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It is well known that various characteristics in risk and queuing process can be formulated as Markov Renewal function. We study the max-position of finitely many Markov Renewal Reward process with countable state space. We define Markov Renewal equation associated with max-posed process. The solutions of the Markov Renewal Reward equation are derived and the asymptotic behaviors of the equations are studied. Also we discuss the idea of delayed Markov renewal reward with state space.

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Genetic algorithms have been used tor solving the problem of scheduling die tasks represented by a task graph onto parallel computing architectures to nunimize the schedule length of the task graph. Due to the random nature of the initial population they, however, face the local extrema problem which could make the resulting schedules sub-optimal. To nunimize this problem, an Aided Genetic Algorithm(AGA) is proposed in this paper, in which a member of the initial population of the Genetic algorithm is obtained from a heuristic pre-scheduler. It is found that the AGA achieves the required convergence in (a) lesser number of iterations, and, (b) lesser number of trials in obtaining the near-optimal solution compared to the conventional genetic algorithm. The proposed AGA also takes the inter-task communication into account while scheduling. The method is then applied to the problem of optimally scheduling the Kalman filtering algorithm onto a multi-transputer network. The results are experimentally verified on a network of T-805 transputers. Keywords .• Genetic algorithms, multiprocessor scheduling, parallel Kalman algorithm, message-passing multiprocessors, transputers. 423 Parallel Process. Lett. 1999.09:423-436. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/21/15. For personal use only. Parallel Process. Lett. 1999.09:423-436. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/21/15. For personal use only.Processor Model: A parallel architecture is characterized by a processor graph GP * (V P , E P ), where V P = {p : p = 1,2,....M} is the set of vertices denoting the processors with associated weights {mi}, where m* represents the service rate of the processors. Ep = {(p.q) : p.q = 1,2,...,M; p * q} is the set of communication links among the processors, and the associated weights on the links {cjj} represent the capacity of the communication links. A processor node is assumed to perform both computation and communication at the same time. The communication channels are assumed to be fullduplex. The oDramunication in the different links are asynchronous. Communication between any two processors which are connected by a direct link is synchronous. Communication Parallel Process. Lett. 1999.09:423-436. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/21/15. For personal use only.