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]
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.[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]
^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. JSTOR25768286.
^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.
^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. JSTOR1426040.
^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. JSTOR2629773.