Mathematical physics questions:

1. A system is put into operation at time t=0. Its time of failure is a random variable X with cdf Fx(x) and pdf fx(x). Let B(t)dt be the probability that it will fail in (t,t+dt) given that it has not failed until time t. (a) Find fx(x) if we are given that B(t)=kt. [5] (b) If X is uniformly distributed in the interval (0,T), then what would be B(t) in this interval. [5] 2. The instructor for EE679, keeps the door of the lecture hall L-5 open for a random time interval Y before his lecture where the pdf, cdf and L.S.T of Y are given as f, (v), Fy (y) and Fy (s), respectively. Students arrive from a Poisson Process with rate ) and can enter L-5 only while the door is open. Let N be the (random) number of students attending the lecture (i.e. those who can enter!). (a) Find the Generating Function G, (=) of the number of students N attending the EE679 lecture. [5] (b) Find the first and seconds moments of N, in terms of the moments of Y using (a) [5] (c) If f, (y) is exponentially distributed with mean , find the distribution of N using GN (=) from (a). [5]1. A M/-/1/2 queue has a service time distribution given by 0.54 0.5uz The Stu (s+ 1 ) 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 [6] (c) What will be the average departure rate from this queue? [3] 2. Consider a MIX/-/1/3 queue where the batch arrival rate is A 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 0.54 0.5u (a) If the service time distribution is draw the state transition diagram Stu (stu ) of the queue with an appropriately defined system state. [3] (b) For a service time distribution of -, do the following - s+ H i. Draw the state transition diagram. [3] ii. Find the state probabilities of the queue. [5] ili. What is the probability that a batch is refused entry into the queue? [2]1. Students enter the mess for breakfast in equally likely groups of either one or two with a group arrival rate of 1. The first member of the group is served in an exponentially distributed time X with pdf b(t) and LST B(s). The second member (if any) orders an extra omelet which requires A seconds more where A is fixed. The mess operates as a Single-Server M"/G/I queue. Find the mean delay that an arriving student will encounter before being served. [10] 2. Consider a 2-priority preemptive resume priority M/G/1 queue with high priority customers of Class 2 and lower priority customers of Class 1. The system enforces the rule that there can be only one Class 2 customer in the system at any time (i.e. there is no buffering for Class 2) - however, there is infinite buffering for Class 1 customers. Let X? (FIXED) be the service time for Class 2 and X, (FIXED) be the service time for Class 1. Arrival processes are Poisson with average arrival rate A/ for Class 1 and 12 for Class 2. For this, consider a Class 1 customer who start service at time f and leaves the system at time /+7. What would be the distribution (or L.S.T.) of the random variable 7? [Hint: You can assume a Poisson distribution at the appropriate place without explicitly deriving it.] [10] 3. Consider the system of Problem 2 once again except that we now assume that Class 2 customers can also be infinitely buffered. For this obtain the average total delays encountered by Class 2 and Class 1 customers. [5]