Explain briefly why it might be reasonable to expect these calls to arrive at random Question 1 The call centre of a major US energy provider (USGEN) expects to be particularly busy during the early shift in the early Winter. During the early shift calls arrive at a mean rate of 180 per hour and are believed to arrive at random. (i) Explain briefly why it might be reasonable to expect these calls to arrive at random. (ii) What would be the probability distribution of the number of calls arriving during a five-minute period in the early shift? (There is no need to calculate any probabilities in this part of the question). (iii) Show (using probability tables or probability formulae) that the probability that there are more than 20 calls in a five-minute period is 0.083. Show your working. (iv) Staffing levels during the early shift are such that they can cope with occasional peaks in arrivals, but service levels deteriorate rapidly when they experience a number of peaks close together. Continuing to assume that calls arrive at random, show that the probability that there are 6 or more five-minute periods in an hour in which the number of calls exceeds 20 is less than 1 in 1000. Show your working. (v) Quiet periods only occur very rarely during the early shift in the USGEN call centre, so when gaps between calls exceed 2 minutes the management takes it as a signal of a telephone system failure and resets the system.