Queues with auto-correlated service times

As empirically observed in restaurants, call centers and intensive care units, service times needed by customers are often related to the delay they experience in queue. Two forms of dependence mechanisms in service systems with customer abandonment immediately come to mind: First, the service requirement of a customer may evolve while waiting in queue, in which case the service time of each customer is endogenously determined by the system's dynamics. Second, customers may arrive (exogenously) to the system with a service and patience time that are stochastically dependent, so that the service-time distribution of the customers that end up in service is different than that of the entire customer population. We refer to the former type of dependence as endogenous, and to the latter as exogenous. Since either dependence mechanism can have significant impacts on a system's performance, it should be identified and taken into consideration for performance evaluation and decision-making purposes. However, identifying the source of dependence from observed data is hard because both the service times and patience times are censored due to customer abandonment. Further, even if the dependence is known to be exogenous, there remains the difficult problem of fitting a joint service-patience times distribution to the censored data. We address these two problems, and provide a solution to the corresponding statistical challenges by proving that both problems can be avoided. We show that, for any exogenous dependence, there exists a corresponding endogenous dependence, such that the queuing dynamics under either dependence have the same law. We also prove that there exist endogenous dependencies for which no equivalent exogenous dependence exists. Therefore, the endogenous dependence can be considered as a generalization of the exogenous dependence. As a result, if dependence is observed in data, one can always consider the system as having an endogenous dependence, regardless of the true underlying dependence mechanism. Since estimating the structure of an endogenous dependence is substantially easier than estimating a joint service-patience distribution from censored data, our approach facilitates statistical estimations considerably.

It is known that a correlation in either the service or interarrival times causes a deterioration in the performance of a queuing system. This study aimed to determine which of the two correlations—in the service times or in the interarrival times—has a stronger influence on the expected queue length, assuming an identical autocorrelation function in both cases. To achieve this goal, a formula for the expected queue length in a system with correlated arrivals was derived first. This new formula, along with a known formula for the expected queue length in a system with correlated service, was used to compare the influence of the two correlations. Various scenarios were studied, such as cases where the common correlation was positive or negative, where the variance of the service or interarrival time was low or high, and where the system load was low or high. Furthermore, both the time-dependent and the steady-state behaviors of the systems were compared. The following two key observations were made. If the impact of other factors on the queue length is minor, then a positive correlation has a worse effect on the queue length when present in service times than in arrival times. On the contrary, a negative correlation has a worse effect on the queue length when present in arrival times than in service times.

Background: Health sector is inseparable from the decentralized system of local autonomy. Health sector is a responsibility of the local government, even though it is frequently included in the political policies of a leader. The direction of healthcare service development, particularly at the level of Health Center, has been maintained in the Mayor's Decree of Singkawang No. 82/2009 on the subsidiary of healthcare in Kota Singkawang. Objective: To find out the quality of healthcare at the Health Centers in relation to the primary healthcare subsidy based on the perception of society, control/supervision of Local Health Office, management, service time, service capacity/type, and attitude of the health center staffs. Method: A descriptive research with case study design was conducted in three Health Centers: Singkawang Tengah, Singkawang Timur, and Singkawang Utara Health Centers. Subjects of the research were 15 health staffs and 111 patients. The data were collected using questionnaire, observation, and interviews. Results: The research found a score of 3.3 for the healthcare in Singkawang Tengah, Singkawang Timur, and Singkawang Utara Health Centers. It means that the Health Center provided relatively high quality healthcare. From the Reliability dimension, a score of 2.92 was found for Point 2 quick examination service with reference to the standard procedure and a score of 2.97 for Point 5, the timeliness of healthcare. From the Responsiveness dimension, a score of 2.77 was found for Point 3 – the patients did not wait long to get the healthcare service – and a score of 2.94 for Point 4 – the working hour of the Health Center. Qualitative analysis showed that the Local Health Office controlled/supervised the Health Centers by means of utilization/ visit reports and management. It was found that service time was frequently ignored and that service type/capacity at the Health Centers was constrained by the availability of reagents and medication. The health staffs tended to ignore service quality and time and there was an indication of deviation in the utilization/visit reports sent by the Health Centers. Conclusion: The Local Health Office did not have adequate tools to control/supervise the Health Centers, as evident from the aspect of management, service time, service type/capacity, and health staff attitude. Procurement of healthcare supplies was hampered by bidding process and the health staffs need continuous training and development. Keywords: Health Office, Health Centers, Public Perception, and Healthcare quality

Delays in a series of queues with correlated service times

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size m and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order m. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order m, but no derivation for the queue length was provided in the general case. This paper introduces an order m^2 phase-type representation (kappa, K) for the queue length distribution in the general case. Moreover, we prove that the order m^2 of the distribution cannot be further reduced in general. Examples for which the order is between m and m^2 are also identified. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute (kappa, K). Moreover, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order m phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the R matrix of a Quasi-Birth-Death Markov chain.

Performance of correlated queues: the impact of correlated service and inter-arrival times

A multi-population algorithm to solve the VRP with stochastic service and travel times

When Service Times Depend on Customers’ Delays: A Relationship Between Two Models of Dependence Service times of customers often depend on the delay they experience in queue, as was recently demonstrated empirically in restaurants, call centers, and intensive care units. Two forms of dependence mechanisms in service systems with customer abandonment are studied in this paper: First, the service requirement of a customer may evolve while waiting in queue. Second, customers may arrive to the system with an exogenous service and patience time that are stochastically dependent. Because either dependence mechanism can have significant impacts on a system's performance, it should be identified and taken into consideration for performance evaluation and decision-making purposes. However, identifying the source of dependence from observed data is hard because both the service times and patience times are censored due to customer abandonment. Further, even if the dependence is known to be the latter exogenous one, there remains the difficult task of fitting a joint service-patience times distribution to the censored data. In “When Service Times Depend on Customers’ Delays: A Relationship Between Two Models of Dependence”, Wu, Bassamboo, and Perry provide a solution to address these statistical challenges.

