | title = Product form in networks of queues with batch arrivals and batch services | journal = Queueing Systems| volume = 6 | pages = 71–87 | year = 1990 }} Under heavy traffic conditions a bulk queue is known to behave like a reflected Brownian motion.{{cite journal | last1 = Iglehart | first1 = Donald L. | last2 = Ward | first2 = Whitt | author-link2 = Ward Whitt | year = 1970 | title = Multiple Channel Queues in Heavy Traffic. II: Sequences, Networks, and Batches | journal = Advances in Applied Probability | volume = 2 | issue = 2 | pages = 355–369 | jstor = 1426324 | access-date = 30 Nov 2012 | url = http://www.columbia.edu/~ww2040/MultipleChannel1970II.pdf | doi=10.1017/s0001867800037435}}{{Cite journal | last1 = Harrison | first1 = P. G. | author-link1 = Peter G. Harrison | last2 = Hayden | first2 = R. A. | last3 = Knottenbelt | first3 = W. | doi = 10.1016/j.peva.2013.08.011 | title = Product-forms in batch networks: Approximation and asymptotics | journal = Performance Evaluation | volume = 70 | issue = 10 | pages = 822 | year = 2013 | url = http://pubs.doc.ic.ac.uk/batches-heavy-traffic-journal/batches-heavy-traffic-journal.pdf | citeseerx = 10.1.1.352.5769 | access-date = 2015-09-04 | archive-url = https://web.archive.org/web/20160303205349/http://pubs.doc.ic.ac.uk/batches-heavy-traffic-journal/batches-heavy-traffic-journal.pdf | archive-date = 2016-03-03 | url-status = dead }}
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.
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{{cite journal | last1 = Downton | first1 = F. | year = 1955 | title = Waiting Time in Bulk Service Queues | journal = Journal of the Royal Statistical Society, Series B | volume = 17 | issue = 2 | pages = 256–261 | publisher = Royal Statistical Society | jstor = 2983959 }}) are served at a rate with independent distribution.{{cite journal|title = On Queueing Processes with Bulk Service | first = Norman T. J. | last = Bailey | journal = Journal of the Royal Statistical Society, Series B | pages = 80–87 | jstor = 2984011 | volume = 61 | number = 1 | year = 1954}} 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.{{cite journal | first1= Rajat K. |last1= Deb | first2= Richard F. |last2 = Serfozo | year = 1973 | title = Optimal Control of Batch Service Queues | journal = Advances in Applied Probability | volume = 5 | issue = 2 | pages = 340–361 | jstor = 1426040|doi= 10.2307/1426040 }}
Bulk arrival
Waiting Time Distribution
The waiting time distribution of bulk Poisson arrival is presented in.{{cite journal | first1= Jyotiprasad |last1= Medhi | year = 1975 | title = Waiting Time Distribution in a Poisson Queue with a General Bulk Service Rule | journal = Management Science | volume = 21 | issue = 7 | pages = 777–782 | jstor = 2629773 | doi=10.1287/mnsc.21.7.777}}