2003 Fall Queuing Theory Midterm Exam(Time limit：2 hours)

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Lecture 5 This lecture is about: Introduction to Queuing Theory

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2003 Fall Queuing Theory Midterm Exam(Time limit：2 hours) (10%) For the Markov Chain in figure 1, which states are: (a) recurrent ? (2%) (b) transient ? (2%) (c) aperiodic ? (2%) (d) periodic ? (2%) (e) Is this chain reducible ? Why or why not? (2%) 1 2 3 4 5 Fig.1

2. ( 6%) Identify the following systems in figure 2 and make complete notations. (eg: X / X / X / X / X) (a) (3%) G G 100 G (b) (3%) M Fig.2

(15%) Consider that the discrete-state,discrete-time Markov chain transition probability matrix is given by . (a) Find the stationary state probability vector . (5%) (b) Find . (5%) (c) Find the general form for . (5%)

(14%) Given the differential-difference equations: Define the Laplace transform . For the initial condition we assume for . Transform the differential-difference equations to obtain a set of linear difference equations in . (a) Show that the solution to the set of equations is: (10%) (b) From (a), find for the case . (4%)

(15%) Consider an M/M/1 system with parameters , in which customers are impatient. Specifically, upon arrival, customers estimate their queuing time and then join the queue with probability or leave with probability . The estimate is when the new arrival finds in the system. Assume . (a) In terms of , find the equilibrium probabilities of finding in the system. Give an expression for in terms of the system parameters. (5%) (b) For , under what conditions will the equilibrium solution hold? (5%) (c) For ,find explicitly and find the average number in the system. (5%)

(15%) Consider an M/M/1 queuing system, the arrival rate is and the service rate is : (a) What three properties would make a Markov chain ergodic? (6%) (b) Prove that the limiting distribution exists only when . (4%) (c) When , argue that the M/M/1 queuing system is ergodic. (5%)

(a) Derive using notations provided above. (10%) (25%) Consider a discrete-time birth-death chain as shown in figure 3. The death rate is p and the birth rate is (1-p). The ratio between birth rate and death rate is . (a) Derive using notations provided above. (10%) (Hint: is the probability that the chain starts at state i and visits state 0 before it visits state m.) (b) Show with derivation that this system is: (5%) 1 2 3 Fig.3