Backpressure-based control protocols: design and computational aspects

Congestion control in packet-based networks is often realized by feedback protocols. In this paper we assess their performance under a back-pressure mechanism that has been proposed and standardized for Ethernet metropolitan networks. In such a mechanism the service rate of an upstream queue is reduced when the downstream queue is congested, in order to protect the downstream queue. We study a Markovian model that captures the essentials of the protocol, but at the same time allows for numerical analysis. We first derive explicit results for the stability condition of the model (which turns out to be nontrivial). Then we present logarithmic estimates of the probability of buffer overflow in the second queue, which are subsequentially used when devising an efficient simulation procedure based on importance sampling. We conclude the paper by presenting a number of numerical results, and some general design guidelines

Pricing strategies under heterogeneous service requirements

This paper analyzes a communication network, used by customers with heterogeneous service requirements. We investigate priority queueing as a way to establish service differentiation. It is assumed that there is an infinite population of customers, who join the network as long as their utility (which is a function of the queueing delay) is larger than the price of the service. We focus on the specific situation with two types of users: one type is delay-sensitive (`voice'), whereas the other is delay-tolerant (`data'); these preferences are reflected in their utility curves. Two models are considered: in the first the network determines the priority class of the users, whereas the second model leaves this choice to the users. For both models we determine the prices that maximize the provider's profit. Importantly, these situations do not coincide. Our analysis uses elements from queueing theory, but also from microeconomics and game theory (e.g., the concept of a Nash equilibrium). We conclude the paper by considering a model in which throughput (rather than delay) is the main performance measure. Again the pricing strategy exploits the heterogeneity in service requirements and willingness-to-pay

Analysis of jitter due to call-level fluctuations

In communication networks used by constant bit rate applications, call-level dynamics (i.e., entering and leaving calls) lead to fluctuations in the load, and therefore also fluctuations in the delay (jitter). By intentionally delaying the packets at the destination, one can transform the perturbed packet stream back into the original periodic stream; in other words: there is a trade off between jitter and delay, in that jitter can be removed at the expense of delay. As a consequence, for streaming applications for which the packet delay should remain below some predefined threshold, it is desirable that the jitter remains small. This paper presents a set of procedures to compute the jitter due to call-level variations. We onsider a network resource shared by a fluctuating set of constant bit rate applications (modelled as periodic sources). As a first step we study the call-level dynamics: supposing that a tagged call sees n0 calls when entering the system, then we compute the probability that at the end of its duration (consisting of, say, i packets) ni calls are present, of which n0,i stem from the original n0 calls. As a second step, we show how to compute the jitter, for given n0, ni, and n0,i; in this analysis generalized Ballot-problems have to be solved. We find an iterative exact solution to these, and explicit approximations and bounds. Then, as a final step, the (packet-level) results of the second step are weighed with the (call-level) probabilities of the first step, thus resulting in the probability distribution of the jitter experienced within the call duration. An explicit Gaussian approximation is proposed. Extensive numerical experiments validate the accuracy of the approximations and bound

Simple and efficient importance sampling scheme for a tandem queue with server slow-down

This paper considers importance sampling as a tool for rare-event simulation. The system at hand is a so-called tandem queue with slow-down, which essentially means that the server of the first queue (or: upstreanm queue) switches to a lower speed when the second queue (downstream queue) exceeds some threshold. The goal is to assess to what extent such a policy succeeds in protecting the first queue, and therefore we focus on estimating the probability of overflow in the downstream queue.\ud It is known that in this setting importance sampling with traditional state-independent distributions performs poorly. More sophisticated state-dependent schemes can be shown to be asymptotically efficient, but their implementation may be problematic, as for each state the new measure has to be computed. This paper presents an algorithm that is considerably simpler than the fully state-dependent scheme; it requires low computational effort, but still has high efficiency

Large deviations for complex buffer architectures: the short-range dependent case

