You have been retained to analyze how many check-in agents should be used at a check-in counter for a large hotel. During normal business hours, customers arrive at a rate of about 24 per hour, and it takes about 7 minutes to complete the check-in process. Both the poisson distribution for arrivals and the exponential distribution for service time apply and a FIFO discipline is used. The check-in counter will accommodate as many as five agents.
Determine the following, utilizing the queuing table D.5:
a. The average utilization rate (p).
b. The average queue length based on 1, 2, 3, 4 or 5 agents.
c. The average waiting time based on 1, 2, 3, 4 or 5 agents.
d. What is your recommendation?
Format your answer as follows:
a. utilization rate:__________
b. queue length: 1 agent__________
2 agents__________
3 agents__________
4 agents__________
5 agents__________
c. waiting time: 1 agent__________
2 agents__________
3 agents__________
4 agents__________
5 agents__________
d. Recommendation:__________________
Work for each of the above
a. To determine the average utilization rate (p), we need to know the ratio of the service time (S) to the interarrival time (1/λ). In this case, the service time is 7 minutes and the arrival rate is 24 customers per hour, thus the interarrival time is 1/λ = 1/(24/60) = 2.5 minutes.
The average utilization rate (p) can be calculated using the formula:
p = (S / 1/λ) * 100
Substituting the values:
p = (7 / 2.5) * 100 = 280
So, the average utilization rate is 280%.
b. The average queue length can be determined using the queuing table D.5. The queuing table provides values for various traffic intensities (ρ) and number of servers (N) combinations.
For each possible number of agents (1, 2, 3, 4, 5), we need to calculate the traffic intensity (ρ) using the formula:
ρ = λ * S / N
Where:
λ = arrival rate (24 customers per hour)
S = service time (7 minutes)
N = number of servers (1, 2, 3, 4, 5)
Using the queuing table D.5, find the corresponding average queue length for each value of ρ and the respective number of agents.
c. The average waiting time can also be determined using the queuing table D.5. We need to find the corresponding average waiting time for each value of ρ and the respective number of agents.
d. Based on the analysis of the average queue lengths and waiting times for each number of agents, you should recommend the number of agents that minimizes the average queue length and waiting time. This would typically be the number of agents that balances the workload and maximizes efficiency.
Remember to consult table D.5 to determine the average queue length and waiting time for each combination of ρ and N. Choose the combination with the lowest values for both metrics as your recommendation.