SOME NEW RESULTS ON WAITING TIME AND BUSY TIME IN M/G/1 QUEUE

This paper considers an M/G/1 queue with Poisson rate lambda > 0 and service time distribution G(t) which is supposed to have finite mean 1/mu. The following questions are first studied: (a) The closed bounds of the probability that waiting time is more than a fixed value; (b)The total busy time of the server, which including the distribution, probability that are more than a fixed value during a given time interval (0, t], and the expected value. Some new and important results are obtained by theories of the classes of life distributions and renewal process.

Emergency Department Waiting Times and Determinants of Prolonged Length of Stay in a Botswana Referral Hospital

Introduction: Emergency medicine is a critical component of quality public health service. In fact length of stay and waiting times in the Emergency department are key indicators of quality. The aim of this study was to determine waiting times and determinants of prolonged length of stay (LOS) in the Princess Marina Hospital Emergency Department. Methods: This was a retrospective observational study. It was done at Princess Marina, a referral hospital in Gaborone, Botswana. Triage forms of patients who presented between 19/11/ 2018 and 18/12/2018 were reviewed. Data from patient files was used to determine time duration from triage to being reviewed by a doctor, time duration from review by emergency doctor to patients’ disposition and the time duration from patient’s triage to disposition (length of stay). Prolonged length of stay was defined as duration > 6 hours. Results: A total of 1052 files repre- senting patients seen over a 1-month period were reviewed. 72.5% of the patients had a prolonged length of stay. The median emergency doctor waiting time was 4.5 hours (IQR 1.6 - 8.3 hours) and the maximum was 27.1 hours. The median length of stay in the emergency department was 9.6 hours (IQR 5.8 - 14.6 hours) and the maximum was 45.9 hours. Patient’s age (AOR 1.01), mental status (AOR 0.61), admission to internal medicine service (AOR 5.12) and pediatrics admissions (AOR 0.11) were significant predictors of prolonged length of stay in the emergency department. Conclusion: Princess Marina Hospital emergency department waiting times and length of stay are long. Age, normal mental status and internal medicine admission were independent predictors of prolonged stay (>6 hours). Admission to the pediatrics service was associated with shorter length of stay. There is a need for interventions to address the long waiting times and length of stay. Interventions should particularly focus on the identified predictors.

Waiting Times and Length of Stay of Trauma Patients in a Botswana Referral Hospital Emergency Department

Background: Mortality and morbidity due to trauma are a significant public health challenge. There is paucity of data on the waiting times and length of stay (LOS) of trauma patients in emergency departments in Botswana. The aim of this study was to determine the Emergency Department (ED) waiting times and LOS of trauma patients at Princess Marina Hospital in Gaborone, Botswana. Methods: This was a retrospective medical records review of waiting times (time from triage to review by ED medical officer) and LOS (time from triage to disposition from the emergency department). The waiting times for the different assigned acuities were assessed against the South African Triage System (SATS) standards. All trauma patients seen from 19/11/2018 to 18/12/2018 were included in the study. Prolonged length of stay was defined as duration > 6 hours. Categorical data was summarized with frequencies while numeric data was summarized with medians and interquartile ranges. Results: A total of 187 trauma patients’ files were analyzed. Of these, 72 (38.5%) were females. The median waiting time was 3.8 hours and the maximum was 19.2 hours. The median length of stay (LOS) was 8.8 hours with a maximum of 37.2 hours. Only 53 (28.3%) of the participants had a LOS of less than 6 hours. None of the emergent patients were seen immediately. Only 5 (4.0%) of the very urgent patients were seen within the target of 10 minutes. Finally, only 10 (20.4%) of urgent patients were seen within the target time of 1 hour. Conclusion: The waiting times and length of stay in Princess Marina Hospital were mostly above the recommended standards. Urgent interventions are needed to reduce waiting times and length of stay for trauma patients. More studies are needed to explore the sources of delay and investigate possible solutions to this public health challenge.

