\absone
{Transient Queueing Approximations for Computer\\Networks}
{December 1986}
{William A. Baker}
{B.S., Rutgers University}
{Dr. P.\ E.\ Cantrell}
{ The objective of this thesis was to evaluate the performance of several
transient queue approximations. The approximations were tested and
characterized for a single M/M/1 queue and a tandem queue (two node)
network.

The five approximations tested in this thesis used a closure
assumption to obtain the probability of an empty system. Then, depending
on the method, equations were integrated to obtain the mean and, in 
some cases, the variance. Johnston's and Rider's methods solved for just the 
mean. Rothkopf/Oren's and Chang/Wang's methods obtained mean and variance 
values, and Clark's method produced several quantities which were used to
find mean and variance statistics.

For the M/M/1 case, the approximations by Clark and Chang were very accurate 
over a wide range of input patterns and initial conditions. Rothkopf's was
accurate over all conditions but never as accurate as Chang or Clark.
Johnston's and Rider's approximations performed acceptably only over some
of the cases. 
The hardest
conditions to follow, based on relative error, were low utilization cases with 
a large number in the queue at $t=0$.

For nonstationary arrival patterns into the M/M/1 queue, Clark's method was
superior to all others; mean and variance values were always within
three percent of the exact. 

For the tandem queue, equations for $dM/dt$ and $dV/dt$ were derived
to observe dependencies on joint probabilities between the queues. While 
the rate of change of the mean was only a function of the marginal 
probabilities of each queue, the rate of change for the variance
included joint probability terms. An assumption of queue independence
was made in order to implement the closure assumptions for the tandem
queue.

The approximations by Chang and Clark were very accurate in producing
the mean. For low utilization cases, the methods experienced difficulties
in following the true variance values. This was due to inaccuracies in the
assumption that the two queues were independent of each other.

In conclusion, the methods by Chang/Wang and Clark hold promise for
use in modeling computer networks, particularly for the mean in each queue.}