Continuous feedback fluid queues

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsContinuous feedback fluid queues ABSTRACT We investigate a fluid buffer which is modulated by a stochastic background process, while the momentary behavior of the background process depends on the current buffer level in a continuous way. Loosely speaking the feedback is such that the background process behaves as a continuous-time Markov chain' with generator Q(y) at times when the buffer level is y, where the entries of Q(y) are continuous functions of y. Moreover, the fluid-flow rates for the buffer may also depend continuously on the current buffer level. First we define the feedback behavior precisely. Then we deduce the Kolmogorov forward equations for the joint background/buffer-process under some regularity assumptions. After presenting the differential equations and boundary conditions for the stationary distributions, we find an explicit solution when the background process has two states. AbstractWe investigate a fluid buffer which is modulated by a stochastic background process, while the momentary behavior of the background process depends on the current buffer level in a continuous way. Loosely speaking the feedback is such that the background process behaves 'as a continuous-time Markov chain' with generator Q(y) at times when the buffer level is y, where the entries of Q(y) are continuous functions of y. Moreover, the fluid-flow rates for the buffer may also depend continuously on the current buffer level.First we define the feedback behavior precisely. Then we deduce the Kolmogorov forward equations for the joint background/buffer-process under some regularity assumptions. After presenting the differential equations and boundary conditions for the stationary distributions, we find an explicit solution when the background process has two states.

On/Off Storage Systems With State-Dependent Input, Output, and Switching Rates

“…(rather than state dependent) on and off time pairs and state dependent production and release rates. The paper also bears some relation to [14], which considers a feedback fluid queue with state-dependent release rates. Scheinhardt et al [14] allow a finite number N ≥ 2 of background states, but restrict themselves to a finite buffer content.…”

“…The paper also bears some relation to [14], which considers a feedback fluid queue with state-dependent release rates. Scheinhardt et al [14] allow a finite number N ≥ 2 of background states, but restrict themselves to a finite buffer content. The paper is also related to [4].…”

The workload-dependent MAP/PH/1 queue with infinite/finite workload capacity

“…In MRMFQs, the infinitesimal generator of the background CTMC as well as the drift of the buffer process are allowed to depend on the regime in which the buffer level resides; see [16,20,10] for more detailed description of MRMFQs as well as the boundary conditions that arise in the solution of the steady-state distribution of the buffer level. In continuous-feedback Markov fluid queues (CFMFQs), both the infinitesimal generator of the background CTMC and the drift of the buffer process are allowed to continuously depend on the instantaneous buffer level [21]. Explicit solutions for the steady-state distribution of CFMFQs are available only for specific subcases [21], and numerical solutions based on discretization are generally the basic tool for their numerical analysis [22].…”

“…In continuous-feedback Markov fluid queues (CFMFQs), both the infinitesimal generator of the background CTMC and the drift of the buffer process are allowed to continuously depend on the instantaneous buffer level [21]. Explicit solutions for the steady-state distribution of CFMFQs are available only for specific subcases [21], and numerical solutions based on discretization are generally the basic tool for their numerical analysis [22].…”

A feedback fluid queue with two congestion control thresholds

“…In the second half of the 1990s models emerged in which the source behavior was influenced by the buffer content; see for instance [1,4,15,17]; in [9,19], Q and R depend continuously on the buffer level.…”