By Gordon F. Newell (auth.)
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Extra resources for Approximate Behavior of Tandem Queues
T) J was, for reasons of symmetry, simpler than a representation in terms of queue lengths. It is possible, of course, to describe the evolution in terms of one cumulative arrival function, DO(t) for example, plus the queue lengths. - D. (t) J- and j 1, 2, ••• , n g(xO' ~l' ~2' ••• , ~n) be the joint probability density of Q. 1) DO(t), 28 Substitution of this into the equations of section 4 gives the fo11owing equations for g t:. n + L j=l [(-fl. ) J o ax o a£. j n-1 for Lt:.. J j=l 0 < £. j < c.
6. Soft Boundaries. We saw in the last two sections that the boundary condi- tions for the diffusion equation are rather complicated and perhaps even questionable. 2) 1 , one encounters further complications at other edges due to the non-analytic behavior of f near edges. To avoid some of these problems, it is possible to formulate the problem in another way, which is actually more accurate than the above. One can eliminate the boundary conditions at the expense of making the diffusion equation more complicated (and almost impossible to solve explicitly).
11) is a surface term from the upper limits of integration from the lower limits. 6), describe consequences of servers k and/or k + 1 serving so many customers as to either dissipate the queue of customers upstream or fill the storage downstream, while possibly interfering with each other. 8). The lower limit terms and/or k + 1 B(k,k+l) describe the consequences of servers k serving too few customers so as to fill the upstream stor- age or cause the downstream queue for may influence the behavior of k + 2 f(k,k+l).
Approximate Behavior of Tandem Queues by Gordon F. Newell (auth.)