Distributional properties of fluid queues busy period and first passage times

In this paper we analyze the distributional properties of a busy period in an on-off fluid queue and the a first passage time in a fluid queue driven by a finite state Markov process. In particular, we show that in Anick-Mitra-Sondhi model the first passage time has a IFR distribution and the busy period has a DFR distribution

Random Fluid Limit of an Overloaded Polling Model

In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. Additionally, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.Comment: 36 pages, 2 picture

Analysis of Fluid Queues Using Level Crossing Methods ID: 11563

This dissertation is concerned with the application of the level crossing method on fluid queues driven by a background process. The basic assumption of the fluid queue in this thesis is that during the busy period of the driving process, the fluid content fills at net rate r_1, and during the idle period of the driving process, the fluid content, if positive-valued, empties at a rate r_2. Moreover, nonempty fluid content, leaks continuously at a rate r_2. The fluid models considered are: the fluid queue driven by an M/G/1 queue in Chapter 2, the fluid queue driven by an M/G/1 queue with net input and leaking rate depending on fluid level, and type of arrivals in the driving M/G/1 queue, in chapter 3, and the fluid queue driven by an M/G/1 queue with upward fluid jumps in Chapter 4. In addition, a triangle diagram has been introduced in this thesis as a technique to visualize the proportion of time that the content of the fluid queue is increasing or decreasing during nonempty cycles. Finally, we provide several examples on how the probability density function of the fluid level is related to the probability density function of the waiting time of M/G/1 queues with different disciplines

Transient analysis of Markov-fluid-driven queues

In this paper we study two transient characteristics of a Markov-fluid-driven queue, viz., the busy period and the covariance function of the workload process. Both metrics are captured in terms of their Laplace transforms. Relying on sample-path large deviations we also identify the logarithmic asymptotics of the probability that the busy period lasts longer than t, as t \to\infty. Examples are included that illustrate the theory

Fluid queues and mountain processes

This paper is devoted to the analysis of a fluid queue with a buffer content that varies linearly during periods that are governed by a three-state semi-Markov process. Two cases are being distinguished: (i) two upward slopes and one downward slope, and (ii) one upward slope and two downward slopes. In both cases, at least one of the period distributions is allowed to be completely general. We obtain exact results for the buffer content distribution, the busy period distribution and the distribution of the maximal buffer content during a busy period. The results are obtained by establishing relations between the fluid queues and ordinary queues with instantaneous input, and by using level crossing theory

Analysis of stochastic fluid queues driven by local time processes

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a certain Markov process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is always singular with respect to the Lebesgue measure which in many applications is ``close'' to reality. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a L\'evy process (a subordinator) hence making the theory of L\'evy processes applicable. Another important ingredient in our approach is the Palm calculus coming from the point process point of view.Comment: 32 pages, 6 figure

On a generic class of two-node queueing systems

This paper analyzes a generic class of two-node queueing systems. A first queue is fed by an on–off Markov fluid source; the input of a second queue is a function of the state of the Markov fluid source as well, but now also of the first queue being empty or not. This model covers the classical two-node tandem queue and the two-class priority queue as special cases. Relying predominantly on probabilistic argumentation, the steady-state buffer content of both queues is determined (in terms of its Laplace transform). Interpreting the buffer content of the second queue in terms of busy periods of the first queue, the (exact) tail asymptotics of the distribution of the second queue are found. Two regimes can be distinguished: a first in which the state of the first queue (that is, being empty or not) hardly plays a role, and a second in which it explicitly does. This dichotomy can be understood by using large-deviations heuristics

Congestion management in traffic-light intersections via Infinitesimal Perturbation Analysis

We present a flow-control technique in traffic-light intersections, aiming at regulating queue lengths to given reference setpoints. The technique is based on multivariable integrators with adaptive gains, computed at each control cycle by assessing the IPA gradients of the plant functions. Moreover, the IPA gradients are computable on-line despite the absence of detailed models of the traffic flows. The technique is applied to a two-intersection system where it exhibits robustness with respect to modeling uncertainties and computing errors, thereby permitting us to simplify the on-line computations perhaps at the expense of accuracy while achieving the desired tracking. We compare, by simulation, the performance of a centralized, joint two-intersection control with distributed control of each intersection separately, and show similar performance of the two control schemes for a range of parameters

Duality relations in finite queueing models

Motivated by applications in multimedia streaming and in energy systems, we study duality relations in fi nite queues. Dual of a queue is de fined to be a queue in which the arrival and service processes are interchanged. In other words, dual of the G1/G2/1/K queue is the G2/G1/1/K queue, a queue in which the inter-arrival times have the same distribution as the service times of the primal queue and vice versa. Similarly, dual of a fluid flow queue with cumulative input C(t) and available processing S(t) is a fluid queue with cumulative input S(t) and available processing C(t). We are primarily interested in finding relations between the overflow and underflow of the primal and dual queues. Then, using existing results in the literature regarding the probability of loss and the stationary probability of queue being full, we can obtain estimates on the probability of starvation and the probability of the queue being empty. The probability of starvation corresponds to the probability that a queue becomes empty, i.e., the end of a busy period. We study the relations between arrival and departure Palm distributions and their relations to stationary distributions. We consider both the case of point process inputs as well as fluid inputs. We obtain inequalities between the probability of the queue being empty and the probability of the queue being full for both the time stationary and Palm distributions by interchanging arrival and service processes. In the fluid queue case, we show that there is an equality between arrival and departure distributions that leads to an equality between the probability of starvation in the primal queue and the probability of overflow in the dual queue. The techniques are based on monotonicity arguments and coupling. The usefulness of the bounds is illustrated via numerical results.1 yea