Waiting times and stopping probabilities for patterns in Markov chains

Suppose that C is a finite collection of patterns. Observe a Markov chain until one of the patterns in C occurs as a run. This time is denoted by τ. In this paper, we aim to give an easy way to calculate the mean waiting time E(τ) and the stopping probabilities P(τ = τA)with A ∈ C, where τA is the waiting time until the pattern A appears as a run.

VIRTUAL WAITING TIMES IN PRIORITY-M/G/1 QUEUES WITH VACARIONS

In this paper exhaustive-service priority-M/G/1 queueing systems with multiple vacations, single vacation and setup times are studied under the nonpreemptive and preemptive resume priority disciplines. For each of the six models analysed, the Laplace-Stieltjes transform of the virtual waiting time Wk(t) at time t of class k is derived by the method of collective marks. A sufficient condition for , where U has the standard normal distribution, is also given.

Reaction Subdiffusion with Random Waiting Time Depending on the Preceding Jump Length

To describe the energy-dependent characteristics of the reaction-subdiffusion process, we analyze the simple reaction A--→B under subdiffsion with waiting time depending on the preceding jump length, and derive the corresponding master equations in the Fourier Laplace space for the distribution of A and B particles in a continuous time random walk scheme. Moreover, the generalizations of the reaction-diffusion equation for the Gaussian jump length with the probability density function of waiting time being quadratically dependent on the preceding jump length are obtained by applying the derived master equations.

Stochastic Model of Waiting Time: A Case of Two Selected Banks in the Sekondi-Takoradi Metropolis

Queue is an act of joining a line to be served and it is part of our everyday human involvement. The objectives of the study focused on using a mathematical model to determine the waiting time of two selected banks as well as compare the average waiting time between the banks. The study uncovered the extent of usage of queuing models in achieving customer satisfaction as well as permitting to make better decisions relating to potential waiting times for customers. The study adopted a case study and observational research with the source of data being primary. Purposive sampling technique was used to select the two banks under study with the target population comprising of all the customers who intended to transact businesses with the banks within the period of 11 am to 12 pm. The sample sizes for the first, second and third day of the first bank are twenty-eight (28), seventeen (17) and twenty (20) respectively with three servers on each day whereas that for the first, second and third day of the second bank is twenty (20), nine (9) and seventeen (17) with two servers on each day. A multiple server (M/M/s) Model was adopted, and Tora Software was the statistical tool used for the analysis. Findings of the study revealed that the second bank had a higher utilization factor than the first bank. Also, the number of customers in the banking hall of the second bank was higher than that of the first bank during the entire period of observation. Finally, it takes customers of the first bank lesser minutes to complete their transaction than the second bank. In conclusion, the three days observations revealed different banking situations faced by customers in both banks which had effect on waiting time of customer service. The waiting time of customer service has effect on the number of customers in the queue and system, the probability associated with the emptiness of the system and the utilization factor. Based on the results, the study recommended, inter alia, that the management of the second bank should adopt a three-server (M/M/3) model.

An approach towards estimating critical value of waiting time at transit stops

Waiting time at transit stops is found to be an influential policy variable for a passenger’s decision on whether to undertake a given transit service. With regard to policy framework for improvement of operational service headway of a transit service and thereby its waiting time, the necessity to have knowledge on its critical value becomes inevitable. The critical value of waiting time for passengers waiting at transit stops is that duration beyond which passengers are found to be no more interested to wait for a that transit service. The paper demonstrates an approach for estimating the critical value of waiting time at urban transit stops with reference to public transport services such as city bus and shared-auto operational in Bhubaneswar, India. The critical value of waiting time is estimated from the point on cumulative distribution curve of waiting time frequency distribution, at which the maximum rate of change of the slope of curve occurs. The work assumes two positively skewed distributions such as gamma and log-normal for observed distributional pattern of waiting time. The work identifies that gamma distribution is comparatively fitting the observed data better than log-normal distribution. The study reveals that the critical value of waiting time for city bus passengers is about twice than that of shared auto passengers.Though, the study presents new information on critical values of waiting time with reference to an urban area of a developing country, it also demonstrates an experience on application of probability distribution functions for understanding distributional pattern of waiting time.

