Question

 

Frost Bank finds that 24 customers arrive at the single drive-through per hour. The teller is able to complete 32 transactions per hour. Assume M/M/1 operating characteristics.

1. What is the average time a customer spends waiting?

2. What is the probability that the system is idle?

3. What is the probability that there are more than 4 customers in line?

4. What is the average number of customers in line?

5. How long does the average customer spend in the system?

6. What is the utilization factor?

7. What is the probability that a customer will have to wait?

Answer 1

arrivals/time period =			λ=	24	/ Hour
served/time period=			μ=	32.00	/ Hour

1)

average time spend in queue Wq                   =

λ/(μ(μ-λ))=

0.09375 Hr~ 5.625 minute

2)

probability that the system is idle =(1-λ/μ) =0.25

3)

probability that there are more than 4 customers in line =1-P(at most 4 customer) =1-(1-(λ/μ)⁵) =0.237

4)

average number of customers in queue Lq =

λ2/(μ(μ-λ))=

2.250

5)

average time spend in system W                   =

1/(μ-λ)=

0.125 Hour =7.5 minute

6)

utilization factor =      ρ                                        =

λ/μ             =

0.75

7)

probability that a customer will have to wait =λ/μ   =0.75