Essay About Pic And Number Of Servers

The Megacard Case
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The Megacard CaseQuestion 1:Let’s explain in detail how we computed the number of servers needed for Indiana, and then only report the final solution for Kansas City and DallasBetween 8:00 and 8:29, we can model the time between call arrivals as a Poisson variable  with rate , the service time as a random variable  with  and .[pic 1][pic 2][pic 3][pic 4][pic 5]To use the ggm.xls file to compute the number of servers needed to achieve a probability of delay , we first need to compute  and .[pic 6][pic 7][pic 8]So we have  and .[pic 9][pic 10]Putting this numbers into the ggm.xls file to use the GGm model, we find that the minimal number of servers needed to achieve a probability of delay  is : there need to be at least 8 TCs between 8:00 and 8:29.[pic 11][pic 12]We compute the number of TCs needed between 8:30 and 8:59 using the same method but with the Poisson variable  having a rate of . We find that we need 15 TCs to achieve the same service quality.[pic 13][pic 14]By using the same method for Kansas City and Dallas we find the following results:Number of TCsIndianaKansas CityDallas8:00-8:298988:30-8:59151211Question 2:This time, instead of looking for the minimum number of servers  such that <2%, we look for the minimum number of servers  such that .[pic 15][pic 16][pic 17][pic 18]This time we find the following results:Number of TCsIndianaKansas CityDallas8:00-8:296768:30-8:591299Question 3:Like in Question 1 the arrival rate is modelled as a Poisson variable  with rate , the service time as a random variable  with  and .[pic 19][pic 20][pic 21][pic 22][pic 23]The number of busy servers  is such that  and .[pic 24][pic 25][pic 26]According to the square root model, the loss rate (the percentage of clients who leave before being served because there is a delay) is equal to ) with  being the standard normal distribution. Because we want this loss rate to be , we are looking for  such that  (because  ).[pic 27][pic 28][pic 29][pic 30][pic 31][pic 32]Computing  and the minimum integer  such that  for each city and each time slot, we find the following results:[pic 33][pic 34][pic 35] Number of TC’sIndianaKansas CityDallas8:00-8:297 8 7 8:30-8:5914 11 10 9:00-9:2913 9 10 9:30-9:5917 8 7 10:00-10:2917 7 9 10:30-10:5915 10 9 11:00-11:2918 7 8 11:30-11:5913 8 10 12:00-12:2910 12 6 12:30-12:599 9 5 1:00-1:2910 7 6 1:30-1:5917 8 7 2:00-2:2913 10 8 2:30-2:5915 7 6 3:00-3:2910 6 5 3:30-3:5914 8 4 4:00-4:299 9 4 4:30-4:597 5 3 5:00-5:296 7 2 5:30-5:592 3 2 Total236 159 128 As we can see, the number of TCs needed using the square root model is lower than the one found in Question 1, where we used the GGm model. This is coherent: in the square-root model, clients who arrive and find all the servers busy leave the system, rather than waiting in a queue like in the GGm model. So in the square-root model the TCs won’t serve all the clients contrarily to the GGm model and therefore need less TCs.