scholarly journals Some reflections on the Renewal-theory paradox in queueing theory

1998 ◽  
Vol 11 (3) ◽  
pp. 355-368 ◽  
Author(s):  
Robert B. Cooper ◽  
Shun-Chen Niu ◽  
Mandyam M. Srinivasan
Keyword(s):  
Waiting Time ◽  
Queueing Theory ◽  
Waiting Times ◽  
Renewal Theory ◽  
Vacation Models ◽  
Polling Models ◽  
Mean Waiting Time ◽  
The Mean ◽  
Expository Paper ◽  
Set Up

The classical renewal-theory (waiting time, or inspection) paradox states that the length of the renewal interval that covers a randomly-selected time epoch tends to be longer than an ordinary renewal interval. This paradox manifests itself in numerous interesting ways in queueing theory, a prime example being the celebrated Pollaczek-Khintchine formula for the mean waiting time in the M/G/1 queue. In this expository paper, we give intuitive arguments that “explain” why the renewal-theory paradox is ubiquitous in queueing theory, and why it sometimes produces anomalous results. In particular, we use these intuitive arguments to explain decomposition in vacation models, and to derive formulas that describe some recently-discovered counterintuitive results for polling models, such as the reduction of waiting times as a consequence of forcing the server to set up even when no work is waiting.

On the stochastic ordering of waiting times for patterns in sequences of random digits

1984 ◽  
Vol 21 (4) ◽  
pp. 730-737 ◽  
Author(s):  
Gunnar Blom
Keyword(s):  
Waiting Time ◽  
Waiting Times ◽  
Stochastic Ordering ◽  
Mean Waiting Time ◽  
Coin Tossing ◽  
The Mean

Random digits are collected one at a time until a pattern with given digits is obtained. Blom (1982) and others have determined the mean waiting time for such a pattern. It is proved that when a given pattern has larger mean waiting time than another pattern, then the waiting time for the former is stochastically larger than that for the latter. An application is given to a coin-tossing game.

THEORETICAL ANALYSIS OF MEAN WAITING TIME FOR MESSAGE DELIVERY IN LATTICE AD HOC NETWORKS

2010 ◽  
Vol 19 (08) ◽  
pp. 1711-1741
Author(s):  
AKIRA OTSUKA ◽  
KEISUKE NAKANO ◽  
KAZUYUKI MIYAKITA
Keyword(s):  
Ad Hoc Networks ◽  
Waiting Time ◽  
Ad Hoc ◽  
Waiting Times ◽  
Traffic Patterns ◽  
Mobile Nodes ◽  
Approximate Methods ◽  
Mean Waiting Time ◽  
The Mean ◽  
Hoc Networks

In ad hoc networks, the analysis of connectivity performance is crucial. The waiting time to deliver message M from source S to destination D is a measure of connectivity that reflects the effects of mobility, and some approximate methods have been proposed to theoretically analyze the mean waiting time in one-dimensional ad hoc networks that consist of mobile nodes moving along a street. In this paper, we extend these approximate methods to analyze the mean waiting time in two-dimensional networks with a lattice structure with various flows of mobile nodes. We discuss how the mean waiting times behave in such complicated street networks and how to approximate two kinds of mean waiting times. We show that our approximate methods can successfully compute the mean waiting times for even traffic patterns and roughly estimate them for uneven traffic patterns in two-dimensional lattice networks. In these analyses, we consider two shadowing models to investigate how shadowing affects the waiting time. We also discuss the effect of different positions of S on the mean waiting time.

scholarly journals Managing Queues with Heterogeneous Servers

2011 ◽  
Vol 48 (2) ◽  
pp. 435-452 ◽  
Author(s):  
Jung Hyun Kim ◽  
Hyun-Soo Ahn ◽  
Rhonda Righter
Keyword(s):  
Waiting Time ◽  
Waiting Times ◽  
Structural Result ◽  
Batch Arrivals ◽  
Optimal Thresholds ◽  
Mean Waiting Time ◽  
Server Problem ◽  
The Mean ◽  
The Impact ◽  
Intuitive Proof

