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lec22 - The M/M/1 Queue Example Printer Queue(continued A...

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The M/M/ 1 Queue: Example Printer Queue (continued) A certain printer in the Stat Lab gets jobs with a rate of 3 per hour. On average, the printer needs 15 min to finish a job. Let X ( t ) be the number of jobs in the printer and its queue at time t . We know already: X ( t ) is a Birth & Death Process with constant arrival rate λ = 3 and constant death rate μ = 4 . The properties of interest for this printer system then are: L = E [ X ( t )] = a 1 - a = 0 . 75 0 . 25 = 3 W s = 1 μ = 0 . 25 hours = 15 min W = L λ = 3 3 = 1 hour W q = W - W s = 0 . 75 hours = 45 minutes L q = W q λ q = 0 . 75 · 3 = 2 . 25 1

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The M/M/ 1 Queue: Example (cont’d) On average, a job has to spend 45 min in the queue. What is the probability that a job has to spend less than 20 min in the queue? We denoted the waiting time in the queue by q ( t ) . q ( t ) has the cumulative distribution function 1 - ae y ( μ - λ ) . The probability asked for is P ( q ( t ) < 2 / 6) = 1 - 0 . 75 · e - 2 / 6 · (4 - 3) = 0 . 4626 . 2
The M/M/ 1 /K queue An M/M/ 1 queue with limited size K is a lot more realistic than the one with infinite queue.

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lec22 - The M/M/1 Queue Example Printer Queue(continued A...

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