HW3

In [31]: import import pandas pandas as as pd pd import import numpy numpy as as np np import import matplotlib.pyplot matplotlib.pyplot as as plt plt import import math math In [32]: def def lambda_ (k): return return 0.5 + ( 0.5/120 ) * k Part a) Use simulation to compute the expectation of the number of customers at time 1PM In [33]: #variables specified by description of problem time_start = 11*60 time_end = 13*60 simulations = 100 rate_at_start = 0.5 / 60 rate_at_end = 1 / 60 rate_of_service = 1 / 35 def def simulation (): customers = 0 arrivals = [] departures = [] for for i in in range (time_start, time_end): y = rate_at_start + (rate_at_end - rate_at_start) * (i - time_start) / (time_end - time_start) if if np . random . rand() < y: arrivals . append(i) customers = customers + 1 time_of_service = np . random . exponential( 1/ rate_of_service) time_of_departure = i + time_of_service departures . append(time_of_departure) customers = customers - 1 customers_1 = sum ( 1 for for d in in departures if if d >= time_end) return return customers_1 ran_simulations = [simulation() for for x in in range (simulations)] expected_value = np . mean(ran_simulations) std_dev = np . std(ran_simulations) difference = ( 1/ np . sqrt( 100 )) *1.96* std_dev lower_ci = round (expected_value - difference, 3 ) upper_ci = round (expected_value + difference, 3 ) print print ( 'We can expect' , expected_value, 'customers at 1pm.' ) print print ( '95 % c % c onfidence interval is' , [lower_ci,upper_ci]) Part b) Use simulation to compute the expectation of averaged waiting time for all those customers that arrive between 12:45 PM to 1 PM. In [34]: services = [] wait_times = [] lambda2 = 1 cust_arrive = [] wait_times_105120 = [] for for _ in in range (simulations): We can expect 0.46 customers at 1pm. 95% confidence interval is [0.338, 0.582]

other_arrive = [] time_depart = [] one_arrival = np . random . poisson(lambda2 * time_start) cust_arrive . append(one_arrival) times_of_arrivals = np . random . uniform( 0 ,time_start,one_arrival) for for x in in range (one_arrival): y = 0.5+ (times_of_arrivals[x] /240 ) z = np . random . uniform( 0 , 1 ) if if z < y: other_arrive . append(times_of_arrivals[x]) other_arrive . sort() time_of_service = np . random . exponential( 1 , size = len (other_arrive)) services . append(time_of_service) for for k in in range ( len (other_arrive)): if if k ==0 : departed = other_arrive[k] + time_of_service[k] else else : departed = max (other_arrive[k], departed) + time_of_service[k] time_depart . append(departed) waited = time_depart[k] - other_arrive[k] - time_of_service[k] wait_times . append(waited) if if 105 <= other_arrive[k] <=120 : waiting = time_depart[k] - other_arrive[k] - time_of_service[k] wait_times_105120 . append(waiting) wait_time_avg = sum (wait_times_105120) / len (wait_times_105120) wait_time_std_105120 = np . std(wait_times_105120) diff = ( 1/ np . sqrt( 100 )) *1.96* wait_time_std_105120 lower_ci1 = round (wait_time_avg - diff, 3 ) upper_ci1 = round (wait_time_avg + diff, 3 ) print print ( 'The average wait time between 12:45pm and 1:00pm is' , wait_time_avg) print print ( '95 % c % c onfidence interval is' , [lower_ci1,upper_ci1]) Part c) Use simulation to compute the expectation of averaged waiting time for all those customers that arrive between 11:45 AM to 12:00 PM. In [35]: services = [] wait_times = [] lambda2 = 1 cust_arrive = [] wait_times_1530 = [] for for _ in in range (simulations): other_arrive = [] time_depart = [] one_arrival = np . random . poisson(lambda2 * time_start) cust_arrive . append(one_arrival) times_of_arrivals = np . random . uniform( 0 ,time_start,one_arrival) for for x in in range (one_arrival): y = 0.5+ (times_of_arrivals[x] /240 ) z = np . random . uniform( 0 , 1 ) if if z < y: other_arrive . append(times_of_arrivals[x]) other_arrive . sort() time_of_service = np . random . exponential( 1 , size = len (other_arrive)) services . append(time_of_service) for for k in in range ( len (other_arrive)): if if k ==0 : departed = other_arrive[k] + time_of_service[k] else else : departed = max (other_arrive[k], departed) + time_of_service[k] time_depart . append(departed) waited = time_depart[k] - other_arrive[k] - time_of_service[k] The average wait time between 12:45pm and 1:00pm is 5.490142732423501 95% confidence interval is [4.322, 6.659]