scholarly journals ASYMPTOTIC BEHAVIORS OF WAITING TIME DISTRIBUTION FUNCTION (WTDF) ψ(t) AND ASYMPTOTIC SOLUTIONS OF CONTINUOUS-TIME RANDOM WALK (CTRW) PROBLEMS

1986 ◽  
Vol 35 (11) ◽  
pp. 1436
Author(s):  
DENG HUI-FANG
Keyword(s):  
Distribution Function ◽  
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.

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.

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

1998 ◽  
Vol 35 (2) ◽  
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 {(Xn,Jn)} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {Fij}, Fij(B) = ℙ[X1 ∈ B, J1 = j | J0 = i], B ∈ B(ℝ), i, j ∈ E. If Fij([x,∞))/(1-H(x)) → Wij ∈ [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 𝔼Xn < 0, at least one Wij > 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[supn≥0Sn > x] → (−𝔼Xn)−1 ∫x∞ ℙ[Xn > u]du as x → ∞, where Sn = ∑1nXk, S0 = 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.

On exponential bounds for the waiting-time distribution function in GI/G/1

1976 ◽  
Vol 13 (2) ◽  
pp. 411-417 ◽  
Author(s):  
R. Bergmann ◽  
D. Stoyan
Keyword(s):  
Distribution Function ◽  
Waiting Time ◽  
Time Distribution ◽  
Arrival Time ◽  
Distribution Functions ◽  
Integral Inequalities ◽  
Arrival Time Distribution ◽  
Waiting Time Distribution ◽  
Exponential Bounds ◽  
Stationary Waiting Time

Exponential bounds for the stationary waiting-time distribution of the type ae–θt are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.

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