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TexPoint fonts used in EMF. CDA6530: Performance Models of Computers and Networks Examples of Stochastic Process, Markov Chain, M/M/* Queue TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

Queuing Network: Machine Repairman Model c machines Each fails at rate ¸ (expo. distr.) Single repairman, repair rate ¹ (expo. distr.) Define: N(t) – no. of machines working 0· N(t) · c

Utilization rate? E[N]? =P(repairman busy) = 1-¼c We can use Complicated

E[N] Alternative: Little’s Law Little’s law: N =¸ T Here: E[N] = arrival ¢ up-time Arrival rate: Up time: expo. E[T]=1/¸ Thus

Markov Chain: Gambler’s Ruin Problem A gambler who at each play of the game has probability p of winning one unit and prob. q=1-p of losing one unit. Assuming that successive plays are independent, what is the probability that, starting with i units, the gambler’s fortune will reach N before reaching 0?

Define Xn: player’s fortune at time n. Question: Xn is a Markov chain Question: How many states? State transition diagram? Note: there is no steady state for this Markov chain! Do not try to use balance equation here

Construct recursive equation Pi: prob. that starting with fortune i, the gambler reach N eventually. No consideration of how many steps Can treat it as using infinite steps End prob.: P0=0, PN=1 Construct recursive equation Consider first transition Pi = p¢ Pi+1 + q¢ Pi-1 Law of total probability This way of deduction is usually used for infinite steps questions Similar to the routing failure example 1 in lecture notes “random”

Another Markov Chain Example Three white and three black balls are distributed in two urns in such a way that each urn contains three balls. At each step, we draw one ball from each urn, and place the ball drawn from the first urn into the second urn, and the ball drawn from the second urn into the first urn. Let Xn denote the state of the system after the n-th step. We say that the system is in state i if the first urn contains i white balls. Draw state transition diagram. Calculate steady-state prob.

Poisson Process Patients arrive at the doctor's office according to a Poisson process with rate ¸=1/10 minute. The doctor will not see a patient until at least three patients are in the waiting room. What is the probability that nobody is admitted to see the doctor in the first hour?