Question

Problem 11-11 (Algorithmic)

Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2.9 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. The consultant averages 10 minutes with each customer.

1. Compute the operating characteristics of the customer waiting line, assuming Poisson arrivals and exponential service times. Round your answers to four decimal places. Do not round intermediate calculations.
   
   L_q =
   
   L =
   
   W_q =  minutes
   
   W =  minutes
   
   P_w =
   
2. Service goals dictate that an arriving customer should not wait for service more than an average of 6 minutes. Is this goal being met? If not, what action do you recommend?
   
   No. Firm should increase the mean service rate u for the consultant or hire a second consultant.
   
3. If the consultant can reduce the average time spent per customer to 8 minutes, what is the mean service rate? Round your answer to four decimal places. Do not round intermediate calculations.
   
   µ =  customers per hour
   
   W_q =  minutes
   
   Will the service goal be met?
   
   Yes

Answer 1

arrival rate lambda = 2.9 per hour

service rate mu = 60/10 =6 per hour

utilisation rho =lambda /mu = 2.9/6 = 0.4833

Lq = (lambda)^2/ mu ( mu-lambda)

= 2.9x2.9 /6x3.1 =0.4521

Wq = Lq /lambda = 0.4521/2.9 =0.1559 hours =9.358 minutes

L = lambda /mu-lambda = 2.9/3.1 =0.9354

W = 1/mu-lambda = 1/3.1 =0.3225 hours =19.358 min.

(ii)Waiting time os 9.358 minutes hence the goal of 6 min is not met.

new service rate to meet the target of 6 min ( or 1/10 hour)

0.1 = lambda / mu ( mu-lambda)

0.1 = 2.9 /mu ( mu-2.9)

mu^2-2.9*mu-29=0

mu = [2.9+ (2.9)^2 + 4*29]/2

= 7.026

(iii) If W=8 minutes that is mu = 7.5  per customer, the new values

Wq = lambda / mu ( mu-lambda) = 2.9 /7.5(7.5-2.9) = 0.084 hours = 5.04 min

The service goals will be met.