# Elementary Properties of Clock Regulated Queues

Elementary Properties of Clock Regulated Queues
O E Percus
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22) ^0) = iif^(l-f).
As for u'(j), either the queue has length j and the message is taken first, or the length is ; — 1 and it is taken second. Together, then, w{j) = (1 — |)p(j) + |p(j — 1): (2. 23) w(j) = ^-f i-dl-f=\ for J > 0. It readily follows from (2. 16) that (2. 24) „■(. , = f. , .. I^l_£LfI.
The average waiting time is thus ^ dx '^-^ 4(1 -p)' The fact that this is identical to the mean queue length (2. IS) divided by the mean arrival rate p is an immediate consequence of Little's form
...ula [4]. However. (2. 24) allows all of the moments to be found just as easily.
3. The Output Process of the Queue A queue transforms an input time process into an output time process. Since the i. I. D. Input assumption proliferates throughout the literature, it is of interest to find the characteristics of the associated output, which of course serves in part as input to the next stage. As a first step, let us examine the probability ^„ of returning to the origin for the first time at the nth step (i.

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