Fast simulation of the leaky bucket algorithm

We use fast simulation methods, based on importance sampling, to efficiently estimate cell loss probability in queueing models of the Leaky Bucket algorithm. One of these models was introduced by Berger (1991), in which the rare event of a cell loss is related to the rare event of an empty finite buffer in an "overloaded" queue. In particular, we propose a heuristic change of measure for importance sampling to efficiently estimate the probability of the rare empty-buffer event in an asymptotically unstable GI/GI/1/k queue. This change of measure is, in a way, "dual" to that proposed by Parekh and Walrand (1989) to estimate the probability of a rare buffer overflow event. We present empirical results to demonstrate the effectiveness of our fast simulation method. Since we have not yet obtained a mathematical proof, we can only conjecture that our heuristic is asymptotically optimal, as k/spl rarr//spl infin/

Optimal Prefix Codes for Infinite Alphabets with Nonlinear Costs

Let $P = \{p(i)\}$ be a measure of strictly positive probabilities on the set of nonnegative integers. Although the countable number of inputs prevents usage of the Huffman algorithm, there are nontrivial $P$ for which known methods find a source code that is optimal in the sense of minimizing expected codeword length. For some applications, however, a source code should instead minimize one of a family of nonlinear objective functions, $\beta$-exponential means, those of the form $\log_a \sum_i p(i) a^{n(i)}$, where $n(i)$ is the length of the $i$th codeword and $a$ is a positive constant. Applications of such minimizations include a novel problem of maximizing the chance of message receipt in single-shot communications ($a<1$) and a previously known problem of minimizing the chance of buffer overflow in a queueing system ($a>1$). This paper introduces methods for finding codes optimal for such exponential means. One method applies to geometric distributions, while another applies to distributions with lighter tails. The latter algorithm is applied to Poisson distributions and both are extended to alphabetic codes, as well as to minimizing maximum pointwise redundancy. The aforementioned application of minimizing the chance of buffer overflow is also considered.Comment: 14 pages, 6 figures, accepted to IEEE Trans. Inform. Theor

Delay-Energy lower bound on Two-Way Relay Wireless Network Coding

Network coding is a novel solution that significantly improve the throughput and energy consumed of wireless networks by mixing traffic flows through algebraic operations. In conventional network coding scheme, a packet has to wait for packets from other sources to be coded before transmitting. The wait-and-code scheme will naturally result in packet loss rate in a finite buffer. We will propose Enhanced Network Coding (ENC), an extension to ONC in continuous time domain. In ENC, the relay transmits both coded and uncoded packets to reduce delay. In exchange, more energy is consumed in transmitting uncoded packets. ENC is a practical algorithm to achieve minimal average delay and zero packet-loss rate under given energy constraint. The system model for ENC on a general renewal process queuing is presented. In particular, we will show that there exists a fundamental trade-off between average delay and energy. We will also present the analytic result of lower bound for this trade-off curve, which can be achieved by ENC

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