Queuing Theory with Excel pt1

Capacity Planning makes extensive use of Queuing theory to predict future response times of a system when the workload increases. There are many tools available that will automatically calculate the queues and can do multi-variate planning. However, if you have a really simple system and no money to buy a dedicated toolset, what can you do?

The formulas are reproduced across the internet and in books, so anyone can use them, however using a calculator/pen&paper is quite a challenge for this piece of mathematics. Lets have a look at the formulas involved.

We're interested in the M/M/c queue which is described in wonderful detail in Wikipedia here: (Queuing theory)
(Multi-server queue model)

I will assume that you either understand those pages, or have read them and are happy NOT to understand them!

Just to bring out the essential equations from those pages:

M/M/c Queue
$\lambda$ = arrival rate per unit of time
$\mu$ = service rate per unit of time
$c$ = number of servers

Note:
Most people are familiar with a "Service Time", e.g. how many seconds does it take to service a particular request. Service rate per unit of time is calculated from the “Service Time” as in the following example:
If the service time is 20 seconds, then the service rate per minute is 60/20 = 3
The service rate per hour is 3600/20 = 180
So. The “Service Rate” is effectively the “number of services that can be performed in the given unit of time”.

The Average Utilisation ( $U$ ) is also known as the Traffic Intensity ( $\rho$ )given as:
$U= \frac {\lambda}{(c\mu)} = \rho$

The system is stable if the Utilisation is less than 1 (100%). Well, if the utilisation is above 100%, then that would be pretty "unstable"... unlike in sport - computer systems can't give 110% percent!!

The formulas continue:
Probability of there being ZERO jobs in the system

$\wp_0 = \biggl [1 +\frac{(c\rho)^c}{c! (1-\rho)} + \sum\limits_{n=1}^{c-1} \frac{(c\rho)^n}{n!} \biggr]^{-1}$

Probability of there being “n” jobs in the system

If n<c

$\wp_n = \wp_0 \frac{(c\rho)^n}{n!}$

If n>=c

$\wp_n = \wp_0\frac{\rho^n c^c}{c!}$

The probability of queuing is
$\varrho = \wp_0 \frac{(c\rho)^n}{c!(1-\rho)}$

Mean response time is
$r = \frac{1}{\mu} (1+\frac{\varrho}{c(1-\rho)})$

So $\rho$ depends on $\wp_0$ and in fact, so does almost everything… so how can Excel work this out?

$\wp_0$ is a long summation that has a $\sum$ in it. Excel doesn't seem to have that as a simple function.

We'll find out how, in the next post... probably best to make sure that the above formulas are either understood, or at least accepted!

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