Impact of adjustment measures on reducing outpatient waiting time in a community hospital： application of a computer simulation

Background As an important determinant of patient satisfaction, waiting time, has gained increasing attention in the field of health care services. The present study aimed to illustrate the distribution characteristics of waiting time in a community hospital and explore the impact of potential measures to reduce outpatient waiting time based on a computer simulation approach. Methods Dudng a one-month study period in 2006, a cross-sectional study was conducted in a community hospital located in Shanghai, China. Baseline data of outpatient waiting time were calculated according to the records of registration time and payment time. A simulation technique was adopted to investigate the impact of perspective reform methods on reducing waiting time. Results Data from a total of 10 092 patients and 26 816 medical consultations were collected in the study and 19 947 medical consultations were included. The average of the total visit time for outpatients in this hospital was 43.6 minutes in the morning, 19.1 minutes in the afternoon, and 34.3 minutes for the whole day studied period. The simulation results suggested that waiting time for outpatients could be greatly reduced through the introduction of appointment system and flexible demand-orientated doctor scheduling according to the numbers of patients waiting at different time of the workday. Conclusion Adoption of an appointment system and flexible management of doctor scheduling may be effective way to achieve decreased waiting time.

Predictive-reactive Strategy for Flowshop Rescheduling Problem:Minimizing the Total Weighted Waiting Times and Instability

Due to the fourth revolution experiencing,referred to as Industry 4.0,many production firms are devoted to integrating new technological tools to their manufacturing process.One of them,is rescheduling the tasks on the machines responding to disruptions.While,for static scheduling,the efficiency criteria measure the performance of scheduling systems,in dynamic environments,the stability criteria are also used to assess the impact of jobs deviation.In this paper,a new performance measure is investigated for a flowshop rescheduling problem.This one considers simultaneously the total weighted waiting time as the efficiency criterion,and the total weighted completion time deviation as the stability criterion.This fusion could be a very helpful and significant measure for real life industrial systems.Two disruption types are considered:jobs arrival and jobs cancellation.Thus,a Mixed Integer Linear Programming(MILP)model is developed,as well as an iterative predictive-reactive strategy for dealing with the online part.At last,two heuristic methods are proposed and discussed,in terms of solution quality and computing time.

Quantifying out-of-station waiting time in oversaturated urban metro systems

Metro systems in megacities such as Beijing,Shenzhen,and Guangzhou are under great passenger demand pressure.During peak hours,it is common to see oversaturated conditions(i.e.,passenger demand exceeds network capacity)and a popular control intervention is to restrict the entering rate by setting up out-of-station queueing with crowd control barriers.The out-of-station waiting can make up a substantial proportion of total travel time but is often ignored in the literature.Quantifying out-of-station waiting is important to evaluating the social benefit and cost of metro services;however,out-of-station waiting is difficult to estimate because it leaves no trace in smart card transactions of metros.In this study,we estimate the out-of-station waiting time by leveraging the information from a small group of transfer passengers—those who transfer from nearby bus routes to the metro station.Based on the transfer interval of this small group,we infer the out-of-station waiting time for all passengers by a Gaussian Process regression and then use the estimated out-of-station waiting time to build queueing diagrams.We apply our method to the Tiantongyuan North station of Beijing metro;results show that the maximum out-of-station waiting time can reach 15 min,and the maximum queue length can be over 3000 passengers.We find out-of-station waiting can cause significant travel costs and thus should be considered in analyzing transit performance,mode choice,and social benefits.To the best of our knowledge,this paper is the first quantitative study for out-of-station waiting time.

