# Sp Ans Essay

1639 Words 7 Pages
(a) First, determine the Queuing Models

1. Queuing Models Poisson Arrivals Standard (Infinite Queue) Exponential Service Times Single Server (M/M/1)

Standard M/M/1 Model
P0 = 1 – P
(Probability of 0 customer in the system)

P (n ≧k ) = ρk
(Probability of

Pn = P0ρn
(Probability of exactly n customers in the system)

Ls = λ / ( μ – λ )
(Mean no. of customers in the system)

Lq = ρλ / ( μ – λ )
(Mean no. of customers in queue)

Lb = λ / ( μ – λ )
(Mean no. of customers in queue for a busy system)

Ws = 1 / ( μ – λ )
(Mean time customer spends in the system)

Wq = ρ / ( μ – λ )
(Mean time customers spends in the queue)

Wb = 1 / ( μ – λ )
(Mean time customers spends in queue for a busy system)
Although the additional renting cost added, the new machine’s total cost each hour is \$14 while the old one is \$18.
Therefore, the company should rent a new machine to replace the old one.

(b) For the old copying machine, what is the probability when a person arrives that he or she will encounter people already waiting in line for service? (Be careful to identify properly the number of customers who might be present for this situation to arise.)

In this situation, there should be 1 person is copying, at least 1 person is waiting and 1 person has just arrived.
Therefore, The number of people in the system (n) = 3.

Pn = P0ρn
(Probability of exactly n customers in the system)

P3 = P0ρ3
Where P0 = ( 1-ρ )
= (1- (15/30)) (15/30)3
=0.0625

Therefore, the probability when a person arrives that he/she will encounter people already waiting in line for service is 0.0625.

(c) Suppose now that the company plans to rent one more (old) machine with an additional cost \$15 per hour. Is it a good decision? Which method is better in terms of saving the cost, renting a new copying machine or renting two old copying machines? Explain your answer with the assumption that there is only one single waiting line and no balking or reneging.

(i) If the company rent one more (old) machine, the Queuing Model should be changed to standard M/M/c (Multiple Service) Model.
Now, creates an M/M/2 system in this case, the arrival rate is 15/hr.
Because

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