Question

Donna Shader, the manager of the Winter Park Hotel, is considering how to restructure the front desk to reach an optimum level of staff efficiency and guest service. At present, the hotel has five clerks on duty, each with a separate waiting line, during the peak check-in time of 3:00 P.M. to 5:00 P.M. Observation of arrivals during this time show that an average of 90 guests arrive each hour (although there is no upward limit on the number that could arrive at any given time). It takes an average of 3 minutes for the front-desk clerk to register each guest.

Ms. Shader is considering a new plan. The new plan is to implement a single-line system. All guests could form a single waiting line to be served by whichever of five clerks became available. This option would require sufficient lobby space for what could be a substantial queue. For new plan, let Po = 0.005.

The clerk is paid $25 per hour, but because of lost goodwill and sales, the Winter Park Hotel losses about $60 per hour of customer time spent waiting for the clerk to take an order.

1. What is the average number of customers waiting to place a check-in service under the current system?

2. Determine the average amount of time that a guest spends checking in under the current system.

3. What is the total cost per hour under the current system?

4. What is the average number of customers waiting to place a check-in service under the new plan?

5. Determine the average amount of time that a guest spends checking in under the new plan.

6. What is the total cost per hour under the new layout?

7. Which option do you recommend and Why?

Answer 1

1.

Average arrival rate in a single queue, λ = 90 / 5 = 18 per hour
Average service rate in a single queue, μ = 1 in 3 minutes = 20 per hour

Average no. of customers waiting in each queue, L_q = λ² / {μ.(μ - λ)} = (18^2) / (20*(20 - 18)) = 8.1

So, average no. of customers waiting in all five queues = 8.1*5 = 40.5

2.

The average time a guest wait, W_q = L_q / λ = 8.1 / 18 = 0.45 hours = 27 minutes

The average time a guest spends while checking in, W_s = W_q + service time = 27+3 = 30 minutes

3.

Total cost of servers = $25 * 5 servers = $125 per hour
Total cost of waiting = $60 * 40.5 = $2,430 per hour

So, total cost per hour = 125 + 2430 = $2,555 per hour

4.

The new system is an M/M/s system with s = 5

Average arrival rate in a single queue, λ = 90 per hour
Average service rate for one server, μ = 1 in 3 minutes = 20 per hour

P₀ = 0.005

So, L_q = 90*20*(90/20)^5 / (FACT(5 - 1)*(5*20 - 90)^2)*0.005 = 6.92

5.

The average time a guest wait, W_q = L_q / λ = 6.92 / 90 = 0.0777 hours = 4.6 minutes

The average time a guest spends while checking in, W_s = W_q + service time = 4.6+3 = 7.3 minutes

6.

Total cost of servers = $25 * 5 servers = $125 per hour
Total cost of waiting = $60 * 6.92 = $415.2 per hour

So, total cost per hour = 125 + 415.2 = $540 per hour

7.

The second system is better as it lets the customers wait a substantially shorter period and the total hourly cost is also lower.