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Unformatted text preview: time is
still exponential. Finally, G/G/1 means both interarrival time and service time are nonexponential (not
necessarily the same distribution). More about this types of queues are left to courses such as CS 594 and
CS 694. CS 522, v 0.94, d.medhi, W’99 24 20. An example of a nonexponential distribution: hyperexponential
The pdf for twostage hyperexponential distribution (H2 ) is given by
f x = ,1 x + ,2 x 1 1e 2 2e 1 The mean is 1 E X = 2 + 1 2 2 0 and 1+ 2=1 : X ! 2
2
1
2
i
, + :
V X = 2
2 The variance is i i=1 1 2 The distribution is considered to be balanced if:
1 1 = 2 2 : The squared coefﬁcient of variation (c2 ) is given by: c2 =
(note: V X
E X 2 = E X 2
, 1:
E X 2 E X 2 is the second moment). How about generating hyperexponential distribution? Given
anced mean can be generated as follows:
Calculate 1 and 2 as follows:
1= 1
2
1 , 2=1 and 1 and 2 as follows: r
, c2 , 1
c2 + 1 c2 and , a hyperexponential with bal ! 1 1 = 2 1
2 = 2 2 : CS 522, v 0.94, d.medhi, W’99 25 21. Considering M/M/1 and M/D/1 together
In the ﬁgures below, we plot N and T for different values of
Notice the difference as
1. for the systems M/M/1 and M/D/1. ! M/M/1 and M/D/1
Average number of customers in steady state mu = 0.1 10 8 6 4 2 0
0.1 0.2 0.3 0.4 0.5 0.6 utilization, ρ 0.7 N_mm1 0.8 0.9 N_md1 M/M/1 and M/D/1
µ = 0.1 Average Delay
120 100 80 60 40 20 0
0.1 0.2 0.3 0.4 0.5 utilization, ρ 0.6 0.7 T_mm1 CS 522, v 0.94, d.medhi, W’99 0.8 0.9 T_md1 26...
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 Fall '08
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 Computer Networks

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