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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]
In the study of queue networks one typically tries to obtain the equilibrium distribution of the network, although in many applications the study of the transient state is fundamental. 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 ...
where as above is the Laplace–Stieltjes transform of the service time distribution function. This relationship can only be solved exactly in special cases (such as the M/M/1 queue ), but for any s {\textstyle s} the value of ϕ ( s ) {\textstyle \phi (s)} can be calculated and by iteration with upper and lower bounds the distribution function ...
In queueing theory, a discipline within the mathematical theory of probability, the G/M/1 queue represents the queue length in a system where interarrival times have a general (meaning arbitrary) distribution and service times for each job have an exponential distribution. [1]
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).
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 stationary distribution is the limiting distribution for large values of t. Various performance measures can be computed explicitly for the M/M/1 queue. We write ρ = λ/μ for the utilization of the buffer and require ρ < 1 for the queue to be stable. ρ represents the average proportion of time which the server is occupied.
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.