How can I solve for x in the equation z = y/x * 1/(x-y)?

[onequestion321]

Homework Statement

Not a homework problem. I'm trying to solve an equation that will help me with some code I'm writing but I'm stuck. I haven't done much math in a while and I'm not sure if I'm missing something or if solving for x here is even possible, though the equation works so... it should be right?

Homework Equations

The title actually contains the result of one of my solution attempts. The original equation is:

z = y/x * 1/(x-y)

This is actually the equation to find the average wait time in a queue in a mono-server system.
I'm hoping to get an equation I can use to determine the ideal number of servers to use given a desired average wait time (z) and the arrival rate (y). x here should equal the service rate needed to attain the desired average wait time. I know that solving for x with those two numbers won't directly give the number of servers needed but I THINK I'll be able to calculate that using the actual service rate and the value of x.

The Attempt at a Solution

As you can see from the title I got to:

x(x-y) = y/z

And here I'm stuck because I can't figure out what to do about (x-y). Every solution I've attempted has been vexed by an inability to get that subtraction out of the equation. If there's some algebraic trick I'm not aware of I'd love to pointed in the right direction.

Homework Statement

PS. There's a bug in your preview viewer that causes the template to be appended to the text.

 

[Mark44]

Homework Statement

Not a homework problem. I'm trying to solve an equation that will help me with some code I'm writing but I'm stuck. I haven't done much math in a while and I'm not sure if I'm missing something or if solving for x here is even possible, though the equation works so... it should be right?

Homework Equations

The title actually contains the result of one of my solution attempts. The original equation is:

z = y/x * 1/(x-y)

This is actually the equation to find the average wait time in a queue in a mono-server system.
I'm hoping to get an equation I can use to determine the ideal number of servers to use given a desired average wait time (z) and the arrival rate (y). x here should equal the service rate needed to attain the desired average wait time. I know that solving for x with those two numbers won't directly give the number of servers needed but I THINK I'll be able to calculate that using the actual service rate and the value of x.

The Attempt at a Solution

As you can see from the title I got to:

x(x-y) = y/z

And here I'm stuck because I can't figure out what to do about (x-y). Every solution I've attempted has been vexed by an inability to get that subtraction out of the equation. If there's some algebraic trick I'm not aware of I'd love to pointed in the right direction.

Homework Statement

PS. There's a bug in your preview viewer that causes the template to be appended to the text.

Expand the left side of your equation and bring the y/z term over to get
x² - xy - y/z = 0

This is a quadratic equation in x. Use the quadratic formula to solve for x in terms of y and z.

The bug you mentioned has been known for quite a while. I don't know if the owner of the site has attempted to fix it.

 

[onequestion321]

Its working very well, thank you.