1.(10%) Let two independent, exponentially distributed random variables and denote the service time and the inter-arrival time of a system with parameters and, respectively. And denotes the busy period length of the system. Define event D: { }. (a)Derive, where. (2%) (b)Derive. (2%) (c)Derive p(A) and p(D). (2%) (d)Derive in terms of p(A), p(D) and. (2%) (e)Derive in terms of and. (2%) 2003 Fall Queuing Theory Final Exam (Time limit ： 2 hours)

2.(20%) Consider an M/M/1 queuing system with parameters and. At each of the arrival instants one new customer will enter the system with probability ½ or two new customers will enter simultaneously with probability ½. (a)Draw the state-transition-rate diagram for this system. (5%) (b)Using the method of non-nearest-neighbor systems write down the equilibrium equations for p k. (5%) (c)Find P(z) and also evaluate any constants in this expression so that P(z) is given in terms only of and. If possible eliminate any common factors in the numerator and denominator of this expression. (5%) (d)From (c) find the expected number of customers in the system. (5%)

3.(20%) There are 10 students and 4 TAs in the final exam of Queuing Theory. The students are always busy doing one of the two activities: writing (work state) or asking questions (service state); no other activities are allowed. Each student is initially in the work state for an exponential, rate, period of time. Each student then attempts to ask one of the TAs a question. If all TAs are busy, then the student is blocked and returns to the work state. Assume that the time each student takes to ask a question is exponentially distributed with rate and then returns to the work state : (a)Define an appropriate state space for this service system. Draw a state diagram for this system showing all transition rates. (4%) (b)Write the balance equations for the system. (4%) (c)Specify a method of computing the ergodic blocking probability for the system—that is, the proportion of attempts to join the service system that will be blocked—in terms of the system parameters and the ergodic state probabilities. (4%) (d)Specify a formula to compute the average question generation rate. (4%) (e)Let = 1/3 questions per minute. Compute the blocking probability as a function of for (0,30). (4%)

4. (2%) (3%) (5%)

5.(25%) Consider the M/M/2 queuing system which has Poisson arrivals, exponential service, two parallel servers and an infinite waiting room capacity. (a)Determine the expected first passage time from state 2 to state 1. (4%) (b)Determine the expected length of the busy period for the ordinary M/M/2 system using the result from (a). (4%) (c)Define as the length of time between successive entries into busy periods, that is, as the length of one busy / idle cycle. Determine the probability that the system is idle at an arbitrary point in time. Determine the probability that there is exactly one customer in the system. (4%) (d)Determine the total expected amount of time the system spends in state 1 during a busy period. (4%) (e)Check (c) and (d) using classical birth-death analysis. (4%) (f)Determine the expected sojourn time for an arbitrary customer by the Little’s Formula. (5%)

6.(15%) Let denote a random period of time and is the Laplace transform of. Let be a Poisson process with rate and let denote the number of arrivals from occurred during the period of time. Derive the relationship between the Z transform of,, and the Laplace transform of.