An "almost-exact" Solution to the N-processor, M Memory Bandwidth Problem
Digital Systems Laboratory, Department of Electrical Engineering, Stanford Univ., 1976 - Computer storage devices - 27 pages
A closed-form expression is derived for the memory bandwidth obtained when N processors are permitted to generate requests to M memory modules. Use of generating functions is made, in a rather unusual fashion, to obtain this expressio n. The one approximation involved is shown to result in only a very small error -- and that, too, only for small values of M and N. This expression, which is asymptotically exact, is shown to be more accurate than existing closed form approximations. Lastly, a family of asymptotically exact solutions are presented which are easier to evaluate than is the first one. Although these expressions are less accurate than the previously derived closed-form solution, they are, nevertheless, better than existing solutions.
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2m n n accurate than existing ALMOST-EXACT apply L'Hospital's rule Approximation Simple Asymptotic Asymptot ic Solut Asymptotic Binomial Approximation asymptotically exact approximations asymptotically exact solutions Binomial Approximation Simple Closed form Improved closed form solution conditional marginal cycle definition of GM Digital Systems Laboratory employ generating functions evaluate family of asymptotically form Improved Asymptotic go to zero i-th posit identical Improved Asymptotic Binomial Improved Asymptotic Solution k l's M-l M-l M-lv M-MEMORY BANDWIDTH PROBLEM M-memory system M2+ N2+ marginal distribution marginal generating function Markov chain memory bandwidth obtained memory module modules M-j+1 n_l M-l N-processor number of processors operator order greater Percentage Error processors queued putting x=l quadratic equation queue size Ramakrishna recurrence relation released processor request satisfies the recurrence server is idle serviced Simple Asymptotic Expressions simple asymptotic solution Solution M=N steady state solution symmetric Systems Laboratory Stanford Table U. S. Energy Research x-VM