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 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 contraints, 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

↑ 1.0 1.1 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.
↑ 4.0 4.1 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.
↑ 5.0 5.1 5.2 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". 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". 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". 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.

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 process Markovian arrival process Rational arrival process

Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network

Service policies	FIFO LIFO Processor sharing Shortest job first Shortest remaining time

Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method

Limit theorems	Fluid limit Mean field theory Heavy traffic approximation Reflected Brownian motion

Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue

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Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itō diffusion Itō process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market SABR volatility Vašíček

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itō integral Itō's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prohorov theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

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