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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]
[6] [7] For an example of the notation, the M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process (where inter-arrival durations are exponentially distributed) and have exponentially distributed service times (the M denotes a Markov process).
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.
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 .
The average response time or sojourn time (total time a customer spends in the system) does not depend on scheduling discipline and can be computed using Little's law as 1/(μ − λ). The average time spent waiting is 1/(μ − λ) − 1/μ = ρ/(μ − λ). The distribution of response times experienced does depend on scheduling discipline.
A M/M/1 queue means that the time between arrivals is Markovian (M), i.e. the inter-arrival time follows an exponential distribution of parameter λ. The second M means that the service time is Markovian: it follows an exponential distribution of parameter μ. The last parameter is the number of service channel which one (1).
In queueing theory, a discipline within the mathematical theory of probability, an M/G/1 queue is a queue model where arrivals are Markovian (modulated by a Poisson process), service times have a General distribution and there is a single server. [1]
In queueing theory, a discipline within the mathematical theory of probability, a rational arrival process (RAP) is a mathematical model for the time between job arrivals to a system. It extends the concept of a Markov arrival process , allowing for dependent matrix-exponential distributed inter-arrival times.