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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]
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 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 ...
Let (,) be a / / queue with arrival times (,) that have interarrival distribution A.Define the size of the queue immediately before the nth arrival by the process =.This is a discrete-time Markov chain with stochastic matrix:
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.
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.
A vivid example is the bus waiting time paradox: For a given random distribution of bus arrivals, the average rider at a bus stop observes more delays than the average operator of the buses. The resolution of the paradox is that our sampled distribution at time t is size-biased (see sampling bias ), in that the likelihood an interval is chosen ...