Example $(1)$:

Ocala Software Systems operates a technical support center for its software customers. If customers have installation or use problems with Ocala software products, they may telephone the technical support center and obtain free consultation. Currently, Ocala operates its support center with one consultant. If the consultant is busy when a new customer call arrives, the customer hears a recorded message stating that all consultants are currently busy with other customers.

The customer is then asked to hold and is told that a consultant will provide assistance as soon as possible.

The customer calls follow a Poisson probability distribution, with an arrival rate of five calls per hour. On average, it takes $7\mathrm{.}5$ minutes for a consultant to answer a customer$$s questions. The service time follows an exponential probability distribution.

Answer the following:

a$)$ What is the service rate in terms of customers per hour?

b$)$ What is the probability that no customers are in the system and the consultant is idle?

c$)$ What is the average number of customers waiting for a consultant?

d$)$ What is the average time a customer waits for a consultant?

e$)$ What is the probability that a customer will have to wait for a consultant?

f$)$ What is the probability of $3$ customers waiting in the line?

g$)$ Ocala$$s customer service department recently received several letters from customers complaining about the difficulty in obtaining technical support. If Ocala$$s customer service guidelines state that no more than $35\%$ of all customers should have to wait for technical support and that the average waiting time should be two minutes or less, does your waiting line analysis indicate that Ocala is or is not meeting its customer service guidelines? What action, if any, would you recommend?

Example $(2)$:

In reference to example $(1),$ Ocala decides to operate its support center by $2$ consultants. Assume again the same arrival rate $=5,$ and the service rate $=8.$

Answer the following given that $($P$0=0\mathrm{.}21)$:

a$)$ What is the average number of customers waiting in line for $2$ consultants?

b$)$ What is the average number of customers in the system $(2$ consultants$)?$

c$)$ What is the average time a customer waits in line for $2$ consultants?

d$)$ What is the average time of customers in the system $(2$ consultants$)?$

e$)$ What is the probability that a customer will have to wait for service $(2$ consultants$)?$

f$)$ What is the probability of $3$ customers waiting in the system?

g$)$ Does your waiting line analysis after adding another consultant indicate that Ocala is meeting its customer service guidelines? What action, if any, would you recommend?

Example $(3)$:

In reference to examples $(1)$ & $(2),$ the following cost information is available; the customer waiting time is valued at $$16$ per hour and the cost to operate the checkout counter is $$22$ per hour.

Answer the following:

a$)$ What is the total cost for the waiting line in example $(2)[1$ consultant$]$

b$)$ What is the total cost for the waiting line in example $(3)[2$ consultants$]$

c$)$ Should Ocala managers use $1$ or $2$ consultants? Justify your answer.