| Actually this example makes a couple of other (very important)
assumptions as well. The most important is that there a a potentialy
infinite number of customers and that their arrivals are independent of
each other or the state of the service center.
>Hence my questions:
>1) is the number of customers waiting the *total* number summed
>on all queues, or is that customers per queue?
The number of customers waiting (according to standard definitions) is
the total of all queues. By the way, a customer is counted as being in
a queue, even if it is actually in service. Thus the queue includes
the service center.
>2) what precisely is meant by "exponential serving"?
Precisely:
Let X be a continuous random variable that can assume any
nonnegative value. X is an exponential random variable if:
/
| 0, t < 0
pdf.X(t) = <
| L*e^(-L*t), t >= 0
\
Where:
t = Time...
pdf.X = the probability density function(pdf) of X
pdf.X(t) = the pdf of X over time...
L = Lambda, the average arrival rate of
customers. (a positive real number)
>3) how are these figures derived? in particular, suppose I have a
>distribution that is not Poisson-based, or that the time for serving
>is not exponential (eg, constant); what would the formula be?
The *formula* is already extremely complicated. Take away these
simplifing assumptions and it either becomes *incredibly* complex
(except in some special cases) or an unsolved problem.
These figures are derived using Probability theory, statistical theory,
the theory of Stochastic functions, Markov Chain theory and the
indispensible Little's Law. It can get REAL messy.
These figures (and related stuff) are crucial to several important
fields, including Operations Analysis and Computer Performance Analysis
(which uses Queueing Network Analysis).
One of the definitive references for the mathematical formula (and the
one I use heavily) is "Computer Performance Modeling Handbook" by
Steven Lavenberg (Academic Press, 1983).
-- Barry
|