On Parametric Steady State Analysis of a Generalized Stochastic Petri Net with a Fork-Join Subnet
On Parametric Steady State Analysis of a Generalized Stochastic Petri Net with a Fork-Join Subnet
复制标题
具有Fork-Join子网的广义随机Petri网的参数稳态分析
DOI：
--
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
G. E. Gallasch
中科院分区：
文献类型：
--
作者：
J. Billington;G. E. Gallasch
同步并行系统的性能分析是一项重要而又具有挑战性的任务。这是因为当需要同步时，产品形式解决方案不可用。所研究的系统是一个由两个并行进程组成的可控分叉连接网络。每个过程都由指数分布的延迟控制。当前系统容量N不能超过。因此，只能并发处理N个服务请求，从而阻塞进一步的请求。请求的到达也受指数分布的支配。我们用广义随机Petri网（GSPN）对这类系统建模，GSPN包含一个由强制容量约束的环境控制的双分支分叉连接结构。我们推导了任意N的参数化约简可达性图，并证明了它具有（N + 1）2个标记和一个强连通分量。我们也证明了GSPN N是有界的,从而表明一个进程不能带领其他超过N得到连续时间马尔可夫链的有关家庭,和全球平衡方程推导出家庭任意N我们解决这些方程的稳态概率为N = 1和2,然后提出一个定理的一般形式的解任意N > 2的比率的多项式过渡率。还得到了这些多项式之间21种关系的格式。最后，我们探讨了稳态概率的一些渐近行为，并将它们与先前得到的闭形式近似解联系起来。
The performance analysis of parallel systems which are synchronised is an important but challenging task. This is because product form solutions are not available when synchronisation is required. The system of interest is a controlled fork-join network, comprising two parallel processes. Each process is governed by an exponentially distributed delay. The system has a capacity, N, which cannot be exceeded. Thus only N requests for service can be handled concurrently, so that further requests are blocked. The arrival of requests is also governed by an exponential distribution. We model this class of system with a Generalized Stochastic Petri Net (GSPN) which includes a two branch fork-join structure controlled by an environment that enforces the capacity constraint. This GSPN is thus parameterised by the capacity, N. We derive the parametric reduced reachability graph for arbitrary N, and show that it has (N + 1)2 markings and one strongly connected component. We also prove that the GSPN is bounded by N, and thus show that one process cannot lead the other by more than N. We obtain the associated family of continuous time Markov chains, and derive the family of global balance equations for arbitrary N. We solve these equations for the steady state probabilities for N = 1 and 2, and then present a theorem for the general form of the solution for arbitrary N > 2 in terms of ratios of polynomials in the transition rates. A scheme of 21 relationships between these polynomials is also obtained. Finally we explore some asymptotic behaviour of the steady state probabilities and relate them to previously obtained closed form approximate solutions.