Use the following uniform 0 1 observations to

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Use the following uniform .0; 1/ observations to generate the interarrival times .53 .37 .08 .62 .53 .56 .14 .67 .82 .68 and the following N.0; 1/ numbers to generate the respective processing times 0:23 0:17 0:43 0:02 2:13 0:04 0:18 0:42 0:24 1:17 Round each service time to the nearest integer. In both cases, read from left to right. (a) What is the average time in queue for the 10 new jobs? (b) What is the (average) utilization of the machine? 6. Short questions. (a) The discrete random variable X has probability (mass) function Pr .X D 1/ D 0:45 , and Pr .X D 2/ D 0:55 . Give the Simio expression that generates realizations of X . (b) Which distribution does the Simio expression Random.Exponential(2) generate data from? Give its parameter(s). (c) Suppose X is geometric .0:8/ . Find P.X > 5 j X > 3/ . (d) Suppose X 1 ; X 2 ; : : : ; X 8 are i.i.d. uniform .0; 2/ . Use the central limit theorem to approxi- mate P.5 < P 8 i D 1 X i < 11/ . (e) Suppose X binomial .3; 0:75/ and Y binomial .2; 0:75/ are independent. Find P.X C Y 3/ . 7. Consider a single-server queuing system with FIFO service discipline and let X.t/ be the number of customers in the system (including the customer in service). A simulation run for 180 minutes produced the following sample path (output) t -interval OE0; 40/ OE40; 72/ OE72; 83/ OE83; 96/ OE96; 129/ OE129; 136/ O X.t/ 0 1 2 1 2 1 t -interval OE136; 148/ OE148; 165/ OE165; 175/ OE175; 180/ OE180; 1 / O X.t/ 2 3 2 1 0 (a) Compute an estimate of the mean number of customers in the system during OE0; 180Ł .
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3 (b) Compute an estimate of the mean server utilization during OE0; 180Ł . (c) Compute an estimate of the mean customer delay in queue during OE0; 180Ł . 8. Parts at a machine (with an infinite-capacity buffer) according to a Poisson process at the rate of 2 parts per minute. They are processed one-at-a-time by the machine. The processing times are i.i.d. uniform between 20 and 30 seconds.
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  • Spring '08
  • ALEXOPOULOS
  • Probability theory, discrete random variable, mean customer delay

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