Finite-Source Queueing Systems and their Applications
Finite-Source Queueing Systems and their Applications
Finite-Source Queueing Systems and their Applications
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János Sztrik 2001/08/05<br />
is called the machine availability in machine interference models, since it<br />
represents the expected fraction of the time that a machine remains in<br />
working condition, E is the the machine efficiency, because it is the ratio of<br />
the total actual production to what would have been achived had no stoppage<br />
taken place. From (9) through (11, 12), it is clear that performance measures<br />
such as ρ ′ , γ, E[T ], <strong>and</strong> E[L] can be obtained once we have evaulated P0.<br />
Let E[Θ] be the mean length of a busy period. Since the state of the system<br />
repeats regenerative cycles of a busy period of mean length E[Θ] <strong>and</strong> an idle<br />
period of mean length E[I] = 1/(Nλ), the probability P0 that the server is<br />
idle at an arbitary time is given by<br />
P0 =<br />
E[I]<br />
E[Θ] + E[I] =<br />
1/(Nλ)<br />
E[Θ] + 1/(Nλ)<br />
If π0 denotes the probability that the service facility is empty after a service<br />
completion, 1/π0 is the mean number of messages that are served during<br />
each busy period. This can be seen by considering a long period of time<br />
during wich a large number of (say N) messages are served. Such a period<br />
<strong>Finite</strong>-<strong>Source</strong> <strong>Queueing</strong> <strong>Systems</strong> <strong>and</strong> <strong>their</strong> <strong>Applications</strong><br />
(14)
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