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.

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 Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies

2015 ◽  
Vol 47 (04) ◽  
pp. 989-1014 ◽  
Author(s):  
P. Vis ◽  
R. Bekker ◽  
R. D. van der Mei
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Waiting Time Distribution ◽  
Asymptotic Results ◽  
Local Service ◽  
Waiting Time Distributions ◽  
Polling Models ◽  
The Impact

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS (first-come-first-served) local service orders, namely last-come-first-served with and without preemption, random-order-of-service, processor sharing, the multi-class priority scheduling with and without preemption, shortest-job-first, and the shortest remaining processing time policy. For each of these policies, we first express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-traffic assumptions, i.e. when the system tends to saturate. The results show that in all cases the asymptotic waiting-time distribution at queueiis fully characterized and of the form Γ Θi, with Γ and Θiindependent, and where Γ is gamma distributed with known parameters (and the same for all scheduling policies). We derive the distribution of the random variable Θiwhich explicitly expresses the impact of the local service order on the asymptotic waiting-time distribution. The results provide new fundamental insight into the impact of the local scheduling policy on the performance of a general class of polling models. The asymptotic results suggest simple closed-form approximations for the complete waiting-time distributions for stable systems with arbitrary load values.

scholarly journals Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies

2015 ◽  
Vol 47 (4) ◽  
pp. 989-1014 ◽  
Author(s):  
P. Vis ◽  
R. Bekker ◽  
R. D. van der Mei
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Waiting Time Distribution ◽  
Asymptotic Results ◽  
Local Service ◽  
Waiting Time Distributions ◽  
Polling Models ◽  
The Impact

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS (first-come-first-served) local service orders, namely last-come-first-served with and without preemption, random-order-of-service, processor sharing, the multi-class priority scheduling with and without preemption, shortest-job-first, and the shortest remaining processing time policy. For each of these policies, we first express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-traffic assumptions, i.e. when the system tends to saturate. The results show that in all cases the asymptotic waiting-time distribution at queue i is fully characterized and of the form Γ Θi, with Γ and Θi independent, and where Γ is gamma distributed with known parameters (and the same for all scheduling policies). We derive the distribution of the random variable Θi which explicitly expresses the impact of the local service order on the asymptotic waiting-time distribution. The results provide new fundamental insight into the impact of the local scheduling policy on the performance of a general class of polling models. The asymptotic results suggest simple closed-form approximations for the complete waiting-time distributions for stable systems with arbitrary load values.

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.

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.

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