By Rao C.R.
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Extra resources for Handbook of Statistics, Vol. 9. Computational Statistics
That is, an arriving customer sees the random observer's distribution with himself removed. The proof of this result is a little involved and will not be presented. Note that in an open network, equation (19) states that the arriver's distribution must be the same as that of the random observer's. Coupled with equation (16), this means that all three distributions are identical in an open network that exhibits flow balance and one step behavior. The departer's distribution can also be related to random observer's distribution (without any assumptions), and yields PDi(n IN) = p~i(n + 1)Pi(n + 1 I N)/Ai(N).
The problem now is that the numerator of ~b(zl, Zz) may involve not just individual probabilities, but also functions like ~b(z1, 0) and 4}(0, z2), none of which can be determined from the balance equations alone. , if any portion of the space c u r v e D ( z 1 , z2) = 0 lies inside or on the 4-dimensional unit hypersphere, then N(zl, z 2 ) = 0 along this portion of the curve. Obtaining appropriate equations is usually intractable or exceedingly difficult in such cases. In view of the difficulties above, it is useful to look for other solution or modeling methods.
The new definition is needed because it is meaningless to speak of 'fraction of time the station is busy' when the station has multiple servers. As an example, consider a multiserver station with k servers. If only n ~