This paper considers Gaussian flows multiplexed in a queueing network, where the underlying correlation structure is assumed to be short-range dependent. Whereas previous work mainly focused on the FIFO setting, this paper addresses overflow characteristics of more complex buffer architectures. We subsequently analyze the tandem queue, a priority system, and generalized processor sharing. In a many-sources setting, we explicitly compute the exponential decay rate of the overflow probability. Our study relies on large-deviations arguments, e.g., Schilder's theore

Continuous feedback fluid queues

We investigate a fluid buffer which is modulated by a stochastic background process, while the momentary behavior of the background process depends on the current buffer level in a continuous way. Loosely speaking the feedback is such that the background process behaves `as a Markov process' with generator $Q(y)$ at times when the buffer level is $y$, where the entries of $Q(y)$ are continuous functions of $y$. Moreover, the flow rates for the buffer may also depend continuously on the current buffer level. Such models are interesting in the context of closed-loop telecommunication networks, in which sources interact with network buffers, but may also be deployed in the study of certain production systems. \u

A duopoly model with heterogeneous congestion-sensitive customers

This paper analyzes a model with multiple firms (providers), and two classes of customers. These customers classes are characterized by their attitude towards `congestion' (caused by other customers using the same resources); a firm is selected on the basis of both the prices charged by the firms, and the `congestion levels'. The model can be represented by a two-stage game: in the first providers set their prices, whereas in the second the customers choose the provider (or to not use any service at all) for given prices. We explicitly allow the providers to split their resources, in order to serve more than just one market segment. This enables us to further analyze the Paris metro pricing ({\sc Pmp}) proposal for service differentiation in the Internet. \u

State-dependent Importance Sampling for a Slow-down Tandem Queue

In this paper we investigate an advanced variant of the classical (Jackson) tandem queue, viz. a two-node system with server slow-down. The slow-down mechanism has the primary objective to protect the downstream queue from frequent overflows, and it does so by reducing the service speed of the upstream queue as soon as the number of jobs in the downstream queue reaches some pre-specified threshold. To assess the efficacy of such a policy, techniques are needed for evaluating overflow metrics of the second queue. We focus on the estimation of the probability of the following rare event: overflow in the downstream queue before exhausting the system, starting from any given state in the state space.\ud Due to the rarity of the event under consideration, naive, direct Monte Carlo simulation is often infeasible. We therefore rely on the application of importance sampling to obtain variance reduction. The principal contribution of this paper is that we construct an importance sampling scheme that is asymptotically efficient. In more detail, the paper addresses the following issues. (i) We rely on powerful heuristics to identify the exponential decay rate of the probability under consideration, and verify this result by applying sample-path large deviations techniques. (2) Immediately from these heuristics, we develop a proposal for a change of measure to be used in importance sampling. (3) We prove that the resulting algorithm is asymptotically efficient, which effectively means that the number of runs required to obtain an estimate with fixed precision grows subexponentially in the buffer size. We stress that our method to prove asymptotic efficiency is substantially shorter and more straightforward than those usually provided in the literature. Also our setting is more general than the situations analyzed so far, as we allow the process to start off at any state of the state space, and in addition we do not impose any conditions on the values of the arrival rate and service rates, as long as the underlying queueing system is stable

State-dependent importance sampling for a Jackson tandem network

This paper considers importance sampling as a tool for rare-event simulation. The focus is on estimating the probability of overflow in the downstream queue of a Jacksonian two-node tandem queue – it is known that in this setting ‘traditional’ state-independent importance-sampling distributions perform poorly. We therefore concentrate on developing a state-dependent change of measure, that we prove to be asymptotically efficient.\ud More specific contributions are the following. (i) We concentrate on the probability of the second queue exceeding a certain predefined threshold before the system empties. Importantly, we identify an asymptotically efficient importance-sampling distribution for any initial state of the system. (ii) The choice of the importance-sampling distribution is backed up by appealing heuristics that are rooted in large-deviations theory. (iii) Our method for proving asymptotic efficiency is substantially more straightforward than some that have been used earlier. The paper is concluded by simulation experiments that show a considerable speed up