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

Abstract

We consider a storage model which can be on or off. When on, the content increases at some state-dependent rate and the system can switch to the off state at a state-dependent rate as well. When off, the content decreases at some statedependent rate (unless it is at zero) and the system can switch to the on position at a state-dependent rate. This process is a special case of a piecewise deterministic Markov process. We identify the stationary distribution and conditions for its existence and uniqueness.

Queues with waiting time dependent service

“…For positive values, the FIL-process is a special case of the model considered in [6], i.e., an on/off storage system with state-dependent rates restricted to up intervals. Applying [6, Theorem 1] combined with [6,Sect.…”

“…Applying [6, Theorem 1] combined with [6,Sect. 6] and taking (in the notation of [6]) r 0 (x) ≡ 1, λ 0 (x) = μr(x) and λ 1 (x)/r 1 (x) ≡ λ with λ 1 (x) and r 1 (x) tending to infinity, directly yields the FIL-density represented in (3). Furthermore, combining results on expected excursion times [6, Theorem 2] with standard renewal arguments provides the remaining constants.…”

“…2, we study the single-server model, where the service speed can be continuously adapted based on the waiting time of the first customer in line. This model is related to the study of dams and queueing systems with workload-dependent service rates; see for example [4,6,16] or [25]. The difference is that the service speed here depends on the waiting time instead of the amount of work present.…”

Quality of service modeling and analysis for carrier ethernet

“…As an aside we mention that, as argued in [10,72], expressions (7.2) and (7.3) should match, since for any level the mean number of upcrossings per unit time equals the mean number of downcrossings per unit time.…”

“…The signaling frequency is de…ned as the expected number of phase transitions per unit time, and is a measure for the signaling overhead. With fì (x) := dFì (x)=dx, we …rst observe that the expected number of upcrossings per unit time through level x is, reasoning as in, e.g., [10,72],…”