Rogers Communications in Canada operates a call center for customer service. As- sume that each caller...

  

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Rogers Communications in Canada operates a call center for customer service. As- sume that each caller speaks either English or French, but not both. Suppose that the arrival for each type of calls follows a Poisson process. The arrival rates for English and French calls are 2 and 1 calls per minute, respectively. Assume that call arrivals for different types are independent. (a) Suppose that the call center is staffed by bilingual operators that speak both English and French. Let N(t) be the total number of calls of both types that arrive during (0, t]. What kind of a process is N(t)? What is the mean and variance of N(t)? (b) What is the probability that no calls arrive in a 10 minute interval? (c) What is the probability that at least two calls arrive in a 10 minute interval? (d) What is the probability that two calls arrive in the interval 0 to 10 minutes and three calls arrive in the interval from 5 to 15 minutes? (e) Given that 4 calls arrived in 10 minutes, what is the probability that all of these calls arrived in the first 5 minutes? (f) Find the probability that, in the first 2 minutes, there are at most 1 call, and, in the first 4 minutes, there are exactly 3 calls? (g) What is the probability that the 2nd English call will arrive after minute 5? (h) Find the probability that, in first 2 minutes, there are at least 2 English calls and exactly 1 French call? (i) What is the expected time for the 1st call that can be answered by a bilingual operator (that speaks both English and French) to arrive? Rogers Communications in Canada operates a call center for customer service. As- sume that each caller speaks either English or French, but not both. Suppose that the arrival for each type of calls follows a Poisson process. The arrival rates for English and French calls are 2 and 1 calls per minute, respectively. Assume that call arrivals for different types are independent. (a) Suppose that the call center is staffed by bilingual operators that speak both English and French. Let N(t) be the total number of calls of both types that arrive during (0, t]. What kind of a process is N(t)? What is the mean and variance of N(t)? (b) What is the probability that no calls arrive in a 10 minute interval? (c) What is the probability that at least two calls arrive in a 10 minute interval? (d) What is the probability that two calls arrive in the interval 0 to 10 minutes and three calls arrive in the interval from 5 to 15 minutes? (e) Given that 4 calls arrived in 10 minutes, what is the probability that all of these calls arrived in the first 5 minutes? (f) Find the probability that, in the first 2 minutes, there are at most 1 call, and, in the first 4 minutes, there are exactly 3 calls? (g) What is the probability that the 2nd English call will arrive after minute 5? (h) Find the probability that, in first 2 minutes, there are at least 2 English calls and exactly 1 French call? (i) What is the expected time for the 1st call that can be answered by a bilingual operator (that speaks both English and French) to arrive?