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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. [2]
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. [7]
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 probability theory and statistics, a Hawkes process, named after Alan G. Hawkes, is a kind of self-exciting point process. [1] It has arrivals at times 0 < t 1 < t 2 < t 3 < ⋯ {\textstyle 0<t_{1}<t_{2}<t_{3}<\cdots } where the infinitesimal probability of an arrival during the time interval [ t , t + d t ) {\textstyle [t,t+dt)} is
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]
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. [1]
An M/M/1 queueing node. 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 ...
Queueing theory is the mathematical study of waiting lines, or queues. [1] A queueing model is constructed so that queue lengths and waiting time can be predicted. [1] Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a ...