examples - CDA6530: Performance Models of Computers and...

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CDA6530: Performance Models of Computers and Networks Examples of Stochastic Process, Markov Chain, M/M/* Queue
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2 Queuing Network: Machine Repairman Model c machines Each fails at rate λ (expo. distr.) Single repairman, repair rate μ (expo. distr.) Define: N(t) – no. of machines working 0 N(t) c
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3 π k 1 μ = k λπ k π k = 1 k ! ( μ λ ) k π 0 c X i =0 π i =1 π 1 0 = c X k =0 1 k ! ( μ λ ) k
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4 Utilization rate? =P(repairman busy) = 1- π c E[N]? We can use Complicated E [ N ]= c X i =1 i π i
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5 E[N] Alternative: Little’s Law Little’s law: N = λ T Here: E[N] = arrival · up-time Arrival rate: Up time: expo. E[T]=1/ λ Thus ημ +(1 η ) · 0 E [ N ]= ημ λ
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6 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?
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examples - CDA6530: Performance Models of Computers and...

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