Calls to a customer service at a certain company are handled by three processors

Question

Calls to a customer service at a certain company are handled by three processors. Calls are handled by Processor A 25% of the time, Processor B 35% of the time, and Processor C 40% of the time. For each processor, the time (in seconds) until he/she answers a call follows an Exponential distribution with a mean of mu_A = 4, mu_B = 7, and mu_C = 10, respectively. Let X denote the time until a call is answers. a. Determine E[X]. b. Determine the probability that it takes less than 4 seconds until a call is answered.

Explanation / Answer

E(X) = Sum( P * x ) = 0.25 * 4 + 0.35*7 + 0.4*10 = 7.45

P( Less than 4 ) = 0.25 * P (Less than 4 seconds if answered by Processor A ) + 0.35 * P (Less than 4 seconds if answered by Processor b ) + 0.25 * P (Less than 4 seconds if answered by Processor C )

Exponential distribution probability is given by

Pr(X4)=1e4/4=0.6321

Similarly for other two,

Pr(Y4)=1e4/7=0.4353

Pr(z4)=1e4/10=0.3297

Hence , probability that it takes less than 4 seconds = 0.25*0.6321 +0.35*0.4353+0.4*0.397 = 0.46918

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