Mr. Chips manufactures an automated vending machine that can prepare a single or

Question

Mr. Chips manufactures an automated vending machine that can prepare a single order of french fries in an average of 40 seconds. The company has placed one of its machines in the Student Center, where it believes customers will arrive according to a Poisson process with a mean rate of 1 per minute. (i) (3 pts.) Mr. Chips is considering placing a second french-fry vending machine in the Student Center. Given the current vending machine service level, would it be worthwhile for Mr. Chips to do this? Justify your answer quantitatively.

Explanation / Answer

= 60 per hour µ = 90 per hour (40 seconds per order) Average Utilization of the system p = / µ = 60/90 = 0.67 Probability that there are 0 orders in the system P0 = 1 - p = 1 - 0.67 = 0.33 Average number of customers in waiting line = Lq = pL = / µ * / µ - = 60/90 * 60/90-60 = 0.67 * 2 = 1.34 customers Average waiting time of customers in line = pW = / µ * 1 / µ - = 60/90 * 1/90-60 = 0.0222 hours or 1.33 minutes = 60 per hour µ = 90 per hour (40 seconds per order) s = 2 Average Utilization of the system p = / Sµ = 60/(2*90) = 0.33 Probability that there are 0 orders in the system (P0 = 1 / [1 + / µ + {( / µ)^s / s!} * (1 / 1-p)] = 1 / [ 1 + 60/90 + (60/90^2 / 2!) * (1 / 1 - 0.33) ] = 1 / [ 1 + 0.67 + 0.33) = 1 / 2 = 0.50 Average number of customers in waiting line = Lq = P0*(/µ)^s*p / s! (1 -p)^2 = (0.50 * 60/90^2 * 0.33) / (2! * 0.67 ^2) = 0.074074 / 0.88889 = 0.0833 customers Average waiting time of customers in line = Lq / = 0.0833/60 = 0.0014 hours or 0.0833 minutes Yes - it is worth introducing second automatic machine as service levels increase drastically

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