A room has 10 cotton spinning looms. Once the looms are set up, they run automat

Question

A room has 10 cotton spinning looms. Once the looms are set up, they run automatically. The setup time is exponentially distributed, with mean 10 minutes. The machines run for an average of 40 minutes, also exponentially distributed. Loom operators are paid $10 an hour, and looms not running incur a cost of $40 an hour. How many loom operators should be employed to minimize the total cost of the loom room? If the objective becomes “on the average, no loom should wait more than 1 minute for an operator” how many people should be employed? How many operators should be employed to ensure that, on average, at least 7.5 looms are running at all times?

Explanation / Answer

The model is M/M/c/10/10 queue

Given arrival rate = = 1/40

Service rate = = 1/10

N=K=10

operators paid = $10 per hour

idle rooms running cost = $40 per hour

Mean cost per hour = $10c + $40L

The no of operators to be employed to minimize the total cost of the room is 3, resulting in a total cost of $120.4

Four operators should be employed for, no loom to wait for more than
one minute for an operator (i.e., to ensure wQ 1 min.).

Here,a loom will only have to wait an average of wq = 0.3 min. = 18 seconds for a cost of $122

Three operators should be employed for mean of at least 7.5 looms are running at all
times (i.e., for K L 7.5 looms).

c Lq L wq(min) K-L Cost 1 5.03 6.02 50.6 3.98 250.8 2 1.46 3.17 8.55 6.83 146.8 3 0.32 2.26 1.65 7.74 120.4 4 0.06 2.05 0.3 7.95 122 5 0.01 2.01 0.05 7.99 130.4

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