# Models for Queue Length in Clocked Queueing Networks

Models for Queue Length in Clocked Queueing Networks
O E Percus
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19) + (l-p)J2i^ic. I -^o = !)-/>), 1 = 1 7 the infinite series representing tlie memory of past queue length.
Unrortunatoly, (2. 19) as it stands makes use not only of the presumably known time series {a, }, but also of the specific process in the form of the condition So = 1. It is hence directly useful only for sufficiently restricted time series that the condition becomes innocuous, and this is precisely what we will take advantage of in the sequel.
->i/„, +tJ^' Tr/lr + l^/^, 3. Output-Input
... Transformation We are now prepared to start a detailed analysis of the time series transformation represented by Fig. 1. Of the two components, collection of inputs (2. 1). And passage through the queue, the former is far simpler to analyze. Since /„ and /^ represent outputs of previous switches, this is basically an input-output transformation, and is most readily handled by the use of generating functions.
Suppose that r times r^i.^T "r are selected, and that (3. 1) {■'. = 0, 1} — l^\, ~ni • • .

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