Question

Little’s Law Firm has just one lawyer. Customers arrive randomly at an average rate of 6 per 8 hour workday. Service times have a mean of 50 minutes and a standard deviation of 20 minutes. How long does a customer spend at Little’s Law Firm on average?

Answer 1

The average time a customer spends at Little’s Law Firm can be calculated using Little’s Law, which takes into account the average arrival rate and the average service time. In this case, the average time is approximately 31.25 minutes.

Step-by-step explanation:

Little’s Law is used to determine the average time a customer spends at a facility based on the average arrival rate and the average service time. In this case, the average arrival rate is 6 customers per 8-hour workday, which can be simplified to 0.75 customers per hour. The average service time is 50 minutes.

To calculate the average time spent by a customer at Little’s Law Firm, we can use the formula:

Average Time = Average Number of Customers * Average Service Time

Since the average arrival rate is 0.75 customers per hour, we can calculate the average number of customers in 50 minutes by multiplying the arrival rate by the time:

Average Number of Customers = (0.75 customers/hour) * (50 minutes / 60 minutes) = 0.625 customers

Now, we can calculate the average time spent by a customer at Little’s Law Firm:

Average Time = 0.625 customers * 50 minutes = 31.25 minutes

Therefore, on average, a customer spends approximately 31.25 minutes at Little’s Law Firm.

Answer 2

On average, customer spends approximately 98 min at Little’s Law Firm

Step-by-step explanation:

Given the data in the question;

Arrival rate λ = 6 per 8 hours = 6 / ( 8 × 60 )min = 6 / 480 = 0.0125 per minute

Service rate δ = 1 / 50 min = 0.02 per minute

Standard deviation σ = 20 min

Now,

Utilization rate U = Arrival rate / Service rate

U = 0.0125 / 0.02

Utilization rate = 0.625

Number of people in Queue will be;

⇒ ( (λ² × σ²) + U² ) / ( 2 × ( 1 - U )

we substitute

⇒ ( (0.0125² × 20²) + 0.625² ) / ( 2 × ( 1 - 0.625 )

⇒ ( 0.0625 + 0.390625 ) / ( 2 × 0.375 )

⇒ 0.453125 / 0.75

Number of people in Queue = 0.6042

Now

Wait in the Queue = Number of people in Queue / λ

= 0.6042 / 0.0125 = 48.336

Wait Time in Office = Wait in the Queue + ( 1 / δ )

= 48.336 + ( 1 / 0.02 )

= 48.336 + 50

Wait Time in Office = 98.336 ≈ 98 min

Therefore, On average, customer spends approximately 98 min at Little’s Law Firm