A queueing theoretic approach to set staffing levels in time-dependent dual-class service systems

This article addresses the optimal staffing problem for a nonpreemptive priority queue with two customer classes and a time-dependent arrival rate. The problem is related to several important service settings such as call centers and emergency departments where the customers are grouped into two classes of “high priority” and “low priority,” and the services are typically evaluated according to the proportion of customers who are responded to within targeted response times. To date, only approximation methods have been explored to generate staffing requirements for time-dependent dual-class services, but we propose a tractable numerical approach to evaluate system behavior and generate safe minimum staffing levels using mixed discrete-continuous time Markov chains (MDCTMCs). Our approach is delicate in that it accounts for the behavior of the system under a number of different rules that may be imposed on staff if they are busy when due to leave and involves explicitly calculating delay distributions for two customer classes. Ultimately, we embed our methodology in a proposed extension of the Euler method, coined Euler Pri, that can cope with two customer classes, and use it to recommend staffing levels for the Welsh Ambulance Service Trust (WAST)

A simulation of a police patrol service system with multi-grade time-varying incident arrivals

Due to the squeeze on public expenditure, the funding cuts imposed on the police provide a great impetus to find an efficient incident response sequence with limited resources. This is especially the case for police response systems which exhibit the characteristics of time-varying volume of demand. In this paper, we investigate two types of priority queues in the patrol service system. Both the incident arrival rate and the scheduled staff level change with time. For such a system, there is no analytical model available to give close-form performance, so simulation is used for the study. Although dynamic priority queues which enable more flexibility in setting the sequence of service requests are widely applied in many service systems, such as the NHS service system, the simulation model results show that in police patrol service systems static priority queue performs better

Modelling of queuing process at airport check-in system: a case study of Manchester and Leeds-Bradford airports

This study built a Simulation Model (SM) using SimEvents toolbox in MATLAB for implementing Analytical Models (AM) of queuing process at airport check - in system. Air travel demand data for Manchester and Leeds - Bradford airports in 2014 were adopted for validation of the model. There was no statistical differenc e between utilisation factor (UF) and service times of AM and SM outputs. Differences in AM and SM outputs for average queue length, average waiting time on queue and average number of arrivals and throughputs were attributed to variations in discrete time events considered by SM in contrary to the AM which assumed constant values for the process. The SM exhibited stochastic behaviour which actually depicts reality hence produces more reliable results. Stochastic analysis methods are therefore recommended for queuing analysis to achieve accurate results. The SM is therefore recommended to give Airport managers prior knowledge of system performance for planning and improved level of service (LOS) at airports. Key words: Airport check - in system , discrete tim e events, analytical models, simulation model, SimEvents toolbo

Recent Advances in Accumulating Priority Queues

This thesis extends the theory underlying the Accumulating Priority Queue (APQ) in three directions. In the first, we present a multi-class multi-server accumulating priority queue with Poisson arrivals and heterogeneous services. The waiting time distributions for different classes have been derived. A conservation law for systems with heterogeneous servers has been studied. We also investigate an optimization problem to find the optimal level of heterogeneity in the multi-server system. Numerical investigations through simulation are carried out to validate the model. We next focus on a queueing system with Poisson arrivals, generally distributed service times and nonlinear priority accumulation functions. We start with an extension of the power-law APQ in Kleinrock and Finkelstein (1967), and use a general argument to show that there is a linear system of the form discussed in Stanford, Taylor, and Ziedins (2014) which has the same priority ordering of all customers present at any given instant in time, for any sample path. Beyond the power-law case, we subsequently characterize the class of nonlinear accumulating priority queues for which an equivalent linear APQ can be found, in the sense that the waiting time distributions for each of the classes are identical in both the linear and nonlinear systems. Many operational queuing systems must adhere to waiting time targets known as Key Performance Indicators (KPIs), particularly in health care applications. In the last aspect, we address an optimization problem to minimize the weighted average of the expected excess waiting time (WAE), so as to achieve the optimal performance of a system operating under KPIs. We then find that the Accumulating Priority queuing discipline is well suited to systems with KPIs, in that each class of customers progresses fairly towards timely access by its own waiting time limit. Due to the difficulties in minimizing the WAE, we introduce a surrogate objective function, the integrated weighted average excess (IWAE), which provides a useful proxy for WAE. Finally, we propose a rule of thumb in which patients in the various classes accumulate priority credit at a rate that is inversely proportional to their time limits

