Customers arrive at Rich Dunn's Styling Shop at a rate of 3 per hour, distributed in a Poisson fashion. Rich's service times follow a negative exponential distribution, and Rich can complete an average of 5 haircuts per hour. a) Find the average number of customers waiting for haircuts. b) Find the average number of customers in the shop. c) Find the average time a customer waits until it is his or her turn. d) Find the average time a customer spends in the shop. e) Find the percentage of time that Rich is busy.

a) 0.9, b) 1.5, c) 0.3hrs, d) 0.5hrs, e) 60%

Step-by-step explanation:

Given dа ta:

rate of arrival = 3customers/hr;

rate of service = 5 haircuts/hr;

a)

Average number of customers = La = λ²/[μ (μ-λ) ]

= 3²/[ (5 (5-3) ]

Average number of customers = La = 0.9

b)

Number of customers in system = Ls = λ / (μ-λ)

= 3 / (5-3)

Number of customers in system = Ls = 1.5

c)

Average waiting time = Ta = λ/[μ (μ-λ) ]

= 3/[ (5 (5-3) ]

Average waiting time = Ta = 0.3hrs or 18mins

d)

Average time spent by customer = Ts = 1 / (μ-λ)

= 1 / (5-3)

Average time spent by customer = Ts = 0.5hrs or 30mins

e)

% of time = Tr = λ/μ

= 3/5

% of time = Tr = 0.6 or 60%