Question

2. Courtney Newell, manager of the Silver Park Hotel, is considering how to restructure the front desk to improve guest service during the peak check-in hours of 3:00 to 5:00 p.m. At present, the hotel has 5 clerks on duty each with a separate waiting line.

Courtney is considering two plans for reducing the guest’s waiting time. The first proposal would be to implement a single waiting line in which guests would be served by whichever of the 5 clerks becomes available first. Observations of arrivals during the peak check-in time show that a guest arrives on average every 40 seconds. It takes an average of 3 minutes for the front-desk clerk to register each guest.

The second proposal would designate one employee as a “quick-service” clerk for guests registering under corporate accounts, a market segment that comprises about 30% of Silver Park’s guests. Since these guests would be pre-registered, it would take an average of only 0.5 minutes for the front-desk clerk to register them. Under this plan, the non-corporate guests would form a single line and proceed to the first available of the 4 remaining clerks. The average time for registering a non-corporate guest is 3.4 minutes.

Which proposal should Courtney implement? Provide appropriate quantitative evidence to support your recommendation.

Answer 1

Current system:

In the current system we have five M/M/1 queue system. Here λ = 60*60/40 = 90 per hour and µ = 60/3 = 20 per hour. However since there are 5 different queues, effectively it is λ = 18 per hour. We use the formulas shown below.

This means the average waiting time of the guests in the system is

W = 1/(µ - λ) = 1/(20-18) = 0.5 hours or 30 minutes.

First proposal:

In this system it becomes a M/M/c queue system where λ = 90, µ = 20 and c = 5. We use the formulas shown below.

This means that the probability of empty queue P0 = 0.005

L = 11.3624 customers and W = L/ λ = 11.3624/90 = 0.1262 hours or 7.57 minutes

Second proposal

This system will have two queues. One is M/M/1 where λ = 90*30% = 27 and µ = 60/0.5 = 120. The second system will have M/M/c queue where λ = 90*70% = 63, µ = 60/3.4 = 17.64 and c = 4.

In the M/M/1 system the average waiting time W = 1 / (120-27) = 0.0107 hours or 0.645 minutes

In the M/M/c queue system P0 = 0.0122

L = 10.0161 and W = 10.0161/63 = 0.159 hours or 9.54 minutes

The overall average waiting time becomes (27*0.645 + 63*9.54)/90 = 6.87 minutes

The second proposal lowers the overall time spent by guests in the front desk check-in system the most.