Statistical Applications in Healthcare Systems

This thesis consists of three contributing manuscripts related to waiting times with possible applications in health care. The first manuscript is inspired by a practical problem related to decision making in an emergency department (ED). As short-run predictions of ED censuses are particularly important for efficient allocation and management of ED resources we model ED changes and present estimations for short term (hourly) ED censuses at each time point. We present a Markov-chain based algorithm to make census predictions in near future. Considering the variation in arrival pattern and service requirements, we apply and compare three models which best describe our data. We provide hourly predictions up to 24 hours in a day which will provide suggestions to ED managers on how to prevent over-crowding in their system. We illustrate our approach using 22 months data obtained from the ED of a hospital in south western Ontario. The next two manuscripts extend the theory underlying the Accumulating Priority queues (APQs). We focus on the queues with two classes of customers and Poisson arrivals. The first work in this topic derives the stationary waiting time distributions for the class of lowest priority customers in an Affine Accumulating Priority queues (Affine APQs). APQs were first studied by Kleinrock (1964) and later revisited by Stanford et al (2014) where they obtained explicit solution for the Laplace Stieltjes Transform (LST) of the stationary waiting times for all classes of customers. All subsequent publications on APQs, have assumed that all arriving customers accumulate priority credits over time starting from the same initial value (assumed, without loss of generality, to be 0). Whereas, our model studies Affine APQs which assume different initial priorities (without loss of generality in a two-class setting we assume the lowest class starts with 0 credit and the higher class customers with positive credit a. In this work we determine the waiting time distributions for the lower class of customers with Poisson arrivals and general service and present some numerical results for special cases of M/M/1, M/M/c and M/D/1. Inspired by health care applications, we have also considered a particular optimization problem related to the Affine APQ model, in order to select the optimum accumulation rate which allows for the lowest class customers to meet their associated KPIs. We next focus on the Analysis of the Maximum priority processes in the context of Affine APQ. Maximum Priority Processes were first introduced in the context of APQs in Stanford et al (2014). We derive the LST of the stationary steady state distributions of the Maximum Priority Processes as recursive functions and derive the explicit solutions for the LSTs in classical APQ (i.e. a = 0). We employ this argument to present a new approach to determine the LST of waiting time distribution for an APQ with two-classes of customers under the M=M=1 discipline. Since the Analysis of the Maximum Priority Processes in this work is done for the general class of Affine APQs, it has provided the grounds for future researches to obtain the LST of the waiting time distributions in the Affine APQs

EUROPEAN CONFERENCE ON QUEUEING THEORY 2016

International audienceThis booklet contains the proceedings of the second European Conference in Queueing Theory (ECQT) that was held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT, Toulouse, France. ECQT is a biannual event where scientists and technicians in queueing theory and related areas get together to promote research, encourage interaction and exchange ideas. The spirit of the conference is to be a queueing event organized from within Europe, but open to participants from all over the world. The technical program of the 2016 edition consisted of 112 presentations organized in 29 sessions covering all trends in queueing theory, including the development of the theory, methodology advances, computational aspects and applications. Another exciting feature of ECQT2016 was the institution of the Takács Award for outstanding PhD thesis on "Queueing Theory and its Applications"

Optimal resource allocation for multi-queue systems with a shared server pool

We study optimal allocation of servers for a system with multiple service facilities and with a shared pool of servers. Each service facility poses a constraint on the maximum expected sojourn time of a job. A central decision maker can dynamically allocate servers to each facility, where adding more servers results in faster processing speeds but against higher utilization costs. The objective is to dynamically allocate the servers over the different facilities such that the sojourn-time constraints are met at minimal costs. This situation occurs frequently in practice, e.g., in Grid systems for real-time image processing (iris scans, fingerprints). We model this problem as a Markov decision process and derive structural properties of the relative value function. These properties, which are hard to derive for multi-dimensional systems, give a full characterization of the optimal policy. We demonstrate the effectiveness of these policies by extensive numerical experiments