What determines the average length of a queue, which stretches in front of a service station? The answer to this question clearly depends on the average rate at which jobs arrive at the queue and on the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations in service and arrival times are also important, and that these are a major source of backlogs and delays. Strategies that could mitigate fluctuations-induced delays are, thus in high demand as queue structures appear in various natural and man-made systems. Here, we demonstrate that a simple service resetting mechanism can reverse the deleterious effects of large fluctuations in service times, thus turning a marked drawback into a favorable advantage. This happens when stochastic fluctuations are intrinsic to the server, and we show that service resetting can then dramatically cut down average queue lengths and waiting times. Remarkably, this strategy is also useful in extreme situations where the variance, and possibly even mean, of the service time diverge—as resetting can then prevent queues from “blowing up.” We illustrate these results on the M/G/1 queue in which service times are general and arrivals are assumed to be Markovian. However, the main results and conclusions coming from our analysis are not specific to this particular model system. Thus, the results presented herein can be carried over to other queueing systems: in telecommunications, via computing, and all the way to molecular queues that emerge in enzymatic and metabolic cycles of living organisms.

The objective of this paper is to consider the vehicle routing problem with time windows under two uncertainties: service and travel times. We introduce new resolution approaches for the robust problem and an efficient parallel procedure for the generation of all possible scenarios. The best robust solution of each scenario can be achieved by using a parallel adaptive large neighborhood search metaheuristic. Through our analysis, we expect to find the best compromise between the reduced running time and a best good solution, which leads to four distinct combinations of parallel/sequential approaches. The computational experiments are performed and tested on Solomon’s benchmark and large randomly generated instances. Furthermore, our results can be protected against delay in service time in a reasonable running time especially for large instances.

We consider the stability of N-model systems that consist of two customer classes and two server pools. Servers in one of the pools can serve both classes, but those in the other pool can serve only one of the classes. The standard fluid models in general are not sufficient to establish the stability region of these systems under static priority policies. Therefore, we use a novel and a general approach to augment the fluid model equations based on induced Markov chains. Using this new approach, we establish the stability region of these systems under a static priority rule with thresholds when the service and interarrival times have phase-type distributions. We show that, in certain cases, the stability region depends on the distributions of the service and interarrival times (beyond their mean), on the number of servers in the system, and on the threshold value. We also show that it is possible to expand the stability region in these systems by increasing the variability of the service times (without changing their mean) while keeping the other parameters fixed. The extension of our results to parallel server systems and general service time distributions is also discussed.

Queueing systems with bulk-service and vacation policy have become one of the pivotal interest for the researchers due to their widespread application in food processing technology, manufacturing systems, power consumption in small cell base stations etc. This article analyzes a single server versatile bulk-service queueing system wherein the customers arrive according to a compound Poisson process and the service time is dependent on the batch size of undergoing service. Moreover, single and multiple vacation policies have been incorporated along with queue-length-dependent vacation. After providing the steady-state system equations, the bivariate probability generating function of queue and server content distribution together has been derived at departure epoch. After the evaluation of the unknown probabilities, complete joint distributions have been extracted in terms of roots of the denominator of the bivariate probability generating function. The discussed procedure and the reported results have been depicted through some numerical examples for different service and vacation time distributions. Some significant observation about the model has been sketched graphically.

For the stationary waiting time process of a GI/G/1 queue we prove the empirically obvious fact that if the service times are increased and the interarrival times decreased, then the correlation of waiting times of successive customers is increased. If the service and interarrival times are respectively larger or smaller than for given exponential service or interarrival times, then the serial covariances of waiting times are bounded by the known covariances for the given exponential case. Also, some general bounds for the covariances are given, and the heavy traffic case is considered. The results are very useful in simulation problems where the mean stationary waiting time is estimated by a sample and it is sought to determine the mean square error of the sample mean.

Long waiting time in the out-patient clinic is a major cause of dissatisfaction in Eye care services. This study aimed to assess patients' waiting and service times in the out-patient Ophthalmology clinic of UITH. This was a descriptive cross-sectional study conducted in March and April 2019. A multi-staged sampling technique was used. A timing chart was used to record the time in and out of each service station. An experiencebased exit survey form was used to assess patients' experience at the clinic. The frequency and mean of variables were generated. Student t-test and Pearson's correlation were used to establish the association and relationship between the total clinic, service, waiting, and clinic arrival times. Ethical approval was granted by the Ethical Review Board of the UITH. Two hundred and twenty-six patients were sampled. The mean total waiting time was 180.3± 84.3 minutes, while the mean total service time was 63.3±52.0 minutes. Patient's average total clinic time was 243.7±93.6 minutes. Patients' total clinic time was determined by the patients' clinic status and clinic arrival time. Majority of the patients (46.5%) described the time spent in the clinic as long but more than half (53.0%) expressed satisfaction at the total time spent at the clinic. Patients' clinic and waiting times were long, however, patients expressed satisfaction with the clinic times. Self-funded.

A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues