On the asymptotic execution time of multi-tasked processes on tandem processors

Consider an ordered set of processes, each consisting of tasks, to be processed by a system of K processors in tandem. Each process, immediately after havings its i-th (1≤i<K) task served by the i-th processor, is queued up on a first-come-first-served basis for processing of its (i+1)-st task by the (i+1)-st processor, until all its tasks have been served. Define the execution time to be the time to serve all the processes in the set.

Given that the processes form a stationary and ergodic sequence, as far as the processing times of the tasks are concernec, the asymptotic execution time is first explicitly computed, as the number of processes tends to infinity. This is done by using subadditive ergodic theory and certain results from the stability theory of tandem queueing networks. Then, design issues of such systems are discussed.     

On the Convergence of Global Rational Approximants for Stochastic Discrete Event Systems

Difficulties often arise in analyzing stochastic discrete event systems due to the so-called curse of dimensionality. A typical example is the computation of some integer-parameterized functions, where the integer parameter represents the system size or dimension. Rational approximation approach has been introduced to tackle this type of computational complexity. The underline idea is to develop rational approximants with increasing orders which converge to the values of the systems. Various examples demonstrated the effectiveness of the approach. In this paper we investigate the convergence and convergence rates of the rational approximants. First, a convergence rate of order O(1/ ) is obtained for the so-called Type-1 rational approximant sequence. Secondly, we establish conditions under which the sequence of [n/n] Type-2 rational approximants has a convergence rate of order .