At the Qatar 2022 FIFA World Cup’s opening game on 20 November 2022, the iconic Al Bayt Stadium was going to make history. It was another battle royal of sorts between the hosts and debutants, the ‘Al Annabi’ (the maroons) against a resurgent Ecuadorian brigade that was out to assert its place in the football world arena after missing out on Russia 2018. Fans seeking to purchase tickets started arriving from 9:00 am running through 1:12 pm, when the gates closed for the final preparations for the long-awaited inaugural game. The fans were observed to arrive at an average of 3,851 each hour, and were being served at a rate of 819 every twelve minutes. Some fans had already purchased their tickets earlier via the ticket sales agency, #Qatar2022 e-commerce portal. The Al Bayt Stadium has a 60,000 seating capacity. In the end, the stadium was 92.5% filled for this game. Given the enthusiasm of the fans that were trickling in, this was going to be the game that would make history, with the first world cup tournament to ever be hosted on Asian soil. From first principles, derive the formulae for the various performance measures, viz: i). the probability that there are n fans in the match ticketing system. ii). the average number of fans in the match ticketing system. iii). the average number of fans awaiting service. iv). the average time a fan spends in the game ticketing system. v). the average time a fan has to queue.
At the Qatar 2022 FIFA World Cup’s opening game on 20 November 2022, the iconic Al Bayt
Stadium was going to make history. It was another battle royal of sorts between the hosts and
debutants, the ‘Al Annabi’ (the maroons) against a resurgent Ecuadorian brigade that was out to
assert its place in the football world arena after missing out on Russia 2018. Fans seeking to purchase tickets started arriving from 9:00 am running through 1:12 pm, when the gates closed for the final preparations for the long-awaited inaugural game. The fans were observed to arrive at an average of 3,851 each hour, and were being served at a rate of 819 every twelve minutes. Some fans had already purchased their tickets earlier via the ticket sales agency, #Qatar2022 e-commerce portal. The Al Bayt Stadium has a 60,000 seating capacity. In the end, the stadium was 92.5% filled for this game. Given the enthusiasm of the fans that were trickling in, this was going to be the game that would make history, with the first world cup tournament to ever be hosted on Asian soil.
From first principles, derive the formulae for the various performance measures, viz:
i). the probability that there are n fans in the match ticketing system.
ii). the average number of fans in the match ticketing system.
iii). the average number of fans awaiting service.
iv). the average time a fan spends in the game ticketing system.
v). the average time a fan has to queue.
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