ESTIMATING WAITING TIMES WITH THE TIME-VARYING LITTLE'S LAW

2013 ◽  
Vol 27 (4) ◽  
pp. 471-506 ◽  
Author(s):  
Song-Hee Kim ◽  
Ward Whitt
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Arrival Rate ◽  
Rate Function ◽  
Time Varying ◽  
Waiting Time Distribution ◽  
Mean Waiting Time ◽  
Little's Law ◽  
Little’S Law

When waiting times cannot be observed directly, Little's law can be applied to estimate the average waiting time by the average number in system divided by the average arrival rate, but that simple indirect estimator tends to be biased significantly when the arrival rates are time-varying and the service times are relatively long. Here it is shown that the bias in that indirect estimator can be estimated and reduced by applying the time-varying Little's law (TVLL). If there is appropriate time-varying staffing, then the waiting time distribution may not be time-varying even though the arrival rate is time varying. Given a fixed waiting time distribution with unknown mean, there is a unique mean consistent with the TVLL for each time t. Thus, under that condition, the TVLL provides an estimator for the unknown mean wait, given estimates of the average number in system over a subinterval and the arrival rate function. Useful variants of the TVLL estimator are obtained by fitting a linear or quadratic function to arrival data. When the arrival rate function is approximately linear (quadratic), the mean waiting time satisfies a quadratic (cubic) equation. The new estimator based on the TVLL is a positive real root of that equation. The new methods are shown to be effective in estimating the bias in the indirect estimator and reducing it, using simulations of multi-server queues and data from a call center.

Convex ordering of the attained waiting times in single-server queues and related problems

1991 ◽  
Vol 28 (02) ◽  
pp. 433-445 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Service Discipline ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Convex Order ◽  
Convex Ordering ◽  
Single Server Queues ◽  
Service Disciplines

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

scholarly journals Waiting Time Problems for Patterns in a Sequence of Multi-State Trials

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1893
Author(s):  
Bara Kim ◽  
Jeongsim Kim ◽  
Jerim Kim
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Probability Generating Function ◽  
Finite Collection ◽  
Numerical Inversion ◽  
Analytic Method ◽  
Waiting Time Distribution ◽  
Mean Waiting Time ◽  
The Matrix ◽  
The Mean

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability generating function of the waiting time. From this, we can obtain the stopping probabilities and the mean waiting time, but it also enables us to compute the waiting time distribution by a numerical inversion.

scholarly journals M/G/1/K system with push-out scheme under vacation policy

1996 ◽  
Vol 9 (2) ◽  
pp. 143-157 ◽  
Author(s):  
Shoji Kasahara ◽  
Hideaki Takagi ◽  
Yutaka Takahashi ◽  
Toshiharu Hasegawa
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Numerical Results ◽  
The Other ◽  
Waiting Time Distribution ◽  
Stieltjes Transform ◽  
Multiple Vacations ◽  
Push Out

We consider an M/G/1/K system with push-out scheme and multiple vacations. This model is particularly important in situations where it is essential to provide short waiting times to messages which are selected for service. We analyze the behavior of two types of messages: one that succeeds in transmission and the other that fails. We derive the Laplace-Stieltjes transform of the waiting time distribution for the message which is eventually served. Finally, we show some numerical results including the comparisons between the push-out and the ordinary blocking models.

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.

scholarly journals The Waiting-Time Distribution for the GI/G/1 Queue under the D-Policy

1992 ◽  
Vol 6 (3) ◽  
pp. 287-308 ◽  
Author(s):  
Jingwen Li ◽  
Shun-Chen Niu
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Busy Period ◽  
Waiting Times ◽  
Optimization Models ◽  
Arrival Process ◽  
Waiting Time Distribution ◽  
Service Times

We study a generalization of the GI/G/l queue in which the server is turned off at the end of each busy period and is reactivated only when the sum of the service times of all waiting customers exceeds a given threshold of size D. Using the concept of a “randomly selected” arriving customer, we obtain as our main result a relation that expresses the waiting-time distribution of customers in this model in terms of characteristics associated with a corresponding standard GI/G/1 queue, obtained by setting D = 0. If either the arrival process is Poisson or the service times are exponentially distributed, then this representation of the waiting-time distribution can be specialized to yield explicit, transform-free formulas; we also derive, in both of these cases, the expected customer waiting times. Our results are potentially useful, for example, for studying optimization models in which the threshold D can be controlled.

