Queueing Theory Formulas
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$L_s$ (Length of the system, Number of customers in the shop) $= \frac{\lambda}{\mu - \lambda}$
$W_s$ (Time spent in the system) $= \frac{1}{\mu - \lambda}$
$L_q$ (Length of the queue) $= \frac{\lambda^2}{\mu(\mu - \lambda)}$
$W_q$ (Time spent in the queue) $= \frac{\lambda}{\mu(\mu - \lambda)}$
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Hi Merry, let's solve this together. The image shows several fundamental formulas for an M M 1 queue system, including the length and time for both the queue and the entire system.
Queuing Theory Formulas (M/M/1)
Let's start by defining our variables. In queuing theory, lambda represents the arrival rate, and mu represents the service rate.
Variables
For a stable system, we assume \lambda < \mu.
The first formula is for L sub s, which represents the expected length of the system. This is the average number of customers in the shop, including those being served.
1. Length of the System ($L_s$)
This represents the average number of customers in the entire system.
Next, we have W sub s, the expected time spent in the system. This includes both the time waiting in the queue and the time being served.
2. Time Spent in the System ($W_s$)
Note that by Little's Law, $L_s = \lambda W_s$.
Now, let's look at the queue specifically. L sub q is the expected length of the queue, or the average number of customers waiting for service.
3. Length of the Queue ($L_q$)
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