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The Pollaczek–Khinchine formula gives the mean queue length and mean waiting time in the system. [ 9 ] [ 10 ] Recently, the Pollaczek–Khinchine formula has been extended to the case of infinite service moments, thanks to the use of Robinson's Non-Standard Analysis.
In queueing theory, a discipline within the mathematical theory of probability, a Markovian arrival process (MAP or MArP [1]) is a mathematical model for the time between job arrivals to a system. The simplest such process is a Poisson process where the time between each arrival is exponentially distributed. [2] [3]
The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed inter-arrival and service time distributions to be considered. [18] Systems with coupled orbits are an important part in queueing theory in the application to wireless networks and signal processing. [19]
Service times have an exponential distribution with rate parameter μ in the M/M/1 queue, where 1/μ is the mean service time. All arrival times and services times are (usually) assumed to be independent of one another. [2] A single server serves customers one at a time from the front of the queue, according to a first-come, first-served ...
In mathematical queueing theory, Little's law (also result, theorem, lemma, or formula [1] [2]) is a theorem by John Little which states that the long-term average number L of customers in a stationary system is equal to the long-term average effective arrival rate λ multiplied by the average time W that a customer spends in the system.
Lindley's integral equation is a relationship satisfied by the stationary waiting time distribution F(x) in a G/G/1 queue. = ()Where K(x) is the distribution function of the random variable denoting the difference between the (k - 1)th customer's arrival and the inter-arrival time between (k - 1)th and kth customers.
It is an extension of an M/M/1 queue, where this renewal process must specifically be a Poisson process (so that interarrival times have exponential distribution). Models of this type can be solved by considering one of two M/G/1 queue dual systems, one proposed by Ramaswami and one by Bright.
Kingman's formula gives an approximation for the mean waiting time in a G/G/1 queue. [6] Lindley's integral equation is a relationship satisfied by the stationary waiting time distribution which can be solved using the Wiener–Hopf method .