Staffing many‐server queues with autoregressive inputs

arXiv (Cornell University), 2021

The Erlang A model-an M/M/s queue with exponential abandonment-is often used to represent a service system with impatient customers. For this system, the popular square-root staffing rule determines the necessary staffing level to achieve the desirable QED (quality-and-efficiency-driven) service regime; however, the rule also implies that properties of large systems are highly sensitive to parameters. We reveal that the origin of this high sensitivity is due to the operation of large systems at a point of singularity in a phase diagram of service regimes. We can avoid this singularity by implementing a congestion-based control (CBC) scheme-a scheme that allows the system to change its arrival and service rates under congestion. We analyze a modified Erlang A model under the CBC scheme using a Markov chain decomposition method, derive non-asymptotic and asymptotic normal representations of performance indicators, and confirm that the CBC scheme makes large systems less sensitive than the original Erlang A model.

Queueing theory is concerned with developing and investigating mathematical models of systems where "customers" wait for "service." The terms "customers" and "servers" are generic. Customers could, for example, be humans waiting in a physical line or waiting on hold on the telephone, jobs waiting to be processed in a factory, or tasks waiting for processing in a computer or communication system. Examples of "service" include a medical procedure, a phone call, or a commercial transaction. Queueing theory started with the work of Danish mathematician A. K. Erlang in 1905, which was motivated by the problem of designing telephone exchanges. The field has grown to include the application of a variety of mathematical methods to the study of waiting lines in many different contexts. The mathematical methods include Markov processes, linear algebra, transform theory, and asymptotic methods, to name a few. The areas of application include computer and communication systems, manufacturing systems, and health care systems. Introductory treatments of queueing theory can be found, for example, in [1] and [2].

Probability in the Engineering and Informational Sciences, 2016

Motivated by non-Poisson stochastic variability found in service system arrival data, we extend established service system staffing algorithms using the square-root staffing formula to allow for non-Poisson arrival processes. We develop a general model of the non-Poisson non-stationary arrival process that includes as a special case the non-stationary Cox process (a modification of a Poisson process in which the rate itself is a non-stationary stochastic process), which has been advocated in the literature. We characterize the impact of the non-Poisson stochastic variability upon the staffing through the heavy-traffic limit of the peakedness (ratio of the variance to the mean in an associated stationary infinite-server queueing model), which depends on the arrival process through its central limit theorem behavior. We provide simple formulas to quantify the performance impact of the non-Poisson arrivals upon the staffing decisions, in order to achieve the desired service level. We c...

2011

Many-Server Queues with Time-Varying Arrivals, Customer A bandonment and Non-Exponential Distributions Yunan Liu This thesis develops deterministic heavy-traffic fluid appr oximations for many-server stochastic queueing models. The queueing models, with many homogenous servers working independently in parallel, are intended to model largescale service systems such as call centers and health care systems. Such models also have been e mploy d to study communication, computing and manufacturing systems. The heavytraffic approximations yield relatively simple formulas for quantities describing syst em performance, such as the expected number of customers waiting in the queue. The new performance approximations are valuable because, i n the generality considered, these complex systems are not amenable to exact mathem atical analysis. Since the approximate performance measures can be computed quite rap idly, they usefully complement more cumbersome computer simulation. Thus these heavy -t...

Queues are common scenario faced in the modern day Banks and other financial Institutions. Queuing theory is the mathematical study of waiting lines; this can also be applicable queues in the banking system. This study examine the queuing system at Guarantee Trust Bank (GTB), putting into consideration the waiting time spend by Customers, Service time spend by a Customer and the average cost a customer loses while in queue and the service cost of each server in order to optimize the system. The First Come First Serve (FCFS) Multi-Server queuing model was used to model the queuing process. The waiting time was assumed to follow a Poisson distribution while the service rate follows an Exponential distribution. This study adopted a case study approach by randomly administering questionnaires, interviews and observation of the participants. The data were collected at the GTB cash deposit unit for four days period. The data collected were analyzed using TORA optimization window based software as well as standard queuing formula. The results of the analysis showed that the average queue length, waiting time of customer with a minimum Total Cost that utilize the system is by using five Servers against the present server level of Three Servers which incur a high total cost to both the Customers and the system.

Operations Research, 2018

Analytic formulas are developed to set the time-dependent number of servers to stabilize the tail probability of customer waiting times for the Gt/GI/st + GI queueing model, which has a nonstationary non-Poisson arrival process (the Gt), nonexponential service times (the first GI), and allows customer abandonment according to a nonexponential patience distribution (the +GI). Specifically, for any delay target w > 0 and probability target α ∈ (0, 1), we determine appropriate staffing levels (the st) so that the time-varying probability that the waiting time exceeds a maximum acceptable value w is stabilized at α at all times. In addition, effective approximating formulas are provided for other important performance functions such as the probabilities of delay and abandonment, and the means of delay and queue length. Many-server heavy-traffic limit theorems in the efficiency-driven regime are developed to show that (i) the proposed staffing function achieves the goal asymptotically...

Journal of the Operational Research Society, 2009

Queueing theory continues to be one of the most researched areas of operational research, and has generated numerous review papers over the years. The phrase 'queue modelling' is used in the title to indicate a more practical emphasis. This paper uses work taken predominantly from the last 50 years of pages of the Operational Research Quarterly and the Journal of the Operational Research Society to offer a commentary on attempts of operational researchers to tackle real queueing problems, and on research foci past and future. A new discipline of 'queue modelling' is proposed, drawing upon the combined strengths of analytic and simulation approaches with the responsibility to derive meaningful insights for managers.

Servers in many real queueing systems do not work at a constant speed. They adapt to the system state by speeding up when the system is highly loaded or slowing down when load has been high for an extended time period. Their speed can also be constrained by other factors, such as geography or a downstream blockage. We develop a state-dependent queueing model in which the service rate depends on the system "load" and "overwork." Overwork refers to a situation where the system has been under a heavy load for an extended time period. We quantify load as the number of users in the system and we operationalize overwork with a state variable that is incremented with each service completion in a high-load period and decremented with each service completion in a low-load period. Our model is a quasi-birth-and-death process with a special structure that we exploit to develop efficient and easy-to- implement algorithms to compute system performance measures. We use the analytical model and simulation to demonstrate how using models that ignore adaptive server behavior can result in inconsistencies between planned and realized performance and can lead to suboptimal, unstable, or oscillatory staffing decisions.

Operations Research, 2012

An algorithm is developed to determine time-dependent staffing levels to stabilize the time-dependent abandonment probabilities and expected delays at positive target values in the Mt/GI/st + GI many-server queueing model, which has a nonhomogeneous Poisson arrival process (the Mt), has general service times (the first GI), and allows customer abandonment according to a general patience distribution (the +GI). New offered-load and modified-offered-load approximations involving infinite-server models are developed for that purpose. Simulations show that the approximations are effective. A many-server heavy-traffic limit in the efficiency-driven regime shows that (i) the proposed approximations achieve the goal asymptotically as the scale increases, and (ii) it is not possible to simultaneously stabilize the mean queue length in the same asymptotic regime.