A Robust Queueing Network Analyzer Based on Indices of Dispersion

In post-industrial economies, modern service systems are dramatically changing the daily lives of many people. Such systems are often complicated by uncertainty: service providers usually cannot predict when a customer will arrive and how long the service will be. Fortunately, useful guidance can often be provided by exploiting stochastic models such as queueing networks. In iterating the design of service systems, decision makers usually favor analytical analysis of the models over simulation methods, due to the prohibitive computation time required to obtain optimal solutions for service operation problems involving multidimensional stochastic networks. However, queueing networks that can be solved analytically require strong assumptions that are rarely satisfied, whereas realistic models that exhibit complicated dependence structure are prohibitively hard to analyze exactly. In this thesis, we continue the effort to develop useful analytical performance approximations for the single-class open queueing network with Markovian routing, unlimited waiting space and the first-come first-served service discipline. We focus on open queueing networks where the external arrival processes are not Poisson and the service times are not exponential. We develop a new non-parametric robust queueing algorithm for the performance approximation in single-server queues. With robust optimization techniques, the underlying stochastic processes are replaced by samples from suitably defined uncertainty sets and the worst-case scenario is analyzed. We show that this worst-case characterization of the performance measure is asymptotically exact for approximating the mean steady-state workload in G/G/1 models in both the light-traffic and heavy-traffic limits, under mild regularity conditions. In our non-parametric Robust Queueing formulation, we focus on the customer flows, defined as the continuous-time processes counting customers in or out of the network, or flowing from one queue to another. Each flow is partially characterized by a continuous function that measures the change of stochastic variability over time. This function is called the index of dispersion for counts. The Robust Queueing algorithm converts the index of dispersion for counts into approximations of the performance measures. We show the advantage of using index of dispersion for counts in queueing approximation by a renewal process characterization theorem and the ordering of the mean steady-state workload in GI/M/1 models. To develop generalized algorithm for open queueing networks, we first establish the heavy-traffic limit theorem for the stationary departure flows from a GI/GI/1 model. We show that the index of dispersion for counts function of the stationary departure flow can be approximately characterized as the convex combination of the arrival index of dispersion for counts and service index of dispersion for counts with a time-dependent weight function, revealing the non-trivial impact of the traffic intensity on the departure processes. This heavy-traffic limit theorem is further generalized into a joint heavy-traffic limit for the stationary customer flows in generalized Jackson networks, where the external arrival are characterized by independent renewal processes and the service times are independent and identically distributed random variables, independent of the external arrival processes. We show how these limiting theorems can be exploited to establish a set of linear equations, whose solution serves as approximations of the index of dispersion for counts of the flows in an open queueing network. We prove that this set of equations is asymptotically exact in approximating the index of dispersion for counts of the stationary flows. With the index of dispersion for counts available, the network is decomposed into single-server queues and the Robust Queueing algorithm can be applied to obtain performance approximation. This algorithm is referred to as the Robust Queueing Network Analyzer. We perform extensive simulation study to validate the effectiveness of our algorithm. We show that our algorithm can be applied not only to models with non-exponential distirbutions but also to models with more complex arrival processes than renewal processes, including those with Markovian arrival processes

A Worst-Case Approximate Analysis of Peak Age-of-Information Via Robust Queueing Approach

A new timeliness metric, called Age-of-Information (AoI), has recently attracted a lot of research interests for real-time applications with information updates. It has been extensively studied for various queueing models based on the probabilistic approaches, where the analyses heavily depend on the properties of specific distributions (e.g., the memoryless property of the exponential distribution or the i.i.d. assumption). In this work, we take an alternative new approach, the robust queueing approach, to analyze the Peak Age-of-Information (PAoI). Specifically, we first model the uncertainty in the stochastic arrival and service processes using uncertainty sets. This enables us to approximate the expected PAoI performance for very general arrival and service processes, including those exhibiting heavy-tailed behaviors or correlations, where traditional probabilistic approaches cannot be applied. We then derive a new bound on the PAoI in the single-source single-server setting. Furthermore, we generalize our analysis to two-source single-server systems with symmetric arrivals, which involves new challenges (e.g., the service times of the updates from two sources are coupled in one single uncertainty set). Finally, through numerical experiments, we show that our new bounds provide a good approximation for the expected PAoI. Compared to some well-known bounds in the literature (e.g., one based on Kingman's bound under the i.i.d. assumption) that tends to be inaccurate under light load, our new approximation is accurate under both light and high loads, both of which are critical scenarios for the AoI performance.Comment: Published in IEEE INFOCOM 202

A QoS-aware workload routing and server speed scaling policy for energy-efficient data centers: a robust queueing theoretic approach

Maintaining energy efficiency in large data centers depends on the ability to manage workload routing and control server speeds according to fluctuating demand. The use of dynamic algorithms often means that management has to install the complicated software or expensive hardware needed to communicate with routers and servers. This paper proposes a static routing and server speed scaling policy that may achieve energy efficiency similar to dynamic algorithms and eliminate the necessity of frequent communications among resources without compromising quality of service (QoS). We use a robust queueing approach to consider the response time constraints, e.g., service level agreements (SLAs). We model each server as a $G/G/1$ processor sharing (PS) queue and use uncertainty sets to define the domain of random variables. A comparison with a dynamic algorithm shows that the proposed static policy provides competitive solutions in terms of energy efficiency and satisfactory QoS

Robust Multiclass Queuing Theory for Wait Time Estimation in Resource Allocation Systems