Partitioning of Independent Tasks for Minimizing Completion Time and Total Waiting Time

Parallel processors provide fast computing environments for various users.But the real efficiencies ofparallel processors intensively depend on the partitioning strategies of tasks over the processors.In thispaper,the partitioning problems of independent tasks for homogeneous system of parallel processors arequantitatively studied.We adopt two criteria,minimizing the completion time and the total waiting time,to determine the optimal partitioning strategy.

Two-stage scheduling on batch and single machines with limited waiting time constraint

This study addresses the problem of two-stage scheduling on batch and single machines with limited waiting time constraint; thus, the makespan is minimized.A mixed-integer linear programming model is proposed for this problem. Three tight lower bounds and a heuristic algorithm are developed. The worst-case performance of the proposed algorithm is discussed. A hybrid differential evolution algorithm is also developed to improve the solution quantity. Numerical results show that the hybrid algorithm is capable of obtaining high-quality solutions and exhibits a competitive

LSTM Based Spectrum Prediction for Real-Time Spectrum Access for IoT Applications

In the Internet of Things(IoT)scenario,many devices will communi-cate in the presence of the cellular network;the chances of availability of spec-trum will be very scary given the presence of large numbers of mobile users and large amounts of applications.Spectrum prediction is very encouraging for high traffic next-generation wireless networks,where devices/machines which are part of the Cognitive Radio Network(CRN)can predict the spectrum state prior to transmission to save their limited energy by avoiding unnecessarily sen-sing radio spectrum.Long short-term memory(LSTM)is employed to simulta-neously predict the Radio Spectrum State(RSS)for two-time slots,thereby allowing the secondary node to use the prediction result to transmit its information to achieve lower waiting time hence,enhanced performance capacity.A frame-work of spectral transmission based on the LSTM prediction is formulated,named as positive prediction and sensing-based spectrum access.The proposed scheme provides an average maximum waiting time gain of 2.88 ms.The proposed scheme provides 0.096 bps more capacity than a conventional energy detector.

A DISCRETE TIME TWO-LEVEL MIXED SERVICE PARALLEL POLLING MODEL

We present a discrete time single-server two-level mixed service polling systems with two queue types, one center queue and N normal queues. Two-level means the center queue will be successive served after each normal queue. In the first level, server visits between the center queue and the normal queue. In the second level, normal queues are polled by a cyclic order. Mixed service means the service discipline are exhaustive for center queue, and parallel 1-limited for normal queues. We propose an imbedded Markov chain framework to drive the closed-form expressions for the mean cycle time, mean queue length, and mean waiting time. Numerical examples demonstrate that theoretical and simulation results are identical the new system efficiently differentiates priorities.

ATS:A Novel Time-Sharing CPU Scheduling Algorithm Based on Features Similarities

Minimizing time cost in time-shared operating systems is considered basic and essential task,and it is the most significant goal for the researchers who interested in CPU scheduling algorithms.Waiting time,turnaround time,and number of context switches are themost time cost criteria used to compare between CPU scheduling algorithms.CPU scheduling algorithms are divided into non-preemptive and preemptive.RoundRobin(RR)algorithm is the most famous as it is the basis for all the algorithms used in time-sharing.In this paper,the authors proposed a novel CPU scheduling algorithm based on RR.The proposed algorithm is called Adjustable Time Slice(ATS).It reduces the time cost by taking the advantage of the low overhead of RR algorithm.In addition,ATS favors short processes allowing them to run longer time than given to long processes.The specific characteristics of each process are;its CPU execution time,weight,time slice,and number of context switches.ATS clusters the processes in groups depending on these characteristics.The traditionalRRassigns fixed time slice for each process.On the other hand,dynamic variants of RR assign time slice for each process differs from other processes.The essential difference between ATS and the other methods is that it gives a set of processes a specific time based on their similarities within the same cluster.The authors compared between ATS with five popular scheduling algorithms on nine datasets of processes.The datasets used in the comparison vary in their features.The evaluation was measured in term of time cost and the experiments showed that the proposed algorithm reduces the time cost.

