scholarly journals The moderate deviations principle for the trajectories of compound renewal processes on the half-line

2021 ◽  
Vol 18 (2) ◽  
pp. 1189-1200
Author(s):  
A. V. Logachov ◽  
A. A. Mogulskii
Keyword(s):  
Moderate Deviations ◽  
Half Line ◽  
Renewal Processes ◽  
Moderate Deviations Principle ◽  
Compound Renewal Processes

Invariance principles in queueing theory

1989 ◽  
Vol 26 (04) ◽  
pp. 845-857
Author(s):  
Michael Alex ◽  
Josef Steinebach
Keyword(s):  
Convergence Rate ◽  
Stochastic Processes ◽  
Queueing Theory ◽  
Counting Process ◽  
Heavy Traffic ◽  
Renewal Processes ◽  
Light Traffic ◽  
Invariance Principles ◽  
Renewal Counting Process ◽  
Compound Renewal Processes

Several stochastic processes in queueing theory are based upon compound renewal processes . For queues in light traffic, however, the summands {Xk }and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for the GI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.

Large deviations, moderate deviations, and queues with long-range dependent input

1999 ◽  
Vol 31 (1) ◽  
pp. 254-278 ◽  
Author(s):  
Cheng-Shang Chang ◽  
David D. Yao ◽  
Tim Zajic
Keyword(s):  
Brownian Motion ◽  
Long Range ◽  
Sample Path ◽  
Transient Behavior ◽  
Moderate Deviations ◽  
Long Range Dependence ◽  
Standard Process ◽  
Liouville Type ◽  
Stationary Increments ◽  
Moderate Deviations Principle

Long-range dependence has been recently asserted to be an important characteristic in modeling telecommunications traffic. Inspired by the integral relationship between the fractional Brownian motion and the standard Brownian motion, we model a process with long-range dependence, Y, as a fractional integral of Riemann-Liouville type applied to a more standard process X—one that does not have long-range dependence. When X takes the form of a sample path process with bounded stationary increments, we provide a criterion for X to satisfy a moderate deviations principle (MDP). Based on the MDP of X, we then establish the MDP for Y. Furthermore, we characterize, in terms of the MDP, the transient behavior of queues when fed with the long-range dependent input process Y. In particular, we identify the most likely path that leads to a large queue, and demonstrate that unlike the case where the input has short-range dependence, the path here is nonlinear.

scholarly journals On the convergence of quadratic variation for compound fractional Poisson processes

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Enrico Scalas ◽  
Noèlia Viles
Keyword(s):  
Stochastic Process ◽  
Random Variable ◽  
Quadratic Variation ◽  
Poisson Processes ◽  
Renewal Processes ◽  
The Relationship ◽  
Compound Renewal Processes

AbstractThe relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of quadratic variation is investigated both as a random variable (for given t) and as a stochastic process.

Sign in / Sign up

Export Citation Format

Share Document