Perturbation analysis of queueing systems with a time-varying arrival rate

This talk considers loss-related performance measures in high-speed networks from the standpoint of gradient estimation by the Infinitesimal Perturbation Analysis (IPA) technique. The underlying model consists of an interconnected system of continuous flow models (CFMs) processing data in a fluid form, and the performance measures of interest are related to the average loss volume caused by buffer overflow over a period of time. The main results presented are: (i) an easily implementable, distributed algorithm for computing the IPA derivative, and (ii) analysis of the algorithm that proves the unbiasedness of the IPA estimator. 'Me distributed algorithm can naturally be implemented in real-time at a network's nodes, and hence potentially can be used for on-line control. The analysis highlights some of the special structure of CFMs, and suggests a general scope of 1PRs unbiasedness, far beyond the specific performance measures discussed. The talk will conclude with remarks about the potential applicability of IPA as a design and analysis tool for networks of CFMs.

Infinitesimal Perturbation Analysis (IPA) estimators of the response gradient for a discrete event stochastic simulation are typically developed within the framework of Generalized semi-Markov processes (GSMPs). Unfortunately, while mathematically rigorous, GSMPs are not particularly useful for modeling real systems. We describe a procedure that allows IPA gradient estimation to be easily and automatically implemented in the more general and intuitive modeling context of event graphs. The intent is to make IPA gradient estimation more easily understood and more widely accessible. The pictorial nature of Event Graphs also provides insights into the basic IPA calculations and alternative descriptions of conditions under which the IPA estimator is known to be unbiased.

Stochastic Flow Models (SFMs) form a class of hybrid systems used as abstractions of complex Discrete Event Systems (DES) for the purpose of deriving performance sensitivity estimates through Infinitesimal Perturbation Analysis (IPA) techniques when these cannot be applied to the original DES. In this paper, we establish explicit connections between gradient estimators obtained through a SFM and those obtained in the underlying DES, thus providing analytical evidence for the effectiveness of these estimators which has so far been limited to empirical observations. We consider DES for which analytical expressions of IPA (or finite difference) estimators are available, specifically G/G/1 and G/G/1/K queueing systems. In the case of the G/G/1 system, we show that, when evaluated on the same sample path of the underlying DES, the IPA gradient estimators of states, event times, and various performance metrics derived through SFMs are, under certain conditions, the same as those of the associated DES or their expected values are asymptotically the same under large traffic rates. For G/G/1/K systems without and with feedback, we show that SFM-based derivative estimates capture basic properties of finite difference estimates evaluated on a sample path of the underlying DES.

Using Infinitesimal Perturbation Analysis of Stochastic Flow Models to Recover Performance Sensitivity Estimates of Discrete Event Systems

In this paper, we develop a new simulation algorithm that generates unbiased gradient estimators for the steady-state workload of a stochastic fluid network, with respect to the throughput rate of each server. Our algorithm is based on the perfect sampling algorithm developed in Blanchet and Chen (2014), and the infinitesimal perturbation analysis (IPA) method. We illustrate the performance of our algorithm with two multidimensional examples, including its formal application in the case of multidimensional reflected Brownian motion.

We investigate the use of a distributed asynchronous algorithm utilizing infinitesimal perturbation analysis (IPA) gradient estimators for on-line optimization of tandem networks of queues. In our scheme, each queue has a processor that updates a control parameter associated with the queue according to a stochastic gradient algorithm driven by IPA estimates of the gradient of the performance measure. The update times of the processors are not synchronized. The processors also communicate results of computations with each other, and this communication involves delay. We give conditions under which the algorithm converges with probability one. In our proof of convergence we analyze a particular subsequence of the sequence of control parameters, and show that this subsequence behaves like a sequence generated by a centralized synchronous gradient algorithm which updates before the start of certain busy periods of the network, and with gradient estimates that are asymptotically unbiased.

Alternative methods such as likelihood ratios (LRs), finite perturbation analysis (FPA), smoothed perturbation analysis (SPA), and rare perturbation analysis (RPA) have been proposed and could be used when infinitesimal perturbation analysis (IPA) does not apply directly. The authors discuss some links that exist between these methods and explain them by showing how each of them can be applied to a specific example, namely to estimate the derivative of the expected number of customers per regenerative cycle in a GI/G/1 queue, with respect to parameters of the interarrival and/or service-time distributions. Two efficient SPA estimators for estimating the derivative of the expected number of customers per regenerative cycle in a GI/G/1 queue are introduced. For the examples examined, these estimators outperformed those that had been proposed previously for the same problem. >

We consider a queueing system with C customer classes under a nonpreemptive service discipline. The goal is to find gradient estimators for the stationary average sojourn time per customer of each class under admission control. Due to the service discipline, IPA (infinitesimal perturbation analysis) estimators are not applicable. We present the idea of harmonic gradient (HG) estimation, based on the Fourier decomposition of periodic functions. The canonical estimators can be used to obtain consistent estimators for all the control variables in a single run. However, the large number of values for each parameter required in the estimation can greatly affect the performance. We then describe the implementations of the phantom RPA (rare perturbation analysis) method. This method requires evaluating, in parallel, the dynamics of as many phantom systems as customers in each busy period. Since this number is random, the implementation of the method can be rather complex. We use the Fourier decomposition ideas to construct a hybrid estimator that we call the phantom HG method. We then give simulation results to compare the performance of the estimators and their complexity.

