Question

At the start of football season, the ticket office gets very busy the day before the first game. Customers arrive at the rate of five every 15 minutes, and the average time to transact business is 2 minutes.

1. What is the average number of people waiting for tickets?

2. What is the average time that a person will spend at the ticket office?

3. What proportion of time is the server busy?

4. What is the average number of people receiving and waiting to receive tickets?

5. What is the average time that a person will spend waiting in line at the ticket office?

Answer 1

Answer:

Arrival Rate = 5 customer in 15 min => 20 customers in 60min

We'll go with M/M/1 model

	Serving time=	2.00 min
	Lamda, λ = arrival rate (poisson) =	20.00 per hour
	μ = mean service rate (exponential) = 60/serving time	30.00 per hour
	Serving time= 1/μ	0.033 hours
	Utilization = ρ = λ /μ =	0.6667
Answer a:	Average number of people waiting for tickets:
	Average no queue =Lq = (ρ)2 / (1-ρ)
	Average no of customers waiting in line =Lq =	1.3333 customers
Answer e	Average time spent in queue:
	Wq = waiting time in the line for service = Lq / lambda = λ^2/μ(μ-λ)	0.0667 hours	OR	4.00 min
Answer b:	average time that a person will spend at the ticket office:
	W = average time in the system = 1/(μ-λ)	0.1000 hours	OR	6.00 min
Answer c:	The proportion of time server is busy = Arrival rate/service rate = λ /μ	0.6667	OR	66.7%
Answer d:	average number of people receiving and waiting to receive tickets
	L = no. of customers in the system = W *Lambda = W*λ	2.00 customers