We consider several versions of the job assignment problem for an M/M/m queue with servers of different speeds. When there are two classes of customers, primary and secondary, the number of secondary customers is infinite, and idling is not permitted, we develop an intuitive proof that the optimal policy that minimizes the mean waiting time has a threshold structure. That is, for each server, there is a server-dependent threshold such that a primary customer will be assigned to that server if and only if the queue length of primary customers meets or exceeds the threshold. Our key argument can be generalized to extend the structural result to models with impatient customers, discounted waiting time, batch arrivals and services, geometrically distributed service times, and a random environment. We show how to compute the optimal thresholds, and study the impact of heterogeneity in server speeds on mean waiting times. We also apply the same machinery to the classical slow-server problem without secondary customers, and obtain more general results for the two-server case and strengthen existing results for more than two servers.

scholarly journals Delay Analysis of anM/G/1/KPriority Queueing System with Push-out Scheme

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Yutae Lee ◽  
Bong Dae Choi ◽  
Bara Kim ◽  
Dan Keun Sung
Keyword(s):  
Communication Networks ◽  
Waiting Time ◽  
Queueing System ◽  
Waiting Times ◽  
Loss Probability ◽  
Mean Waiting Time ◽  
The Mean ◽  
Recursive Equations ◽  
Push Out ◽  
Stieltjes Transforms

This paper considers anM/G/1/Kqueueing system with push-out scheme which is one of the loss priority controls at a multiplexer in communication networks. The loss probability for the model with push-out scheme has been analyzed, but the waiting times are not available for the model. Using a set of recursive equations, this paper derives the Laplace-Stieltjes transforms (LSTs) of the waiting time and the push-out time of low-priority messages. These results are then utilized to derive the loss probability of each traffic type and the mean waiting time of high-priority messages. Finally, some numerical examples are provided.

On the stochastic ordering of waiting times for patterns in sequences of random digits

1984 ◽  
Vol 21 (04) ◽  
pp. 730-737 ◽  
Author(s):  
Gunnar Blom
Keyword(s):  
Waiting Time ◽  
Waiting Times ◽  
Stochastic Ordering ◽  
Mean Waiting Time ◽  
Coin Tossing ◽  
The Mean

Random digits are collected one at a time until a pattern with given digits is obtained. Blom (1982) and others have determined the mean waiting time for such a pattern. It is proved that when a given pattern has larger mean waiting time than another pattern, then the waiting time for the former is stochastically larger than that for the latter. An application is given to a coin-tossing game.

scholarly journals An audit of waiting times at a specialist psychotherapy service

2006 ◽  
Vol 30 (5) ◽  
pp. 182-184 ◽  
Author(s):  
Ged Garry ◽  
Graham Paley
Keyword(s):  
Service Delivery ◽  
Waiting Time ◽  
Action Plan ◽  
Waiting Times ◽  
Clinical Implications ◽  
Audit Process ◽  
Average Waiting Time ◽  
Mean Waiting Time ◽  
The Mean ◽  
Using Data

Aims and MethodReferrals to a specialist psychotherapy service were audited to measure the average waiting time for a first appointment and the proportion of patients waiting longer than 13 weeks. Recommendations for improving service delivery were made, an action plan implemented and the audit repeated.ResultsIn 2003, an initial audit of 355 referrals was completed using data from 2002. This found a mean waiting time to first appointment of 11.5 weeks with 30% of patients waiting longer than 13 weeks. In 2004, following implementation of the action plan, a re-audit of 200 patients found that the mean waiting time from receipt of referral to first appointment had reduced to 6.7 weeks with only 2.3% waiting more than 13 weeks.Clinical ImplicationsAudit can improve the efficiency of service delivery in a specialist psychotherapy service. However, this may require that psychotherapists review traditional ways of working. Also, it is important that they feel personally involved in the audit process.

scholarly journals The impact of the COVID-19 pandemic on waiting times for elective surgery patients: A multicenter study

PLoS ONE ◽  
2021 ◽  
Vol 16 (7) ◽  
pp. e0253875
Author(s):  
Mikko Uimonen ◽  
Ilari Kuitunen ◽  
Juha Paloneva ◽  
Antti P. Launonen ◽  
Ville Ponkilainen ◽  
...  
Keyword(s):  
Health Care ◽  
Waiting Time ◽  
Multicenter Study ◽  
Elective Surgery ◽  
Waiting Times ◽  
Public Hospitals ◽  
Mean Waiting Time ◽  
The Mean ◽  
The Impact ◽  
Surgery Incidence

