Question

The sandwich maker of the EE-Cole-Eye Sandwich Truck was just fired (for a reason described below) and EE-Cole-Eye needs to find a replacement. They want to make sure that the replacement is fast enough so that customers don't have to wait too long on line. Customers arrive according to a Poisson process with arrival rate 2 customers/min and sandwich makers take an exponentially distributed length of time to make a sandwich, with mean time service time of 29 seconds. There are no limits to the number of customers that wait for sandwiches. Consider the queue of customers and analyze it assuming it is in steady state.

Part 1 What is the probability there are 20 customers or fewer in the system? (Round to 3 decimal places)

What is the probability there are exactly 5 customers in the system? (Round to 3 decimal places)

What is the average number of customers in the system?

The performance described in the previous part is unsatisfactory, and that is why the sandwich maker was fired. How fast must the new sandwich maker work (that is, what is the required μ) so that the probability that there are 5 customers or fewer in the line is greater than 0.9? (Round to 3 decimal places)

How long does it take now to make a sandwich on average? (Answer in seconds, round to 3 decimal places)

Part 4 Continued from Part 3 What will be the average number of customers in the system with this new sandwich maker? (Round to 2 decimal places)

Answer 1

Given,

Arrival rate () = 2 customers per minute

Mean service time = 29 seconds

=> Service rate () = 1/29 x 60 = 60/29 customers per minute

Part 1)

a) Probability of k or more customers in the system is given by

=> Probability that 21 or more customers in the system is

Hence, probability that there are 20 or fewer customers in the system = 1 - 0.4907 = 0.5093 0.509

b) Probability that there are exactly n customers in the system is given by

=> Probability that there are exactly 5 customers in the system is

=

=

c) Average number of customers in the system is given by

=

=

Part 2) - After the old sandwich maker is fired...

Let service rate of new sandwich maker be

Probability of 6 or more customers in system is

=> Probability that 5 or fewer customers in system =

As per question, this is greater than 0.9

=>

=>

=>

=>

=>

Time taken now to make a sandwich = 60 / 2.9356 = 20.439 seconds

Part 4) Average number of customers in system with new sandwich maker =