Suppose you are managing a service system in which customers arrive at a rate of two per minute according to a Poisson process. You are investing in a single machine that provides the service and does so with exponential service time. Your objective is that the average time a customer spends in the system be less than or equal to five minutes. How fast of a machine do you need?
I googled "exponential service time" the first choice (5.2 exponential service) may have the equations you need in appendix A. as I am clueless in this area I hope I have been helpfull :)
To determine how fast of a machine you need for the average time a customer spends in the system to be less than or equal to five minutes, you'll need to use the formulas from Appendix A of the resource you mentioned. Let's go through the steps to calculate this:
1. The arrival rate of customers, λ, is given as two customers per minute in your scenario since it follows a Poisson process.
2. The service time follows an exponential distribution, which means it has a parameter μ representing the average service rate.
3. To ensure the average time a customer spends in the system is less than or equal to five minutes, you can use the Little's Law formula:
- Average time in the system (W) = Average number of customers in the system (L) / Arrival rate of customers (λ)
- W ≤ 5 minutes
4. You can also define the average service time (S) as:
- S = 1 / μ
5. In this scenario, there is a single machine providing the service, so the average number of customers in the system (L) can be expressed as:
- L = λ * W + 1
(The additional "+ 1" accounts for the customer being served by the machine.)
6. Combining the above equations, you get:
- W = L / λ
- W = (λ * W + 1) / λ
7. Rearranging the equation, you have:
- λ * W = W - 1
8. Since λ = 2 customers per minute, you can substitute the value and solve for W:
- 2 * W = W - 1
- W = 1
9. Now that you have the average time in the system, W = 1, you can use Little's Law to find the required average number of customers in the system (L):
- L = λ * W + 1
- L = 2 * 1 + 1
- L = 3 customers
10. Finally, to find the required average service rate μ, you can use the formula S = 1 / μ:
- S = 1 / μ
- 1 = μ
Therefore, to achieve an average time a customer spends in the system of less than or equal to five minutes, you would need a machine that provides service at a rate of at least one customer per minute.