Background A concern has been that health care reorganizations during the first COVID-19 wave have led to delays in elective surgeries, resulting in increased complications and even mortality. This multicenter study examined the changes in waiting times of elective surgeries during the COVID-19 pandemic in Finland. Methods Data on elective surgery were gathered from three Finnish public hospitals for years 2017–2020. Surgery incidence and waiting times were examined and the year 2020 was compared to the reference years 2017–2019. The mean annual, monthly, and weekly waiting times were calculated with 95% confidence intervals (CI). The most common diagnosis groups were examined separately. Findings A total of 88 693 surgeries were included during the study period. The mean waiting time in 2020 was 92.6 (CI 91.5–93.8) days, whereas the mean waiting time in the reference years was 85.8 (CI 85.1–86.5) days, resulting in an average 8% increase in waiting times in 2020. Elective procedure incidence decreased rapidly in the onset of the first COVID-19 wave in March 2020 but recovered in May and June, after which the surgery incidence was 22% higher than in the reference years and remained at this level until the end of the year. In May 2020 and thereafter until November, waiting times were longer with monthly increases varying between 7% and 34%. In gastrointestinal and genitourinary diseases and neoplasms, waiting times were longer in 2020. In cardiovascular and musculoskeletal diseases, waiting times were shorter in 2020. Conclusion The health care reorganizations due to the pandemic have increased elective surgery waiting times by as much as one-third, even though the elective surgery rate increased by one-fifth after the lockdown.

scholarly journals P.032 The impact of waiting time on the assessment of the first seizure onset in pediatrics

2017 ◽  
Vol 44 (S2) ◽  
pp. S21-S21
Author(s):  
AA Khan ◽  
J Lim ◽  
B Janzen ◽  
A Amiraslany ◽  
S Almubarak
Keyword(s):  
Medical Care ◽  
Waiting Time ◽  
Waiting Times ◽  
Seizure Outcome ◽  
Seizure Onset ◽  
Pediatric Neurology ◽  
Mean Waiting Time ◽  
The Mean ◽  
The Impact

Background: Childhood epilepsy has increased in global incidence. Children with epilepsy require immediate healthcare evaluation and monitoring. Waiting times between first seizure onset and pediatric neurology assessment may impact seizure outcome at follow-up. Quality of medical care for children with first seizure onset will be assessed and the impact of pediatric neurology clinic waiting times on seizure outcomes will be determined Methods: This retrospective study, based on chart review, includes patients with first seizure evaluation at the Royal University Hospital in Saskatoon between January 2012 and December 2015. The interim period before first assessment and other factors were studied in relation to seizure outcome on follow-up. Results: 1158 patients were assessed. 378 (32.6%) patients had first seizure clinic assessment. 197 (52%) had epileptic events. 181 (48%) had non-epileptic events. The mean age of patients was 8.8 years. The mean waiting time for assessment by a pediatric neurologist was 4.33 months. The mean duration of follow-up was 20.9 months. At the last seizure assessment, 132 patients were free of seizures and 65 patients had a recurrence of seizures. Conclusions: First seizure assessment is crucial for management of children with epilepsy. Waiting time and other factors may influence seizure outcome, representing opportunities to improve standard medical care.

scholarly journals Managing Queues with Heterogeneous Servers

2011 ◽  
Vol 48 (02) ◽  
pp. 435-452 ◽  
Author(s):  
Jung Hyun Kim ◽  
Hyun-Soo Ahn ◽  
Rhonda Righter
Keyword(s):  
Waiting Time ◽  
Waiting Times ◽  
Structural Result ◽  
Batch Arrivals ◽  
Optimal Thresholds ◽  
Mean Waiting Time ◽  
Server Problem ◽  
The Mean ◽  
The Impact ◽  
Intuitive Proof

We consider several versions of the job assignment problem for an M/M/m queue with servers of different speeds. When there are two classes of customers, primary and secondary, the number of secondary customers is infinite, and idling is not permitted, we develop an intuitive proof that the optimal policy that minimizes the mean waiting time has a threshold structure. That is, for each server, there is a server-dependent threshold such that a primary customer will be assigned to that server if and only if the queue length of primary customers meets or exceeds the threshold. Our key argument can be generalized to extend the structural result to models with impatient customers, discounted waiting time, batch arrivals and services, geometrically distributed service times, and a random environment. We show how to compute the optimal thresholds, and study the impact of heterogeneity in server speeds on mean waiting times. We also apply the same machinery to the classical slow-server problem without secondary customers, and obtain more general results for the two-server case and strengthen existing results for more than two servers.

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