This is under Operations Management or Operations Research of Industrial Enginee

Question

This is under Operations Management or Operations Research of Industrial Engineering. Topic is called Queueing Theory. Manufacturing Systems Modeling and Analysis Second Edition. Please do process.

Let S be a random variable representing the sum of the dots showing on the top of a pair of dice. What is P(S S 4)? If we have an arrival process with a mean rate of 5 jobs/hr, what are the inter-arrival time characteristics (E[A], C2IA], Var[A]) under the following distributional assumptions: I. I1. a. Poisson arrivals b. Erland-3 arrivals c. Exponentially distributed inter-arrival times !

Explanation / Answer

I. The number of combinations in case of 2 dices is 36 (6 X 6) though a sum could be same for different combinations. For e.g. On one side, 2 on Dice 1 and 2 on Dice 2 will return the sum as 4 and on the other side, 1 on Dice 1 and 3 on Dice 3 will also return the sum as 4.

Minimum Sum possible = 2 ( 1 on both dice)

Maximum Sum possible = 12 ( 6 on both dice)

All numbers between 2 and 12 (both included) are possible returns [as sum] when the 2 dice are rolled. However, getting <=4 as sum is possible in the following conditions:

Therefore, there are a total of 6 combinations which can provide sum as <=4.

On the other side, total combinations [for Sum] possible when 2 dice are rolled are 36.

Hence P(S <= 4) = 6 / 36 (or 6 divided by 36) = 1 / 6 = 0.167

Therefore, P(S <= 4) = 0.167

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