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.

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.

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.

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.

On higher-order properties of compound geometric distributions

2002 ◽  
Vol 39 (02) ◽  
pp. 324-340 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Equilibrium Distribution ◽  
Risk Model ◽  
Higher Order ◽  
Residual Lifetime ◽  
Classical Risk Model ◽  
Time Of Ruin ◽  
The Laplace Transform ◽  
The Time Of Ruin ◽  
Stop Loss ◽  
The Classical Risk Model

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to thenth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of thekth moment of the time of ruin in the classical risk model.

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.

On higher-order properties of compound geometric distributions

2002 ◽  
Vol 39 (2) ◽  
pp. 324-340 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Equilibrium Distribution ◽  
Risk Model ◽  
Higher Order ◽  
Residual Lifetime ◽  
Classical Risk Model ◽  
Time Of Ruin ◽  
The Laplace Transform ◽  
The Time Of Ruin ◽  
Stop Loss ◽  
The Classical Risk Model

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to the nth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of the kth moment of the time of ruin in the classical risk model.

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 VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 85 ◽  
Author(s):  
Mohamed Lkabous ◽  
Jean-François Renaud
Keyword(s):  
Risk Model ◽  
Risk Measures ◽  
Risk Measure ◽  
Short Paper ◽  
Classical Risk Model ◽  
Parisian Ruin ◽  
The Classical Risk Model

In this short paper, we study a VaR-type risk measure introduced by Guérin and Renaud and which is based on cumulative Parisian ruin. We derive some properties of this risk measure and we compare it to the risk measures of Trufin et al. and Loisel and Trufin.

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.

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