Risk Factors for Mortality of Patients Waiting for Elective Coronary Artery Bypass Graft Surgery

Background: Insufficient capacity for cardiac surgery results in extensive waiting time for patients requiring coronary artery bypass grafting (CABG). Previous studies have reported a consequence of an increased risk of mortality while waiting for CABG. Identification of risk factors for mortality is important in patients waiting for CABG. Objectives: To assess mortality rates and identify risk factors for mortality of patients waiting for CABG. Methods: This retrospective cohort study was done on patients waiting for elective CABG in dr. Kariadi General Hospital from January 2018 to December 2020. Identification of risk factors associated with mortality was done on patients who were waiting for CABG using logistic regression methods. Results: There were 162 patients fulfilling the criteria, with a mean waiting time for surgery of 9.8 months. While waiting for CABG surgery, 32 (19.7%) patients died of any cause. Independent risk factors for death while waiting for CABG included left ventricular ejection fraction ≤ 45% (OR 4.75;95% CI 1.76 - 12.78;p = 0.002), left main disease (OR 4.12;95% CI 1.50 - 11.27;p = 0.006), serum creatinine ≥ 1.5 mg/dl (OR 3.71;95% CI 1.41 - 9.74;p = 0.008), and a number of coronary artery disease risk factors ≥ 3 (OR 3.34;95% CI 1.24 - 8.99;p = 0.017). Conclusions: Long waiting time for CABG is associated with a high mortality rate which is influenced by left ventricular ejection fraction ≤ 45%, left main disease, serum creatinine ≥ 1.5 mg/dl, and a number of coronary arteries disease risk factors ≥ 3.

Probability Distributions Arising in Connection with the Inspection Paradox for the Poisson Process

The Inspection Paradox refers to the fact that in a Renewal Process, the length of the interarrival period which contains a fixed time is stochastically larger than the length of a typical interarrival period. To provide a more complete understanding of this phenomenon, conditioning arguments are used to obtain the distributions and moments of the lengths of the interarrival periods other than the one containing this fixed time for the case of the time-homogeneous Poisson Process. Distributions of the waiting times for events that occur both before and after this fixed time are derived. This provides a fairly complete probabilistic analysis of the Inspection Paradox.

Probability Distributions Arising in Connection with the Inspection Paradox for Bernoulli Trials

In renewal theory, the Inspection Paradox refers to the fact that an interarrival period in a renewal process which contains a fixed inspection time tends to be longer than one for the corresponding uninspected process. We focus on the paradox for Bernoulli trials. Probability distributions and moments for the lengths of the interarrival periods are derived for the inspected process, and we compare them to those for the uninspected case.

Glasgow coma scale and APACHE-II scores affect the liver transplantation outcomes in patients with acute liver failure

BACKGROUND:The timing and selection of patients for liver transplantation in acute liver failure are great challenges.This study aimed to investigate the effect of Glasgow coma scale(GCS)and APACHE-II scores on liver transplantation outcomes in patients with acute liver failure.METHOD:A total of 25 patients with acute liver failure were retrospectively analyzed according to age,etiology,time to transplantation,coma scores,complications and mortality.RESULTS:Eighteen patients received transplants from live donors and 7 had cadaveric whole liver transplants.The mean duration of follow-up after liver transplantation was 39.86±40.23 months.Seven patients died within the perioperative period and the 1-,3-,5-year survival rates of the patients were72%,72%and 60%,respectively.The parameters evaluated for the perioperative deaths versus alive were as follows:the mean age of the patients was 33.71 vs 28 years,MELD score was 40 vs32.66,GCS was 5.57 vs 10.16,APACHE-II score was 23 vs 18.11,serum sodium level was 138.57 vs 138.44 mmol/L,mean waiting time before the operation was 12 vs 5.16 days.Low GCS,high APACHE-II score and longer waiting time before the operation(P【0.01)were found as statistically significant factors for perioperative mortality.CONCLUSION:Lower GCS and higher APACHE-II scores are related to poor outcomes in patients with acute liver failure after liver transplantation.