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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]
The arrivals in the process whose intensity is () are the "daughters" of the arrival at time . The integral ∫ 0 ∞ ϕ ( t ) d t {\displaystyle \int _{0}^{\infty }\phi (t)\,dt} is the average number of daughters of each arrival and is called the branching ratio .
The system is described in Kendall's notation where the G denotes a general distribution for both interarrival times and service times and the 1 that the model has a single server. [ 3 ] [ 4 ] Different interarrival and service times are considered to be independent, and sometimes the model is denoted GI/GI/1 to emphasise this.
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]
and F(u) is the service time distribution and λ the Poisson arrival rate of jobs to the queue. Markov chains with generator matrices or block matrices of this form are called M/G/1 type Markov chains, [ 13 ] a term coined by Marcel F. Neuts .
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 probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
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.