A gradient-estimation procedure for a general class of stochastic discrete-event systems is developed. In contrast to most previous work, the authors focus on performance measures whose realizations are inherently discontinuous (in fact, piecewise constant) functions of the parameter of differentiation. Two broad classes of finite-horizon discontinuous performance measures arising naturally in applications are considered. Because of their discontinuity, these important classes of performance measures are not susceptible to infinitesimal perturbation analysis (IPA). Instead, the authors apply smoothed perturbation analysis, formalizing it and generalizing it in the process. Smoothed perturbation analysis uses conditional expectations to smooth jumps. The resulting gradient estimator involves two factors: the conditional rate at which jumps occur, and the expected effect of a jump. Among the types of performance measures to which the methods can be applied are transient state probabilities, finite-horizon throughputs, distributions on arrival, and expected terminal cost.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Quality of service (QoS) is very important issue for today's Internet users, due to the development of the Internet multimedia application such as voice over IP, and video streaming. In such applications QoS is required, especially the delay. End-to-end delay is the summation of the packet queueing delay in the routers and the propagation delays. In this paper we introduce a new approach for controlling the packet queueing delay in a finite buffer at a specified value. We concentrate on the packet queueing delay rather than propagation delay because in many cases queueing delay is the most important component of end-to-end delay over the internet. In this mechanism we use a continuous-time framework that operates on a router buffer and dynamically adjusts a queue threshold with the aim of constraining the delay in the buffer at a specified value as the arrival rate changes over time.

Simulation of IPA gradients in hybrid network systems

In this paper, a manufacturing system composed of a single-product machine, a buffer and a stochastic demand is considered. Two models are presented: continuous and discrete flow models including constant delivery times, machine failures and random demands. The objective is to determine the value of the optimal buffer level, for a hedging point policy which minimises the total average cost function. The cost function is the sum of inventory, transportation and lost sales costs. Infinitesimal perturbation analysis is used for optimisation of the failure-prone manufacturing system. The trajectories of buffer level are studied for the continuous and discrete cases and the infinitesimal perturbation analysis estimators are evaluated. These estimators are shown to be unbiased and then they are implemented in an optimisation algorithm which determines the optimal buffer level in the presence of constant delivery time. Numerical results are presented for continuous and discrete flow models and then compared in order to evaluate the application of the infinitesimal perturbation analysis on the discrete flow model.

A simple example of a discrete event dynamic system is used to show that given that an interevent time has a distribution function that is a function of t/ tau , where tau is the mean interarrival time and t is the interdeparture time, one cannot conclude that a perturbation of delta tau , which is the parameter perturbation, would cause every interevent time to stretch by the factor delta tau / tau . This example is used to illustrate the fact that, to ensure that the constructed perturbed path is physically possible, when performing general perturbation analysis one must keep track of events in both the nominal and perturbed paths; but when performing infinitesimal perturbation analysis (IPA) one must employ interevent distribution functions given the event sequence observed in the nominal path. One implication of the latter observation is that IPA estimates are independent of the representation. >

The authors explore the use of models of continuous tandem (CT) lines for performance analysis of discrete tandem (DT) production lines. They formalize the translation of input parameters and performance measures (PMs) between DT and CT lines. They show that the CT model can be represented as a generalized semi-Markov process. This leads to a concise simulation algorithm for a CT model. The authors show empirically that the CT model provides reasonable estimates for the DT line performance measures. They provide preliminary result on gradient estimation for CT models via infinitesimal perturbation analysis. The aim of this work is to provide a basis for the further exploration of CT models as a means to parameter optimization for DT lines.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A service facility with multiple customer classes is considered. Each class forms an independent Poisson arrival process with admission controlled by a threshold parameter and is characterized by a general service time distribution. Gradient estimators are derived for performance measures of the system (mean system time, throughput, blocking probabilities) using perturbation analysis techniques. The approach is based on exploiting alternative sample path representations of generalized semi-Markov process models of the system, such that the resulting sample functions are continuous with respect to parameters of interest. The unbiasedness of the estimators is proved. Simulation examples are included to illustrate their properties. >