Fluid queue

In queueing theory, a discipline within the mathematical theory of probability, a fluid queue (fluid model, fluid flow model or stochastic fluid model) is a mathematical model used to describe the fluid level in a reservoir subject to randomly determined periods of filling and emptying. The term dam theory was used in earlier literature for these models. The model has been used to approximate discrete models, model the spread of wildfires, in ruin theory and to model high speed data networks. The model applies the leaky bucket algorithm to a stochastic source. In queueing theory, a discipline within the mathematical theory of probability, a fluid queue (fluid model, fluid flow model or stochastic fluid model) is a mathematical model used to describe the fluid level in a reservoir subject to randomly determined periods of filling and emptying. The term dam theory was used in earlier literature for these models. The model has been used to approximate discrete models, model the spread of wildfires, in ruin theory and to model high speed data networks. The model applies the leaky bucket algorithm to a stochastic source. The model was first introduced by Pat Moran in 1954 where a discrete-time model was considered. Fluid queues allow arrivals to be continuous rather than discrete, as in models like the M/M/1 and M/G/1 queues. Fluid queues have been used to model the performance of a network switch, a router, the IEEE 802.11 protocol, Asynchronous Transfer Mode (the intended technology for B-ISDN), peer-to-peer file sharing, optical burst switching, and has applications in civil engineering when designing dams. The process is closely connected to quasi-birth–death processes, for which efficient solution methods are known. A fluid queue can be viewed as a large tank, typically assumed to be of infinite capacity, connected to a series of pipes that pour fluid in to the tank and a series of pumps which remove fluid from the tank. An operator controls the pipes and pumps controlling the rate at which fluid pours in to the buffer and the rate at which fluid leaves. When the operator puts the system in to state i we write ri for the net fluid arrival rate in this state (input less output). When the buffer contains fluid, if we write X(t) for the fluid level at time t, The operator is a continuous time Markov chain and is usually called the environment process, background process or driving process. As the process X represents the level of fluid in the buffer it can only take non-negative values. The model is a particular type of piecewise deterministic Markov process and can also be viewed as a Markov reward model with boundary conditions. The stationary distribution is a phase-type distribution as first shown by Asmussen and can be computed using matrix-analytic methods. The additive decomposition method is numerically stable and separates the eigenvalues necessary for computation using Schur decomposition. For a simple system where service has a constant rate μ and arrival fluctuate between rates λ and 0 (in states 1 and 2 respectively) according to a continuous time Markov chain with generator matrix