Asymptotic behaviors of the waiting-time distribution functionψ(t) and asymptotic solutions of continuous-time random-walk problems

1988 ◽  
Vol 37 (4) ◽  
pp. 2212-2219 ◽  
Author(s):  
Deng Hui-fang
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Continuous Time ◽  
Time Distribution ◽  
Continuous Time Random Walk ◽  
Asymptotic Solutions ◽  
Waiting Time Distribution ◽  
Asymptotic Behaviors

CONTINUOUS-TIME FINANCE AND THE WAITING TIME DISTRIBUTION: MULTIPLE CHARACTERISTIC TIMES

2012 ◽  
Vol 26 (23) ◽  
pp. 1250151 ◽  
Author(s):  
KWOK SAU FA
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Waiting Time ◽  
Continuous Time ◽  
Probability Distribution Function ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Exponential Functions ◽  
Multiple Characteristic ◽  
Sum Of Products

In this paper, we model the tick-by-tick dynamics of markets by using the continuous-time random walk (CTRW) model. We employ a sum of products of power law and stretched exponential functions for the waiting time probability distribution function; this function can fit well the waiting time distribution for BUND futures traded at LIFFE in 1997.

scholarly journals The use of Spitzer's identity in the investigation of the busy period and other quantities in the queue GI/G/1

1962 ◽  
Vol 2 (3) ◽  
pp. 345-356 ◽  
Author(s):  
J. F. C. Kingmán
Keyword(s):  
Random Walk ◽  
Characteristic Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Waiting Time Distribution ◽  
Period Distribution ◽  
Stationary Waiting Time

As an illustration of the use of his identity [10], Spitzer [11] obtained the Pollaczek-Khintchine formula for the waiting time distribution of the queue M/G/1. The present paper develops this approach, using a generalised form of Spitzer's identity applied to a three-demensional random walk. This yields a number of results for the general queue GI/G/1, including Smith' solution for the stationary waiting time, which is established under less restrictive conditions that hitherto (§ 5). A soultion is obtained for the busy period distribution in GI/G/1 (§ 7) which can be evaluated when either of the distributions concerned has a rational characteristic function. This solution contains some recent results of Conolly on the quene GI/En/1, as well as well-known results for M/G/1.

scholarly journals A Directed Continuous Time Random Walk Model with Jump Length Depending on Waiting Time

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Long Shi ◽  
Zuguo Yu ◽  
Zhi Mao ◽  
Aiguo Xiao
Keyword(s):  
Random Walk ◽  
Probability Density Function ◽  
Laplace Transform ◽  
Probability Density ◽  
Density Function ◽  
Waiting Time ◽  
Continuous Time ◽  
Dimensional Space ◽  
Continuous Time Random Walk ◽  
Random Walk Model

In continuum one-dimensional space, a coupled directed continuous time random walk model is proposed, where the random walker jumps toward one direction and the waiting time between jumps affects the subsequent jump. In the proposed model, the Laplace-Laplace transform of the probability density functionP(x,t)of finding the walker at positionxat timetis completely determined by the Laplace transform of the probability density functionφ(t)of the waiting time. In terms of the probability density function of the waiting time in the Laplace domain, the limit distribution of the random process and the corresponding evolving equations are derived.

Subexponential asymptotics of a Markov-modulated random walk with queueing applications

1998 ◽  
Vol 35 (02) ◽  
pp. 325-347 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Autocorrelation Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Length Distribution ◽  
Service Time Distribution ◽  
Waiting Time Distribution ◽  
Matrix Measure ◽  
Markov Modulated

Let {(X n ,J n )} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {F ij }, F ij (B) = ℙ[X 1 ∈ B, J 1 = j | J 0 = i], B ∈ B (ℝ), i, j ∈ E. If F ij ([x,∞))/(1-H(x)) → W ij ∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G +(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼X n < 0, at least one W ij > 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[sup n≥0 S n > x] → (−𝔼X n )−1 ∫ x ∞ ℙ[X n > u]du as x → ∞, where S n = ∑1 n X k , S 0 = 0. Two general queueing applications of this result are given. First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.

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