A relationship between a family of waiting time distributions and the asymptotic residual waiting time distribution

1985 ◽  
Vol 14 (3) ◽  
pp. 709-717 ◽  
Author(s):  
Robert M. Norton
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Time Distribution ◽  
Waiting Time Distributions

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 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 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 (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.

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.

Part Waiting Time Distribution in a Two-Machine Line

2012 ◽  
Vol 45 (6) ◽  
pp. 457-462 ◽  
Author(s):  
Chuan Shi ◽  
Stanley B. Gershwin
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Time Distribution
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