T3T2

Answer these questions well

L A system is put into operation at time t=. Its lime of failure is a random variable X 1with cdf Fxfx] and pdf fxfx]. Let itldt be the probability that it will fail in [t,t+dt} given that it has not failed unlil time t. {:1} Find fx{x} if we are given that (t}=kt. {.5} {b} If X is uniformly distributed in the interval (llT), then what would be Brit) in this interval. {.5} 2. The instructor for EEET'}, keeps the door of the lecture hall LS open for a random time interval Y before his lecture where the pdf, cdf and L.S.T of Y are given as fry}, F1. {y} and E, (s), respectively. Students arrive from a Poisson Process with rate is and can enter LS only while the door is open. Let N be the (random) number of students attending the lecture (Le. those who can enterl}. {a} Find the Genemting Function GM {3) of the number of students N attending the BEE-'9 lecture. {5} {b} Find the rst and seconds moments of N, in terms of the moments of Y using [a] {5} {c} If f]. (y) is exponentially distributed with mean )4\Note: [A] Use r= 1 for convenient notation in both problems [B] Use calculator for numerical answers in Problem 2(b) 1. A M/-/1/2 queue has a service time distribution given by 0.5m 0.5mr . The s + m (s + m) average arrival rate is 1. Note that the queue is limited to a maximum state of 2. Use the method of stages to solve this queue and obtain the following - (a) State Transition Diagram (with a proper definition of system states) (b) Obtain the state probability distribution (c) What will be the average departure rate from this queue? 2. Consider a MIX/-/1/3 queue where the batch arrival rate is I and the generating function of the batch sizes is given by (0.25+0.25z +0.5z'). Note that the queue is limited to a maximum state of 3 and follows the WBAS strategy (a) If the service time distribution is - 0.5m 0.5m s+ m (s+ m)?' 2. draw the state transition diagram of the queue with an appropriately defined system state. 13] (b) For a service time distribution of - m -, do the following - s+ m i. Draw the state transition diagram. 13] ii. Find the state probabilities of the queue. [5] ifi. What is the probability that a batch is refused entry into the queue? [21