UNCONVENTIONAL BEHAVIOR IN HAVING COMPLETELY MONOTONE ARRIVALS

2020 ◽  
Vol 126 (1) ◽  
pp. 33-63
Author(s):  
Fumiaki Machihara ◽  
Taro Tokuda
Keyword(s):  
Completely Monotone

Transient behavior of regulated Brownian motion, I: Starting at the origin

1987 ◽  
Vol 19 (03) ◽  
pp. 560-598 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
First Passage Time ◽  
Cumulative Distribution ◽  
Stochastic Flow ◽  
Completely Monotone ◽  
Constant Diffusion ◽  
Flow Systems ◽  
The Moment ◽  
First Moment

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that thekth-moment c.d.f is thek-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k +1)th moment approaches its steady-state limit more slowly than thekthmoment. It is also easy to derive the asymptotic behavior ast→∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2)c.d.f.&s (mixtures of two exponentials). TheH2approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of theH2approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.

scholarly journals Duality and Asymptotics for a Class of Nonneutral Discrete Moran Models

2009 ◽  
Vol 46 (3) ◽  
pp. 866-893 ◽  
Author(s):  
Thierry Huillet ◽  
Martin Möhle
Keyword(s):  
Transition Matrix ◽  
Random Number ◽  
Dual Process ◽  
Probabilistic Interpretation ◽  
Uhlenbeck Process ◽  
Moran Model ◽  
Completely Monotone ◽  
Finite State ◽  
Selection Mechanisms ◽  
Finite State Space

A Markov chain X with finite state space {0,…,N} and tridiagonal transition matrix is considered, where transitions from i to i-1 occur with probability (i/N)(1-p(i/N)) and transitions from i to i+1 occur with probability (1-i/N)p(i/N). Here p:[0,1]→[0,1] is a given function. It is shown that if p is continuous with p(x)≤p(1) for all x∈[0,1] then, for each N, a dual process Y to X (with respect to a specific duality function) exists if and only if 1-p is completely monotone with p(0)=0. A probabilistic interpretation of Y in terms of an ancestral process of a mixed multitype Moran model with a random number of types is presented. It is shown that, under weak conditions on p, the process Y, properly time and space scaled, converges to an Ornstein–Uhlenbeck process as N tends to ∞. The asymptotics of the stationary distribution of Y is studied as N tends to ∞. Examples are presented involving selection mechanisms. results.

scholarly journals On the Correlation Structure of a Lévy-Driven Queue

2008 ◽  
Vol 45 (4) ◽  
pp. 940-952 ◽  
Author(s):  
Abdelghafour Es-Saghouani ◽  
Michel Mandjes
Keyword(s):  
Random Variable ◽  
Correlation Structure ◽  
Single Server ◽  
Monotone Functions ◽  
Single Server Queue ◽  
Completely Monotone Functions ◽  
Complementary Distribution ◽  
Completely Monotone ◽  
Server Queue ◽  
Heavy Tailed

In this paper we consider a single-server queue with Lévy input and, in particular, its workload process (Qt)t≥0, with a focus on the correlation structure. With the correlation function defined asr(t) := cov(Q0,Qt) / var(Q0) (assuming that the workload process is in stationarity at time 0), we first determine its transform ∫0∞r(t)e-ϑtdt. This expression allows us to prove thatr(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. We also show thatr(·) can be represented as the complementary distribution function of a specific random variable. These results are used to compute the asymptotics ofr(t), for larget, for the cases of light-tailed and heavy-tailed Lévy inputs.

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