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In queueing theory, a discipline within the mathematical theory of probability, the M/M/c queue (or Erlang–C model [1]: 495 ) is a multi-server queueing model. [2] In Kendall's notation it describes a system where arrivals form a single queue and are governed by a Poisson process, there are c servers, and job service times are exponentially distributed. [3]
An M/M/∞ queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers currently being served. Since, the number of servers in parallel is infinite, there is no queue and the number of customers in the systems coincides with the number of customers being served at any moment.
In queueing theory, a discipline within the mathematical theory of probability, Burke's theorem (sometimes the Burke's output theorem [1]) is a theorem (stated and demonstrated by Paul J. Burke while working at Bell Telephone Laboratories) asserting that, for the M/M/1 queue, M/M/c queue or M/M/∞ queue in the steady state with arrivals is a Poisson process with rate parameter λ:
In 1953, David George Kendall solved the GI/M/k queue [15] and introduced the modern notation for queues, now known as Kendall's notation. In 1957, Pollaczek studied the GI/G/1 using an integral equation. [16] John Kingman gave a formula for the mean waiting time in a G/G/1 queue, now known as Kingman's formula. [17]
M X /M Y /1 queue: D: Degenerate distribution: A deterministic or fixed service time. M/D/1 queue: E k: Erlang distribution: An Erlang distribution with k as the shape parameter (i.e., sum of k i.i.d. exponential random variables). G: General distribution: Although G usually refers to independent service time, some authors prefer to use GI to ...
The Erlang B formula (or Erlang-B with a hyphen), also known as the Erlang loss formula, is a formula for the blocking probability that describes the probability of call losses for a group of identical parallel resources (telephone lines, circuits, traffic channels, or equivalent), sometimes referred to as an M/M/c/c queue. [5]
A Jackson network consists of a number of nodes, where each node represents a queue in which the service rate can be both node-dependent (different nodes have different service rates) and state-dependent (service rates change depending on queue lengths). Jobs travel among the nodes following a fixed routing matrix.
An M/M/1 queue, a model which counts the number of jobs in a queueing system with arrivals at rate λ and services at rate μ, has transition-rate matrix = ...