36 research outputs found
Approximations for the Moments of Nonstationary and State Dependent Birth-Death Queues
In this paper we propose a new method for approximating the nonstationary
moment dynamics of one dimensional Markovian birth-death processes. By
expanding the transition probabilities of the Markov process in terms of
Poisson-Charlier polynomials, we are able to estimate any moment of the Markov
process even though the system of moment equations may not be closed. Using new
weighted discrete Sobolev spaces, we derive explicit error bounds of the
transition probabilities and new weak a priori estimates for approximating the
moments of the Markov processs using a truncated form of the expansion. Using
our error bounds and estimates, we are able to show that our approximations
converge to the true stochastic process as we add more terms to the expansion
and give explicit bounds on the truncation error. As a result, we are the first
paper in the queueing literature to provide error bounds and estimates on the
performance of a moment closure approximation. Lastly, we perform several
numerical experiments for some important models in the queueing theory
literature and show that our expansion techniques are accurate at estimating
the moment dynamics of these Markov process with only a few terms of the
expansion
Large deviations analysis for the queue in the Halfin-Whitt regime
We consider the FCFS queue in the Halfin-Whitt heavy traffic
regime. It is known that the normalized sequence of steady-state queue length
distributions is tight and converges weakly to a limiting random variable W.
However, those works only describe W implicitly as the invariant measure of a
complicated diffusion. Although it was proven by Gamarnik and Stolyar that the
tail of W is sub-Gaussian, the actual value of was left open. In subsequent work, Dai and He
conjectured an explicit form for this exponent, which was insensitive to the
higher moments of the service distribution.
We explicitly compute the true large deviations exponent for W when the
abandonment rate is less than the minimum service rate, the first such result
for non-Markovian queues with abandonments. Interestingly, our results resolve
the conjecture of Dai and He in the negative. Our main approach is to extend
the stochastic comparison framework of Gamarnik and Goldberg to the setting of
abandonments, requiring several novel and non-trivial contributions. Our
approach sheds light on several novel ways to think about multi-server queues
with abandonments in the Halfin-Whitt regime, which should hold in considerable
generality and provide new tools for analyzing these systems
Overlapping time of a virtual customer in time-varying many-server queues
Motivated by the ongoing COVID-19 pandemic, this paper investigates
customers' infection risk by evaluating the overlapping time of a virtual
customer with others in queueing systems. Most of the current methodologies
focus on characterizing the risk in stationary systems, which may not apply to
the more practical time-varying systems. As such, we propose an approximation
framework that relies on the fluid limit to compute the expected overlapping
time in time-varying queueing systems. Simulation experiments verify the
accuracy of our approach
On the modelling and performance measurement of service networks with heterogeneous customers
Service networks are common throughout the modern world, yet understanding how their individual services effect each other and contribute to overall system performance can be difficult. An important metric in these systems is the quality of service. This is an often overlooked measure when modelling and relates to how customers are affected by a service. Presented is a novel perspective for evaluating the performance of multi-class queueing networks through a combination of operational performance and service quality—denoted the “flow of outcomes”. Here, quality is quantified by customers moving between or remaining in classes as a result of receiving service or lacking service. Importantly, each class may have different flow parameters, hence the positive/negative impact of service quality on the system’s operational performance is captured. A fluid–diffusion approximation for networks of stochastic queues is used since it allows for several complex flow dynamics: the sequential use of multiple services; abandonment and possible rejoin; reuse of the same service; multiple customers classes; and, class and time dependent parameters. The scalability of the approach is a significant benefit since, the modelled systems may be relatively large, and the included flow dynamics may render the system analytically intractable or computationally burdensome. Under the right conditions, this method provides a framework for quickly modelling large time-dependent systems. This combination of computational speed and the “flow of outcomes” provides new avenues for the analysis of multi-class service networks where both service quality and operational efficiency interact
Strong approximations for time-varying infinite-server queues with non-renewal arrival and service processes
In real stochastic systems, the arrival and service processes may not be renewal processes. For example, in many telecommunication systems such as internet traffic where data traffic is bursty, the sequence of inter-arrival times and service times are often correlated and dependent. One way to model this non-renewal behavior is to use Markovian Arrival Processes (MAPs) and Markovian Service Processes (MSPs). MAPs and MSPs allow for inter-arrival and service times to be dependent, while providing the analytical tractability of simple Markov processes. To this end, we prove fluid and diffusion limits for MAP(t)/MSPt/ queues by constructing a new Poisson process representation for the queueing dynamics and leveraging strong approximations for Poisson processes. As a result, the fluid and diffusion limit theorems illuminate how the dependence structure of the arrival or service processes can affect the sample path behavior of the queueing process. Finally, our Poisson representation for MAPs and MSPs is useful for simulation purposes and may be of independent interest.111sciescopu
Performance analysis of time-dependent queueing systems: survey and classification
Many queueing systems are subject to time-dependent changes in system parameters, such as the arrival
rate or number of servers. Examples include time-dependent call volumes and agents at inbound call
centers, time-varying air traffic at airports, time-dependent truck arrival rates at seaports, and cyclic message volumes in computer systems.There are several approaches for the performance analysis of queueing systems with deterministic parameter changes over time. In this survey, we develop a classification scheme that groups these approaches according to their underlying key ideas into (i) numerical and analytical solutions,(ii)approaches based on models with piecewise constant parameters, and (iii) approaches based on mod-ified system characteristics. Additionally, we identify links between the different approaches and provide a survey of applications that are categorized into service, road and air traffic, and IT systems