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Unformatted text preview: Probability and Random Processes STA 3533 Module 6 Networks of Queues Continue Topics Case Study Solution Introduction to Queuing Open queuing networks Closed queuing networks Introduction to Queuing Problem 1 A data communication line delivers a block of information every 10 microseconds. A decoder checks each block for errors and corrects the errors if necessary. It takes 1 μ s to determine whether a block has any errors. If the block has error, it takes 5 μ s to correct it, and if it has more than one error it takes 20 μ s to correct the error. Blocks wait in a queue when the decoder falls behind. Suppose that the decoder is initially empty and that the numbers of errors in the first 10 blocks are 0, 1, 3, 1, 0, 4, 0, 1, 0, 0. a. Plot the number of blocks in the decoder as a function of time. b. Find the mean number of blocks in the decoder. c. What percentage of the time is the decoder empty? Introduction to Queuing Solution 1: 1. Interarrivals are constant with interarrival times = 10 msec 2. Service time if 0 error 1 μ sec 1 error 1+5 μ sec >1 error1+20 μ sec Arrival time 10 20 30 40 50 60 70 80 90 100 Errors 1 3 1 4 1 Service time 1 6 21 6 1 21 1 6 1 1 Dep. Time 11 26 51 57 58 81 82 88 91 101 Introduction to Queuing a. The plot b. 96 . ] 1 1 6 2 3 20 10 1 12 3 20 10 6 1 [ 100 1 ) ( 100 1 110 10 = + + + + + + + + + + + + + = ∫ dt t N 10 30 20 40 60 50 70 80 90 100 110 N(t) T Observation = 100 msec Introduction to Queuing c. Server is working during 65 msec = Σ service times proportion idle time = 1 – 65/100 = 0.35 Introduction to Queuing Problem 2: Three queues are arranged in the loop as shown below. Assume that the mean service time in the queue i is m i = 1/ μ . a. Suppose the queue has a single customer circulating in the loop. Find the mean time E[T] it takes the customer to cycle around the loop. Deduce from E[T] the mean arrival rate λ at each of the queues. Verify that Little’s formula holds for these two quantities. b. If there are N customers circulating in the loop, how are the mean arrival rate and mean cycle time related? μ 1 μ 2 μ 3 Introduction to Queuing Solution: a. One customer no waiting Little’s Formula 3 2 1 3 2 1 1 ] [ m m m m m m T + + = ⇒ + + = λ ε system in customer one m m m m N i queue in customer time m m m m m N i i i i i i i 1 ] [ % ] [ 3 1 3 1 3 2 1 3 2 1 ⇒ = + + = = + + = = ∑ ∑ = = ε λ ε Introduction to Queuing...
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This note was uploaded on 06/14/2011 for the course HIST 1301 taught by Professor Gonzales during the Spring '11 term at The University of Texas at San Antonio San Antonio.
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