Question

NEED HELP SOLVING (f)!!!

Sharp Discounts Wholesale Club has two service desks (call them A and B) with one server at each desk. There is one desk at each entrance of the store. Customers arrive at each service desk at an average of one every six minutes following a Poisson distribution. The average service time at each service desk is 4 minutes per customer following the exponential service distribution.

a) How many customers, on average, are at service desk A, either being helped or waiting in line for service?

b) How much time does a customer spend at service desk B waiting for help?

c) Looking at service desk A, on average, what is the probability that there are at no more than three customers in line already waiting for service when the next individual arrives at this service desk?

d) Sharp is now considering consolidating its two service desks into one location, staffed by two clerks. The clerks will continue to work at the same individual speed of 4 minutes per customer. • How many customers, on average, are waiting in line? • How much total time does a customer spend at the service desk?

e) Do you think the Sharp Discounts Wholesale Club should consolidate the service desks? Explain and your discussion must consider both the economics and the customer service aspects of this waiting line operation. There would be no extra cost for moving the two operations to one operation. The clerks would continue to be paid the same as they were paid at the single desk operation.

f) Suppose that it is no longer required that the clerks be paid the same under either operation. Considering the consolidated operation and based on past work experience one clerk is paid $45 per hour and the second clerk is paid $35 per hour. The estimated cost to Sharp for time a customer is at the service desk either waiting or being helped is $0.50 per minute. For the two-service desk operation assuming that each clerk will be paid the same hourly rate, what is the maximum hourly rate that each of the clerks can be paid under this two service desk operation so that the total cost of the operation is the same for either arrangement? Assume that there is no cost for consolidating the service.

(a) For each desk, Average arrival rate, λ = 1 in every 6 minutes = 10 per hour Average service rate, μ = 1 customer is 4 minutes = 15 per hour The number of customers in A (both serviced and in wait), L = λ / (λ - μ) = 10 / (15 - 10) = 2 customers

(b) Average waiting time in B, Wq = λ / {μ (λ - μ)} = 10 / (15*(15 - 10)) = 2/15 hours = 8 minutes

(c) Prob(# customers waiting <= 3) = Prob(# customers in the system <= 4) = Prob(# customers in the system > 3) Pn>3 = (λ / μ)3+1 = (10/15)4 = 0.198 (d) Now this becomes a M/M/s quening system where Average arrival rate, λ = 2 in every 6 minutes = 20 per hour Average service rate, μ = 1 customer is 4 minutes = 15 per hour Number of serviers, s = 2

Computations will give us: P0 = 2.0, Lq = 1.067, and Ls = 2.4 So, # customers, on average, waiting in line = 1.067 Total time spent in the system, Ws = Ls / λ = 2.4/20 hrs. = 7.2 minutes

(f) NEED HELP SOLVING!!

Answer 1

The clerks should be paid the average rate.. That average rate will be calculated by adding $35+45/2=40.