please solve this according to given data
Passengers landing at Heathrow airport need to go through passport control checks in order to enter the UK. EU and UK nationals use a fast-track system. where they scan their passport at a self-service station. where queue times average 20 minutes with 5 minutes standard deviation. Non-UKrEU nationals need to have their passports checked by the UK Border Force. a process that is often longer. with queue times averaging 30 minutes and approximately 15 minutes standard deviation. The rate of change of the number of passengers queueing within the borders of Heathrow airport for passport control can be approximated as: dN 3:313452'3'3: where X; is a random variable representing the number of passengers arriving at Heathrow per hour. if; is a rand om variable representing the number of passengers that have gone through passport control per hour. and x, Is a random variable representing the number of transit passengers. In more detail. the random variable X2 can be modelled as: X: = \"osros + \"arm where no; is the number of staffed gates open for overseas passengers and r0; Is the number of passengers a certain gate can service per hour. Similarly. up Is the number of gates open for UKJEU nationals. and r\" is the rate of passengers going through that gate per hour. Table 3 shows a monthly breakdown of the nationalities of passengers arriving at London Heathrow airport throughout 2019. and Table ll shows the total number of terminal and transit passengers at London airports in 2019. a} [5%] FII'Id a SUIIEDIB approximation for the number Of transit passengers X3 at the airport. Show the mean and variance of this random variable. [3} [35%] Estimate the number of gates that need to be open in order to ensure the queues of passengers waiting to go through passport control does not increase with time. Use the data in the tables provided alongside realistic estimates for the rate at which passengers go through the two types of passport control gates {H om evae rseas). c) [35%] Build a mathematical model to predict how passport control queues change as a function of the number and type of tills open. Consider the uncertainty of your model. Based on your model, recommend a till opening strategy for the airport. TABLE 3. HEATHROW MONTHLY TRAFFIC STATISTICS BY NATIONALITY (2019). Non-EU North Latin Middle Asia / Month UK EU Europe Africa America America East Pacific Total January 2019 326,386 1,817,088 426,466 313,497 1,277,923 121,749 628,759 1,015,675 5,927,543 February 2019 320,861 1,859,085 418,782 273,567 1,088,818 106,026 541,573 873,576 5.482,288 March 2019 391,467 2,217,166 487,150 303,704 1,470,992 114,644 597,277 945.089 6,527,489 April 2019 419,327 2,319,492 480,811 301,144 1,557,547 112,837 677,609 929.445 6,798,212 May 2019 435,701 2,404,663 445.300 261,598 1,706,145 113,709 518,061 883,726 6,768,903 June 2019 431,711 2,536,137 504,697 280.538 1,807,271 116,985 607.953 961,305 7,246,597 July 2019 446,608 2,689,689 552,150 302,642 1,833,961 122,240 746,174 1,060,675 7,754,139 August 2019 437,054 2,619,533 535,723 312,796 1,829.388 121,380 790,105 1,034,348 7,680,327 September 2019 388,187 2,333,969 469,720 277,583 1,615,711 109,523 643,594 938,828 6,777,115 October 2019 432,183 2,420,621 478,882 292,290 1,677,327 114,544 643,064 1,005,868 7,064,779 November 2019 414,684 2,090,235 422,161 284,821 1,416,945 111,486 613,100 880,369 6,233,801 December 2019 396,168 2,153,470 471,837 310,426 1,552,509 117,139 743,168 951,362 6,696,079 TABLE 4. HEATHROW TERMINAL AND TRANSIT PASSENGERS STATISTICS OVER A YEAR (2019). Airport | Total Passengers Terminal Transit GATWICK 46576473 46574786 1687 HEATHROW 80890031 80886589 3442 LONDON CITY 5122271 5122271 LUTON 18216207 18213901 2306 SOUTHEND 2035535 2035535 0 STANSTED 28124292 28124292
