hmwk11sol - ISyE 3232 Stochastic Manufacturing and Service...

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Stochastic Manufacturing and Service Systems Spring 2011 H. Ayhan and J. Dai Solutions to Homework 11 1. The arrival rate is λ ( t ) = 10 9 t 11 10( t - 10) 11 < t 12 15 + 5(13 - t ) 12 < t 13 15 13 < t 17 (a) λ (12 . 5) = 17 . 5. (b) E [ N (8)] = R 17 9 λ ( s ) ds = 112 . 5 . (c) The probability of k arrivals between 11:30 and 11:45 is just (d) ( R 11 . 75 11 . 5 λ ( s ) ds ) k e - R 11 . 75 11 . 5 λ ( s ) ds k ! = ( R 11 . 75 11 . 5 10( s - 10) ds ) k e - R 11 . 75 11 . 5 20( s - 10) ds k ! = (4 . 0625) k e - 4 . 0625 /k ! . (e) To do this, note that (here minutes need to be changed into hours, e.g. 10 mins = 1 6 hours) P ( T > 1 6 ) = P ( N (12 1 6 ) - N (12) = 0) and P ( T > 1 3 ) = P ( N (12 1 3 ) - N (12) = 0) . Then P ( T > 1 6 ) = e - R 12 1 6 12 15+5(13 - t ) dt = e - 235 / 72 . Here T is not an exponential random variable, since it’s not memoryless (try comparing P ( T > 15 | T > 5) with P ( T > 10)). 2.
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hmwk11sol - ISyE 3232 Stochastic Manufacturing and Service...

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