Need help with solution and explanation of how you got it, please. THe more deta
ID: 454086 • Letter: N
Question
Need help with solution and explanation of how you got it, please. THe more detail the better! Can't use Excel, only scientific calculator.
Customers arrive at MOD Pizza in Galleria area at rate of one every 90 seconds. It takes 80 seconds to assemble a pizza based on customer’s preference. Assume cook is paid $10 per hour and cost of waiting of customers is $30 per hour.
a- What are the arrival and service rates per hour?
b- Find the average time (in minutes) that a customer must wait until a cook assemble his/her pizza?
c- find the average number of cusotmers waiting to place orders
d-Find the utilization rate of this system. Also find the probability that there are two or more customers in the system.
e- Find the total daily cost of the queue (cost of waiting + cost of providing service) if the store operates 12 hours per day. Hint: cost of waiting = avg. wait in queue in hours*number of arrivals per hour* number of working hours per day* cost of customer waiting per hour. Also cost of providing service = number of hours working per day*hourly pay
Explanation / Answer
One customer arrive at the rate of every 90 second
Customers arrived in one hour = 60 * 60 /90 = 40 customers per hour
Arrival rate = 40 customer per hour
Service rate per hour µ = 60 *60 / 80 = 45 customers per hour
b- Find the average time (in minutes) that a customer must wait until a cook assemble his/her pizza?
Average waiting time in system W = 1 / (µ - )
Where, Arrival rate = 40 customer per hour
Service rate per hour µ = 60 *60 / 80 = 45 customers per hour
W = 1 / (45 -40) = 1 /5 = 0.2 hours = 0.2 * 60 min = 12 minutes
c- find the average number of customers waiting to place orders
that is average waiting time in line Wq = avg. utilization * avg. waiting time in system
where, avg. utilization = /µ = 88.89%
avg. waiting time in system = 12 minutes
therefore Wq = 88.89% * 12 = 10.67 minutes
d-Find the utilization rate of this system. Also find the probability that there are two or more customers in the system.
utilization rate = /µ = 88.89%
Where, Arrival rate = 40 customer per hour
Service rate per hour µ = 60 *60 / 80 = 45 customers per hour
the probability that there are two or more customers in the system
= 1 – probability of less than 2 customers in system = 1 – (1- ) *
= 1 – (1 – 0.8889) * 0.8889
= 0.901
Where utilization rate = 88.89%
d- Find the total daily cost of the queue (cost of waiting + cost of providing service) if the store operates 12 hours per day. Hint: cost of waiting = avg. wait in queue in hours*number of arrivals per hour* number of working hours per day* cost of customer waiting per hour. Also cost of providing service = number of hours working per day*hourly pay
total daily cost of the queue = cost of waiting + cost of providing service
(avg. wait in queue in hours*number of arrivals per hour* number of working hours per day* cost of customer waiting per hour) +( number of hours working per day*hourly pay)
average waiting time in queue Wq = 10. 67 /60 hour
Arrival of customer per hour = 40
number of working hours per day = 12 hour
cost of customer waiting per hour =$ 30
hourly pay = $ 10
now
total daily cost of the queue = (10.67/60 * 40 * 12 * 30 ) + (12 *10)
= $ 2560.8 + $ 120 = $ 2680.8
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