Lec12_MM1k_Queueing System2

Lec12_MM1k_Queueing System2 - 4.2 M/M/1/k Queueing Model...

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OR372-Dr.Khalid Al-Nowibet 1 Characteristics 1. Interarrival time is exponential with rate λ Arrival process is Poisson Process with rate λ 2. Interarrival time is exponential with rate µ Number of services is Poisson Process with rate µ 3. Single Server 4. System size is finite = k 5. Queue Discipline : FCFS Notation M / M / 1 / k / FCFS
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OR372-Dr.Khalid Al-Nowibet 2 Steady-State Distribution State of the system system is in state n if there are n customers in the system (waiting or serviced) Let P n be probability that there are n customers in the system in the steady-state . n = 0 , 1 , 2 , 3 , …, k
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OR372-Dr.Khalid Al-Nowibet 3 Steady-State Distribution Rate Diagram: 1. If system changes state, where to go? 2. How fast the system changes state? 0 1 2 k-1 k . . . λλ λ λ µ µ µµ µ λ
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OR372-Dr.Khalid Al-Nowibet 4 Steady-State Distribution Balance Equations: For each state n: Average Rate out of state n = Average Rate in to state n = Average Rate in to State n Average Rate out of State n = k n} state in Pr{system k) n (rates k k} state in Pr{system n) k (rates
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OR372-Dr.Khalid Al-Nowibet 5 Steady-State Distribution Balance Equations: n = 0 ⇒λ P 0 = µ P 1 n = 1 P 1 + µ P 1 = λ P 0 + µ P 2 ( λ + µ )P 1 = λ P 0 + µ P 1 n = 2 P 2 + µ P 2 = λ P 1 + µ P 3 ( λ + µ )P 2 = λ P 0 + µ P 3 n = 3 P 3 + µ P 3 = λ P 2 + µ P 4 ( λ + µ )P 3 = λ P 2 + µ P 4 . . . . . . . . . . . n = k ⇒µ P k = λ P k-1 ⇔µ P k = λ P k-1 0 1 2 k-1 k . . . λλ λ λ µ µ µµ µ λ Average Rate in to State n Average Rate out of State n =
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OR372-Dr.Khalid Al-Nowibet 6 Steady-State Distribution Solution of Balance Equations: λ P 0 = µ P 1 ( λ + µ )P 1 = λ P 0 + µ P 2 ( λ + µ )P 2 = λ P 1 + µ P 3 . . . . . . . . . . . µ P k = λ P k-1 Eq-1 ⇔λ P 0 = µ P 1 Eq-2 ( λ + µ )P 1 λ ( µ / λ )P 1 = µ P 2 P 1 = µ P 2 Eq-3 ( λ + µ )P 2 λ ( µ / λ )P 2 = µ P 3 P 2 = µ P 3 . . . . . . . . . . .
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This note was uploaded on 06/14/2011 for the course DATABASE 101 taught by Professor - during the Spring '11 term at Aarhus Universitet.

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Lec12_MM1k_Queueing System2 - 4.2 M/M/1/k Queueing Model...

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