By Prof. Dr. W. Everling (auth.)

ISBN-10: 3540074015

ISBN-13: 9783540074014

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**Extra resources for Exercises in Computer Systems Analysis**

**Example text**

W ' however, the convergence (in probabilities) appears to be 'slower'. 41 In fact, the observed values differ so widely that the confidence in observed averages of queuing variables appears as a serious problem. This problem arises with simulations as well as with observations of an existing queuing system ' Confidence intervals for the averages of queuing variables. Some theoretical considerations will be discussed here, on which two rules for practical application can be based. Consider an average of M V(V) random variables = M ~" m=l v 1 .

They m a y be applied to the n u m e r a t o r of ns~T) g(0, T ) , viz• m=l ts, m , if T is large because then the possible truncation of two service times m e n t i o n e d on p a g e 27 b e c o m e s insignificant• When a POISSON arrival process d e t e r m i n e s ns(0, T), pectation Ra T N and v a r i a n c e V(n) equal to = N Ts / T V ( g ( 0 , T) ) : 2 ( N V s + V(n) T s ) / : E(t 2) U ~ Ts N o t e that the v a r i a n c e tends to v e r g e n c e (in probabilities) of g(0, T) (see p a g e E ( g ( 0 , T) ) and bution of this number of summands has the e x - 0 g(0, T) as T ~ to = / 18 ).

U = Tw / ( T w + . 72 Ra = U / Ts . The example 'values lead to = . 8 0 5 / 1 . 68 sec "1 An approximation for the distribution of wait times. Distributions of the queuing variables were considered in chapter 3 , page wait time distribution Fw(X) , the knowledge of the expectation a reasonable approximation. In fact, Fw(0 ) Tw 43 ff. For the is sufficient to find is also known. It is the probability that an item does not wait at all. For POISSON arrivals, this probability is equal to the fraction of time for which the server is idle, due to the 'uniformly probing property' of random arrivals.

### Exercises in Computer Systems Analysis by Prof. Dr. W. Everling (auth.)

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