In this paper, we study systems that allocate different types of scarce resources to heterogeneous allocatees based on predetermined priority rules-the U.S. deceased-donor kidney allocation system or the public housing program. We tackle the problem of estimating the wait time of an allocatee who possesses incomplete system information with regard, for example, to his relative priority, other allocatees' preferences, and resource availability. We model such systems as multiclass, multiserver queuing systems that are potentially unstable or in transient regime. We propose a novel robust optimization solution methodology that builds on the assignment problem. For first-come, first-served systems, our approach yields a mixed-integer programming formulation. For the important case where there is a hierarchy in the resource types, we strengthen our formulation through a drastic variable reduction and also propose a highly scalable heuristic, involving only the solution of a convex optimization problem (usually a second-order cone problem).We back the heuristic with an approximation guarantee that becomes tighter for larger problem sizes. We illustrate the generalizability of our approach by studying systems that operate under different priority rules, such as class priority. Numerical studies demonstrate that our approach outperforms simulation. We showcase how our methodology can be applied to assist patients in the U.S. deceased-donor kidney waitlist. We calibrate our model using historical data to estimate patients' wait times based on their kidney quality preferences, blood type, location, and rank in the waitlist

Routing and Staffing when Servers are Strategic

Traditionally, research focusing on the design of routing and staffing policies for service systems has modeled servers as having fixed (possibly heterogeneous) service rates. However, service systems are generally staffed by people. Furthermore, people respond to workload incentives; that is, how hard a person works can depend both on how much work there is, and how the work is divided between the people responsible for it. In a service system, the routing and staffing policies control such workload incentives; and so the rate servers work will be impacted by the system's routing and staffing policies. This observation has consequences when modeling service system performance, and our objective is to investigate those consequences. We do this in the context of the M/M/N queue, which is the canonical model for large service systems. First, we present a model for "strategic" servers that choose their service rate in order to maximize a trade-off between an "effort cost", which captures the idea that servers exert more effort when working at a faster rate, and a "value of idleness", which assumes that servers value having idle time. Next, we characterize the symmetric Nash equilibrium service rate under any routing policy that routes based on the server idle time. We find that the system must operate in a quality-driven regime, in which servers have idle time, in order for an equilibrium to exist, which implies that the staffing must have a first-order term that strictly exceeds that of the common square-root staffing policy. Then, within the class of policies that admit an equilibrium, we (asymptotically) solve the problem of minimizing the total cost, when there are linear staffing costs and linear waiting costs. Finally, we end by exploring the question of whether routing policies that are based on the service rate, instead of the server idle time, can improve system performance.Comment: First submitted for journal publication in 2014; accepted for publication in Operations Research in 2016. Presented in select conferences throughout 201

Boundary–crossing probabilities for stochastic processes and their applications

In this thesis, we focus on the problem that a stochastic process crossing (or not crossing) upper and/or lower deterministic boundaries and its application in statistics, inventory management, finance, risk and ruin theory and queueing. In Chapter 2, we provide a fast and accurate method based on fast Fourier transform (FFT), to compute the (complementary) cumulative distribution function (CDF) of the Kolmogorov-Smirnov (KS) statistic when the CDF under the null hypothesis, F(x), is purely discrete, mixed or continuous, and thus obtain exact p values of the KS test. Secondly, we developed a C++ and an R implementation of the proposed method, which fills in the existing gap in statistical software. The numerical performance of the proposed FFT-based method, implemented both in C++ and in the R package KSgeneral, available from https://CRAN.R project.org/package=KSgeneral, is illustrated when F(x) is mixed, purely discrete, and continuous. In Chapter 3, we develop an efficient method based on FFT, for computing the probability that a non-decreasing, pure jump (compound) stochastic process stays between arbitrary upper and lower boundaries (i.e., deterministic functions, possibly discontinuous) within a finite time period. We further demonstrate that our FFT-based method is computationally efficient and can be successfully applied in the context of inventory management (to determine an optimal replenishment policy), ruin theory (to evaluate ruin probabilities and related quantities) and double-barrier option pricing or simply computing non-exit probabilities for Brownian motion with general boundaries. In Chapter 4, we give explicit formulas and a numerically efficient FFT-based method for computing the probability that a non-decreasing, pure jump stochastic process will first exit from above the strip between two deterministic, possibly discontinuous, time-dependent boundaries, within a finite-time interval with an overshoot (not) exceeding a positive value. The stochastic process is a compound process with events of interest arriving according to an arbitrary point process with conditional stationary independent increments (PPCSII), and event severities with any possibly dependent joint distribution. The class of PPCSII is rather rich covering point processes with independent increments (among which non homogeneous Poisson processes and negative binomial processes), doubly stochastic Poisson (i.e., Cox processes) including mixed Poisson processes (among which processes with the order statistics property) and Markov modulated point processes. These assumptions make our framework and results generally applicable for a broad range of models arising in insurance, finance, queueing, economics, physics, astronomy and many other fields. We present examples of such applications in queueing, ruin and inventory management optimization, leading to new results in the latter fields, illustrated also numerically. In Chapter 5, we consider the large class of PPCSII and the family GD of random variables with arbitrary, possibly dependent joint distribution. These families are interchangeably used to model customers arrival and service times in the very general framework of GD/PPCSII/1 and its inverse PPCSII/GD/1 queueing models. The latter cover well known models, e.g. the G/M/1 and M/G/1 queues, but also models incorporating dependence in the arrival times, service times and across, either by directly stating their joint distribution, through a copula and appropriate marginals, or through the PPCSII class. We further introduce a double–boundary crossing (DBC) queueing duality that extends the known Cramér–Lundberg – G/M/1 duality. The DBC–queueing duality is used to establish new results with respect to the joint and marginal distributions of the busy period, idle time and the maximum waiting time, including bounds, approximations and closed form formulas. We present a FFT-based method for efficient computation of the latter distributions. We also formulate and solve novel profit optimization problems, e.g., of determining the optimal capacity of the server so as to maximize the worse-case profit margin jointly with its related probability. Results are illustrated numerically