Convex ordering of the attained waiting times in single-server queues and related problems

1991 ◽  
Vol 28 (2) ◽  
pp. 433-445
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Waiting Times ◽  
Service Discipline ◽  
Single Server ◽  
Waiting Time Distribution ◽  
Convex Order ◽  
Convex Ordering ◽  
Single Server Queues ◽  
Service Disciplines

The attained waiting time of customers in service of the G/G/1 queue is compared for various work-conserving service disciplines. It is proved that the attained waiting time distribution is minimized (maximized) in convex order when the discipline is FCFS (PR-LCFS). We apply the result to characterize finiteness of moments of the attained waiting time in the GI/GI/1 queue with an arbitrary work-conserving service discipline. In this discussion, some interesting relationships are obtained for a PR-LCFS queue.

scholarly journals Analysis of MAP/PH(1), PH(2)/2 Queue with Bernoulli Vacations

2008 ◽  
Vol 2008 ◽  
pp. 1-20 ◽  
Author(s):  
B. Krishna Kumar ◽  
R. Rukmani ◽  
V. Thangaraj
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Queueing System ◽  
Arrival Process ◽  
Waiting Time Distribution ◽  
Steady State Probability ◽  
Matrix Geometric Method ◽  
Mean Waiting Time ◽  
The Mean ◽  
Number Of Customers

We consider a two-heterogeneous-server queueing system with Bernoulli vacation in which customers arrive according to a Markovian arrival process (MAP). Servers returning from vacation immediately take another vacation if no customer is waiting. Using matrix-geometric method, the steady-state probability of the number of customers in the system is investigated. Some important performance measures are obtained. The waiting time distribution and the mean waiting time are also discussed. Finally, some numerical illustrations are provided.

scholarly journals M(t)/M/1 Queueing System with Sinusoidal Arrival Rate

2016 ◽  
Vol 11 (1) ◽  
pp. 120-127
Author(s):  
A. P. Pant ◽  
R. P. Ghimire
Keyword(s):  
Mathematical Model ◽  
Waiting Time ◽  
Queueing System ◽  
Arrival Rate ◽  
Rate Function ◽  
Recursive Method ◽  
Mean Waiting Time ◽  
Number Of Customers ◽  
The Mathematical Model ◽  
Mean Time

This paper deals with the study of M (t)/M/1 queueing system with customers arrive to the system with sinusoidal arrival rate function λ (t) and are served exponentially with the rate μ. On formulating the mathematical model, we obtain the expressions for mean waiting time in the queue, mean time spent in the system, mean number of customers in the queue and in the system by using recursive method. Some numerical illustrations are also obtained by using computing software so as to show the applicability of the model under study.Journal of the Institute of Engineering, 2015, 11(1): 120-127

scholarly journals Waiting-time statistics in magnetic systems

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Ivandson Praeiro de Sousa ◽  
Gustavo Zampier dos Santos Lima ◽  
Marcio Assolin Correa ◽  
Rubem Luis Sommer ◽  
Gilberto Corso ◽  
...  
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Domain Walls ◽  
Waiting Times ◽  
Magnetic Domain ◽  
Threshold Level ◽  
Waiting Time Distribution ◽  
Barkhausen Effect ◽  
Magnetic Systems ◽  
Domain Wall Dynamics

Abstract Many complex systems, from earthquakes and financial markets to Barkhausen effect in ferromagnetic materials, respond with a noise consisting of discrete avalanche-like events with broad range of sizes and durations, separated by waiting times. Here we focus on the waiting-time statistics in magnetic systems. By investigating the Barkhausen noise in amorphous and polycrystalline ferromagnetic films having different thicknesses, we uncover the form of the waiting-time distribution in time series recorded from the irregular and irreversible motion of magnetic domain walls. Further, we address the question of if the waiting-time distribution evolves with the threshold level, as well as with the film thickness and structural character of the materials. Our results, besides informing on the temporal avalanche correlations, disclose the waiting-time statistics in magnetic systems also bring fingerprints of the universality classes of Barkhausen avalanches and a dimensional crossover in the domain wall dynamics.

493 Evaluation of Patient Waiting Times for Plain Radiographs Whilst Attending Outpatient Clinics in A Tertiary Orthopaedic Centre

2021 ◽  
Vol 108 (Supplement_2) ◽  
Author(s):  
Z Hayat ◽  
E Kinene ◽  
S Molloy
Keyword(s):  
Waiting Time ◽  
Waiting Times ◽  
Time Frame ◽  
Outpatient Clinics ◽  
Contributing Factors ◽  
Time Data ◽  
Work Related ◽  
Mean Waiting Time ◽  
Appointment Time ◽  
Related Stress

Abstract Introduction Reduction of waiting times is key to delivering high quality, efficient health care. Delays experienced by patients requiring radiographs in orthopaedic outpatient clinics are well recognised. Method To establish current patient and staff satisfaction, questionnaires were circulated over a two-week period. Waiting time data was retrospectively collected including appointment time, arrival time and the time at which radiographs were taken. Results 84% (n = 16) of radiographers believed patients would be dissatisfied. However, of the 296 patients questioned, 56% (n = 165) were satisfied. Most patients (89%) felt the waiting time should be under 30 minutes. Only 36% were seen in this time frame. There was moderate negative correlation (R=-0.5); higher waiting times led to increased dissatisfaction. Mean waiting time was 00:37 and the maximum 02:48. Key contributing factors included volume of patients, staff shortages (73.7%), equipment shortages (57.9%) and incorrectly filled request forms. Eight (42.1%) had felt unwell from work related stress. Conclusions A concerted effort is needed to improve staff and patient opinion. There is scope for change post COVID. Additional training and exploring ways to avoid overburdening the department would benefit. Numerous patients were open to different days or alternative sites. Funding requirements make updating equipment, expanding the department and recruiting more staff challenging.

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