A company in Singapore operates a call center for customer service. Assume that each caller speaks either English or Chinese, but not both. Suppose that the arrival for each type of calls follows a Poisson process. The arrival rates for English and Chinese calls are 2 and 1 call per minute, respectively. Assume that call arrivals for different types are independent.
(a) What is the probability that the second English call will arrive after minute 5?
(b) Find the probability that, during the first 2 minutes, there are at least 2 English calls and exactly 1 Chinese call?
(c) What is the expected time for the 1st call that can be answered by a bilingual operator (who speaks both English and Chinese) to arrive?
(d) Suppose that the call center is stat ed by bilingual operators that speak both English and Chinese. 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 of N(t)?
(e) What is the probability that at least two calls arrive during a 10 minute interval?
(f) What is the probability that two calls arrive within the interval 0 to 10 minutes and three calls arrive in the interval from 5 to 15 minutes?
(g) Given that 4 calls arrived in 10 minutes, what is the probability that all of these calls arrived in the fi rst 5 minutes?
(h) Find the probability that, in the first 2 minutes, there are at most 1 call, and, in the fi rst 4 minutes, there are exactly 3 calls?