Sengupta's invariant relationship and its application to waiting time inference

1997 ◽  
Vol 34 (03) ◽  
pp. 795-799 ◽  
Author(s):  
Hiroshi Toyoizumi
Keyword(s):  
Waiting Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Service Process ◽  
Switching Systems ◽  
Waiting Time Distribution ◽  
Call Processing ◽  
Virtual Waiting Time ◽  
Fifo Queue ◽  
Invariant Relationship

This paper presents a new proof of Sengupta's invariant relationship between virtual waiting time and attained sojourn time and its application to estimating the virtual waiting time distribution by counting the number of arrivals and departures of a G/G/1 FIFO queue. Since this relationship does not require any parametric assumptions, our method is non-parametric. This method is expected to have applications, such as call processing in communication switching systems, particularly when the arrival or service process is unknown.

Sengupta's invariant relationship and its application to waiting time inference

1997 ◽  
Vol 34 (3) ◽  
pp. 795-799 ◽  
Author(s):  
Hiroshi Toyoizumi
Keyword(s):  
Waiting Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Service Process ◽  
Switching Systems ◽  
Waiting Time Distribution ◽  
Call Processing ◽  
Virtual Waiting Time ◽  
Fifo Queue ◽  
Invariant Relationship

This paper presents a new proof of Sengupta's invariant relationship between virtual waiting time and attained sojourn time and its application to estimating the virtual waiting time distribution by counting the number of arrivals and departures of a G/G/1 FIFO queue. Since this relationship does not require any parametric assumptions, our method is non-parametric. This method is expected to have applications, such as call processing in communication switching systems, particularly when the arrival or service process is unknown.

scholarly journals The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

2002 ◽  
Vol 39 (03) ◽  
pp. 619-629 ◽  
Author(s):  
Gang Uk Hwang ◽  
Bong Dae Choi ◽  
Jae-Kyoon Kim
Keyword(s):  
Discrete Time ◽  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Autoregressive Process ◽  
Waiting Time Distribution ◽  
Tail Probabilities ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Asymptotic Decay Rate

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

On applications of residual lifetimes of compound geometric convolutions

2004 ◽  
Vol 41 (03) ◽  
pp. 802-815
Author(s):  
Gordon E. Willmot ◽  
Jun Cai
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Risk Model ◽  
Residual Lifetime ◽  
Waiting Time Distribution ◽  
Classical Risk Model ◽  
Virtual Waiting Time ◽  
Compound Geometric Distribution ◽  
Renewal Risk ◽  
The Classical Risk Model

We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.

scholarly journals The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

2002 ◽  
Vol 39 (3) ◽  
pp. 619-629 ◽  
Author(s):  
Gang Uk Hwang ◽  
Bong Dae Choi ◽  
Jae-Kyoon Kim
Keyword(s):  
Discrete Time ◽  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Autoregressive Process ◽  
Waiting Time Distribution ◽  
Tail Probabilities ◽  
Waiting Time Distributions ◽  
Virtual Waiting Time ◽  
Asymptotic Decay Rate

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.

scholarly journals A Note on the Waiting-Time Distribution in an Infinite-Buffer GI[X]/C-MSP/1 Queueing System

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
A. D. Banik ◽  
M. L. Chaudhry ◽  
James J. Kim
Keyword(s):  
Probability Density ◽  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Linear Equations ◽  
Computational Procedure ◽  
Service Process ◽  
Model Parameters ◽  
Single Server ◽  
Waiting Time Distribution

This paper deals with a batch arrival infinite-buffer single server queue. The interbatch arrival times are generally distributed and arrivals are occurring in batches of random size. The service process is correlated and its structure is presented through a continuous-time Markovian service process (C-MSP). We obtain the probability density function (p.d.f.) of actual waiting time for the first and an arbitrary customer of an arrival batch. The proposed analysis is based on the roots of the characteristic equations involved in the Laplace-Stieltjes transform (LST) of waiting times in the system for the first, an arbitrary, and the last customer of an arrival batch. The corresponding mean sojourn times in the system may be obtained using these probability density functions or the above LSTs. Numerical results for some variants of the interbatch arrival distribution (Pareto and phase-type) have been presented to show the influence of model parameters on the waiting-time distribution. Finally, a simple computational procedure (through solving a set of simultaneous linear equations) is proposed to obtain the “R” matrix of the corresponding GI/M/1-type Markov chain embedded at a prearrival epoch of a batch.

On applications of residual lifetimes of compound geometric convolutions

2004 ◽  
Vol 41 (3) ◽  
pp. 802-815 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jun Cai
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Risk Model ◽  
Residual Lifetime ◽  
Waiting Time Distribution ◽  
Classical Risk Model ◽  
Virtual Waiting Time ◽  
Compound Geometric Distribution ◽  
Renewal Risk ◽  
The Classical Risk Model

We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.

Some analyses on the control of queues using level crossings of regenerative processes

1980 ◽  
Vol 17 (3) ◽  
pp. 814-821 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing Systems ◽  
Density Functions ◽  
Waiting Time Distribution ◽  
Regenerative Processes ◽  
Stieltjes Transform ◽  
Expected Number ◽  
Control Of Queues ◽  
Special Case

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

scholarly journals Role of the Solar Minimum in the Waiting Time Distribution Throughout the Heliosphere

2021 ◽  
Author(s):  
Yosia I Nurhan ◽  
Jay Robert Johnson ◽  
Jonathan R Homan ◽  
Simon Wing
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Solar Minimum ◽  
Waiting Time Distribution

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.

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